Thread: The Ontological Status of Mathematical Entities Board: Oblivion / Ship of Fools.
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Posted by Jack o' the Green (# 11091) on
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I've been reading quite a lot about the Philosophy of Mathematics recently (I know, my Rock 'n' Roll lifestyle is going to catch up on me soon at some point). For those with a social life, the philosophy of mathematics examines the 'ontological status' (the existence) of abstract objects like numbers and geometric shapes (via their abstract mathematical properties).
http://en.wikipedia.org/wiki/Philosophy_of_mathematics
The main bone of contention appears to be that for some mathematicians, (usually described as mathematical platonists) many of these ideas e.g. 2+2=4, the value of pi etc seem to be true necessarily, which for some means they must exist necessarily in a non-physical, non-temporal 'platonic third realm' which we access via our 'mathematical intuition'.
The question vexes the Christian philosopher William Lane-Craig quite a lot because of the possibility that if mathematical entities exist necessarily then this contradicts the scriptural statements that everything was created by God and impacts on the doctrine of divine aseity i.e. the view that God is self-existent and self-sufficient, and that by implication, everything else is dependant on him.
http://en.wikipedia.org/wiki/Aseity
http://www.reasonablefaith.org/god-and-abstract-objects-oct-2013
Some convinced mathematical platonists have also been strong theists e.g. Kurt Gödel. Christian Philosophers like Edward Feser and Keith Ward seem to consider mathematical entities to exist necessarily in the mind of God as ideas abstracted by the intellect.
My own position is that even if mathematical ideas are necessarily true, that doesn't mean that they necessarily exist, and that for them to exist, they must be conceived by an intellect or mind. The fact that they are necessarily true simply means that even God can't conceive of 2+2 equalling anything other than 4. I don't find the concept of a 'third realm' convincing at all as it leaves so many questions unanswered and also seems superfluous.
Does anyone else have any thoughts?
Posted by Lamb Chopped (# 5528) on
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My doubtless naively expressed view (I'm a staid creature, not up to your wild and crazy lifestyle!) is that mathematics, ethics, and any other abstract foundational always-true thing is rooted in God's own nature and is a reflection of it, though we may not see how. Yet. And if this is true, then the only way to have a different self-consistent and true to the universe mathematics/ethics/whatever would be to have a different God, first. Which is nonsense.
Posted by Enoch (# 14322) on
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I'm with Lamb Chopped on this.
I also cannot see how the ontological status of mathematical concepts reflects on the existence of God. I'm not even convinced that it's a useful question whether they are created or uncreated. That question is usually only asked of the divine light as distinct from the ordinary light produced by the sun, an electric light or whatever.
Before or outside creation, is it even useful, relevant or even interesting to speculate whether a mathematical concept exists or not?
Posted by itsarumdo (# 18174) on
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Since I can't imagine a world in which numeration (i.e. saying there are 3 people in the room or I have 3 oranges) is not a way of perceiving (i.e. there is no conscious concept of numbers), the question seems a bit moot.
But the conceptualisation of (e.g.) 3 as an entity in its own right is, I think, implicit in how the universe came about during creation, and in one sense there is a hierarchy of integers.
Is there a practical implication for all of this, or is it an arcane philosophical point with no particular application?
[ 31. May 2015, 14:28: Message edited by: itsarumdo ]
Posted by Doc Tor (# 9748) on
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Integers aside, I'd argue that the value of pi (being the ratio of the circumference of a circle to its diameter) is a constant based on the physical properties of the universe, and thus is as much part of creation as Planck's constant.
Posted by mousethief (# 953) on
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quote:
Originally posted by itsarumdo:
Is there a practical implication for all of this, or is it an arcane philosophical point with no particular application?
Very often in mathematics, or philosophy, or as in this case both, practical applications come along decades or centuries after the intellectual groundwork is laid. "Imaginary" numbers were an intellectual plaything until it was found they perfectly describe electromagnetic fields and nothing else really works to do so. Then they went from being arcana to an integral part of science.
Posted by Honest Ron Bacardi (# 38) on
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More or less the same thing happened with group theory.
Posted by Humble Servant (# 18391) on
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quote:
Originally posted by Doc Tor:
Integers aside, I'd argue that the value of pi (being the ratio of the circumference of a circle to its diameter) is a constant based on the physical properties of the universe, and thus is as much part of creation as Planck's constant.
The value of pi is defined (or at least implied) in the bible at 1 kings 7:23. pi is thus a biblical concept.
Posted by Eutychus (# 3081) on
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A lot of ink has been spilled on explaining why it gives an inaccurate value, though.
(welcome, by the way!)
Posted by quetzalcoatl (# 16740) on
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Jack o' the Green, abstract objects have been similarly controversial, since they have odd qualities, e.g. not existing in space. In fact, they are quite spooky, and arouse disquiet on various grounds. Will return, cricket demanding attention. (In fact, numbers are abstract objects, you don't see them in the street).
[ 31. May 2015, 15:57: Message edited by: quetzalcoatl ]
Posted by agingjb (# 16555) on
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Pi is independent of the physical universe, It is a number that turns up in many non-geometric ways.
I would say that in maths we pose, even invent, questions, and discover the answers - when there are answers.
Mathematically adept theologians, I imagine, have tremendous fun with independence proofs in set theory.
Posted by Doc Tor (# 9748) on
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quote:
Originally posted by agingjb:
Pi is independent of the physical universe, It is a number that turns up in many non-geometric ways.
The value of pi does. What that numerical value is, however, is fixed in this universe.
Posted by Doc Tor (# 9748) on
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And as an aside, one of the most elegant shorthands for "we're not in Kansas anymore" was in Bob's Shaw's The Ragged Astronauts where the protagonist scientist is musing as to why, no matter what size circle he drew, the ratio of its circumference to its diameter was always only, and exactly, 3.
Posted by Eutychus (# 3081) on
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Woah. I wonder if it got knocked to 3.14.... as a result of the Fall?
Posted by Jack o' the Green (# 11091) on
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Thank you for your replies. I don't have any problem with accepting that mathematical entities are abstract and 'exist' outside space and time. I think the question is in what way they can be said to exist if at all if not conceived by a mind. Although abstract objects are usually taken to be a causal, some philosophers and scientists think they are so fundamental in understanding the universe, that they could be its ultimate cause, and because they are necessarily true (and therefore necessarily exist) can explain the universe without reference to God. Peter Atkins is supposed to have said as much in a conversation with the philosopher Keith Ward.
Abstracts such as pi exist in the 'abstract realm' far more precisely than in the physical universe ie we could never prove its value to the same extent simply by measuring circles in the real world. Mousethief's comment about imaginary numbers makes a good point and is fundamental to mathematical realists' arguments that there must be a fundamental reason as to why abstract ideas can describe the physical universe so well.
Posted by itsarumdo (# 18174) on
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quote:
Originally posted by Jack o' the Green:
Thank you for your replies. I don't have any problem with accepting that mathematical entities are abstract and 'exist' outside space and time. ...,.
well - maybe not. You have to have at least 2 dimensions for pi or phi to have a meaning. And since pretty well everything material is transient, if time doesn't exist, would any numbers really exist at all? Groupings and agglomerations would not be definable in the same ways. The Planck constant is not necessarily a constant (qv Sheldrake) in the same way that 1 is.
Posted by Jack o' the Green (# 11091) on
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Those are good points. However, does pi need 2 dimensions as an abstract concept since pi is now calculated without reference to physical circles, but instead uses mathematics. I should've been more precise and said that I don't have a problem with abstract entities existing outside space and time as long as they are said to be residing in a mind which is also beyond space and time.
Posted by maryjones (# 13523) on
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I am not a mathematician.
The discussion reminds me of such philiosophical gems as creative science
My favourite in this genre is God in the quad
Posted by maryjones (# 13523) on
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Sorry, everybody , the links aren't working. I shall try again later. My apologies
Posted by Jack o' the Green (# 11091) on
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I like that. I agree with quite a lot of what Berkeley says about things only existing if they are perceived.
Posted by Schroedinger's cat (# 64) on
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I am with Doc Tors first comment, in that these mathematical concepts are fundamentally a part of our universe and our interpretation of it.
So the fact that the definition we give to the quantity of oranges we experience is given the title "3" does not mean that "3" has any concept or existence outside this universe or even outside our perception of it.
What this means is that there could be a universe (not a world in this universe) where the value of pi (as in the ratio of a circles circumference to the radius) is 2.6. Of course, in their universe, this might be referred to as something different - the core structure of the universe defined the ontological nature of these entities, including numbers.
So I am not sure they are "abstract entities", because the core aspects of mathematics are fundamentally related to our universe. Mathematics as we understand it is in this universe, not least as it is at the heart of how this universe is constructed. A different construction might have a different mathematics.
Posted by Jack o' the Green (# 11091) on
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quote:
Originally posted by maryjones:
Sorry, everybody , the links aren't working. I shall try again later. My apologies
The second one worked for me ![[Smile]](smile.gif)
Posted by Lamb Chopped (# 5528) on
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If God exists as usually defined (omniscient, omnipresent, eternal, uncreated creator, etc.) then the perception problem goes away, as there will always have been a mind to perceive whatever-it-is. Plus that mind will be logically prior even to eternal concepts/truths, because those have their root in his own nature. So even where there is not a priority-in-time there can be a priority-in-logic. One is the source of the other.
The real problem, it seems to me, is discerning which truths about our universe could have been different (God could have created them differently, perhaps HAS created them differently somewhere else) and which are necessarily fixed by the fact that they flow out of God's own nature, and cannot be different, anywhere or anywhen, as long as God is God.
Some are obvious--truth will always be preferable to error, sense to nonsense, and so on, because God IS truth, is wisdom, and etc. So we will see that fact reflected in any and all possible versions of creation. For God to create otherwise he would have to uncreate himself.
But mathematics? I just don't know. It may be that the version of mathematics we have is the only possible one, because it is in some way rooted in and reflecting God's own nature. Or it could be that this is something he made up creatively, and could have made differently. I don't have enough data or understanding to call that one.
Posted by Dafyd (# 5549) on
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quote:
Originally posted by Doc Tor:
And as an aside, one of the most elegant shorthands for "we're not in Kansas anymore" was in Bob's Shaw's The Ragged Astronauts where the protagonist scientist is musing as to why, no matter what size circle he drew, the ratio of its circumference to its diameter was always only, and exactly, 3.
Much as I enjoyed The Ragged Astronauts trilogy - who doesn't like interplanetary balloon travel? - this makes no sense (to me).
I think agingjb is overstating the case when he says that pi crops up in numerous non-geometric contexts: most of them turn out to have geometric analogues. Nevertheless, he's right that properties of the Euclidean circle, such that its equation is x^2 + y^2 = k, mean that pi can be defined and calculated from purely arithmetic analytical considerations without going through anything explicitly geometrical. If pi is a property of our physical universe you'll need to find some point in algebra or arithmetic or analysis where an empirical constant is smuggled in, and I don't think that can be done.
Posted by balaam (# 4543) on
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quote:
Originally posted by maryjones:
I am not a mathematician.
The discussion reminds me of such philiosophical gems as creative science
My favourite in this genre is God in the quad
Fixed it for you.
Posted by Dafyd (# 5549) on
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quote:
Originally posted by Jack o' the Green:
The main bone of contention appears to be that for some mathematicians, (usually described as mathematical platonists) many of these ideas e.g. 2+2=4, the value of pi etc seem to be true necessarily, which for some means they must exist necessarily in a non-physical, non-temporal 'platonic third realm' which we access via our 'mathematical intuition'.
I would agree with this, although I think it's important to put in some qualifications.
In particular, whatever mathematical truths are they are not analagous to physical objects: they cannot be perceived by mathematical intuition in the way we perceive physical objects by physical senses. While they obviously affect the physical world (because a good reason for believing in mathematical realism / platonism is that predictions about the physical realm based on mathematics turn out true where the premises are true), they cannot do so by efficient causation.
As for the relation to God - I am not sure that the claim that God created everything is ever stated explicitly in the Bible. (Genesis 1 mentions the waters that the spirit of God moves over - it doesn't explicitly say that God made them. Theologically I think we're obliged to say God did make them, but scripture alone doesn't compel that directly.) I think classical theology does require us to believe that God is not determined by anything outside God; so presumably truths of logic are derived from God's nature, but how that works is perhaps impossible to say.
Posted by itsarumdo (# 18174) on
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You wouldn't go with the system of numerology then, whereby (at least the smaller) integers each have a symbolic meaning of their own that transcends their mathematical usage?
wrt pure maths (with no required connection to real things) - the numbers used as a basis for that maths all began life as real things - so pure maths is in some ways akin to impressionism and the flights of fancy of Picasso and Dali and their ilk. As did the manipulative symbology - I don't see that it makes a difference how many meta levels you step up - one meta level for the {+} addition sign, maybe only 2 or 3 more for even the most complex mathematical constructions. They are all based initially on real experience.
Posted by balaam (# 4543) on
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Mathematics is not perfect. Quantum Mechanics contradicts General Relativity on a mathematical level.
Yet a Sat-Nav works.
Posted by Jack o' the Green (# 11091) on
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quote:
Originally posted by Dafyd:
As for the relation to God - I am not sure that the claim that God created everything is ever stated explicitly in the Bible. (Genesis 1 mentions the waters that the spirit of God moves over - it doesn't explicitly say that God made them. Theologically I think we're obliged to say God did make them, but scripture alone doesn't compel that directly.) I think classical theology does require us to believe that God is not determined by anything outside God; so presumably truths of logic are derived from God's nature, but how that works is perhaps impossible to say.
The text Craig Lane quotes is John 1:3 "Through him all things were made; without him nothing was made that has been made."
He acknowledges that it doesn't say all things in total ie there could be uncreated 'entites' such as a platonic realm of necessary mathematical truths, but he raises the problem of that implying 2 'ultimates' - God and mathematical entities.
Posted by Enoch (# 14322) on
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quote:
Originally posted by Jack o' the Green:
... He acknowledges that it doesn't say all things in total ie there could be uncreated 'entites' such as a platonic realm of necessary mathematical truths, but he raises the problem of that implying 2 'ultimates' - God and mathematical entities.
I'm not persuaded that is either right or follows. The theological arguments about uncreated light, are that if it is so, it is the light of God, part of his nature. The usual example is debate as to whether the light the disciples saw on the mount of Transfiguration was created or uncreated. The debate then moves on to light that mystics and others experience. If it is uncreated, then they are experiencing the divine nature.
If there could be uncreated entities, then by being uncreated, they would be part of God's nature, his personality, who he is. But if that's the case, then could they be entities? They would not be separate from God.
There is, though, no fundamental reason why things that are unseen, are not part of the created order. Angles are for a start. So, actually, is gravity. If platonic ideals have some objective or even abstract ontological existence, then they would be created too. There is no need to posit two different sorts of ultimates, God and mathematical entities. It is an illogical answer to a question that really doesn't need to be asked.
[ 31. May 2015, 20:57: Message edited by: Enoch ]
Posted by mousethief (# 953) on
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quote:
Originally posted by balaam:
Mathematics is not perfect. Quantum Mechanics contradicts General Relativity on a mathematical level.
I would say that's far more likely evidence that either Quantum Mechanics or General Relativity is not perfect, than that Mathematics is not perfect.
Posted by Enoch (# 14322) on
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Sorry, I missed the correction slot. 'Angles' in my post should of course be 'Angels'.
Posted by balaam (# 4543) on
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quote:
Originally posted by mousethief:
quote:
Originally posted by balaam:
Mathematics is not perfect. Quantum Mechanics contradicts General Relativity on a mathematical level.
I would say that's far more likely evidence that either Quantum Mechanics or General Relativity is not perfect, than that Mathematics is not perfect.
Not either, both.
Quantum Mechanics does not work for large objects such as the size of a sat nav satellite. General Relativity does not work for small objects such as the sub atomic size.
The two sets of maths are incompatible, yet the humble sat nav needs both sets of maths to work.
I love paradox.
Posted by itsarumdo (# 18174) on
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quote:
Originally posted by Enoch:
Sorry, I missed the correction slot. 'Angles' in my post should of course be 'Angels'.
Non Angli, sed angeli
![[Cool]](cool.gif)
Posted by Pulsator Organorum Ineptus (# 2515) on
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quote:
Originally posted by balaam:
quote:
Originally posted by mousethief:
quote:
Originally posted by balaam:
Mathematics is not perfect. Quantum Mechanics contradicts General Relativity on a mathematical level.
I would say that's far more likely evidence that either Quantum Mechanics or General Relativity is not perfect, than that Mathematics is not perfect.
Not either, both.
Quantum Mechanics does not work for large objects such as the size of a sat nav satellite. General Relativity does not work for small objects such as the sub atomic size.
The two sets of maths are incompatible, yet the humble sat nav needs both sets of maths to work.
I love paradox.
The fact that we haven't yet found a single set of equations that combine general relativity with quantum mechanics doesn't mean there's something wrong with mathematics. It merely means we haven't found the right equations yet.
Posted by orfeo (# 13878) on
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quote:
Originally posted by Doc Tor:
Integers aside, I'd argue that the value of pi (being the ratio of the circumference of a circle to its diameter) is a constant based on the physical properties of the universe, and thus is as much part of creation as Planck's constant.
Yep, I agree with you. Pi relates to a physical thing.
Whereas 2 + 2 = 4 doesn't. Heck, I've done maths where 2 + 2 doesn't equal 4. In the right cyclic ring, 2 + 2 = 1.
Mathematics relies on axioms - on unprovable things that are just asserted to create a base for doing more complicated mathematics.
I certainly wouldn't assert that much of our counting system reflects some inherent property of the universe, because what it actually reflects is that we have 10 fingers and find base 10 convenient.
Posted by agingjb (# 16555) on
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I would always say that that there are physical "things" that turn out to be effectively described by mathematics. That mathematics was originally approached by the properties of the physical world, but there is a vast amount of abstract pure maths now with no immediate correspondence with physics.
We live in a world which turns out not to be precisely described as a Euclidian space; but Euclidian geometry remains as a theory sufficient to itself.
Posted by Golden Key (# 1468) on
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If God exists and made everything, and math is intrinsic (rather than a human interpretation), then She made math. Simples.
If God exists but didn't make everything, She still might have math.
If She doesn't exist and didn't make anything, then either it's our interpretation or the Universe built math into itself.
We're only in trouble if She doesn't exist, but DID make everything.
![[Biased]](wink.gif)
Posted by Dave W. (# 8765) on
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quote:
Originally posted by balaam:
Quantum Mechanics does not work for large objects such as the size of a sat nav satellite.
In what way does QM "not work" for such objects?
Posted by mousethief (# 953) on
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quote:
Originally posted by orfeo:
Whereas 2 + 2 = 4 doesn't. Heck, I've done maths where 2 + 2 doesn't equal 4. In the right cyclic ring, 2 + 2 = 1.
Then you're just redefining "plus" (addition). On the integers, 2+2 never equals anything but 4. But that's a poor example anyway because it's largely a matter of definition. What is marvelous is that that definition gives us a system so minbogglingly useful in describing, understanding, and predicting the real world.
And the base 10 thing is beneath you. Nobody thinks our notation has an ontological status similar to pi or phi or e or any of the other mathematical constants. That's an outrageously straw straw man.
