Ship of Fools: i

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Posted by Alan Cresswell (# 31) on :

Inspired by another thread, I got thinking ... speculating may be a better word.

i (or for engineers, j) is the imaginary unit, the square root of -1. Since any real number multiplied by itself is positive, on that basis i shouldn't exist. It's a logically fallacious, impossible entity. Well, my speculation went, what else gets called a logically fallacious, impossible entity? Well, God gets called that by many atheists.

But, as many of you will know, i turns out to be incredibly useful. First, they were first seriously considered as they were needed to solve cubic and quartic polynomials, as such maybe just a mathematical curiosity. But, it was also discovered that trigonometric functions can be expressed as exponentials involving i, allowing complex trigonometrical functions to be expressed as much simpler exponential functions. Complex numbers (combining real and imaginary numbers) form the basis of Fourier Transforms, essential in signal processing. Complex numbers are well suited to describing oscillators and waves (with all sorts of applications in seismology, vibration analysis, fluid dynamics, electromagnetism and other fields). The mathematics of quantum mechanics, the Schrodinger equation, Hilbert spaces, the matrix mechanics of Heisenberg all utilise complex numbers. Multiplication of time by i creates a space like dimension, and thus we have space-time essential to Relativity. If you remember the beauty of Mandelbrot sets, then you can thank i since they are plotted on a complex plane. It makes you wonder, if something is that useful in such a wide range of situations ... could it be a real, rather than imaginary, entity.

So, as I continue to speculate, if the atheists are right to describe God as a logically fallacious, impossible entity just as i appears what does our experience of i tell us about whether a logically fallacious and impossible entity may, nonetheless, be real?

Posted by mr cheesy (# 3330) on :

Problem is, Alan, you're probably the only person here who could follow beyond the first sentence of your post.

[I really wish I had brought up a different example other than imaginary numbers. I hate imaginary numbers..]

Posted by LeRoc (# 3216) on :

One of the most fascinating thing about complex numbers I find to be the Residue Theorem. This is the one mathematical theorem that make me think "How can these things be related to each other? This is so useful, surely this can't be true??"

To me, the Residue Theorem is more than sufficient proof that God created i.

Posted by hatless (# 3365) on :

Yes, I think it's a good analogy. God does not exist. If you could list all the things in the universe, God would not be on the list. But there are things you cannot (easily) say without God. God is a word that increases our ability to communicate and express ourselves. It gives us power, in particular, to speak of things like a person's worth, or of forgiveness, great beauty, moral imperatives, etc.

God, like i, is a term that only makes sense when doing some work.

Posted by no prophet's flag is set so... (# 15560) on :

When i saw the thread's title, my first thought was something Apple had invented, maybe iGod, who maybe comes with an app iJesus.

Posted by Beeswax Altar (# 11644) on :

quote:
Originally posted by mr cheesy:
Problem is, Alan, you're probably the only person here who could follow beyond the first sentence of your post.

[I really wish I had brought up a different example other than imaginary numbers. I hate imaginary numbers..]

Give us some credit. Even I got nearly halfway through the second complete paragraph before getting lost and I'm horrible at math. Alan was making perfect sense until Fourier Transforms.

Posted by HCH (# 14313) on :

Some people have commented that God invented the counting numbers (1, 2, 3, ...) and all other numbers are human inventions. The only problem with i is that someone dubbed it "imaginary one" and the name stuck.

The beauty of i is that once we have it, we have the complex numbers and any polynomial of degree n has n roots. (Yes, i itself has a square root in the complex numbers.)

Posted by Baptist Trainfan (# 15128) on :

It was the complexity of Fourier Series (all done by hand and with slide rules or log tables) which led me to abandon maths for my final year of Mechanical Engineering.

I like your analogy, though ... and Hatless's comment. (Although I don't want God to be either "imaginary" or "negative"!)

Posted by Nick Tamen (# 15164) on :

quote:
Originally posted by no prophet's flag is set so...:
When i saw the thread's title, my first thought was something Apple had invented, maybe iGod, who maybe comes with an app iJesus.

I thought Alan was just having a moment of Hispanophonic excitement.

Posted by Sipech (# 16870) on :

quote:
Originally posted by mr cheesy:
Problem is, Alan, you're probably the only person here who could follow beyond the first sentence of your post.

