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Source: (consider it)
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Thread: The Ontological Status of Mathematical Entities
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Jack o' the Green
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# 11091
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Posted
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?
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Lamb Chopped
Ship's kebab
# 5528
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Posted
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.
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Enoch
Shipmate
# 14322
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Posted
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?
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itsarumdo
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# 18174
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Posted
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 ]
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Doc Tor
Deepest Red
# 9748
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Posted
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.
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mousethief
Ship's Thieving Rodent
# 953
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Posted
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.
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Honest Ron Bacardi
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# 38
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Posted
More or less the same thing happened with group theory.
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Anglo-Cthulhic
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Humble Servant
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# 18391
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Posted
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.
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Eutychus
From the edge
# 3081
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Posted
A lot of ink has been spilled on explaining why it gives an inaccurate value, though.
(welcome, by the way!)
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quetzalcoatl
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# 16740
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Posted
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 ]
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agingjb
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# 16555
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Posted
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.
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Doc Tor
Deepest Red
# 9748
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Posted
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.
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Doc Tor
Deepest Red
# 9748
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Posted
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.
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Eutychus
From the edge
# 3081
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Posted
Woah. I wonder if it got knocked to 3.14.... as a result of the Fall?
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Jack o' the Green
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# 11091
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Posted
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.
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itsarumdo
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# 18174
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Posted
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.
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Jack o' the Green
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# 11091
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Posted
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.
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maryjones
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# 13523
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Posted
Sorry, everybody , the links aren't working. I shall try again later. My apologies
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Jack o' the Green
Shipmate
# 11091
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Posted
I like that. I agree with quite a lot of what Berkeley says about things only existing if they are perceived.
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Schroedinger's cat
Ship's cool cat
# 64
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Posted
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.
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Jack o' the Green
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# 11091
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Posted
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)
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Lamb Chopped
Ship's kebab
# 5528
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Posted
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.
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Dafyd
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# 5549
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Posted
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.
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balaam
Making an ass of myself
# 4543
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Posted
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.
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Dafyd
Shipmate
# 5549
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Posted
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.
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itsarumdo
Shipmate
# 18174
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Posted
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.
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balaam
Making an ass of myself
# 4543
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Posted
Mathematics is not perfect. Quantum Mechanics contradicts General Relativity on a mathematical level.
Yet a Sat-Nav works.
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Jack o' the Green
Shipmate
# 11091
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Posted
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.
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Enoch
Shipmate
# 14322
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Posted
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 ]
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mousethief
Ship's Thieving Rodent
# 953
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Posted
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.
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Enoch
Shipmate
# 14322
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Posted
Sorry, I missed the correction slot. 'Angles' in my post should of course be 'Angels'.
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balaam
Making an ass of myself
# 4543
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Posted
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.
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itsarumdo
Shipmate
# 18174
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Posted
quote:
Originally posted by Enoch:
Sorry, I missed the correction slot. 'Angles' in my post should of course be 'Angels'.
Non Angli, sed angeli
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Pulsator Organorum Ineptus
Shipmate
# 2515
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Posted
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.
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orfeo
Ship's Musical Counterpoint
# 13878
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Posted
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.
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agingjb
Shipmate
# 16555
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Posted
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.
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Golden Key
Shipmate
# 1468
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Posted
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)
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Dave W.
Shipmate
# 8765
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Posted
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?
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mousethief
Ship's Thieving Rodent
# 953
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Posted
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.
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orfeo
Ship's Musical Counterpoint
# 13878
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Posted
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 ]
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itsarumdo
Shipmate
# 18174
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Posted
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.
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"Iti sapis potanda tinone" Lycophron
Posts: 994 | From: Planet Zog | Registered: Jul 2014
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Doc Tor
Deepest Red
# 9748
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Posted
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.
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Forward the New Republic
Posts: 9131 | From: Ultima Thule | Registered: Jul 2005
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Dafyd
Shipmate
# 5549
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Posted
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).
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we remain, thanks to original sin, much in love with talking about, rather than with, one another. Rowan Williams
Posts: 10567 | From: Edinburgh | Registered: Feb 2004
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orfeo
Ship's Musical Counterpoint
# 13878
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Posted
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.
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Technology has brought us all closer together. Turns out a lot of the people you meet as a result are complete idiots.
Posts: 18173 | From: Under | Registered: Jul 2008
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itsarumdo
Shipmate
# 18174
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Posted
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
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"Iti sapis potanda tinone" Lycophron
Posts: 994 | From: Planet Zog | Registered: Jul 2014
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IngoB
Sentire cum Ecclesia
# 8700
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Posted
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...
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They’ll have me whipp’d for speaking true; thou’lt have me whipp’d for lying; and sometimes I am whipp’d for holding my peace. - The Fool in King Lear
Posts: 12010 | From: Gone fishing | Registered: Oct 2004
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IngoB
Sentire cum Ecclesia
# 8700
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Posted
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.
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They’ll have me whipp’d for speaking true; thou’lt have me whipp’d for lying; and sometimes I am whipp’d for holding my peace. - The Fool in King Lear
Posts: 12010 | From: Gone fishing | Registered: Oct 2004
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agingjb
Shipmate
# 16555
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Posted
I still wonder to what extent theologicians speculate about the implications of large cardinal axioms in set theory.
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Refraction Villanelles
Posts: 464 | From: Southern England | Registered: Jul 2011
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Dafyd
Shipmate
# 5549
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Posted
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.)
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we remain, thanks to original sin, much in love with talking about, rather than with, one another. Rowan Williams
Posts: 10567 | From: Edinburgh | Registered: Feb 2004
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