Posted by orfeo (# 13878) on
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quote:
Originally posted by mousethief:
quote:
Originally posted by orfeo:
Whereas 2 + 2 = 4 doesn't. Heck, I've done maths where 2 + 2 doesn't equal 4. In the right cyclic ring, 2 + 2 = 1.
Then you're just redefining "plus" (addition). On the integers, 2+2 never equals anything but 4.
I'm not redefining addition, actually, but redefining the set of integers. In a ring they don't go on forever, but bend around.
As for base 10 remarks being "beneath me", the original post did not limit the discussion to physical constants. I am in fact agreeing with a previous post that singled out those kinds of constants as being different from some other parts of mathematics, so I don't see the point of having a go at me.
[ 01. June 2015, 05:28: Message edited by: orfeo ]
Posted by itsarumdo (# 18174) on
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The point about a counting ring is that the universe it exists in is not infinite?
It's a nice point - we can only count to big numbers because we live in a big place.
What I find difficult are the infinities an the infinite-ies. On small numbers, the fact that we can have negative exponents as big as we like (e.g. 10^-3589461029998763) seems to fool some people into thinking that scale dependence is also infinitely regressive. And expansive. So although we live in a largely fractal universe, that has physical upper and lower bounds, which maths doesn't have. The convenience of "effective" infinity and "effective" singularity of space-time in equations kinda makes us take them for granted. Whereas they are actually - weird.
Posted by Doc Tor (# 9748) on
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Maths allows us to easily write a number, say 10^80, and contemplate going higher. Whereas, that number is the largest number you'll ever need - the approximate number of hydrogen atoms in the universe.
We have a mathematical symbol for infinity: I don't think it's necessary for infinity to exist outside of our need for it as an operator within mathematics. Especially when the evidence points to the universe being finite.
Posted by Dafyd (# 5549) on
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quote:
Originally posted by Doc Tor:
We have a mathematical symbol for infinity: I don't think it's necessary for infinity to exist outside of our need for it as an operator within mathematics. Especially when the evidence points to the universe being finite.
We have several mathematical symbols for infinity, depending upon which size of infinity you're after. There are fewer integers than there are real numbers. On the other hand, there are just as many integers as there are rational numbers, or even algebraic numbers.
(The continuum hypothesis is the claim that there's no size of infinity between the integers and the real numbers. AIUI it has been proven that this is arbitrarily true or false depending on what you specify by adding an axiom.)
We can reach numbers of arbitrary sizes, and indeed, start defining infinities, using techniques that lead to true statements where physical quantities are in question; and there's no intrinsic difference in the techniques used that allows us to tell where we've strayed from the physical constraints of the universe by looking at the mathematics alone. Which suggests that if any mathematical statements are true (and the predictive success of hypotheses constructed from empirical statements using mathematics suggest they are) then all of them are true regardless of physical foundations.
I suppose a challenge for a mathematical realist is to decide what to do about the fact that the truth of the continuum hypothesis is underdetermined by the usual axioms. But I think the mathematical realist can just accept that as a real truth (or meta-truth).
Posted by orfeo (# 13878) on
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quote:
Originally posted by itsarumdo:
The point about a counting ring is that the universe it exists in is not infinite?
It's a nice point - we can only count to big numbers because we live in a big place.
No, that's not remotely the point. I'm fairly confident I was still in this universe when I was using a counting ring. Unless the lecture hall in the local university was accessed via a wormhole I didn't notice.
Posted by itsarumdo (# 18174) on
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quote:
Originally posted by orfeo:
quote:
Originally posted by itsarumdo:
The point about a counting ring is that the universe it exists in is not infinite?
It's a nice point - we can only count to big numbers because we live in a big place.
No, that's not remotely the point. I'm fairly confident I was still in this universe when I was using a counting ring. Unless the lecture hall in the local university was accessed via a wormhole I didn't notice.
But you constructed a sub-universe of size=3
Posted by IngoB (# 8700) on
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Both maths and logic are abstractions from "how the world works", so they are both created by God in creating the world. There is some surprise in how well maths and logic describe the world, sure, but that surprise is really about how well the human mind abstracts and extrapolates (works further with its abstractions). That maths and logic can describe physics is as such as surprising as that a blueprint matches the structure of the building it is the blueprint of. Finally, when we say things like "God cannot make square (Euclidean) circles", we are not actually saying that there is some extrinsic mathematical and logical constraint on God. We are saying that God is consistent and unchanging. It is the intrinsic and eternal coherence of God that guarantees that God is not suddenly switching His blueprint of creation on us. God created things in a certain way, and saw that it was good. Updates may be in the works, but they will come (if at all) only when this world ends...
Posted by IngoB (# 8700) on
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To add to the above: human imagination is not limited to dreaming up say a dragon flying through the air. It is also behind the "strange maths" that we have developed. Consider a dragon: it is not simply a random concept. It is an animal, its looks combine those of a lizard, snake and bat, it can fly like some animals can fly, it can spit fire as human can create and throw fire, etc. Lots of aspects of a dragon, indeed even how it is "put together" structurally, are derived from how the world actually works. It is just that we by imagination suspend other aspects of how the world works, or often enough, combine different ways in which the world works such as it would not be possible in the actual world. Bat wings are not going to keep a body wearing tons in the air. Steel-melting fire cannot be generated within biological materials that we know. But none of these elements is inherently at odds with the world.
Likewise, "strange maths" is not just random. It is quite simply human imagination working on the "library" of available mathematical and logical concepts, putting them together freely. Sometimes, as it happens, imagination may dream up things that are later found in physics. Other times this is not the case. But these "mathematical dragons" are not "contrary to physics" in the sense that it is inexplicable how one could arrive at them based on abstractions from the world. Rather, just like the imaginary dragon, they are still derived from how the world works in their parts, but not assembled by the mind into something that we actually find in the world.
Posted by agingjb (# 16555) on
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I still wonder to what extent theologicians speculate about the implications of large cardinal axioms in set theory.
Posted by Dafyd (# 5549) on
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quote:
Originally posted by orfeo:
Whereas 2 + 2 = 4 doesn't. Heck, I've done maths where 2 + 2 doesn't equal 4. In the right cyclic ring, 2 + 2 = 1.
I think though that modular arithmetic depends upon natural arithmetic, whereas the reverse is not true. (For every statement in modular arithmetic, I think you can define an equivalent set of statements in natural arithmetic that corresponds, but not vice versa. Thus, to the above statement in modular arithmetic corresponds in natural arithmetic: 2 + 3a + 2 + 3b = 1 + 3c for all a, b integers, where c is also an integer.)
Posted by Jack o' the Green (# 11091) on
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quote:
Originally posted by IngoB:
Finally, when we say things like "God cannot make square (Euclidean) circles", we are not actually saying that there is some extrinsic mathematical and logical constraint on God. We are saying that God is consistent and unchanging.
Just for clarification, are you saying that in your opinion a square Euclidean circle is possible and that God has simply chosen not to create one in this universe?
Posted by IngoB (# 8700) on
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quote:
Originally posted by Jack o' the Green:
Just for clarification, are you saying that in your opinion a square Euclidean circle is possible and that God has simply chosen not to create one in this universe?
I'm saying that "is possible" is a function of "this universe". If God had created a universe with exclusively square Euclidean circles, then it would be obvious to us that round Euclidean circles are impossible.
It is however not possible to introduce square Euclidean circles to this universe without fundamentally altering it. Basically it would become another kind of universe, rather than this one. (*)
In a way all this is rather uninteresting, because all I'm saying that maths and logic derive from the world. Hence they are whatever (the blueprint of) the world is.
What is interesting is rather the following question: to what extent could God continue "me" from this universe into one that is fundamentally changed? For this is really what we are asking when we say "could God create a square Euclidean circle?" (right in front of me, here and now). Clearly my body would not survive this without being changed to whatever can be maintained in a "square Euclidean circles" world. But would this blow my mind to such an extent as to obliterate "me", or not?
I cannot really answer that question, because while I'm fairly sure of the place of maths and logic in the world, I'm not sure about "me". I don't really know just how separable "I" am from having grown up in a "round Euclidean circle" world. However, the "New Creation" may well be along those lines, if perhaps with milder world updates.
(*) One could critique this by saying that I'm applying logic in this argument, which however is also determined by God. True enough. However, there are limits to how many assumptions I can ditch before my language turns into meaningless gibberish. If it is possible to have "square Euclidean circles" in this world, then I cannot really talk about that now. And the reason I can talk about "square Euclidean circles" in the first place is really simply that you let me get away with gibberish a bit.
In the end all I'm really saying is that we are world-limited, the world is God-limited, and God is unlimited.
[ 01. June 2015, 19:47: Message edited by: IngoB ]
Posted by agingjb (# 16555) on
:
Are we world-limited or does God enable us, in some ways, to apprehend something not so limited.
And I wonder if abstract mathematics is an example of this.
Posted by orfeo (# 13878) on
:
I'm finding myself in full agreement with Ingo here.
The "strange" maths is just as much from this universe as the "ordinary" maths. It's simply that the "ordinary" maths is suitable enough for solving a wide range of problems, whereas the "strange" maths is more helpful for solving particular kinds of problems.
It's just part of a suite of techniques that can be used when a mathematical question of some kind comes up.
Posted by agingjb (# 16555) on
:
OK. If my feeling that maths is independent of the material world is shot down, then I would have to conclude that there is nothing beyond that world - and be a materialist, even an atheist.
Posted by orfeo (# 13878) on
:
Well it depends what you mean by "independent".
Ingo's link is that God created human imagination. That's not quite the same as the notion that maths is some kind of... physical property of the universe.
Posted by IngoB (# 8700) on
:
quote:
Originally posted by agingjb:
OK. If my feeling that maths is independent of the material world is shot down, then I would have to conclude that there is nothing beyond that world - and be a materialist, even an atheist.
Is that so? Maybe the mystery you yearn for has just been moved to its proper location... Note that there are no triangles in the world. There is no physical object that can faithfully make a perfect triangle. Yet an (Euclidean) triangle comfortably exists in your mind, you understand triangularity - and a quick drawing of something roughly like a triangle is all it takes for that. Depending on the quality of your education, you may even be able to manipulate triangles in sophisticated ways, like proving that its inner angles sum to 180 degrees (or even more sophisticatedly, that if they don't this is a measure for non-Euclideaness).
The Platonic realm is in your head. Or is it? It is really quite mysterious how the matter in your head can represent points that have no extent, lines that meet in infinity, and any number of completely clear mathematical concept that are derivable from the world, but not realisable in the world but in some approximation. Is the matter in your head special? Or is there more to you than just the matter in your head?
The traditional view (Thomist at least, in this case deviating from the classical view) is that understanding, while entirely built on sense information, requires operations that go beyond what matter can support. Thus a rational soul (form) - unlike the souls (forms) of stones, plants and animals - must have some immaterial support. It is this part that can sustain activity past the death of the body, whereas the souls (forms) of stones, plants and animals die with the body they shape. It is this part that is made "in the image and likeness of God": it is an immaterial spirt, like God is an immaterial Spirit, and it can "read out of" the world in understanding what God "put into" the world in creation.
Platonism is wrong in this picture not so much in its "grand view", but rather in ignoring the piecemeal construction of the Platonic realm in the understanding of human beings operating on sense data. The realm of triangles and number lines and Fermat's theorem is immaterial indeed, but has its home in the rational understanding of human beings.
(For the specialists: of course one can built a classifier that "machine-learns" to recognise all sorts of imperfect triangles in pictures as "triangle class". But in an unsupervised classifier this amounts to detecting statistical similarities, if in a sophisticated manner. It does not refer to a conceptual realisation that a triangle is a plane figure with three sides and three angles. If such a realisation is part of the classifier, then it is built in by the programmer as a shortcut and thus not machine-learned but taught-to-machine.)
Posted by Dafyd (# 5549) on
:
quote:
Originally posted by IngoB:
If God had created a universe with exclusively square Euclidean circles, then it would be obvious to us that round Euclidean circles are impossible.
It is however not possible to introduce square Euclidean circles to this universe without fundamentally altering it. Basically it would become another kind of universe, rather than this one.
In theory a universe could use a non-standard metric. So instead of distance being: root (x^2 + y^2 + z^2) for arbitrary orthogonal axes; distance could be: maximum (x, y, z) for absolute axes.
If on the other hand Euclidean includes distance as in our universe I have problems making sense of the proposal.
No miracle, not even the incarnation, is logically impossible. For that matter, while the Trinity strains our comprehension it doesn't utterly suspend it. 'Logic and mathematics don't apply to God as God is in God's own being' is not a theologically acceptable defence of the doctrine.
The relation between God and mathematical truths may be beyond our comprehension. But beyond our comprehension isn't the same as within our comprehension but logically incoherent.
Posted by IngoB (# 8700) on
:
Dafyd, I think you are confusing creation-intrinsic properties with God. Maths and logic are creatures (or really something like constructions principles used for all creatures). They do not constrain God other than within creation, i.e., God's creation is coherent so we find that maths and logic are univers(e)al. But our observations allow no conclusions about God Himself other than in the act of creation. We simply have no access to that, and extrapolating beyond creation other than by negation is not warranted.
And please note that in no way, shape or form have I used this to defend the Incarnation, explain miracles or whatever. Those are entirely separate concerns. They are in fact arguments about what is possible within this universe. But my point is that what is possible in this universe is a property of this universe. It is not something extrinsic to the universe that constrains the creativity of God. It is simply a specific aspect of creation. When we talk about this or that being impossible even for God, we are actually making coherence arguments about God. We are world-constraining God. That is fair enough in the sense that God has made the world He has made, and in that way has constrained Himself. But it is false if we project these constraints beyond this world, and then assume God had to make this world in this way because of these constraints. There are no constraints external to God, there are also no constraints internal to God. But God is both eternal (unchanging) and pure act, so His completely free action is perfectly self-constraining. He does what He does, exactly, He never does otherwise. Still, there is a virtual perfect freedom there. He does not do because He can do no other. He can do whatever He wants, but He has done this.
Posted by orfeo (# 13878) on
:
quote:
Originally posted by IngoB:
But my point is that what is possible in this universe is a property of this universe. It is not something extrinsic to the universe that constrains the creativity of God.
Yes. Occasionally I'm getting the sense of people thinking "first there was mathematics, and then the universe came into being and lo and behold, it conformed to mathematics".
Which is putting things the wrong way around. Mathematics is the way it is because it was designed to describe the universe within which mathematics was conceived.
Posted by Dafyd (# 5549) on
:
quote:
Originally posted by IngoB:
That is fair enough in the sense that God has made the world He has made, and in that way has constrained Himself. But it is false if we project these constraints beyond this world, and then assume God had to make this world in this way because of these constraints. There are no constraints external to God, there are also no constraints internal to God.
I think it's wrong to talk about logic or mathematics as if they're constraints.
Such talk is certainly metaphorical.
A chain might constrain a dog from doing things it might otherwise do if the chain were not there. The chain exerts causal force, such that the dog does not do what it might wish.
We can talk in an extended fashion of the law constraining the government or police from acting high-handedly. In an even more extended sense, we can talk of relativity constraining objects from accelerating past the speed of light.
However, the laws of logic and of mathematics are unlike. They have no causal force. They aren't entities of that kind. Nor is there anything that could happen that might happen if it were not for them. To say that a state of affairs cannot co-exist with a contradictory state of affairs is not to assert a constraint upon possible states of affairs. The fact that 2+2=4 does not prevent us from putting two pairs of apples together and getting five.
That is a longer way of putting C.S.Lewis' point that we do not make a meaningful sentence out of nonsense by putting the words 'God can' in front of it; nor indeed is nonsense preceded by 'God can't' meaningful in the way that 'I can't leap over tall buildings' is meaningful.
I don't know that we're able to put the question in an unconfused fashion. But the question of whether logic constrains (or doesn't constrain) God is I think clearly the wrong of way of thinking about it.
Posted by Dafyd (# 5549) on
:
quote:
Originally posted by orfeo:
Which is putting things the wrong way around. Mathematics is the way it is because it was designed to describe the universe within which mathematics was conceived.
There are true statements in mathematics which do not describe the universe in any straightforward fashion. For every prime number there is another larger prime number, would be an example.
Posted by Doc Tor (# 9748) on
:
quote:
Originally posted by Dafyd:
No miracle, not even the incarnation, is logically impossible.
Yes, but.
God became man. Not something we couldn't comprehend, but something we could.
Posted by agingjb (# 16555) on
:
I said "if my feeling that maths is independent of the material world is shot down."
Which was a way of saying that I continue to view maths as independent of the material world. Certainly the world is effectively, but not precisely, described by some mathematics,
Euclidean geometry, and a lot else, is coherent, but not an exact reflection of the world.
As for the actual ontology, I don't know.
Posted by IngoB (# 8700) on
:
quote:
Originally posted by Dafyd:
I think it's wrong to talk about logic or mathematics as if they're constraints. Such talk is certainly metaphorical.
No, it is not at all metaphorical, other than in the remote sense that historically the word may have been abstracted from a physical device. It now simply means a limitation or restriction. If you argue that certain things cannot happen, you are constraining the possibilities. You do argue this way using maths and logic, hence they are actual constraints.
quote:
Originally posted by Dafyd:
They have no causal force.
Does a blueprint have "causal force"? Not in the direct sense of in and by itself bringing about a building. But certainly in the indirect sense of governing the actions that bring about the building. Maths and logic are a significant part of the blueprint God created for the universe we find ourselves in. Or more precisely, maths and logic are derived from what we understand of this blueprint through observing the world. But they form a system, a language, which allows us to go beyond this world imaginatively - just like once one has understood what signifies a door in a blueprint, one can add many doors everywhere to it, even if that is not "realistic" and will not be "realised".
quote:
Originally posted by Dafyd:
Nor is there anything that could happen that might happen if it were not for them. To say that a state of affairs cannot co-exist with a contradictory state of affairs is not to assert a constraint upon possible states of affairs. The fact that 2+2=4 does not prevent us from putting two pairs of apples together and getting five.
It is indeed the mathematical 2+2=4 that stops you from (non-miraculously) adding two pairs of apples together and getting five (and a miracle does not change this, it merely adds an apple without physical cause). It is indeed the logical principle of non-contradiction that stops contradictory states from co-existing. That nothing else can happen is not an indication of "no force", but rather of "perfect force". The force is measured against virtual possibility though, similar to how God's eternal and perfect freedom is measured against virtual possibility.
If I say "What number squared gives two?" and then restrict the answer to the integers, I have performed a real action, even though nothing physical has happened. Something has become impossible by virtue of this action, which if I had allowed real number would not have been. I have in fact stopped you from finding a square root in this case, though I did nothing actively that stayed your hand. Rather I have reduced your action space with a constraint, as compared to the virtual possibilities offered by real numbers. While there is nothing you can point to in physical space that exerts causal force on you, nevertheless my imposition of a constraint is what stops you from finding this square root.
Of course, my analogy works for you, because you can in fact side-step my constraint and work out an answer, and from this grander perspective judge what I'm doing to you by selecting the integers. We cannot similarly step outside of maths and logic, but this inability does not mean that the pattern does not work. It just means that we cannot imagine the virtual space against which the constraints of maths and logic imposed by God would have to be measured.
quote:
Originally posted by Dafyd:
That is a longer way of putting C.S.Lewis' point that we do not make a meaningful sentence out of nonsense by putting the words 'God can' in front of it; nor indeed is nonsense preceded by 'God can't' meaningful in the way that 'I can't leap over tall buildings' is meaningful.