No he's not. I happen to have a Master's degree in maths. [Devil]

The problem with the OP is the idea that i is fallacious. It was thought to be so, but then again, so were negative numbers. The widespread use of negative numbers came from having to account for debts. Neither negative numbers nor complex numbers are logically impossible. They are the very opposite, in fact; they are logically necessary.

It is better to say that they are merely outside of our normal experience. If you are used to using numbers to count how many apples you have, then of course a negative number or a complex number of apples is absurd. What is needed is to broaden your horizon to ask what a number really is.

As an aside, it's a good test to throw to maths students: ask them what a number is.

If they give you a quick answer off the cuff, they're no good. If they suddenly look all confused and pensive, then you've got someone who can actually think through the question and realise the answer is not as simple to give as the it is to phrase the question.

Posted by Tortuf (# 3784) on :

My understanding of the beauty and function of numbers is that they can be used to describe and analyze reality. As in equations help us understand gravity and its effect on objects with mass.

Where we cannot use numbers to fully accurately describe or measure something might just be a function of not knowing what numbers to use, what formula to use, or not knowing what combination of numbers and equations to use. What comes to mind is measuring the content of barrels. Merchants could accurately measure the contents by sliding a measuring stick into a barrel so the bottom of the stick was in the barrel on one side and the top was at the opposite side of the barrel.

Mathematicians of the time had to break the barrel up into ever smaller cubes to measure content and even then could never fully know what was in the barrel because there was always some bit that could not be cubed in.

So, boring the pants off of everyone further, with God, we can use some functions (texts, experiences) to describe God, but we cannot "prove" God, or the lack of "God" using the tools we have now. Will there be tools sufficient to the task in the future? Maybe. Doubt it.

Knowing that not everything can be described with fully rational, non imaginary numbers yet, may be a good start.

Posted by Crœsos (# 238) on :

quote:
Originally posted by Alan Cresswell:
It makes you wonder, if something is that useful in such a wide range of situations ... could it be a real, rather than imaginary, entity.

I think you're getting caught up in the semantics. Imaginary numbers aren't "imaginary" in the sense of having no real existence any more than irrational numbers are unamenable to human reason. They're called these things largely for historic, contingent reasons and because they have to be called something.

Posted by mr cheesy (# 3330) on :

I am humbled by your intelligence and knowledge of mathematics, oh shipmates. [Overused]

Posted by Enoch (# 14322) on :

As a matter of tangential curiosity, and from a position of almost total ignorance of serious maths (this question probably demonstrates that):-

Are 1 and -1 two totally different numbers, largely different but with some things in common. Or are they the same number with different polarity?

Posted by lilBuddha (# 14333) on :

Originally posted by Alan Cresswell:

quote:
So, as I continue to speculate, if the atheists are right to describe God as a logically fallacious, impossible entity just as i appears what does our experience of i tell us about whether a logically fallacious and impossible entity may, nonetheless, be real?

God(s)/mystical forces exist because it/they do or because they fill the gap between known and unknown. They serve a purpose, so "logically fallacious" is less than accurate. IMO, but what does a non-theist know?

Posted by Dafyd (# 5549) on :

quote:
Originally posted by Enoch:
Are 1 and -1 two totally different numbers, largely different but with some things in common. Or are they the same number with different polarity?

I think the former is correct in the way you're asking. Although you could probably set up your terms in such a way that it's the latter.

But 9+1 does not = 9 + (-1), which generally means they're different. Or if you're bothered by the sum 9 + (-1): 9+ (5-4) is not the same as 9 + (3-4).

Posted by Jack o' the Green (# 11091) on :

quote:
Originally posted by Enoch:
Are 1 and -1 two totally different numbers, largely different but with some things in common. Or are they the same number with different polarity?

They have the same absolute value. A negative number is any number whose value is less than zero, and a positive number is any number whose value is greater than zero. In other words, they are sort of 'opposites' to each other, or each other's additive inverse i.e. two numbers which added together equal 0.

Posted by Carex (# 9643) on :

quote:
Originally posted by HCH:
Some people have commented that God invented the counting numbers (1, 2, 3, ...) and all other numbers are human inventions.