You are reiterating something that I have already dismissed, and I think successfully. We are simply not discussing whether God can create mathematical or logical impossibilities in this world. We are discussing whether mathematics and logic are something that reigns over this world from the outside, or is an aspect of this world on the inside. In both cases one can argue against God creation square Euclidean circles and the like in this universe. Since that is true for both cases, such argument cannot distinguish between the cases. There simply is no mileage in these kind of considerations for the actual question I'm discussing.
quote:
Originally posted by Dafyd:
But the question of whether logic constrains (or doesn't constrain) God is I think clearly the wrong of way of thinking about it.
If you think so, then you are clearly mistaken.
Posted by orfeo (# 13878) on
:
quote:
Originally posted by Dafyd:
quote:
Originally posted by orfeo:
Which is putting things the wrong way around. Mathematics is the way it is because it was designed to describe the universe within which mathematics was conceived.
There are true statements in mathematics which do not describe the universe in any straightforward fashion. For every prime number there is another larger prime number, would be an example.
Prime numbers describe situations in which an attempt to divide a cache of things evenly is doomed to failure unless you can give everyone exactly 1 each.
Are we guaranteed to avoid this problem by adding extra stuff to the cache? No.
[ 02. June 2015, 16:07: Message edited by: orfeo ]
Posted by Dafyd (# 5549) on
:
quote:
Originally posted by Doc Tor:
quote:
Originally posted by Dafyd:
No miracle, not even the incarnation, is logically impossible.
Yes, but.
God became man. Not something we couldn't comprehend, but something we could.
I can't see why that merits a 'but'.
Posted by Dafyd (# 5549) on
:
quote:
Originally posted by orfeo:
quote:
Originally posted by Dafyd:
There are true statements in mathematics which do not describe the universe in any straightforward fashion. For every prime number there is another larger prime number, would be an example.
Prime numbers describe situations in which an attempt to divide a cache of things evenly is doomed to failure unless you can give everyone exactly 1 each.
Are we guaranteed to avoid this problem by adding extra stuff to the cache? No.
The mathematical statement remains true even once the number of items in the cache is larger than the number of items in the physical universe.
Or consider: the square root of two is not a rational number. For any physical purpose you can find rational numbers that approximate the square root of two sufficiently closely to make no practical difference. That being the case, what does it mean to say it isn't rational, assuming one is making statements about the physical universe?
Posted by Doc Tor (# 9748) on
:
quote:
Originally posted by Dafyd:
quote:
Originally posted by Doc Tor:
quote:
Originally posted by Dafyd:
No miracle, not even the incarnation, is logically impossible.
Yes, but.
God became man. Not something we couldn't comprehend, but something we could.
I can't see why that merits a 'but'.
Because God didn't come to us as a logical impossibility? He could have, reasonably, chosen to manifest as a hyperintelligent shade of blue, and expected us to have a relationship with that, but no.
So a miracle has to be comprehensible in order for it to be comprehended. Logical impossibilities aren't in that category. (Though, I suppose that depends on how logical you are. Coming back from the dead after 3 days is a big ask, but it is comprehensible.)
Posted by Enoch (# 14322) on
:
It's over fifty years since I did any maths, but from recollection 'rational' in the context 'rational number' is being used in a peculiar mathematical or Humpty Dumpty sense that has nothing much to do with its normal meaning.
From memory, I also seem to recall that even in creaking, antiquated, Euclidean geometry, it's easy to draw √2, even though it cannot be expressed as a rational number.
Posted by Lamb Chopped (# 5528) on
:
"Rational" AFAIK means "capable of being expressed as a fraction/ratio" (or as a terminating or repeating decimal, something like 3.14 or 7/22 or 3/141414141414...
but NOT pi, which never repeats and never terminates. Making it irrational)
Posted by Dafyd (# 5549) on
:
quote:
Originally posted by IngoB:
quote:
Originally posted by Dafyd:
I think it's wrong to talk about logic or mathematics as if they're constraints. Such talk is certainly metaphorical.
No, it is not at all metaphorical, other than in the remote sense that historically the word may have been abstracted from a physical device. It now simply means a limitation or restriction. If you argue that certain things cannot happen, you are constraining the possibilities. You do argue this way using maths and logic, hence they are actual constraints.
No, properly I'm arguing that states of affairs that involve logical contradictions do not even reach the status of 'certain things'. They are not certain things that cannot happen because logic prevents them from happening. The reason they cannot happen is that they do not qualify as certain things at all.
For example, the English phrase 'a rock God cannot lift' is not a thing. It is not even a possible but not actual thing. It is not even a logically impossible thing. It just is not a candidate thing at all. It is a shadow cast by a string of words.
quote:
quote:
Originally posted by Dafyd:
They have no causal force.
Does a blueprint have "causal force"? Not in the direct sense of in and by itself bringing about a building. But certainly in the indirect sense of governing the actions that bring about the building. Maths and logic are a significant part of the blueprint God created for the universe we find ourselves in. Or more precisely, maths and logic are derived from what we understand of this blueprint through observing the world. But they form a system, a language, which allows us to go beyond this world imaginatively - just like once one has understood what signifies a door in a blueprint, one can add many doors everywhere to it, even if that is not "realistic" and will not be "realised".
I deny that mathematics has any indirect causal force, even in the sense of governing actions. There is no stage in the operation of physical laws in which mathematics corresponds to consulting the blueprint, or even to looking at the blueprint before starting.
If I understand your analogy correctly, I do not think maths and logic are a blueprint in the sense you're using in the analogy. A blueprint I take to be the schematic diagram that is followed when constructing an object. That corresponds to the empirical facts of our universe. But I believe there is a sharp distinction between empirical facts and logical considerations. (It's all in Chesterton. Does nobody else read the Ethics of Elfland these days?)
quote:
quote:
Originally posted by Dafyd:
Nor is there anything that could happen that might happen if it were not for them. To say that a state of affairs cannot co-exist with a contradictory state of affairs is not to assert a constraint upon possible states of affairs. The fact that 2+2=4 does not prevent us from putting two pairs of apples together and getting five.
It is indeed the mathematical 2+2=4 that stops you from (non-miraculously) adding two pairs of apples together and getting five (and a miracle does not change this, it merely adds an apple without physical cause).
Nothing stops me from adding two apples to two apples to get five apples. The reason I can't do that (and here I am stretching language in saying that 'that' is a thing I can't do) is that no action, not even actions ruled out by the make-up of this universe, can count as adding two apples to two apples to get five. Just because 'adding two apples to two apples to get five' obeys the rules of English syntax, grammar, and vocabulary, does not entail that it therefore describes something that I am stopped from doing. Language is here just spinning its wheels without doing work.
quote:
If I say "What number squared gives two?" and then restrict the answer to the integers, I have performed a real action, even though nothing physical has happened. Something has become impossible by virtue of this action, which if I had allowed real number would not have been. I have in fact stopped you from finding a square root in this case, though I did nothing actively that stayed your hand. Rather I have reduced your action space with a constraint, as compared to the virtual possibilities offered by real numbers.
You are rather vague about what you mean when you say 'restrict the answer to the integers'? Do you successfully fool me into thinking that the answer must be an integer? That would be an action with causal force. But not comparable with the fact that the square root of two has only one positive answer, which is not an integer.
Do you tell me that you will not accept any answer that isn't an integer? That has a causal effect only in a somewhat different sense. (Just as a jury's declaration that someone is innocent or guilty certainly affects what happens to that person but does not affect the facts of the matter or necessarily what anyone else believes.) But again, not comparable with the facts about the square root of two.
[ 02. June 2015, 23:27: Message edited by: Dafyd ]
Posted by orfeo (# 13878) on
:
quote:
Originally posted by Dafyd:
quote:
Originally posted by orfeo:
quote:
Originally posted by Dafyd:
There are true statements in mathematics which do not describe the universe in any straightforward fashion. For every prime number there is another larger prime number, would be an example.
Prime numbers describe situations in which an attempt to divide a cache of things evenly is doomed to failure unless you can give everyone exactly 1 each.
Are we guaranteed to avoid this problem by adding extra stuff to the cache? No.
The mathematical statement remains true even once the number of items in the cache is larger than the number of items in the physical universe.
This is a misleading claim. The mathematical statement isn't about a specific prime number, so saying "aha, I can find a prime number larger than the number of items in the physical universe" actually has nothing to do with the mathematical statement.
I am merely illustrating to you the basic point that mathematics isn't actually about abstract ideas, it's about practical problem-solving. It's designed that way. The reason there's an interest in prime numbers is because knowing when and how you can subdivide is useful for a range of problems.
Many years ago in my school days I was involved in a very high-level extracurricular maths 'class', run by people at the local university, which was basically preparation for an international maths competition. That class/competition really opened my eyes up to what mathematics is for. The competition, which was incredibly challenging, was fundamentally about throwing a problem at you and NOT giving you clues on how to solve it. The whole point was that you had to pick from the range of mathematical techniques you knew about, and work out which ones would assist you towards solving the problem.
This, I now understand, is what actual professional mathematicians do. When Shipmates declare their love of mathematics as an abstraction, it feels to me like they're completely missing the purpose of the mathematics. It's somewhat akin to admiring the prettiness of the Golden Gate Bridge while ignoring that its purpose is to make a journey from one side of the bay to the other possible.
Posted by Ricardus (# 8757) on
:
quote:
Originally posted by orfeo:
I am merely illustrating to you the basic point that mathematics isn't actually about abstract ideas, it's about practical problem-solving. It's designed that way. The reason there's an interest in prime numbers is because knowing when and how you can subdivide is useful for a range of problems.
I dunno, there were a few jokes that went around my university, such as:
A farmer goes to a mathematician and says: 'My hens have stopped laying eggs: can you help me?' The mathematician promises to go away and think about it. A few days later the mathematician returns and says: 'I've got good news and bad news. The good news is I've found a solution to get your hens laying again. The bad news is that it only works for spherical hens on a frictionless surface.'
Difference between a mathematician and an engineer: An engineer says, 'Ooh, this is an interesting algorithm for solving vector equations, I wonder if I can use it to design flight controls?' The mathematician says, 'Ooh, this is an interesting algorithm for solving vector equations, I wonder if I can generalise it to n dimensions?'
Serious point is: there seems to be a lot of stuff in maths that has no obvious purpose (calculating pi to a bazillion decimal places for example), without this automatically being a sign of bad maths. Whereas an expensively designed gadget that doesn't solve any practical problem would be seen as a sign of bad engineering.
Posted by Enoch (# 14322) on
:
Whereas an expensive machine which could do nothing but calculate π to an infinite number of places would be a waste of money.
Posted by IngoB (# 8700) on
:
quote:
Originally posted by orfeo:
I am merely illustrating to you the basic point that mathematics isn't actually about abstract ideas, it's about practical problem-solving. It's designed that way.
This is false. And I say this as a theoretical physicist working in an engineering department on biological questions. While there is a branch of mathematics that is about "practical problem-solving" - namely Applied Mathematics - that is just a branch of mathematics. And arguably Applied Mathematics is the interface to other disciplines, not the core of mathematics.
quote:
Originally posted by orfeo:
The reason there's an interest in prime numbers is because knowing when and how you can subdivide is useful for a range of problems.
Not really, no.
quote:
Originally posted by orfeo:
The whole point was that you had to pick from the range of mathematical techniques you knew about, and work out which ones would assist you towards solving the problem. This, I now understand, is what actual professional mathematicians do.
I know a large number of professional mathematicians, physicists and engineers, and this describes the intentions of engineers foremost, of physicists / applied mathematicians less so, and of pure mathematicians least. Of course, everybody who works with maths uses the mathematical techniques they have at their disposal in order make progress with their work. Considered as a general description this is hence always true. However, I assume you mean in the above a "concrete problem in the world", i.e., to prove a lemma is not the kind of problem you had in mind. Then I have to say that many a mathematician couldn't care less whether his maths solves anything concrete in the world. And in fact, the kind of solutions many mathematicians arrive at in their work are most remarkably useless for real world applications. For example, they may prove that certain equations in certain space have a solution, without being capable of actually providing any such solution, or for that matter without knowing or caring whether the mathematical space matches any part of nature in a straightforward manner.
Posted by orfeo (# 13878) on
:
No, I didn't particularly mean a "concrete problem", if what you're talking about is something that is immediate and wants a solution right now. I fully accept that mathematicians are capable of solving things without yet knowing how the solution will be useful.
Nevertheless, I see mathematics as a science, not an art, and that has implications for its purpose.
And we were getting along so well...
Posted by IngoB (# 8700) on
:
Mathematics is a science in the old sense of science as representing an organised body of knowledge, but it is not a science in the modern sense of a systematic study of the physical and natural world through observation and experiment. It is the language of much of modern science, but not a modern science itself.
Much of modern mathematics is indeed mathematics for mathematics' sake, and thus more closely corresponds to our modern concept of art.
However, I think this is really irrelevant to the topic we are discussing here. That mathematics is "world-derived" is not proven or disproven by how applied it is in practice. Just like Jabberwocky does not demonstrate that English is not dependent on the world for its semantics.
Posted by Dafyd (# 5549) on
:
quote:
Originally posted by orfeo:
quote:
Originally posted by Dafyd:
The mathematical statement remains true even once the number of items in the cache is larger than the number of items in the physical universe.
This is a misleading claim. The mathematical statement isn't about a specific prime number, so saying "aha, I can find a prime number larger than the number of items in the physical universe" actually has nothing to do with the mathematical statement.
I fail to see why the second half of the sentence follows from the first; nor is the statement the claim that one can find a prime number larger than the number of items in the physical universe (merely that there are an infinite number of them); and I'm pretty sure the mathematical statement is indeed exactly to do with the claim that one can prove the existence of a prime number larger than any number one cares to specify. There are many linguistic statements that do not do what they appear to do on the surface, but mathematical claims are not among them.
quote:
The reason there's an interest in prime numbers is because knowing when and how you can subdivide is useful for a range of problems.
I've studied maths at university level; the interest in prime numbers goes way beyond anything required for those problems. (In the last few decades it became relevant to cryptography, but that's more recent than the work done.)
I've been in a conversation in which it was discussed whether it was mathematically possible to cut a cake in two in such a way that both parts have the same shape and mass as the original. The answer is yes, if you accept the axiom of choice. I am pretty sure that will never be a physical possibility.
Posted by mousethief (# 953) on
:
quote:
Originally posted by orfeo:
Nevertheless, I see mathematics as a science, not an art, and that has implications for its purpose.
But, forgive me, is how YOU see mathematics at all relevant? Surely the relevant question is how mathematicians see mathematics. And there is a fair split among mathematicians as to whether pure mathematics or applied mathematics is the real deal (to put it colloquially). It's not quite as simple as you make it out to be.
Posted by Jack o' the Green (# 11091) on
:
quote:
Originally posted by mousethief:
quote:
Originally posted by orfeo:
Nevertheless, I see mathematics as a science, not an art, and that has implications for its purpose.
But, forgive me, is how YOU see mathematics at all relevant? Surely the relevant question is how mathematicians see mathematics. And there is a fair split among mathematicians as to whether pure mathematics or applied mathematics is the real deal (to put it colloquially). It's not quite as simple as you make it out to be.
Absolutely. Mathematicians like Kurt Gödel or Sir Roger Penrose would see mathematics as something which has practical applications, but also contains 'truths' beyond its practical application, and even beyond the physical universe. Something to be enjoyed for its own sake, and which provides aesthetic pleasure e.g. the Mandelbrot Set.
One of the ideas which I mentioned in my opening post was the equating of necessary truth with necessary existence which means that some would argue that the numbers which are the mathematical foundation of the world are self existent and don't require any further explanation. Roger Penrose more or less equates the concepts of necessary truth and necessary existence in his book "The Emperor's New Mind". I don't find the the idea convincing, but was wondering what others thought.
Posted by itsarumdo (# 18174) on
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Interesting how Penrose Tiling was started as a fun thing and then it became useful
Posted by Jack o' the Green (# 11091) on
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Originally posted by itsarumdo:
Interesting how Penrose Tiling was started as a fun thing and then it became useful
I did write to him asking if he'd do my bathroom, but he never responded.
Posted by mousethief (# 953) on
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Originally posted by Jack o' the Green:
Roger Penrose more or less equates the concepts of necessary truth and necessary existence in his book "The Emperor's New Mind". I don't find the the idea convincing, but was wondering what others thought.
I read "The Emperor's New Mind" so long ago I don't remember much about it. ![[Frown]](frown.gif)
Posted by orfeo (# 13878) on
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Originally posted by mousethief:
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Originally posted by orfeo:
Nevertheless, I see mathematics as a science, not an art, and that has implications for its purpose.
But, forgive me, is how YOU see mathematics at all relevant? Surely the relevant question is how mathematicians see mathematics.
Are you proposing we close the thread until we can get some actual mathematicians in here?
Plenty of people, meanwhile, are posting about how they see mathematics. My views aren't any less relevant than those of other Shipmates.
Posted by LeRoc (# 3216) on
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orfeo: Are you proposing we close the thread until we can get some actual mathematicians in here?
(Here I am, you can go on.)
Posted by LeRoc (# 3216) on
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Suppose there is a universe V that is in some way the same as ours, but the ratio of the circumference to the diameter of a circle is 4.37505944089... there. The people in universe V call this number κ. Does π exist in this universe? I guess it depends on how you define π:- If you define π as the number 3.14159265... then yes, this number still exists in universe V. But it's a pretty ordinary number. Chances are, no-one has ever come across it.
- If you define π as the ratio of the circumference to the diameter of a circle, then yes it exists in universe V and its value is 4.37505944089...
- If you define π as "the number that people call π" then perhaps it doesn't exist in universe V.
Suppose there is a universe W that is in some way the same as ours, but there is no fixed ratio between the circumference and the diameter of a circle there. Does π exist in universe W?- If you define π as the number 3.14159265... then yes, this number still exists in universe W. But it's a pretty ordinary number. Chances are, no-one has ever come across it.
- If you define π as the ratio of the circumference to the diameter of a circle, then it doesn't exists in universe W. Or perhaps it does exist, but it is variable (say, a function of temperature).
- If you define π as "the number that people call π" then perhaps it doesn't exist in universe W.
Suppose there is a universe X that is in some way the same as ours, but its structure is such that numbers and mathematics don't really help to explain it. Perhaps poetry is a much better way to make sense of this universe. Does π exist in universe X?- If you define π as the number 3.14159265... I don't know. Does it still exist? Maybe some people in universe X have invented mathematics as something esoterical that has nothing to do with the universe, because they still find it interesting. I guess than then the number exists. But if no-one has ever thought of numbers, because they have no use for them? Does a falling tree make a sound if no-one hears it?
- If you define π as the ratio of the circumference to the diameter of a circle, then I think you can forget about it.
- If you define π as "the number that people call π", hmmm... Perhaps if someone from universe X writes a poem saying "If a ryeiapt would exist or a number (neither of these words have meaning in universe X)/ I would call it π to do you wonder" (I didn't say he was a good poet )
So to me, the question "does π exist independently of our universe?" depends on how you define π.
Posted by agingjb (# 16555) on
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But our universe is locally non-Euclidean (not by much, of course).
Posted by LeRoc (# 3216) on
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agingjb: But our universe is locally non-Euclidean (not by much, of course).
I know. I guess that in our universe, it makes sense to define π as "the ratio between the circumference and the diameter of a circle if our universe were Euclidian, ergo 3.14159265..."
This makes historical and practical sense. Centuries ago, when we first defined π we thought our universe was Euclidian. And for most practical purposes, the difference is negligible. So, in practice we just say "π the ratio between the circumference and the diameter of a circle" and leave the last part to the pedants.
Things become different whhen we describe things like the geometry around a black hole of course. However, in this case it still makes practical sense to keep π=3.14159265... We just need to remember that it isn't the ratio between the circumference and the diameter of a circle there.
Just for fun, suppose we discovered aliens living very close to a black hole (in our universe) and we had intensive contact with them. Suppose that the ratio between the circumference and the diameter of a circle would be 4.37505944089... where they live.
This would change a bit what we would consider "practical", and hence we'd need to take another look at how we define π. My guess is that we wouldn't just say "π is the ratio between the circumference and the diameter of a circle" anymore, but we'dd add "on Earth". And we'd choose another Greek letter for the number 4.37505944089...
Posted by agingjb (# 16555) on
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I suspect that e is a more interesting case.
Of course for Athanasians the first three non-negative integers necessarily exist, which, I wonder, implies maths up to the reals, and the first real question becomes the Continuum Hypothesis.
Posted by LeRoc (# 3216) on
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Originally posted by agingjb:
I suspect that e is a more interesting case.
Yes! A much more interesting number, in various ways!
It's fun to speculate about universes where a number exists with some (all?) of the characteristics of e but with a different numerical value. I can think of various ways of doing this.
Posted by IngoB (# 8700) on
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If you want interesting, consider
e^(iπ)+1=0
the two most famous transcendental numbers, e and π, the imaginary unit i, the multiplicative identity 1, and the identity element of addition 0, all in one neat equation. I can totally see that one as a creative act of God...
Posted by LeRoc (# 3216) on
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Originally posted by IngoB:
If you want interesting, consider
e^(iπ)+1=0
the two most famous transcendental numbers, e and π, the imaginary unit i, the multiplicative identity 1, and the identity element of addition 0, all in one neat equation. I can totally see that one as a creative act of God...
Oh yes. I don't see why you give such complex proofs of the existence of God when this would have sufficed
Posted by Dafyd (# 5549) on
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Originally posted by LeRoc:
It's fun to speculate about universes where a number exists with some (all?) of the characteristics of e but with a different numerical value. I can think of various ways of doing this.
The problem with supposing unvierses with alternative values of pi and e is that their definitions do not determine their values arbitrarily. For example, the Basel identity (pi^2 / 6 = the sum of 1/x^2 over all the integers x) is an arithmetic formula whose value Euler proved by reference to geometric concepts). Likewise, e is the sum of x^n/n! over all the integers n where x = 1. Supposing a universe in which they have different values requires us to suppose either that the proofs of those identities are false in that universe, or that those sums come to a different result. My mind isn't up to supposing either.
Posted by agingjb (# 16555) on
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Originally posted by IngoB:
If you want interesting, consider
e^(iπ)+1=0
the two most famous transcendental numbers, e and π, the imaginary unit i, the multiplicative identity 1, and the identity element of addition 0, all in one neat equation. I can totally see that one as a creative act of God...
not to mention the operations of addition, multiplication, exponentiation, and equality in there as well.
Posted by LeRoc (# 3216) on
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Dafyd: Supposing a universe in which they have different values requires us to suppose either that the proofs of those identities are false in that universe, or that those sums come to a different result. My mind isn't up to supposing either.
Mine is ![[Big Grin]](biggrin.gif)
Posted by agingjb (# 16555) on
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Well yes, can we have some clues about this alternate maths?
Posted by LeRoc (# 3216) on
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agingjb: Well yes, can we have some clues about this alternate maths?
I don't know. It's not easy to think about other universes, our minds are so much geared to this one.
As a (relatively) easy example for myself, I try to think about universes where if you put one apple "o" and another apple "o" together, you have three apples "ooo". The third apple doesn't spontaneously appear from somewhere; such a conclusion would be based on how our universe works: assumptions about time, about causality, about where things come from ... that don't work in the other universe. It's simply how that other universe ticks.
(PS I tried if I could post an Apple character here but sadly it didn't work. Maybe it works for Shipmates with a Mac.)
Posted by Jack o' the Green (# 11091) on
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Originally posted by LeRoc:
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agingjb: Well yes, can we have some clues about this alternate maths?
I don't know. It's not easy to think about other universes, our minds are so much geared to this one.
As a (relatively) easy example for myself, I try to think about universes where if you put one apple "o" and another apple "o" together, you have three apples "ooo". The third apple doesn't spontaneously appear from somewhere; such a conclusion would be based on how our universe works: assumptions about time, about causality, about where things come from ... that don't work in the other universe. It's simply how that other universe ticks.
I think that is the fundemental sticking point. Although I can see how breaking some 'rules' of mathematics can be both intelligible and constructive eg imaginary numbers, I can't see how 1 positive value plus another positive value can equal anything other than 2 positive values ie 1+1=2.
Posted by Ricardus (# 8757) on
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Originally posted by LeRoc:
Suppose there is a universe V that is in some way the same as ours, but the ratio of the circumference to the diameter of a circle is 4.37505944089... there.
On what grounds would it be called a circle?
Posted by LeRoc (# 3216) on
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Ricardus: On what grounds would it be called a circle?
For example, on the ground that it is the set of points that have the same distance to a fixed point (the easiest way to do this is to change the concept of 'distance').
Posted by mark_in_manchester (# 15978) on
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The value of pi is defined (or at least implied) in the bible at 1 kings 7:23. pi is thus a biblical concept.
I'm a (very) late-comer to this thread, but I do some bronze casting in my back garden. This bronze tank of Huram's is a f*cking monster. He must have had hundreds of men blowing fires with little ceramic crucibles in them, pouring the mold in a constant stream of men running about in sandals with said orange-hot crucibles so as to avoid it going off mid-pour and cracking. I don't think this got made without a lot of death! And how the hell did he get the massive fancy pattern out of the sand without it all breaking up?
Posted by IngoB (# 8700) on
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Originally posted by Jack o' the Green:
I think that is the fundemental sticking point. Although I can see how breaking some 'rules' of mathematics can be both intelligible and constructive eg imaginary numbers, I can't see how 1 positive value plus another positive value can equal anything other than 2 positive values ie 1+1=2.
But what exactly does that prove? If, as I believe, the rules of logic and mathematics are arbitrary in principle, but eternally fixed by God for creation in practice, then you have here simply declared that you are part of this universe, and your cognition has been shaped by it. You cannot really suggest a test that would decide between maths and logic as unchangeable per se, and unchangeable as willed by God, other than hoping that God will create something governed by other maths and logic and that you will somehow experience that. An observational test is hence at least delayed till the Second Coming (which may or may not include such changes in the "New Creation").
What you can however think about is what the claim that maths and logic are unchangeable per se would mean for God. I think that this is incompatible with His status as Creator. If the Creator is limited by anything in any way, we can always say: and who or what established those limits? For that will be the true Creator, with whatever operates within those limits being demoted to the status of demiurge...
One could propose maths and logic as God's means of panentheism and say that they are what they are because God is what He is. But still, that doesn't work for me. If we find structure, any structure, that requires explanation. And tracking this back to God we must find that He does the explaining, rather than being explained.
Posted by mark_in_manchester (# 15978) on
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And another thing...
Dutch Christian philosopher Hermann Dooyeweerd has something to say about all this when he decomposes reality into his 'modal aspects'.
It's a bit like Fourier decomposition of complex waveforms into orthogonal basis functions (sine waves, usually - e^jwt and all that), the decomposition of taste into salt / sweet / sour / bitter / umami (did I get that right?) or the decomposition of space into x-y-z or r-phi-theta coordinates.
Dooyeweerd suggests doing this with *everything*, into about 15 orthogonal dimensions which start out with basic things like numerical aspect 'how many x', physical aspect 'what is x Young's modulus', then moves 'up' to more complex properties like biological aspect 'is x alive, and how', then things like aesthetic aspect 'is x nice to look at', juridical aspect 'who does x belong to' etc etc.
The idea is that you can change any property in any aspect without changing any of the others - which is the idea in any orthogonal decomposition.
The importance to this discussion includes the fact that sitting at the 'top' of (the Christian) Dooyeweerd's list is the 'pistic' aspect - that to do with faith, and relating the aspects (the created order) to the uncreated and eternal God who created them. But a key thing for me was realising that the temptation to move beyond a useful decomposition to an attempt to make one aspect *responsible* for the creation of other aspects, or existing outside the created order (as OP) makes as much sense as saying that x is necessary before you can have y and z - which is clearly daft. It is, basically, a temptation to idolatry, which idea in this context excites me in a way that bronze calves (not withstanding my last post) has never done.
Well, my kids are up and I have to admire their hair styles, so no more chance to think about this unless anyone else finds it interesting.
Posted by LeRoc (# 3216) on
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Jack o' the Green: I think that is the fundemental sticking point. Although I can see how breaking some 'rules' of mathematics can be both intelligible and constructive eg imaginary numbers, I can't see how 1 positive value plus another positive value can equal anything other than 2 positive values ie 1+1=2.
The thing is that I don't trust "what I can see" as a limit of what is possible in a hypothetical alternative universe.
My physical brain is obviously part of this universe: it follows its physical laws and it has evolved mostly to make sense of this universe. In the hypothetical case that there is some other universe, I see no intrinsic reason why that universe would follow the same laws. So my brain probably wouldn't be geared towards it, or the right instrument to understand it.
I do believe that my mind (which in my view isn't the same thing) can transcend this universe a bit. Through reasoning, through spirituality, through art, through contemplation ... I feel that we can reach out of this universe somewhat. However, I have no reason to believe that this goes very far.
So, I do not accept "what my mind can think" of as an upper limit of what is possible in a hypothetical alternative universe.
Posted by Dafyd (# 5549) on
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Originally posted by LeRoc:
So, I do not accept "what my mind can think" of as an upper limit of what is possible in a hypothetical alternative universe.
Whereof we cannot speak thereof we must be silent.
If you can't think it you can't think it may be possible in another universe.
Posted by LeRoc (# 3216) on
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Dafyd: Whereof we cannot speak thereof we must be silent.
Of course, I always must be careful when speaking about things outside of our universe. Our language is mainly an instrument to try to understand our universe, so it doesn't have to be applicable outside of it. For example, when I talk about things outside our universe, even the word 'outside' is problematic. It doesn't have to be applicable; it (almost) certainly doesn't have a geographical meaning.
However, in talking about these things I have to use language. Otherwise I might as well join the 8th Day forum about silence. So I use these words and cross my fingers that they will convey my meaning somewhat. I could put them all between scare quotes, but there are so many of them that my posts would become a raison loaf. I can assure you that I'm making scare quotes with my fingers besides my computer screen when I write them.
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Dafyd: If you can't think it you can't think it may be possible in another universe.
I said that I don't accept what I can think of as an upper limit of what is possible in another universe. I didn't say "what I can think of will never be applicable in another universe".
In fact, I explicitly said that I believe that my mind can think some things outside of our universe, somewhat.
(And BTW, your self-referential replies are getting a bit stale. I might stop reacting to them some time in the future.)
Posted by mousethief (# 953) on
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I don't understand at all how Dafyd's post, which I quote here in full, is self-referential. Perhaps somebody can explain this.
quote:
Originally posted by Dafyd:
quote:
Originally posted by LeRoc:
So, I do not accept "what my mind can think" of as an upper limit of what is possible in a hypothetical alternative universe.
Whereof we cannot speak thereof we must be silent.
If you can't think it you can't think it may be possible in another universe.
That's just it. If you can't think it, if it's logically incoherent, then all you're doing is adding "in another universe" to a meaningless concatenation of words.
"Maybe in another universe, God can make a rock so big he can't lift it" is just as meaningless as "Can God make a rock so big he can't lift it?" Putting it in another universe doesn't change the meaninglessness a whit.
Posted by LeRoc (# 3216) on
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Originally posted by mousethief:
I don't understand at all how Dafyd's post, which I quote here in full, is self-referential. Perhaps somebody can explain this.
quote:
Originally posted by Dafyd:
quote:
Originally posted by LeRoc:
So, I do not accept "what my mind can think" of as an upper limit of what is possible in a hypothetical alternative universe.
Whereof we cannot speak thereof we must be silent.
If you can't think it you can't think it may be possible in another universe.
That's just it. If you can't think it, if it's logically incoherent, then all you're doing is adding "in another universe" to a meaningless concatenation of words.
"Maybe in another universe, God can make a rock so big he can't lift it" is just as meaningless as "Can God make a rock so big he can't lift it?" Putting it in another universe doesn't change the meaninglessness a whit.
The last paragraph of Dafyd's post was self-referential.
"Can God make a rock so big He can't lift it?" isn't meaningless (in this universe). It's paradoxical, but that isn't the same as meaningless.
Posted by Dafyd (# 5549) on
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Originally posted by IngoB:
What you can however think about is what the claim that maths and logic are unchangeable per se would mean for God. I think that this is incompatible with His status as Creator. If the Creator is limited by anything in any way, we can always say: and who or what established those limits? For that will be the true Creator, with whatever operates within those limits being demoted to the status of demiurge...
I'm not convinced by this as an argument. One could equally ask what established that God is necessary? If God is logically necessary, then whatever established that necessity will be the true Creator etc etc...
I don't think that argument works obviously, but I think the 'who or what established logic and mathematics as limits' argument doesn't work either. The priority of logic over God or its reverse seems to me to be just as false a dilemma as the priority of good over God or its reverse, or whether God's nature is prior to God's free will or the other way round.
Posted by IngoB (# 8700) on
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Originally posted by mousethief:
That's just it. If you can't think it, if it's logically incoherent, then all you're doing is adding "in another universe" to a meaningless concatenation of words.
That is not true. You assume in this argument that maths and logic (M&L) is something that holds apart from its setting - i.e., irrespective of the universe wherein the logical analysis is being performed. But if M&L does depend essentially on the specific setting of the universe, then the question of what is "meaningless" is dependent on the universe in which that question is being asked. So adding "in another universe" potentially can turn nonsense into something meaningful there. It is true that you cannot think the nonsense to be meaningful as such, since your mind is tied to the local M&L. But you can think that other minds in another universe could think this nonsense to be meaningful.
quote:
Originally posted by mousethief:
"Maybe in another universe, God can make a rock so big he can't lift it" is just as meaningless as "Can God make a rock so big he can't lift it?" Putting it in another universe doesn't change the meaninglessness a whit.
Actually, this is more a case of "God-physics" being coherent across all universes to a degree. Concretely, this claim is really a "God-physics" statement that a created entity must have less "power" than the Creator. From which we can conclude that said rock is not a possible entity in any created universe. But the reason we can make reference here to all actual and potential universes is that we are really talking about God, and we that we are making two assumptions: that our logic can say meaningful things about God, and that this God is the same for all (potential) universes in a relevant sense.
It may be possible to construct "Divine kernel M&L", by which I mean that the claim that we can say something meaningful about God, and that God is the same across all possible and actual universes, could allow us to determine minimal features of M&L that must be shared by all (potential) universes created by God. But I do not think that this extends down to things like "1+1=3" or "square Euclidean circles". So if a "Divine kernel M&L" can be constructed, then it would not comprise all M&L as we know it, leaving the other M&L as universe specs to be set by God in creation. And the "Divine kernel M&L" itself would precisely not limit Divine creativity, since it simply would be a reflection of what it means to be created by God.
Posted by Eliab (# 9153) on
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Originally posted by IngoB:
That is not true. You assume in this argument that maths and logic (M&L) is something that holds apart from its setting - i.e., irrespective of the universe wherein the logical analysis is being performed. But if M&L does depend essentially on the specific setting of the universe, then the question of what is "meaningless" is dependent on the universe in which that question is being asked. So adding "in another universe" potentially can turn nonsense into something meaningful there. It is true that you cannot think the nonsense to be meaningful as such, since your mind is tied to the local M&L. But you can think that other minds in another universe could think this nonsense to be meaningful.
You and LeRoc are treating logic (and maths, as a form of logic-with-numbers) as being a feature that's designed into this universe - a constraint that might conceivably have been different. Dafyd and mousethief are treating it (in my view, correctly) more like a process than a thing - a formal way of checking whether someone is contradicting themselves.
While you are correct to point out that no experiment could discern which of those viewpoints is right, because "arbitrary logic imposed by the unchanging will of an omnipotent God" would look (to us) exactly like "logic as strictly necessary relationships between ideas", it seems to me that the Dafyd/mousethief view is to be preferred. Because we know that at least for us, in our reality, their description of what logic is is spot-on, and further is not derived from any observable physical laws of a sort that we could imagine changing, whereas we have absolutely no warrant for supposing that a reality in which logic does not apply even exists (or could exist).
But suppose I'm wrong about that, and statements like "1+1=2" are in truth dependent on being in a universe like this, and therefore are not valid in absolutely all universes. In that case, I can revise the statement to read "In a universe in which conditions A, B and C are satisfied, then 1+1=2". And that statement, if I knew what conditions A, B and C were in order to articulate them, would be a mathematically true statement in any universe, because conditions A, B and C are precisely that set of exhaustive statements of the feature of reality necessary for 1 plus 1 to be equal to 2. The statement in fact derives from a tautology - in any universe in which 1 plus 1 (always) equals 2, 1 plus 1 equals 2.
Does that challenge the sovereignty or necessary existence of God? No more, it seems to me, than does any other tautology. I don't suppose for a second that God finds the necessary truth of the statement 2=2 to be a constraint on his divine freedom, so I need not suppose that he finds 1+1=2 constraining, since "1+1" is essentially just a different way of expressing the concept "2". All universally true logical and mathematical statements would, to a sufficiently advanced mind, appear as statements of necessary relationships between ideas: one way of expressing a certain concept is equivalent to expressing the same concept in a different way: not as constraints on thinking, but as clarity of thinking. Therefore I do not think it limits God to suppose that logic is a universally applicable process, rather than an arbitrary form of reality-regulation.
Posted by Jack o' the Green (# 11091) on
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I agree with everything which Eliab has said, and couldn't have put it better.
The tautology involved in saying 1+1=2 can easily be demonstrated by writing O + O in a white board and rubbing out the + sign to get your answer. If on the number line each additional number 1,2,3,4..... is simply an addition of another single, positive whole value, then 2 by definition is 1+1. To argue that it is only so in this universe and could be otherwise in another has the danger of leading to a crisis of epistemology. If we are wrong about 1+1 necessarily equalling 2 despite the fact that we can demonstrate it in various (seemingly watertight and cogent) ways, what else are we mistaken about, and how do we distinguish between cogent and faulty arguments? Many of the arguments for God's existence used by thinkers like Aquinas are only valid if we assume that our ability to reason has some objective validity beyond our particular psychological or neurological make-up. A desire to make God free of all constraints (which in my view don't exist anyway) seems to endanger the very foundation of natural philosophy since any reasoned argument could simply be met with the retort that "you are bound to reason this way and are unable to comprehend other alternatives because of the nature of your conditioned existence." Humility in the face of our limitations and the vastness of reality is one thing. Intellectual and rational suicide is quite another.