And there is no reason that God had to create the ordinal numbers, either. A count of things such as animals would be an important concept to communicate as language developed in pre-historic people. The extension of that to convenient representations of larger numbers would be a natural progression as civilization evolved and such numbers were needed. From there it is just a continuation of extending numbers to express more things.

I'm one who uses j regularly (that's the the electrical engineer's version of i to avoid confusion with the use of i to represent electrical current) and don't consider it to be at all imaginary, except in the specialized use of that term to describe certain types of numbers. But then, we also have specialized uses for terms such as admittance and reluctance that aren't shared with most other people either.

Posted by Jack o' the Green (# 11091) on :

1= 1×1
Or -1×-1

-1= -1×1
Or 1×-1

Posted by LeRoc (# 3216) on :

quote:
Enoch: Are 1 and -1 two totally different numbers, largely different but with some things in common. Or are they the same number with different polarity?

It depends a bit on what you want to do with them.

If you look at it purely from a mathematical point of view, then these are two different numbers. There are no degrees in "differentness", you cannot say that one number pair is "largely different" and another number pair is "totally different". -1 ≠ 1, and that's it.

However, in some applications it makes a certain sense to see some number pairs as more different than others.

For example, sometimes you do an algebraic calculation and you're not really interested of the value of the number that comes out. You just want to know whether it's even or odd. In this application, it makes sense to see the numbers 1; 3 as pretty much the same and 1; 2 as much more different. In the former pair, the numbers are both odd, so they are basically the same for the purpose you're using them for.

In other calculations, you're not really interested in the exact result, you just want to know the rough "order of greatness" that comes out of it. In this application, the numbers 1; 2; 3 are all pretty much the same (they're all small) and 1,037,472,274,853 is very different from that (because it's much bigger).

(There are mathematically sound ways to do both things in a formal way. If you want to know, they have to do with algebraic groups and big-O notation respectively.)

So, in some applications it makes sense to consider some number pairs as "more different" and others as "less different".

I can imagine that in calculations where you're not very much interested in the sign of your answer, it would make sense to consider -1 and 1 as pretty much the same. As others have said, you'd probably use the absolute value notation |x| for that.

Posted by Jengie jon (# 273) on :

I have not explored the debate but there is a discussion within the philosophers of mathematics about whether mathematical constructs are invented or discovered.

The problem and it is a real one, is that logic is nothing more than an application of these constructs. (Equivalence is still debated as the challenge to derive the constructs from logic has not been shown). Therefore if mathematical constructs are invented, logic is invented!

Jengie

Posted by balaam (# 4543) on :

quote:
Originally posted by HCH:
Some people have commented that God invented the counting numbers (1, 2, 3, ...) and all other numbers are human inventions.

Modern maths was impossible until the Arabs invented the concept of 0 (zero).

Posted by Chapelhead (# 21) on :

A few odd thoughts...

I recall a university maths lecturer telling me that, when he was an undergraduate, the starting point of the maths he was taught was, "There are positive integers". This, he realised later, was an assumption - there's no proof that "1" and "2" and "3" etc exist.

So I suppose that when you get to i you have something whose existence cannot be proved even more. But it still 'works'.

I quite like i, and last Christmas 'Secret Santa' bought me this. But I prefer the transfinite numbers ("numbers bigger than infinity" would be an inaccurate but 'easy' way to understand them, for those not mathematically inclined). I recall hearing Buzz Lightyear saying "To infinity and beyond", and being slightly puzzled why people thought this funny - it sounds quite reasonable to me.

Posted by LeRoc (# 3216) on :

quote:
Carex: I'm one who uses j regularly

Get behind me! Why don't we have a smiley making a protective cross symbol with its fingers? [Smile]

My view of whether the numbers are discovered or invented is: clearly we invented the numbers 1, 2, 3. But this is because we discovered that it makes sense to do this. Our Universe is such that 1+1=2 makes sense in understanding it.

This was already discussed a couple of weeks ago, but as a Christian, I believe that God created the Universe in this way. We invented mathematics, but God created the Universe such that mathematics makes sense to understand it.

Posted by Schroedinger's cat (# 64) on :

I think the point about mathematical concepts like i is that they do not directly map onto something in the real world. The integers do - we count things using them, we can count 1 cushion, for example - it doesn't make them any more extant, but they relate to something we can see and identify. 1 cushion is a concept that we can relate to, and so we tend to accept that "1" is something extant.