I have absolutely no argument with the idea that an unconditioned, infinite intellect can conceive coherent ideas, definitions, concepts, abstractions etc of which we have absolutely no inkling or understanding, and can create an alternative universe which reflects these. It just seems absurd that 1+1=3 could be part of that.
1+1 necessarily equalling 2 isn't a limitation on God, it is simply an intrinsic result of the freely chosen definitions God (or humans) have ascribed to the terms 1, +, and 2. They are conditioned terms, therefore the result of their interaction is going to be fixed.
Posted by itsarumdo (# 18174) on
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isn't "1+1=2" implicit in the way that 1 is defined?
for instance we know that 1 male + 1 female may = 2, but also may = 3 or more.
And 1 apple + 1 orange = 2 Fruit...
It 's necessary in defining 1 such that it can be added to another 1 that it is as universal and non-specific as possible. Which leads to an interesting question - is there something even more universal than an integer?
Anyhow, numeric 1 is ultimately universal, and also there is also a prohibition that 1 cannot interact with any other 1 outside the limits of addition - so they cannot copulate or otherwise operate on each other to alter their state.
Which leads to mathematics and logic - no matter how large they become - only being a tiny subset of the universe unless they also become capable of describing self-altering interactions.
Posted by IngoB (# 8700) on
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quote:
Originally posted by Eliab:
You and LeRoc are treating logic (and maths, as a form of logic-with-numbers) as being a feature that's designed into this universe - a constraint that might conceivably have been different. Dafyd and mousethief are treating it (in my view, correctly) more like a process than a thing - a formal way of checking whether someone is contradicting themselves.
The contrast you draw there is simply false. There is nothing characteristic of "a formal way of checking for contradiction" that could not also be "a feature designed into this universe". All we are saying is that the process "checking for contradiction" is not independent of the universe in which it is happening.
quote:
Originally posted by Eliab:
Because we know that at least for us, in our reality, their description of what logic is is spot-on, and further is not derived from any observable physical laws of a sort that we could imagine changing, whereas we have absolutely no warrant for supposing that a reality in which logic does not apply even exists (or could exist).
Whether "their" description is spot-on or not, it is not in contradiction to our assertion of universe-dependence. Hence you cannot use that to decide between the claims. And it is simply not true that we cannot imagine any changes to a universe that are incompatible with the maths and logic of this universe. In particular, I very much can imagine a universe in which 1+1=3. That's simple enough, all that requires is that when similar objects are brought in physical proximity (the physical antecedent to addition) another similar object appears as well. (I would add that such a rule would instantly destabilise our universe, but I do not need to be able to imagine a stable universe.)
quote:
Originally posted by Eliab:
In that case, I can revise the statement to read "In a universe in which conditions A, B and C are satisfied, then 1+1=2". And that statement, if I knew what conditions A, B and C were in order to articulate them, would be a mathematically true statement in any universe, because conditions A, B and C are precisely that set of exhaustive statements of the feature of reality necessary for 1 plus 1 to be equal to 2.
Possibly, but if so then only by virtue of (implicit) reference to God (as the origin of A, B and C perhaps). After all, you are attempting a logical argument there, and that has no necessary validity apart from this universe, unless you are discussing a feature that is necessarily the same between this universe and a different universe with potentially different logic (and such a feature would involve God at least implicitly as the only known constant across all potential universes). I'm basically denying that you can logic about other worlds, unless you prove first that you are relying on shared axioms.
quote:
Originally posted by Eliab:
I need not suppose that he finds 1+1=2 constraining, since "1+1" is essentially just a different way of expressing the concept "2".
Only in a trivial sense, if you define it to be so for these symbols. The concept of multiplicity of objects however can exist without the particular operation "addition" that we know as increasing the multiplicity of objects. I can define an operator for which 1+1=3, or let's give it a different symbol for clarity, 1(+)1=3, and then think about what it will do. You may consider this (+) operator to be silly or useless, but that judgement is really a reference to "what is the case in the world when you put objects close together", i.e., in this universe if you put one apple next to another, then you have two apples. It is not priori clear that there could not be a universe wherein the usefulness is reversed, i.e., the (+) operator describes what is happening in the world, and the + operator is silly or useless.
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Originally posted by Eliab:
All universally true logical and mathematical statements would, to a sufficiently advanced mind, appear as statements of necessary relationships between ideas...
There is only one necessary entity, God. Nothing else has necessary being, nothing else can possibly have necessary being. If there are aspects to maths and logic that are necessary in an absolute sense, then they must be in some sense an aspect of God. It may be possible that certain aspects of maths and logic can be shown to be reducible to statements about God. Then they could be necessary. Otherwise, however, they cannot be. And I think maths and logic are not entirely implicit in God. I would say that to a considerable part they are implicit in this universe, hence created, hence not necessary.
Posted by IngoB (# 8700) on
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Originally posted by Jack o' the Green:
Many of the arguments for God's existence used by thinkers like Aquinas are only valid if we assume that our ability to reason has some objective validity beyond our particular psychological or neurological make-up. A desire to make God free of all constraints (which in my view don't exist anyway) seems to endanger the very foundation of natural philosophy since any reasoned argument could simply be met with the retort that "you are bound to reason this way and are unable to comprehend other alternatives because of the nature of your conditioned existence."
This is confused. The following are two distinct claims:
- Maths and logic are the same in all possible created universes.
- The maths and logic of this universe allow us human beings to draw true conclusions about God (based on our human observations of this same universe).
They are not the same, and one doesn't follow from the other. The traditional proofs of God's existence rely on the second claim, not on the first.
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Originally posted by Jack o' the Green:
1+1 necessarily equalling 2 isn't a limitation on God, it is simply an intrinsic result of the freely chosen definitions God (or humans) have ascribed to the terms 1, +, and 2. They are conditioned terms, therefore the result of their interaction is going to be fixed.
This is true but trading on an ambiguity. Obviously you can define symbols, and operators on symbols, entirely as you like. But you are precisely not treating "1" and "2" as arbitrary symbols. You are treating them as a signifiers for specific multiplicities of objects in the world. Likewise, you are not treating "+" as an arbitrary operator defined simply by how it maps symbols to each other. You are treating it as a signifier of "heaping up" actions in this world.
Arbitrary definitions are "fixed" by themselves. But when you (falsely, as it turns out) claim that one cannot imagine any other definition to hold, you are not actually referring to an arbitrary definition (which obviously can be changed arbitrarily). You are referring to a definition that has meaning for this world. You mean that when you put one apple next to another apple, it makes sense that there are two apples. It is this sense that you "borrow" for declaring a particular mathematical definition as meaningful. But this sense is not independent of the world, it is created through and through. I see no a priori reason why this should be "fixed" by anything else than the creative will of God.
Posted by Dafyd (# 5549) on
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Originally posted by IngoB:
I can define an operator for which 1+1=3, or let's give it a different symbol for clarity, 1(+)1=3, and then think about what it will do. You may consider this (+) operator to be silly or useless, but that judgement is really a reference to "what is the case in the world when you put objects close together", i.e., in this universe if you put one apple next to another, then you have two apples.
The + operator applies in a far wider variety of contexts than merely what happens when you put two objects in close proximity. In order to say that there's two planets closer to the sun than the earth there is no need to specify that they're anywhere near each other.
Likewise, measuring a piece of wood as constructed of two 1m segments; or as two 1cm segments for that matter; or a field as consisting of two 1 acre portions...
But there's a lot more questions. For example, what happens when you put one baker's dozen of apples next to one baker's dozen of apples? Do you get thirty nine apples? If not, then 1 isn't the multiplicative identity. What happens if you put twelve apples next to fourteen? What happens if you take away one apple from the fourteen, and then take away another apple? How many apples have you taken away?
What if you add one apple and then add another apple? Have you got two or three or four more than you started with?
What is 1(+)2? Can you define (-) as an operator? Can you find a number x such that 1 (+) x = 2?
Perhaps you can construct a system around (+) as an operator. But I suspect that the digit 1 in such a system would be incommensurable with 1 in our system in such a way that it's not clear that calling it 1 is anything other than a homonym.
[ 06. June 2015, 15:23: Message edited by: Dafyd ]
Posted by LeRoc (# 3216) on
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Eliab: You and LeRoc are treating logic (and maths, as a form of logic-with-numbers) as being a feature that's designed into this universe - a constraint that might conceivably have been different. Dafyd and mousethief are treating it (in my view, correctly) more like a process than a thing - a formal way of checking whether someone is contradicting themselves.
While you are correct to point out that no experiment could discern which of those viewpoints is right, because "arbitrary logic imposed by the unchanging will of an omnipotent God" would look (to us) exactly like "logic as strictly necessary relationships between ideas", it seems to me that the Dafyd/mousethief view is to be preferred.
But that's exactly my point: we don't know. We don't know if the IngoB/LeRoc view is true, or if the Dafyd/mousethief view is true. Like you said, there's no way of knowing that.
I'll try to explain why I think the L/I position is true. Note, this is not proof. I can't prove it. But I want to try to explain why I think this way.
It is interesting to note that both IngoB and me are trained particle physicists (my life took a different turn afterwards). I feel that when you look at the universe in such a close way, it is surprising how much the structure of the universe and mathematics seem to be connected. They seem to be interwoven in a very intimate way. I'm sure I'm not the first physicist who as been in awe at this. To many people, the connection between math and the universe seems to have something transcendent.
So, what I feel (once again, I cannot prove it), is that God created the universe and mathematics together. They are linked in such an intricate way.
Sure, if there are other universes, He could have created all of them using math. But He could have used something else to create another universe.
I can imagine different ways of creating another universe. And if I, humble limited little LeRoc, can create other ways of doing it, then surely God can. (And let's face it, if God imagines creting another universe, it's already there.)
This is why I feel that other universes needn't be such that our logic is a valid way to describe them.
But once again: there's no way of knowing that. Let me explain why this is important.
I believe that God exists partly outside of our universe. I think of this in an panentheistic way. If you want to, we can see Heaven as outside of our universe (that seems like orthodox Christian thinking to me) and in some way, part of God lives there. This is very simplistic and there are lot of scare quotes in here, but I hope this will do.
So, does our logic apply to Heaven? And to God, who partly lives there? Is the I/L position true or is the D/m position true? We have no way of knowing.
This means that every time someone on the Ship uses logical reasoning to draw a conclusion about God, this conclusion depends on whether the D/m position is true. These conclusions — which sometimes are stated with a lot of conviction — depend on something we don't know. That's not very strong, is it? These conclusions might be valid. But that's as far as I can go.
That's why every time anyone uses logic to draw a conclusion about God, my answer is: we don't know. Your conclusion might be valid, it might not be. It depends on something we don't know. And that's it.
Posted by Dafyd (# 5549) on
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Originally posted by IngoB:
The following are two distinct claims:
- Maths and logic are the same in all possible created universes.
- The maths and logic of this universe allow us human beings to draw true conclusions about God (based on our human observations of this same universe).
They are not the same, and one doesn't follow from the other. The traditional proofs of God's existence rely on the second claim, not on the first.
I am not convinced that the first doesn't follow from the second...
The second requires us to believe that there isn't and could not be any possible universe lying between us and God in which the logical principles we are using to reason to God do not apply. So we cannot be downstream of a universe in which contingent events happen without explanation. But if we can rule that out we can rule out there being any possible universe in which events happen without explanation.
Now, if you argue that those particular logical principles apply to all possible universes, but that others may not, you need to have an a priori explanation of how you know which is which - 'the ones that we need to argue to the existence of God must apply generally' is not a valid criterion if the argument is to be sound.
Posted by IngoB (# 8700) on
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Originally posted by Dafyd:
The second requires us to believe that there isn't and could not be any possible universe lying between us and God in which the logical principles we are using to reason to God do not apply.
No, it doesn't. It requires only the former (there isn't), not the latter (there could not be any). I do not believe that there is any way of asserting confidence in human reasoning based on human reasoning, since that is inherently circular (and at best one can cleverly hide the circularity). I say rather that it is an essential part of my faith that God created humans so as to be capable of correct reasoning about the universe within their limits. (And as it happens, at the edge of these limits we can argue about God - and thus according to my faith, correctly so.) Basically, this is what I think the bible means when it says that we are created "in the image and likeness" of God, namely that the immaterial rational powers of our soul are sufficiently similar to the immaterial creative powers of God for us to find and understand the truth about His creation (in principle).
Consequently, whenever I talk about the classical proofs of God, I stress that they are based on a fundamental optimism about human mental powers, and that these proofs can be refuted - or more accurately, side-stepped - by asserting that the universe at some point becomes fundamentally impenetrable to human understanding (leaving us with "brute facts" that we can only accept but not understand). It seems obvious that the human mind can see much, perhaps most, of what the universe is like. That is practically proven by our scientific and technological success. But I do not believe that we can establish the full capability of human reasoning other than by assertion based on for example faith.
So I fully accept that it could be true that the universe bottoms out of human reason somehow and somewhere, and indeed also that it could be true that this compromises my cherished proofs of God fatally. I just do not believe that this is the case. And frankly, I see no point in believing this to be the case. If you wish, this is a kind of "Pascal's wager". Though I think psychologically this is really a part of the "Zweckoptimismus" (German: optimism in order to achieve something) that has been drilled into me as scientist...
Posted by Eliab (# 9153) on
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Originally posted by IngoB:
The contrast you draw there is simply false. There is nothing characteristic of "a formal way of checking for contradiction" that could not also be "a feature designed into this universe". All we are saying is that the process "checking for contradiction" is not independent of the universe in which it is happening.
[…]
Whether "their" description is spot-on or not, it is not in contradiction to our assertion of universe-dependence. Hence you cannot use that to decide between the claims.
I concede both points. If your view is correct, logic and maths would look (in this universe) exactly as they do now, so their functionality here doesn’t prove their (meta-)universal applicability.
My argument different – and is meant to be persuasive rather than compelling. It’s that given that we can see that logic works as a universally valid process here, and doesn’t appear to be dependent on any physical peculiarities of our reality, there is no reason to doubt the general applicability of logic in other any other realities that there might be, and no warrant for assuming that a different ‘logic’ would be workable anywhere, since we can’t really conceive what that would mean.
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And it is simply not true that we cannot imagine any changes to a universe that are incompatible with the maths and logic of this universe. In particular, I very much can imagine a universe in which 1+1=3. That's simple enough, all that requires is that when similar objects are brought in physical proximity (the physical antecedent to addition) another similar object appears as well. (I would add that such a rule would instantly destabilise our universe, but I do not need to be able to imagine a stable universe.)
This is an aside, because your argument does not depend on you being able to imagine anything (and, indeed, if you could convincingly articulate a complete ‘alternative logic’, it would probably not be different enough to make your point). However you have not successfully conceptualised a universe in which 1+1=3 with that example. You have conceptualised a universe with an unusual physical law which governs how objects come into existence.
The reason is that addition is not limited to “similar objects in physical proximity”. Addition works on anything – it’s not a physically dependent process. Suppose I’m trying to communicate a secret numerical code to you and, believing I might be overheard, say “Add the first digit of my birthday to the number of girlfriends I’ve had”, if you knew me well enough you could translate that to “1+1=2”, without thinking that the things being added were physically similar or close together. In the universe you are imagining, where similar objects breed others, 1 plus 1 still equals 2, it’s just that there are complications that apply when physically enacting that fact. In this universe, bringing ‘1’ object, say, a lump of enriched uranium, into physical proximity with ‘1’ similar object, can have complications that mean you don’t end up with ‘2’ of those objects, but with something else, such as an explosion. This physical difficulty does not challenge the general view that 1+1=2.
For 1 plus 1 to equal 3 in any universe in the way that it equals 2 here, then the expression ‘1+1’ would have to be simply another way of saying ‘3’, not merely a way of generating a third of something when two of them are brought into proximity. And the problem with that is that ‘1+1’ isn’t another way of saying ‘3’ in any universe that we can conceive.
As accepted above, the failure of our minds to conceive of such a universe isn’t a proof that there no mind (or God’s mind) ever could. It doesn’t win the argument for me. It’s just that I’ll need a lot of persuading before I accept a view which must inevitably involve the possible existence of something which is meaningless and (to me, unavoidably) self-contradictory.
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Possibly, but if so then only by virtue of (implicit) reference to God (as the origin of A, B and C perhaps). After all, you are attempting a logical argument there, and that has no necessary validity apart from this universe, unless you are discussing a feature that is necessarily the same between this universe and a different universe with potentially different logic (and such a feature would involve God at least implicitly as the only known constant across all potential universes). I'm basically denying that you can logic about other worlds, unless you prove first that you are relying on shared axioms.
But it has yet to be established that logic is invalid anywhere. That’s the point under discussion.
However my argument is not limited to the logic of this universe. If your thesis is right, then I can say:
"In a universe in which conditions A, B and C are satisfied, then 1+1=2” and it will be true in any universe whatever.
The proof of that is that in this universe, where 1 plus 1 does equal 2, I could also say (again assuming that you are right):
“In a universe in which conditions D, E and F are satisfied, then 1+1=3”, and that will be true here – and everywhere else.
The statements aren’t only true in the universes where the conditions are satisfied – they are true by definition, because the conditions are defined as the conditions to be satisfied for 1+1 to equal whatever it equals under those conditions.
So even if you are right, there are universally true mathematical statements. And I think it’s stretching the point beyond breaking to say that the conditions are implicit references to God, because he’s the origin of those conditions. That’s true, because God is the origin of everything, but the point that, in a universe with all the conditions necessary for 1 plus 1 always to equal [whatever-it-is-that-it-equals] then 1+1=[that] is a tautology which is unavoidably true without reference to whatever it was that set those conditions up.
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quote:
Originally posted by Eliab:
All universally true logical and mathematical statements would, to a sufficiently advanced mind, appear as statements of necessary relationships between ideas...
There is only one necessary entity, God. Nothing else has necessary being, nothing else can possibly have necessary being. If there are aspects to maths and logic that are necessary in an absolute sense, then they must be in some sense an aspect of God. It may be possible that certain aspects of maths and logic can be shown to be reducible to statements about God. Then they could be necessary. Otherwise, however, they cannot be. And I think maths and logic are not entirely implicit in God. I would say that to a considerable part they are implicit in this universe, hence created, hence not necessary.
I think the word “necessary” is a source of confusion here.
I agree entirely that there’s a metaphysical concept of “necessary being” which God has, and must have, in order to be what we understand as God, and which nothing but God ever could have.
That’s not at all the quality that I think mathematical truth has. 1+1=2 is not “necessary” in that sense. It’s not an "entity" which "necessarily" exists. It’s not “necessary” that any mind in creation should ever have formulated that expression, or that any objects should ever have existed by which its truth could be demonstrated to the sceptical. It merely cannot be false (given the usual definitions of each of its components). It’s a tautology.
Tautologies are “necessarily true”, if you like, but they do not “necessarily exist” nor do they constitute “necessary being” in a metaphysical sense. A tautology is simply two ways of saying the same thing. To a sufficiently advanced mind, any tautology, even the most complex mathematical proof, is trivially true. A trivially true statement is, by definition, unavoidably true, and incapable of being false, but does not have that quality needed to fulfil the role that the concept of “necessary being” fulfils when we are talking about God. God has “necessary being” in contrast to “contingent being” because all contingent facts depend on him. No contingent fact depends on “1+1=2” being true. The statement adds no new information to the universe. It therefore presents no challenge to God’s necessary being, nor does it detract from his status as the creator of all.