Concepts like i do not have a sensory-world-analogue, so we treat these with more suspicion. We instinctively struggle with comprehending this, even if we can accept it as something that is useful, and an idea that exists within a conceptual paradigm (like, the world of Maths). The thing is, it is no more or less extant that those things we can visualise. They are just less intuitive.

In the same way, I think, God is not something/one we can relate to the physical, sensory world, by his nature. That doesn't mean he is any less real, just that his "reality" is not one that is limited by our sensory perception.

Posted by mark_in_manchester (# 15978) on :

If 'j' (I was a kind of electrical engineer, and 'i' was reserved for current) does your head in, what about quaternions?

They are 3D complex numbers (think of a 3D Argand diagram with no real axis, where i, j and k refer to 90deg rotations in each of the three orthogonal planes).

Then
i^2=j^2=k^2=ijk=-1 !!

I really miss teaching this stuff, Fourier transforms and all that - it's not hard, really. But I don't at all miss having to try to find ways to bring in money to a university, on the back of it!

Posted by Crœsos (# 238) on :

quote:
Originally posted by balaam:

quote:
Originally posted by HCH:
Some people have commented that God invented the counting numbers (1, 2, 3, ...) and all other numbers are human inventions.
Modern maths was impossible until the Arabs invented the concept of 0 (zero).

Indians, actually. (Or rather one of the small states in what is now northwestern India.) The system of what we now call 'Arabic numerals' is called thus because the Arabs were the first people to have the brilliant idea of stealing them from India. Credit where credit is due.

Posted by mark_in_manchester (# 15978) on :

Ooooh - to connect with the 'God' ideas in the OP - if anyone attracted by Alan's excellent thread who likes Fourier Transforms would like to discuss Dooyeweerd (Dutch Christian philosopher, died in 1970s) and his Fourier-like decomposition of all reality (man!) into something like a set of orthogonal basis functions he calls the 'modal aspects' - which specifically include the dimension of faith - then do say and maybe I'll start another thread.

It's a bit specialist, but maybe Alan has prised up a rotting plank and revealed scurrying mathematicians within the ship...

Posted by mark_in_manchester (# 15978) on :

Enoch asked:

quote:

Are 1 and -1 two totally different numbers, largely different but with some things in common. Or are they the same number with different polarity?

An engineer would say they have the same magnitude, but opposite _phase_ - which is similar but a slightly more flexible concept to your polarity.

So if my 2 kids were on a pair of swings next to each other, I might swing them both to a given height and call this '1'. If they moved exactly together (same frequency, and peaking at just the same point in time) they both have magnitude 1 swinging, and where they are at *just* a particular point in time we could find write down with a 'sin' function. But very often we don't care - we just need to know how big the swing is (the magnitude). This is like saying mains voltage is 240v...but if you touch it quick, it might be less or even zero...or it might not! A bit like jumping under the swings...time it right!

If my kids move in opposite directions (one peaks back, when one peaks forwards) then we say they are 180deg out of phase. Then an engineer would say one has magnitude 1, and the other has magnitude -1.

It gets more fun if one peaks *just* as the other moves through (say) the rest position...where she'll end up when I get bored and go and read the paper. Then the first is moving back through this position *just* as the second peaks.

We say there is 90deg of phase between the two of them - but they still have the same frequency and magnitude 1.

Imagine we could multiply the motion of kid 2 by a magic something to shift her motion relative to kid 1 by 90deg. If we did it again, she'd be shifted by 180deg. This is a shift of polarity, and magnitude 1 becomes magnitude -1.

Therefore 'magic something'x'magic something' = -1

So 'magic something' ^2 = -1

So 'magic something' = sq root (-1)

Sorry folks - I just don't get the opportunity any more...

[ 15. September 2015, 18:21: Message edited by: mark_in_manchester ]

Posted by Jay-Emm (# 11411) on :

quote:
Originally posted by Crœsos:
The system of what we now call 'Arabic numerals' is called thus because the Arabs were the first people to have the brilliant idea of stealing them from India. Credit where credit is due.

you mean Edison copied even that
[Eek!]