[ 08. June 2015, 13:05: Message edited by: Eliab ]
Posted by IngoB (# 8700) on
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Originally posted by Eliab:
It’s that given that we can see that logic works as a universally valid process here, and doesn’t appear to be dependent on any physical peculiarities of our reality, there is no reason to doubt the general applicability of logic in other any other realities that there might be, and no warrant for assuming that a different ‘logic’ would be workable anywhere, since we can’t really conceive what that would mean.
I think there are no features of "maths and logic" (M&L) that are not "experientially derived" from the physical peculiarities of our reality. None. Some of these features are not "experientially derived" in the sense that we individually have experienced them. But rather, they are "hard-wired" into us by whatever made us become the kind of embodied being that we are (evolution if you wish). Still, natural selection is based on the drastic "experience" of prior generations of procreating faster than dying. So in this quite general sense I will claim that there is nothing at all in M&L that "transcends" some origin in input to organisms. It is an exercise in abstracting specific features from essentially sensory input received from the world, even if we can then play with these abstractions in some sense beyond the direct constraints of this universe.
I see no evidence for some kind of "magic" origin of M&L. Indeed, M&L pretty much falls into two parts: On one hand "intuitive" M&L, whose value for the survival of organisms is obvious (like estimating whether this heap has "more" stuff than that other heap). And on the other hand "non-intuitive" M&L which is learned, and for the most part is learned explicitly by building up from "lower level" concepts (in particular at the bottom from "intuitive" ones). There is no good reason to believe that this is some kind of "voyage of discovery in Platonic M&L space". This is a straightforward constructive learning process based on simple elements which are very clearly "universe-derived". That's how we teach our children, that's how our ancestors discovered these things originally.
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Originally posted by Eliab:
However you have not successfully conceptualised a universe in which 1+1=3 with that example. You have conceptualised a universe with an unusual physical law which governs how objects come into existence.
And that is how we arrived at the concept of addition. That you can do all sorts of wonderful things with addition once you have abstracted it from the physical process of gathering things is irrelevant. That is the power of abstraction. But the key point is that this is an abstraction "from how the world works" in the first place. If the world worked differently, then you would abstract differently. Abstractions are very flexible imagination tools. Just like I can dream up a 1(+)1=3 operation, working away from my prior understanding of the 1+1=2 addition operation, so someone in a different universe where 1(+)1=3 is close to the physical reality could work away from that to a strange 1+1=2 operation. But if there was a universe in which things do not "heap up", but somehow still a sapient entity lived, then they would be highly unlikely to have "addition" as a core concept, and if they did, it would be a rather "high level" concept for them - the sort of thing only mathematicians care about.
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Originally posted by Eliab:
The reason is that addition is not limited to “similar objects in physical proximity”. Addition works on anything – it’s not a physically dependent process.
You are looking at the abstract entity and claim that it has no "physical support". That's true enough, but ignores that this abstract entity was abstracted in the first place from physical reality. Once you have defined your operation as an idea, you can use it for anything. But you would not have defined operation were it not due to how the world works. Open any book teaching maths to infants: "One chicken." "Two sheep." "Three apples." "One elephant is here. Another elephant comes. Two elephants." That is a brain learning counting and addition. You would not get that unless physical objects had characteristic "heaping behaviours". Perhaps you have taught a child to divide. "There are six apples, and two children. How many apples does every child get?" That is a brain learning division. And so on. Yes, at some point in the ramping up cognitive development we start building abstract concepts upon abstract concepts "x^2+3x-4=0, solve for x", but that does not sever the origin in how the world works.
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Originally posted by Eliab:
And I think it’s stretching the point beyond breaking to say that the conditions are implicit references to God, because he’s the origin of those conditions.
I rather think that this is the only possible way of making any "always and everywhere true" statement. Full stop. All else is in principle up for grabs. If you cannot ground your absolute truth value in God somehow, then it is not absolute but conditional.
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Originally posted by Eliab:
That’s true, because God is the origin of everything, but the point that, in a universe with all the conditions necessary for 1 plus 1 always to equal [whatever-it-is-that-it-equals] then 1+1=[that] is a tautology which is unavoidably true without reference to whatever it was that set those conditions up.
Define "condition", define "tautology", define "true", define "reference", etc. I think you are simply not aware how much baggage your argument is carrying around there. But anyway, the statement "things are as they are in that universe" does no work against the point I am making. I am after all not saying that what is real is still arbitrary. It is not, it is what it is, and in being it is not arbitrary at all. What I am saying is rather that (logically) before it became real, what it would become was arbitrary. You may or may not be able to make your nice descriptive list of how things are, but that's postdiction, not prediction.
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Originally posted by Eliab:
It merely cannot be false (given the usual definitions of each of its components). It’s a tautology.
This is just silly. Of course, you can say "If we define 1+1=2, then it follows that 1+1=2, by definition." So what? Nobody cares about that. The real question is whether 1+1=2 in a much more fundamental sense, so that I can say about some other universe: "Whatever else may be the case, still 1+1=2 must be true also there."
And no, the definition is not fundamental in the relevant sense for our discussion here. It is obvious that also in our maths we can define things to be so or so far beyond what we see in our universe. But we can do so only once we stepped into abstraction, and in doing so we are still bound to those abstractions. I define a "strange kind of addition" 1(+)1=3, but I do so because I can imagine modifications of the 1+1=2 addition that I abstracted first. My concept space is spanned by these first level abstractions, and they are world-bound. It is possible that in some other universe maths and logic hold that are not containable in this imaginative generalisation of what I learned from my universe. Then I cannot even think their universe and its math.
Posted by Jack o' the Green (# 11091) on
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I agree that the world in which we live has in some ways influenced our view of mathematics i.e. the abstractions we use in theoretical or pure maths began from empirical observations and empirical applications. However, I still think that 1+1=2 is 'truer' than 1+1=3 regardless of the universe we are in - simply by referencing the number line of positive integers. As I've said before, if every whole number from 1 is simply an addition of another single, whole positive value, then 1+1=2 in a way that it doesn't =3 - even if we lived in a universe where placing two objects in close proximity always caused a third to appear. In this example, the inhabitants of "Weirdmathsicon 5", couldn't trace their maths back to the theoretical world like we (in this instance) can.
Edward Feser states it in a more polemical way than I think is needed, but I think he makes the point well;
"...it is sometimes suggested that if the physical world was set up in such a way that whenever we put two objects together with two other objects, a fifth object magically appeared among them, this would be a case where 2+2=5. People who give such arguments really should listen to themselves more carefully. For by their own account, what they have described is not 2 and 2 equalling 5, but rather the act of placing 2 objects together with 2 other objects (which makes 4 objects total) suddenly and magically causing a fifth one to appear. ("X causes Y" doesn't mean "X equals Y".)"
It may seem a trivial thing to take this view of numbers, and in a sense it is. However, for some mathematicians it is an important point in the sense that it puts mathematical entities in an abstract world of their own. It may be true that human maths began with the abstraction of empirical observations, but according to some of these thinkers, it enabled us to discover a 'world' which is prior, necessary and independent of this contingent, material universe.
I don't go along with the platonic school of maths, so we are in agreement on that. It creates far more problems than it solves - e.g. (1) why do we need to access a special 'platonic realm' world via our 'mathematical intuition', why can't we simply create or discover mathematical ideas from our own intellect/understanding. The implication seems to be that if the platonic realm didn't exist, then we wouldn't be able to do maths, which seems absurd. (2) If mathematical entities are necessary truths/existents, how can we access them in the first place since in our own world, we can only usually interact with things which are contingent and therefore part of our contingent world.
However many very good/exceptional mathematicians like Gödel and Penrose are platonists, and since I'm not naturally gifted where maths is concerned, I was wondering about other people's views.
[ 08. June 2015, 17:48: Message edited by: Jack o' the Green ]
Posted by LeRoc (# 3216) on
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Jack o' the Green: Edward Feser states it in a more polemical way than I think is needed, but I think he makes the point well;
"...it is sometimes suggested that if the physical world was set up in such a way that whenever we put two objects together with two other objects, a fifth object magically appeared among them, this would be a case where 2+2=5. People who give such arguments really should listen to themselves more carefully. For by their own account, what they have described is not 2 and 2 equalling 5, but rather the act of placing 2 objects together with 2 other objects (which makes 4 objects total) suddenly and magically causing a fifth one to appear. ("X causes Y" doesn't mean "X equals Y".)"
Once again, this shows how prejudiced our brains can be, for the fact of living in this universe.
To us, the fifth apple 'magically appeared'. And this was 'caused' by placing 2 and 2 apples together. To them, the fifth apple is there simply because 2+2=5. Dûh.
Just for fun, imagine the Weirdmathsicans thinking about our universe (which is a very strange theoretical construct to them). To them, when we put 2 and 2 apples together, the fifth apple magically disappears. And this is caused in our universe by putting 2 and 2 apples together.
On which basis do you decide which point of view is right?
Posted by Jack o' the Green (# 11091) on
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I would say via appealing to the number line. Couldn't you argue that the Wierdmathiconian's brains are too biased to be able to see the truth? I think I understand the point you're making, it just doesn't seem to go back to first principles in the same way 1+1=2 or 2+2=4 does.
[ 08. June 2015, 18:32: Message edited by: Jack o' the Green ]
Posted by LeRoc (# 3216) on
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Originally posted by Jack o' the Green:
I would say via appealing to the number line.
[I]Our [/] number line. Our number line is the way it is exactly because when we put 2 and 2 apples together, we get 4 apples. They have another number line. How do you decide which is the 'right' one?
Posted by Jack o' the Green (# 11091) on
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Because our number line is based on increments of 1 positive value at each stage and so is internally consistent. By contrast, the Wierdmathsiconian's number line isn't. It has to skip a step which is inconsistent to make it work.
[ 08. June 2015, 18:45: Message edited by: Jack o' the Green ]
Posted by Dafyd (# 5549) on
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quote:
Originally posted by IngoB:
I think there are no features of "maths and logic" (M&L) that are not "experientially derived" from the physical peculiarities of our reality.
I'd say that our knowledge of most mathematical concepts is derived from abstraction from physical reality and constructions from those concepts, together perhaps with some biological hardwiring. Mathematical intuition seems to me both unnecessary and not in accordance with the facts. (We prove mathematical arguments rather than believe them on testimony.)
That does not however imply that the mathematical concepts are contingent upon empirical reality.
The problem here is that our imagination and language are set up to deal with empirical objects, and with things that can be conceived on analogy with empirical objects. So we're inclined to think of mathematical objects as being like empirical objects only existing in a mathematical or Platonic reality rather than empirical reality. But nothing that behaves analogously to empirical objects could have the consequences or significance of mathematical truths. So we're left not being able to say much useful in a positive way about what their manner of existence is.
quote:
I define a "strange kind of addition" 1(+)1=3, but I do so because I can imagine modifications of the 1+1=2 addition that I abstracted first.
1 (+) 1 = 3 doesn't define an operator (+) until you've generalised it. x(+)y=x+y+1 defines an operator, as does x(+)y=3(x+y)/2. But both of those operators depend upon the + operator. (Not so much because the + operator cannot be defined in terms of either of those operators as because the + operator is required to define the number line itself. 3 just is the number you get when you perform the +1 operation on 1 and then perform it again.)
It is possible to raise the theoretical possibility of a universe in which our mathematics doesn't hold. What we can say about it is that it is so outside our ability to think that we cannot even affirm it as a theoretical possibility. All one can really say is here be intellectual dragons.
Posted by Jack o' the Green (# 11091) on
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So if there's a Greek-Syrian mathematician on the Ship to play the part of St George, now would be a good time to enter the fray!
Posted by LeRoc (# 3216) on
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quote:
Jack o' the Green: Because our number line is based on increments of 1 positive value at each stage and so is internally consistent. By contrast, the Wierdmathsiconian's number line isn't. It has to skip a step which is inconsistent to make it work.
I can assure you that their number line is completely consistent to them. Because it agrees with how their universe works.
And there is another prejudice in play here. That more consistent is better. This is because how our universe works. Many times, the simpler, more consistent, more elegant mathematical solution turns out to be the right one for a physical problem. I even have the feeling that this happens more often than it ought to. This isn't necessarily true in another universe.
Even if our number line is more consistent than theirs (I'm still not entirely sure how to define this), so what?
[ 08. June 2015, 19:25: Message edited by: LeRoc ]
Posted by Jack o' the Green (# 11091) on
:
quote:
Originally posted by LeRoc:
quote:
Jack o' the Green: Because our number line is based on increments of 1 positive value at each stage and so is internally consistent. By contrast, the Wierdmathsiconian's number line isn't. It has to skip a step which is inconsistent to make it work.
I can assure you that their number line is completely consistent to them. Because it agrees with how their universe works.
I don't disagree that it's consistent with how their universe works in some sense, but describing it as 1+1+1=3 is also consistent with how their universe works (original1+original1+spontaneous1=3) and is also internally or conceptually consistent and allows it to exist within a consistent framework apart from any material universe. Plus this way of conceptualising it translates in both universes whereas 1+1=3 only works in theirs.
Posted by LeRoc (# 3216) on
:
quote:
Jack o' the Green: I don't disagree that it's consistent with how their universe works in some sense, but describing it as 1+1+1=3 is also consistent with how their universe works (original1+original1+spontaneous1=3) and is also internally or conceptually consistent and allows it to exist within a consistent framework apart from any material universe. Plus this way of conceptualising it translates in both universes whereas 1+1=3 only works in theirs.
(I'm having some difficulties following you. I see that you moved from 2+2=5 to 1+1=3 which is ok, I just need to adjust a bit.)
You are still assuming that one of their apples spontaneously appeared. It didn't.
Posted by Jack o' the Green (# 11091) on
:
quote:
Originally posted by LeRoc:
(I'm having some difficulties following you. I see that you moved from 2+2=5 to 1+1=3 which is ok, I just need to adjust a bit.)
You are still assuming that one of their apples spontaneously appeared. It didn't.
Apologies, the Feser quote used a different example to make a pertinent point so I've mixed my examples somewhat.
Even if the apples caused another apple to appear via two 'apple-morphic fields' coming into contact (we're going to be creating an entire world at this rate!), and so can be represented as 1+1=3, in an applied sense, that still leaves the fundemental problems of my previous post which is the mathematical inconsistency, as well as the fact that our way of adding can be applied to both worlds. Despite the fact that in our world, human reproduction means that 1+1 could be said to equal 3 (or 4 or 5 etc), I don't know anyone who would feel that this is an accurate way of stating it mathematically (earthling prejudice?). Each of the children would be counted as an additional '1', making up the final (internally consistant) total.
Posted by LeRoc (# 3216) on
:
I'm going bit by bit here:
quote:
Jack o' the Green: Even if the apples caused another apple to appear
No apple appeared here. Nothing was caused. There are three apples because 1+1=3.
quote:
Jack o' the Green: via two 'apple-morphic fields' coming into contact
You're making up explanations where none are needed. There are three apples because 1+1=3.
quote:
Jack o' the Green: that still leaves the fundemental problems of my previous post which is the mathematical inconsistency
It's inconsistent with our mathematics, not with theirs. And like I asked before, why is consistency inherently good?
quote:
Jack o' the Green: as well as the fact that our way of adding can be applied to both worlds.
I'm sure that they can find a formulation within their way of adding that can be applied to both worlds too. And why is it important that a way of adding can be applied in two universes?
quote:
Jack o' the Green: I don't know anyone who would feel that this is an accurate way of stating it mathematically (earthling prejudice?).
Exactly. You don't know any Wierdmathsiconians.
Posted by Jack o' the Green (# 11091) on
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Apologies again - double post.
Suppose on Wierdmathsiconia, 3 apples were placed close enough to be counted, but far enough apart to prevent their morphic fields from creating others. The Wierdmathsiconians can count 3 apples thus observing that 1+1+1=3. Then they eat one - leaving 2 - observing that 1+1=2. Then they move the 2 apples together to create a third. That would enable them to observe that while for apples close together 1+1 may cause another one = 3, as a simple mathematical abstraction, without any 'causal interference', 1+1=2 not 3.
Posted by Jack o' the Green (# 11091) on
:
You seem to be wanting it all ways. The third apple wasn't created, it didn't spontaneously appear, yet you still want 1+1=3 based on the Wierdmathsiconian's empirical experience.
It's not enough to say that you're sure they can find a maths consistent with ours. That would need to be demonstrated.
The consistency is important because if one set of axioms can be successfully applied to both worlds, I would think that the implication is that (coupled with its internal consistency), that earthling maths is a better, more accurate description of reality and is more than convention based on our empirical observations. Which leads to the question in what way and for what reason is this the case.
Posted by LeRoc (# 3216) on
:
quote:
Jack o' the Green: Apologies again - double post.
No it isn't
quote:
Jack o' the Green: Suppose on Wierdmathsiconia, 3 apples were placed close enough to be counted, but far enough apart
Whoa. You've introduced the concept of distance here. Who says they have distance on Wierdmathsiconia?
quote:
Jack o' the Green: Then they eat one
'Then'? Now we have a concept of time. And events consequentially following eachother. Does this exist on Wierdmathsiconia?
I thought we were talking about mathematics here, which according to you is decoupled from the structure of the universe. But now it seems that you need the concepts of time and distance to make your point. Both are intrisically part of the structure of our universe.
Posted by LeRoc (# 3216) on
:
quote:
Jack o' the Green: You seem to be wanting it all ways. The third apple wasn't created, it didn't spontaneously appear, yet you still want 1+1=3 based on the Wierdmathsiconian's empirical experience.
I don't want anything. This is just how it works on Wierdmathsiconia.
In our universe, if there is an apple it needs to have come from somewhere. It's appearance needs to have been caused by something. Otherwise it has magically appeared. Those are the rules of our universe. These rules aren't valid on Wierdmathsiconia
quote:
Jack o' the Green: It's not enough to say that you're sure they can find a maths consistent with ours. That would need to be demonstrated.
I know. I guess that if I put my mind to it, I could do it. But this would mean that we would be arguing details about (+) operators, and I prefer to leave this to others on this thread.
quote:
Jack o' the Green: The consistency is important because if one set of axioms can be successfully applied to both worlds, I would think that the implication is that (coupled with its internal consistency), that earthling maths is a better, more accurate description of reality and is more than convention based on our empirical observations.
Earthling maths is a more accurate description of our reality. Wierdmathsiconian maths is a more accurate description of their reality.
Posted by agingjb (# 16555) on
:
Can Christian theologians tell me whether one God and three persons mean that these numbers were created or do they necessarily exist?
Posted by Dafyd (# 5549) on
:
quote:
Originally posted by LeRoc:
'Then'? Now we have a concept of time. And events consequentially following eachother. Does this exist on Wierdmathsiconia?
You've jumped in on a description of putting two apples together with two apples to get five apples, so it appears that there's some sort of sequence.
I think I would say that just as they do not have the concept of time and distance as we understand it; they do not have the concepts of 1, or 2, or 3, or 5. So they do not have the concepts of 1+1=3 or 2+2=5. It's not that they have a contradictory mathematics to ours: they just have a completely different one, about which we cannot even say that it contradicts our mathematics.
That being the case, if our logic doesn't apply to them, we cannot say anything about them; not even that our logic doesn't apply.
Posted by LeRoc (# 3216) on
:
quote:
Dafyd: You've jumped in on a description of putting two apples together with two apples to get five apples, so it appears that there's some sort of sequence.
That's interesting, isn't it?
When we are talking about 1+1=2, we are already taking aboard a lot of assumptions about our universe. First we have two separate apples, then we join them together. There is an element of time there, and possibly an element of space. Those are the most basic building blocks of universe.
We also assume that there is some kind of continuity. The apple that was there before counting is the same apple that is there after counting. And we assume that we can identify this apple.
Going to Wierdmatsiconia, we assume that if there are three apples after counting (that time element again!), one of the three has appeared 'magically'. Which one? Well, we assume that we can identify the two apples that have remained and the 'new one'. And of course, the appearance of this third apple needs to have been caused by something.
So, it seems to me that if we want to argue that 1+1=2 and not 3, we need a lot of stuff:- An element of time, and possibly space
- An element of continuity (the apple is the same before and after counting)
- An element of being able to identify the apples
- An element of causation (why is an apple there?)
All of these have to do with how our universe works, with confirms my point: our maths is linked to the structure of our universe.
quote:
Dafyd: they do not have the concepts of 1, or 2, or 3, or 5. So they do not have the concepts of 1+1=3 or 2+2=5.
But they do. They have all these things.
Of course, you could ask yourself if their 1 is the same as our 1. They call it 1. It is still the lowest integer. It is still the neutral element for multiplication. But it works different in addition than ours. Is it still the same 1? It depends on how you define 1.
Which makes my point again: our mathematics depend on our universe.
quote:
Dafyd: That being the case, if our logic doesn't apply to them, we cannot say anything about them; not even that our logic doesn't apply.
There is a universe, it is rather dark and dampy. I kindly invite you to stick your self-references there.
Posted by George Spigot (# 253) on
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I thought it was all labels.
Posted by Dafyd (# 5549) on
:
quote:
Originally posted by LeRoc:
quote:
Dafyd: You've jumped in on a description of putting two apples together with two apples to get five apples, so it appears that there's some sort of sequence.
That's interesting, isn't it?
When we are talking about 1+1=2, we are already taking aboard a lot of assumptions about our universe. First we have two separate apples, then we join them together.
You and IngoB and Jack O' the Green and Feser. As Eliab and I were pointing out, arithmetic applies to a lot more contexts than merely the greengrocer's. And many of those contexts are quite unlike the greengrocer's.
There's no physical similarity between apples and the motion of stellar bodies. Yet the same basic arithmetic underlies the study of both.
quote:
quote:
Dafyd: they do not have the concepts of 1, or 2, or 3, or 5. So they do not have the concepts of 1+1=3 or 2+2=5.
But they do. They have all these things.
Of course, you could ask yourself if their 1 is the same as our 1. They call it 1. It is still the lowest integer. It is still the neutral element for multiplication. But it works different in addition than ours. Is it still the same 1? It depends on how you define 1.
Which makes my point again: our mathematics depend on our universe.
Despite the multiplicity of languages on one small planet in our universe, in their universe they still call it '1' (even though there's no time in which to say anything and even though there's no distance between the top of the stroke and the bottom, and even though there's no continuity of physical objects). Er... right. (Desp
You're spinning your wheels without saying anything. Or rather, you can say absolutely anything; but if you can say absolutely anything then you're saying nothing at all.
quote:
quote:
Dafyd: That being the case, if our logic doesn't apply to them, we cannot say anything about them; not even that our logic doesn't apply.
There is a universe, it is rather dark and dampy. I kindly invite you to stick your self-references there.
If you would like to explain:
a) why what I'm saying is self-reference? Clearly, it's not just the use of first person pronouns since you're happy to use those.
b) what is wrong with self-reference? I mean, it appears to be some kind of ghastly sin against polite discussion, but I'm really not seeing why.
Posted by Dafyd (# 5549) on
:
quote:
Originally posted by George Spigot:
I thought it was all labels.
Labels with a lot of predictive power.
Posted by agingjb (# 16555) on
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If there is a possible world in which our maths is inapplicable then surely the maths of a world is always independent of other worlds, and also independent of its own world.
But I still think that maths is accessed through the world but is independent of it.
Going back to the ontology, I think of mathematical entities as the answers to carefully posed questions, which, of course, leaves me with the problem of the ontological status of questions and answers.
Posted by LeRoc (# 3216) on
:
quote:
Dafyd: There's no physical similarity between apples and the motion of stellar bodies. Yet the same basic arithmetic underlies the study of both.
Yes, that's rather remarkable isn't it? We count everyday things like apples, and somehow this leads to a system that allows us to understand the furthest galaxies in our universe. There's no intrinsic reason why this should be the case. And it's quite possible that things don't work like this in another universe.
I haven't got a clue though how this is an answer to the part of my post you quoted.
quote:
Dafyd: Despite the multiplicity of languages on one small planet in our universe, in their universe they still call it '1' (even though there's no time in which to say anything and even though there's no distance between the top of the stroke and the bottom, and even though there's no continuity of physical objects). Er... right.
For us, if you take away the concepts of time and distance and continuity, there doesn't seem to be anything left instead of chaos. But that's our prejudice again. There might be other concepts that make sense of their universe.
But you're focussing on something that isn't the crux of my argument. Of course, they won't call this number '1' (although there's no intrinsic reason why this would be impossible). But it is still the lowest integer, it is still the neutral element for multiplication ... When is something in another universe the same as our number 1?
This is what I was talking about with my π example. When is a number in our universe the same as a number in another universe?
quote:
Dafyd: You're spinning your wheels without saying anything. Or rather, you can say absolutely anything; but if you can say absolutely anything then you're saying nothing at all.
I'm saying something rather specific. The problem is that you don't understand it.
quote:
Dafyd:
a) why what I'm saying is self-reference? Clearly, it's not just the use of first person pronouns since you're happy to use those.
b) what is wrong with self-reference? I mean, it appears to be some kind of ghastly sin against polite discussion, but I'm really not seeing why.
a) You assume that I am saying "We can't say anything about another universe" and then you let the word 'anything' apply to the sentence "We can't say anything about another universe". That's self-reference.
b) The problem isn't the self-reference itself, although it looks a bit smug when you repeatedly do it. The problem is that I never said "We can't say anything about another universe". In fact, I'm saying some very specific things about them.
quote:
agingjb: If there is a possible world in which our maths is inapplicable then surely the maths of a world is always independent of other worlds, and also independent of its own world.
Those are two non-sequiturs
Posted by itsarumdo (# 18174) on
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what if we could only experience "processes"(i.e. verbs) instead of "things" (nouns) - would 1 still be perceptible (or even exist)?
Posted by Jack o' the Green (# 11091) on
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I'm not sure. I have heard that in some languages they are adjectives rather than nouns.
Posted by Dafyd (# 5549) on
:
quote:
Originally posted by LeRoc:
quote:
Dafyd: There's no physical similarity between apples and the motion of stellar bodies. Yet the same basic arithmetic underlies the study of both.
Yes, that's rather remarkable isn't it? We count everyday things like apples, and somehow this leads to a system that allows us to understand the furthest galaxies in our universe. There's no intrinsic reason why this should be the case. And it's quite possible that things don't work like this in another universe.
I haven't got a clue though how this is an answer to the part of my post you quoted.
If arithmetic applies in two areas that have no apparent physical similarity, that suggests that arithmetic is valid independent of any physical properties of our universe. The application may be dependent upon physical properties (it doesn't make sense to talk about irrational numbers of apples, although irrational numbers may be derived by considering integers), but the arithmetic itself is valid independent of any application.
quote:
For us, if you take away the concepts of time and distance and continuity, there doesn't seem to be anything left instead of chaos. But that's our prejudice again. There might be other concepts that make sense of their universe.
It's possible. But we do not have those concepts. We cannot make sense of their universe with those concepts since we do not have them.
quote:
quote:
Dafyd: You're spinning your wheels without saying anything. Or rather, you can say absolutely anything; but if you can say absolutely anything then you're saying nothing at all.
I'm saying something rather specific. The problem is that you don't understand it.
A classroom of children who have been taught to recite the times tables from memory do not thereby understand the times tables. They don't understand them until they can use the times tables to do mathematics. A child reared on names and dates of the Kings of England does not understand what they are reciting until they can make use of what they've been to recite by having some grasp of who those Kings were and what happened during their reign (and for that matter some grasp of the powers and responsibilities of the Kings over history).
And of course you are quite right. I cannot understand what you are saying. I cannot place it in context. In particular, just like the child taught to parrot the times table, I cannot make use of your combination of statements to perform calculations. Were I to say that in that universe '1+1=3', where 1 is the lowest integer and the multiplicative identity, I would not be saying anything; I would be merely parroting.
Note of course that the difference between saying and parroting does not lie in an impression of meaningy-ness in the head, but in the ability to make use in discourse.
Now, when you say that in that universe '1+1=3', where 1 is the lowest integer and the multiplicative identity are you sure you're actually saying something more than I would be were I to parrot the words after you?
(I don't believe the inner impression of meaning proves anything one way or another. Meaning is use. If someone can't make use of what they're trying to say, they don't mean anything.)
This is another way of saying that if what you say imposes no constraints upon what you go on to say you've said nothing, only approaching from the other end. The ability to use concepts to do work is indistinguishable from the ability to use concepts to rule things out. If you say something utterly specific it is as meaningless as if you say something utterly unspecific.
quote:
a) You assume that I am saying "We can't say anything about another universe" and then you let the word 'anything' apply to the sentence "We can't say anything about another universe". That's self-reference.
I'm saying that 'We can't say anything about another universe'. I don't say you say that. I say that follows from what you're trying to say about the other universe. It follows from what you're trying to say that what you're saying is too specific to count as saying anything.
Posted by LeRoc (# 3216) on
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quote:
Dafyd: If arithmetic applies in two areas that have no apparent physical similarity, that suggests that arithmetic is valid independent of any physical properties of our universe.
No, that doesn't follow at all. To the contrary, what scientists are finding is that mathematics and the physical properties of our universe are deeply linked.
Logically, there are various possibilities (once again, assuming that multiple universes exist):- Mathematics is at the basis of the structure of all universes.
- Mathematics happens to be the basis of the structure of our universe; other universes can have other concepts at their base.
- Most universes are less ordered than ours, but they cannot sustain intelligent life (the anthropic principle).
- The orderedness of our universe is a freak coincidence.
- The universe is so ordered that a simple concept (the counting of apples) leads to a model that helps to understand it all, because it was designed that way.
We simply cannot know. We're back at the same point: we cannot use what we know about our universe to put an upper limit on what is possible in other universes.
quote:
Dafyd: It's possible. But we do not have those concepts. We cannot make sense of their universe with those concepts since we do not have them.
Yes, that's what I've been saying all along. If another universe has a profoundly different structure from ours, we can't say a lot about it because we lack the concepts. Our brains are wired to deal with our universe, so it's to be expected that they'll fall short. But that doesn't mean that we can't say anything about such a universe at all.
quote:
Dafyd: I'm saying that 'We can't say anything about another universe'. I don't say you say that. I say that follows from what you're trying to say about the other universe.
Well, you're wrong. It doesn't.
Posted by agingjb (# 16555) on
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Non-sequitors? Well yes, my argument could be more carefully spelled out, but how in worlds where sequitors may not exist?
Posted by LeRoc (# 3216) on
:
quote:
agingjb: Non-sequitors? Well yes, my argument could be more carefully spelled out, but how in worlds where sequitors may not exist?
If there are no sequiturs, your argument may be right, or it may be wrong, or it may be something else. You're the one trying to make an argument here. You're not exactly helping yourself.
Posted by Dafyd (# 5549) on
:
quote:
Originally posted by LeRoc:
quote:
Dafyd: It's possible. But we do not have those concepts. We cannot make sense of their universe with those concepts since we do not have them.
Yes, that's what I've been saying all along. If another universe has a profoundly different structure from ours, we can't say a lot about it because we lack the concepts. Our brains are wired to deal with our universe, so it's to be expected that they'll fall short. But that doesn't mean that we can't say anything about such a universe at all.
If you can say something about another universe to which our concepts of mathematics and logic do not apply then demonstrate that you can, bearing in mind the distinction I made between saying and parroting.
(That distinction between saying and parroting is, if you can say something about a universe in which 1+1=3, you have to explain it, or deduce something non-trivial from it, or otherwise use it. Otherwise, you're not really saying anything any more than a pupil taught by Michael Gove to recite the times table without understanding is saying anything.
I'm not insisting you agree with my distinction between saying and parroting. But if you disagree - if you think it's possible to say something that you can't use - give reasons for rejecting it.)
Posted by agingjb (# 16555) on
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I believe that the Peano Axioms are universal and independent of the physical universe.
If there were a universe where the Peano Axioms do not describe anything in that universe, then the Peano Axioms, which do exist, are independent of that universe, therefore they can be independent of a universe.
Why would I suppose, given their utility and fruitfulness both in our universe and in other possible universes, that they are not actually independent of any universe?
Posted by orfeo (# 13878) on
:
quote:
Originally posted by agingjb:
If there were a universe where the Peano Axioms do not describe anything in that universe, then the Peano Axioms, which do exist, are independent of that universe, therefore they can be independent of a universe.
Whoa. Your "then" statement is a load of assertions hiding in a ball of missing relational clauses.
Let me fix it for you. If there were a universe where the Peano Axioms which exist in this universe do not describe anything in that universe, the Peano Axioms, which do exist in this universe are independent of that universe, therefore they can be "independent" of a particular universe but it's not all obvious that they can therefore be independent of every universe because I haven't established they're independent of my own universe.
Your argument is rather like me claiming that the 2-year-old next door is "independent" because I don't provide him with food and shelter. Sure, he's independent of an adult (me), but that doesn't mean he's "independent" as a free-standing concept.
Relational clauses. They matter.
Posted by agingjb (# 16555) on
:
The Peano Axioms exist.
I'm being told i'm wrong (at best) when I say they are not dependent on physics. The reason I'm wrong is because it is possible to imagine a physics to which the axioms are inapplicable. Well there are some odd corners of my universe where the answers to "how many" do not conform to the abstraction in terms of zero and successor.
Oh well.
Posted by orfeo (# 13878) on
:
And this whole insistence of our particular method of counting being inherent... sorry, but you don't even have to go to other universes to show this isn't correct. You can stay right here on this planet. There are cultures where zero doesn't exist as a concept. There are cultures which reach "infinity" before those of us using Arabic numerals have reached triple figures the idea that you can just keep adding 1 and moving along the number line forever is not a universal human truth.
If the concepts of mathematics are not just universal, but multiversal, then it seems very odd indeed that we can actually date and trace the origin of these concepts. Heck, some of these things were completely unknown to the authors of the Bible, and yet we want to claim they're true even in other universes? Sounds problematic to me.
Posted by orfeo (# 13878) on
:
quote:
Originally posted by agingjb:
The Peano Axioms exist.
Where?
Lots of things exist without turning up in my back yard. You're not just making a claim that the Axioms exist, you're making a claim about where they exist.
Again, relational clauses. "The Axioms exist" is simply not the same claim as "The Axioms exist in every universe".
[ 13. June 2015, 05:46: Message edited by: orfeo ]
Posted by agingjb (# 16555) on
:
Well, I thought the question underlying this thread was precisely the sense in which mathematical entities exist.
The Axioms exist because Peano wrote them down. What does it mean to say that something does not exist because there is some other universe in which it is not instantiated?
I've offered my view of maths as the the answers to well posed questions, and admitted that the status of questions and answers may require more thought.
I've tried to say why I think that maths is abstract.
I've asked about the implications of theology.
None of this has sufficed.
Posted by orfeo (# 13878) on
:
quote:
Originally posted by agingjb:
Well, I thought the question underlying this thread was precisely the sense in which mathematical entities exist.
The Axioms exist because Peano wrote them down. What does it mean to say that something does not exist because there is some other universe in which it is not instantiated?
I didn't say that something does not exist. I challenged the view that they existed everywhere. Again, simply not the same thing.
If we're going to talk about "precisely the sense in which mathematical entities exist", isn't this exactly what we're supposed to be talking about? To say that they exist in some way is obvious. We couldn't have this thread without at least knowing about them.
Posted by agingjb (# 16555) on
:
I know enough about the Peano Axioms, and their implications, to be led to the belief that they are universal.
Posted by LeRoc (# 3216) on
:
quote:
Dafyd: If you can say something about another universe to which our concepts of mathematics and logic do not apply then demonstrate that you can, bearing in mind the distinction I made between saying and parroting.
Okay, here goes:- It's a universe
- It has a different structure from ours
- I can't make much sense of it
See? That's already three things I've said about this universe. And I haven't parrotted anything.
If you want more, I guess you can see it as a theoretical exercise. Imagine all theoretically possible universes. Imagine that for at least one of them, the concept of distance still exists (there's no reason why it should only exist in our universe).
Bingo! That's a fourth thing you can say about this universe. You can't see much else. Other rules of mathematics and logic may or may not apply in this universe. We may or may not have the concepts to talk about how the rest of this universe works. But we can say one thing: the concept of distance exists.
It is in this theoretical way that I think of the examples on this thread. Maybe we cannot say much about the Weirdmathsiconian universe. But we can say one thing: 1+1=3. And Weirdmathsiconian children aren't parrotting anyone when they say this. (Except Jackie who sits in the corner of the classroom and never pays attention.)
quote:
agingjb: The Axioms exist because Peano wrote them down. What does it mean to say that something does not exist because there is some other universe in which it is not instantiated?
Suppose there is another universe where there was never any Peano to write down these axioms. Where these axioms aren't useful at all to understand that universe. Where even the concept of axioms is strange to them (they have other 'concepts' we can't really grasp because our minds our wired to understand an axiomatic universe). And in that universe, they'll never have contact with our universe to know that Peano's axioms exist and are useful here.
Do Peano's axioms still exist in that universe? I guess that in some very theoretical sense you could say that they do. But that isn't really saying much, is it?
Let's turn it around for a moment. Suppose there are billions of billions of billions of other universes. And suppose they have billions and billions and billions of 'concepts' (not axioms, I'm deliberately using scare quotes here) that help them make sense of their universes.
Most of these 'concepts' make no sense at all in our universe. They aren't helpful to understand our universe at all. We don't have a way to express these 'concepts'. Indeed, we don't even have a way to express the concept of these 'concepts'. And we'll never have contact with these universes to know that these 'concepts' even exist.
Do all these billions of billions of billions of 'concepts' still exist in our universe? Mwah. We could just as well say that they don't. I don't see how they exist here in any meaningful way.
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agingjb: I've asked about the implications of theology.
None of this has sufficed.
To me, the ontological status of maths and logic has important theological consequences. I believe that they are created entities, put into the structure of our universe by God.
There is another theological consequence. We can probably never prove or disprove the existence of other universes. But as Christians, we believe that at least one other universe exists. We call it Heaven. And we believe that at least in some sense, God comes from there.
The theological consequence is that we have to be very careful drawing logical conclusions about Heaven, or about God.
Posted by agingjb (# 16555) on
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I may well be careful, as a layman, about theology, others have made some very positive assertions.
So, did God create the number three, or is that number co-eternal, and necessary?
Posted by LeRoc (# 3216) on
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agingjb: So, did God create the number three, or is that number co-eternal, and necessary?
I'm a layman when it comes to theology too, but I believe that God created the number three. In the sense that He created our universe such that the integer line is helpful in making sense of it.
Posted by Jack o' the Green (# 11091) on
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Originally posted by agingjb:
I may well be careful, as a layman, about theology, others have made some very positive assertions.
So, did God create the number three, or is that number co-eternal, and necessary?
That would depend on what are saying is necessary ie, whether you can cogently distinguish between necessary truth and nesessary existence. Some mathematicians don't really distinguish between the two and can speak of a platonic world of necessarily existing mathematical entities which necessarily exist because they are necessarily true since both concepts are the same thing. It is this world which (I think) has grave consequences for theism as it posits two 'ultimates' - God and the mathematical realm. Both uncreated, both eternal, both playing a part in the existence of creation. It is this danger which some theist philosophers concerned about.
If you distinguish between necessary truth and necessary existence, then I think you can avoid these problems. The number 1 doesn't necessarily exist, it's simply an abstract, coherent concept conceived in a mind. It isn’t on its own a statement or proposition which is either true or false. Place it in the sum 1+1=2, and it becomes an intrinsic part of a necessary truth - simply because it is a tautology. As an abstract truth (whether discerned via observations from this universe or entirely via abstract reasoning), it is simply true. Meaning that if you are sticking to particular definitions regarding positive intagers then you couldn't conceive 1+1= anything other than 2. This doesn’t mean however that this truth exists necessarily outside the mind conceiving it.
Taking such a view doesn’t constrain God - it fully allows abstract concepts have been created by him, and to have been conceived to describe and be instantiated within this universe which he also created.
I don't deny the possibility that other universes may exist which are better described by different concepts. I simply don't think that this fact prevents 'our' maths from containing concepts/abstract entities which are true (though not existing) by necessity.
Posted by Dafyd (# 5549) on
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Originally posted by LeRoc:
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Dafyd: If you can say something about another universe to which our concepts of mathematics and logic do not apply then demonstrate that you can, bearing in mind the distinction I made between saying and parroting.
Okay, here goes:- It's a universe
- It has a different structure from ours
- I can't make much sense of it
See? That's already three things I've said about this universe. And I haven't parrotted anything.
Do you know it's a universe and not two or three or more universes? There's no mathematics that apply between universes (if you take it together with our universe have you got two universes or three?) so how do you know whether the concepts of singularity or plurarity apply to it. So you can't say it's a universe rather than a multiverse.
How do you know it doesn't have the same structure? It might be that the concepts of same and different don't apply in the same way there. It might be that it doesn't have any structure at all. You don't know.
That it doesn't make sense to us is a statement about us rather than it.
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Imagine all theoretically possible universes. Imagine that for at least one of them, the concept of distance still exists (there's no reason why it should only exist in our universe).
Bingo! That's a fourth thing you can say about this universe. You can't see much else. Other rules of mathematics and logic may or may not apply in this universe. We may or may not have the concepts to talk about how the rest of this universe works. But we can say one thing: the concept of distance exists.
How do you know? Within our universe, using our logic, if you have a set all of whose members have a given property and you pick an item from that set your item has that property. So if you pick one of the blue balls out of a box of coloured balls, you'll get a ball that's blue. But if that logical principle doesn't necessarily apply, you don't know that if you pick one of the universes with the concept of distance then the universe you picked will have the concept of distance in it. You don't know that logical principle applies to universes.
But even so... Earlier, when Jack o'the Green asked you how the mathematics of their universe you said 'you're introduced the concept of distance here. Who says they have distance in Weirdmathsiconia?'. Now you stipulate that they do have distance; Jack o'the Green was right to introduce distance after all. That's an example of not saying anything - in the absence of logic there's no reason to suppose you can't say anything and then unsay it.
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Maybe we cannot say much about the Weirdmathsiconian universe. But we can say one thing: 1+1=3. And Weirdmathsiconian children aren't parrotting anyone when they say this.
How do you know they have children? Or that they're able to say anything?
So, you can speculate that we don't know. But then you can't say anything beyond that.
Posted by LeRoc (# 3216) on
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For someone whose main argument is that we can't say anything about universes where our maths isn't valid, you're saying an awful lot about universes where our maths isn't valid.
I know that your main argument is like this:- If in another universe our maths/logic isn't valid, we can't say anything about it
- If we can't say anything about it, we can't say "our maths/logic isn't valid in that universe"
- Therefore, our maths/logic must be valid in that universe
I'm sorry, but it doesn't fly.
Even in our understanding of maths/logic (which is limited by living in this universe), there are alternatives to the 'standard' maths/logic. If there are alternative unverses, it is entirely reasonable that some of them could be structured in such a way that one of these alternatives may be a better way to understand them. And that's just the beginning. These are just the alternative forms of math/logic we can think of, there might very well be more.
If you believe that you have proven that all universes must be such that they can be understood by our maths/logic, then I wish you good luck with it. I don't believe it.
Posted by agingjb (# 16555) on
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I am being driven inexorably towards what is, I suppose, a Platonic view of the existence of mathematical objects.
Posted by Dafyd (# 5549) on
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Originally posted by LeRoc:
For someone whose main argument is that we can't say anything about universes where our maths isn't valid, you're saying an awful lot about universes where our maths isn't valid.
I know that your main argument is like this:- If in another universe our maths/logic isn't valid, we can't say anything about it
- If we can't say anything about it, we can't say "our maths/logic isn't valid in that universe"
- Therefore, our maths/logic must be valid in that universe
I'm sorry, but it doesn't fly.
It is more that possible universes can be divided into:
Universes we can say are possible to which some maths or logic that we can grasp apply. (We can say that a universe with a chess king metric is possible, but that's not a maths we don't understand.)
Universes which contradict our maths and logic, and about which we can say nothing, not even that they are possible. It would be like trying to say things about Kantian noumena: Kantian reality outwith our concepts.
There may be things outside the limits of our logical thought. I doubt it, but obviously one can't prove it if it's outside the limits of our logical thought. But if there are we can't say so. We cannot step outside the limits even to look at the limits from outside.
What one cannot do is both assert that there's a universe we cannot understand and assert that we can understand it.
That said, I think the question of whether one universe plus one universe makes two universes is of interest if you think mathematics derives from the physical structure within universes. Is there supposed to be a maths that derives from the meta-structure of the multiverse?
Posted by Jack o' the Green (# 11091) on
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Originally posted by agingjb:
I am being driven inexorably towards what is, I suppose, a Platonic view of the existence of mathematical objects.
That's interesting. Existence or truth, or are they the same thing?
Posted by LeRoc (# 3216) on
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Dafyd: Universes which contradict our maths and logic, and about which we can say nothing
Well, you just said something about them.
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Dafyd: not even that they are possible.
I say that they are possible. See, I can. They are possible! Shoot me.
In fact, you just literally said that they are possible too ("Possible universes can be divided into ...")
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Dafyd: What one cannot do is both assert that there's a universe we cannot understand and assert that we can understand it.
Oh we're ok then, because I never asserted these things.
/You seem to have a black-and-white view of things: either we can completely understand them through logical thought, or we can say nothing about them. That's simply not true, not even in our universe. (Exhibit A: women.)
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Dafyd: There may be things outside the limits of our logical thought. I doubt it, but obviously one can't prove it if it's outside the limits of our logical thought. But if there are we can't say so. We cannot step outside the limits even to look at the limits from outside.
Of course there are many things outside of the limits of our logical thought, even in our universe. And we can say things about them.
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Dafyd: Is there supposed to be a maths that derives from the meta-structure of the multiverse?
Maybe there is. Maybe there isn't. Maybe there's something else.
Posted by Dafyd (# 5549) on
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Originally posted by LeRoc:
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Dafyd: Universes which contradict our maths and logic, and about which we can say nothing
Well, you just said something about them.
Not much. In fact, my sentence specified less than no useful information about them whatsoever. It contains negative information.
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Dafyd: not even that they are possible.
I say that they are possible. See, I can. They are possible! Shoot me.
In fact, you just literally said that they are possible too ("Possible universes can be divided into ...")
In the sense that it's impossible to say unicorns don't exist without using the phrase 'unicorns are...'
I could probably have tightened up my language.
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You seem to have a black-and-white view of things: either we can completely understand them through logical thought, or we can say nothing about them. That's simply not true, not even in our universe. (Exhibit A: women.)
(I decline to find your sexist remark funny.}
I feel that the amount of assumptions that I disagree with in the phrase 'completely understand them through logical thought' are so many that I can't get started on them.
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Dafyd: There may be things outside the limits of our logical thought. I doubt it, but obviously one can't prove it if it's outside the limits of our logical thought. But if there are we can't say so. We cannot step outside the limits even to look at the limits from outside.
Of course there are many things outside of the limits of our logical thought, even in our universe. And we can say things about them.
Really? I suspect we disagree on your analysis of what's going on there.
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Dafyd: Is there supposed to be a maths that derives from the meta-structure of the multiverse?
Maybe there is. Maybe there isn't. Maybe there's something else.
[ 14. June 2015, 22:03: Message edited by: Dafyd ]
Posted by mousethief (# 953) on
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Originally posted by Dafyd:
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Originally posted by LeRoc:
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Dafyd: Universes which contradict our maths and logic, and about which we can say nothing
Well, you just said something about them.
Not much. In fact, my sentence specified less than no useful information about them whatsoever. It contains negative information.
Actually this isn't even correct. The real problem here is a conflation of language and meta-language. Talking about another universe, and talking about talking about another universe, are not the same thing. Even if we can't say anything about another universe, we can say things about saying things about another universe. It's unhelpful, confusing, and leads to error to conflate the two.
Posted by orfeo (# 13878) on
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Personally I'm beginning to think we might as well just focus on getting as far as Mars. We can worry about the mathematics of other universes when we're launching the first probe there.
Posted by LeRoc (# 3216) on
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Dafyd: Not much. In fact, my sentence specified less than no useful information about them whatsoever. It contains negative information.
You're not playing fair. Whenever I say something about another universe you thrash it, but you're allowed to build complete logical constructs about them (including a categorisation no less) because "nonono, I've said less than useful information about them." Please define what is 'useful information' about a universe that contradicts our math and logic.
And who made you the arbiter about what we can or cannot say about other universes anyway?
BTW, I just spotted another error in your earlier argument that I missed before. You assume that things (like universes) can be divided in those to which our math and logic apply, and those who contradict our math and logic.
There is an assumption about the completeness of math and logic here that you cannot prove. Logically (even within our universe), it is very much possible that there are things to which our math and logic don't apply, but which don't contradict our math and logic.
"Our math and logic don't apply to X" and "X contradicts our math and logic" aren't synonyms. In fact, I believe that a lot of our misunderstanding rests on this error of yours.
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Dafyd: (I decline to find your sexist remark funny.}
I feel that the amount of assumptions that I disagree with in the phrase 'completely understand them through logical thought' are so many that I can't get started on them.
There's nothing sexist about saying that women cannot be completely be understood by logical thought. (Neither can men of course.) In fact, I find your assertion that I've made a sexist remark rather offensive.
It is you who started the argument about logical thought, not me. You said: "There may be things outside the limits of our logical thought. I doubt it (...)"
There are many things outside of the limits of our logical thought, even in our universe. Yet, I can say something about them.
The obvious example is love of course. There are definitely parts of love that are outside of the limits of logical thought. When I'm in love with someone, it's not logical thinking I'm engaged in. And if you think you can give an explanation of love using logical thought, I'm all ready to hear it.
Of course you'll say "There's nothing about love that contradicts math and logic!" This would be based on the same error you made before (either X contradicts math and logic or it is within the limits of math and logic). And to be honest, I'm not so sure that love never contradicts logic.
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mousethief: The real problem here is a conflation of language and meta-language. Talking about another universe, and talking about talking about another universe, are not the same thing. Even if we can't say anything about another universe, we can say things about saying things about another universe. It's unhelpful, confusing, and leads to error to conflate the two.
That's rather helpful.
Posted by Dafyd (# 5549) on
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Originally posted by LeRoc:
Please define what is 'useful information' about a universe that contradicts our math and logic.
It seems to me that there is no such thing as useful information about a universe that contradicts our maths and logic.
Information isn't information unless you can do something with it. And if the universe contradicts our maths and logic then we don't have any rules for doing anything with anything. Even if you specify some rule you can't specify that there isn't another rule which is exactly the opposite.
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And who made you the arbiter about what we can or cannot say about other universes anyway?
The same guy who made you the arbiter on whether God created maths and logic, surely?
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Logically (even within our universe), it is very much possible that there are things to which our math and logic don't apply, but which don't contradict our math and logic.
Possibly you are right that I've been sloppy about the difference between 'contradicts our maths and logic' and 'outside our maths and logic'.
Let's distinguish between three cases.
1) Maths and logic underdetermine what is the case in other universes. (Yes, unless you believe in a multiverse in which all logical possibilities obtain.) In that sense, almost everything in our universe is beyond our maths and logic, though nothing is inconsistent with them. That's what it means to think that there's a distinction between empirical matters and logical matters.
2) They have a particular axiomatisation of maths (or, say, modal logic) in their universe that we don't have here. I'm happy to say that some other axiomatisation applies to other universes. There might be a universe where distance works on a chess king metric. After all, we thought Euclidean geometry applies to space in our universe and since Einstein we believe it doesn't.
Note that doesn't mean that we can no longer understand Euclidean geometry now we know it doesn't apply directly, nor that we think it is now inconsistent. Euclidean geometry is apparently independent of any physical instantiation.
3) They have the same axiomatisation as we do, yet the results come out different. The axioms are the same, but the maths is inconsistent with ours. I don't think that is possible in any sense which we could affirm to be possible. Trying to affirm it is just stringing words together.
Now I've been arguing against 3); if you're only asserting 2) then I apologise for the misunderstanding.
On the other hand, some of what you're saying about challenging me to explain love using logical thought looks like you think I'm arguing against 1). Which I am not.
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There are many things outside of the limits of our logical thought, even in our universe. Yet, I can say something about them.
The obvious example is love of course. There are definitely parts of love that are outside of the limits of logical thought. When I'm in love with someone, it's not logical thinking I'm engaged in. And if you think you can give an explanation of love using logical thought, I'm all ready to hear it.
Can we agree that all of the above is true, and also that no part of love is outside the limits of logical thought, and when you're in love with someone it's logical thinking you're engaged in?
Herbert McCabe observes somewhere that while some people might argue about whether dropping napalm on children is ever morally justified, dropping napalm on children can never be said to be an act of love. That is not an empirical generalisation, but a logical point about what love means. If someone argues that even so dropping napalm on children might be a loving thing to do because love is not bound by logic, we'd suspect that they don't care what they're saying as long as it's propaganda that suits their side.
Love isn't logical thinking, just as the survival of the natural world isn't logical thinking. But we don't do them justice if we're illogical when we think about them. If you think you ought to love your neighbour as yourself that logically means you oughtn't to burn him or her as a heretic or a pagan or a feminist, because burning your neighbour is logically incompatible with burning them.
Posted by LeRoc (# 3216) on
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Dafyd: It seems to me that there is no such thing as useful information about a universe that contradicts our maths and logic.
Basically, what's happening is this:
When I say something about such a universe, you don't allow me to say it.
When you say something about such a universe, it's ok because you were giving less than no useful information. And oh by the way, there is no such thing as useful information about them.
That seems like wiggling to me. You set up whole constructs, categorising universes (including those that contradict math and logic) into classes and drawing conclusions about them. Yet, I'm not allowed to say anything about them. That's a double standard.
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Dafyd: The same guy who made you the arbiter on whether God created maths and logic, surely?
I'm going to ask you to retract that. I explicitly said that it is my belief that God created maths and logic. That's a whole lot less than setting myself up as His arbiter.
Anyway, I wasn't not talking about theological concepts here, I'm talking about this discussion. I don't think this discussion is going to work if one of us (you) sets himself up as the arbiter of what we can or cannot say about other universes (and then breaks these rules himself). That's not how reasonable discussions work.
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Dafyd: On the other hand, some of what you're saying about challenging me to explain love using logical thought looks like you think I'm arguing against 1). Which I am not.
I'm mostly arguing for 1) (insofar as I understand what you've written about it). I use 3) sometimes to get to 1).
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Dafyd: Can we agree that all of the above is true, and also that no part of love is outside the limits of logical thought, and when you're in love with someone it's logical thinking you're engaged in?
No, I disagree. I strongly disagree in fact. It's utter bullshit. Just because you can say some logical things about love ("Dropping napalm on someone isn't love". Duh) doesn't mean that love is within the limits of logical thought.
Posted by Ricardus (# 8757) on
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Originally posted by LeRoc:
That seems like wiggling to me. You set up whole constructs, categorising universes (including those that contradict math and logic) into classes and drawing conclusions about them. Yet, I'm not allowed to say anything about them. That's a double standard.
I think you are taking Dafyd's 'cannot' a little too personally.
One of the traditional answers - I think it is at least Scholastic - to questions like 'can an omnipotent being create a rock he can't lift', is that the sentence is malformed and meaningless. It follows (at least on this argument) that one cannot say anything meaningful about such a rock, because whatever collection of words you apply to the rock has no referent, not even a hypothetical referent. You might as well talk about pflurge. AIUI, Dafyd's comments are basically a form of this argument.
You may or may not agree with this argument. The point is, it's an argument about what language (and the concept of language) means. It's nothing to do with attempting to lay down the rules of a debate.
Posted by LeRoc (# 3216) on
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Ricardus: It's nothing to do with attempting to lay down the rules of a debate.
That may be, but that isn't what I was arguing against here.
Posted by agingjb (# 16555) on
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If there are other universes, whatever that means, then, if there is a God, then they have the same God. And for Christians that God is the Triune God. Which would mean that at least the integers One and Three are coeval with the God of any universe, and precede our, or any other, material world.
Posted by LeRoc (# 3216) on
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I'm afraid that I'm not a very good Trinitarian, so that argument won't work very well with me.
Posted by agingjb (# 16555) on
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I'm a sceptic, of sorts, but if I entertain belief, it's Athanasian (without the the first clause). So, as I said earlier, if you defeat my intuition that maths is independent of the world, then you would compel me to atheisim - although in practice, I'd simply retain my scepticism.
Posted by LeRoc (# 3216) on
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Hmm, maybe I should refrain from discussing this further then. I wouldn't want to compel you to atheism (it seems so boring to me).
Posted by Jack o' the Green (# 11091) on
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Originally posted by agingjb:
If there are other universes, whatever that means, then, if there is a God, then they have the same God. And for Christians that God is the Triune God. Which would mean that at least the integers One and Three are coeval with the God of any universe, and precede our, or any other, material world.
I think I would differenciate between our concepts of 1 and 3 and God's nature. According to the orthodox view, the divine being (both essence and existence) is what it is due to itself, and is therefore independent of any concepts - mathematical or otherwise. We might find it useful to use the terms 1 or 3 when thinking about God, but I can't see how that would impact one iota on God. The most you could say is that the abstract concepts of 1 and 3 are instantiated in some way in the divine nature. However, like LeRoc, I'm not a good Trinitarian.
Posted by agingjb (# 16555) on
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This thread prompted me to reread "A Mathematician's Apology" by G.H.Hardy.
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