Another way 1 and -1 fundementally differ is that 1 is the identity for multiplication, and -1 isn't. (That is 5*1 is still 5, 5*-1 is no longer 5).
Although because 5*-1*-1=5, -1 is still in some ways like 1 [in particular when dealing with primes and irreducables, 5*2 and -5*-2 don't count as seperate factorisations, but once you deal with fractions everything's a unit, anyway].

[ 15. September 2015, 18:22: Message edited by: Jay-Emm ]

Posted by LeRoc (# 3216) on :

quote:
mark_in_manchester: Dooyeweerd (Dutch Christian philosopher, died in 1970s) and his Fourier-like decomposition of all reality (man!)

I've only vaguely heard of him. This is so brilliantly funny that it must be true [Smile]

Posted by mr cheesy (# 3330) on :

If there are an infinity of integers between 0 and infinity, and an infinity of fractions between 0 and 1.. how come there are not an infinity of infinity of possible fractions?

Posted by LeRoc (# 3216) on :

quote:
mr cheesy: If there are an infinity of integers between 0 and infinity, and an infinity of fractions between 0 and 1.. how come there are not an infinity of infinity of possible fractions?

The "kind of infinity" (mathematicians call it "cardinality") of the integers is actually the same as the "kind of infinity" of fractions. There is a rather beautiful proof of that.

Posted by Jack o' the Green (# 11091) on :

quote:
Originally posted by LeRoc:

quote:
Carex: I'm one who uses j regularly
Get behind me! Why don't we have a smiley making a protective cross symbol with its fingers?

My view of whether the numbers are discovered or invented is: clearly we invented the numbers 1, 2, 3. But this is because we discovered that it makes sense to do this. Our Universe is such that 1+1=2 makes sense in understanding it.

This was already discussed a couple of weeks ago, but as a Christian, I believe that God created the Universe in this way. We invented mathematics, but God created the Universe such that mathematics makes sense to understand it.

I would say something very similar. I think the label which best describes my position is 'Divine Conceptionalist'.

Posted by Humble Servant (# 18391) on :

quote:
Originally posted by mr cheesy:
If there are an infinity of integers between 0 and infinity, and an infinity of fractions between 0 and 1.. how come there are not an infinity of infinity of possible fractions?

It's because infinity is not a number. It is a concept that behaves very differently from the numbers. If infinity were a number then infinity +1 would be a larger number, which would mean infinity was not infinite.

Posted by LeRoc (# 3216) on :

My internet connection is a bit better now, so I can expand a bit more on this.

In a sense, we can say there are just as many integers as there are fractions. This is a bit tricky, since the answer to both is 'infinity', but there exists a mathematical way of comparing different kinds of infinity with each other.

A thing about integers is that we can put them on a line: 1, 2, 3 ... This is how we count. What we can show is that we can put all fractions on a line in the same way.

The way to do this is shown in this image. The arrows form a path in which all fractions can be put after each other on a line. (You may have to skip the ones in red.)

Since we can put all fractions on a line, we can number them 1, 2, 3 ... Every integer corresponds to one fraction. This shows that in a sense, there are as many fractions as there are integers.

Posted by Hedgehog (# 14125) on :

quote:
Originally posted by LeRoc:

quote:
mr cheesy: If there are an infinity of integers between 0 and infinity, and an infinity of fractions between 0 and 1.. how come there are not an infinity of infinity of possible fractions?
The "kind of infinity" (mathematicians call it "cardinality") of the integers is actually the same as the "kind of infinity" of fractions. There is a rather beautiful proof of that.

But there are "larger" infinities out there, as Chapelhead mentioned: the transfinite numbers. A poor university prof tried to make me understand them some 30+ years ago, and, but for Chapelhead's post, I would have forgotten their name. IIRC, the infinite number of decimals between 0 and 1 is a larger infinity than the infinite number of fractions.

And there is a mathematical proof of that, too. Because it isn't the sort of thing you'd say without proof.

But I don't remember it because it was 30+ years ago and I had no use for the knowledge until right now.

Posted by Jay-Emm (# 11411) on :

The version I've seen (and it's a beautiful combination with the other set) follows on from the proofs where you put the natural numbers the odd numbers, the integers, the squares, the fractions, etc in some order (so you can say what the 79th fraction in the infinite list is).

It supposes you could make the list, and then creates a real number not on the list.

the diagonal proof

Posted by LeRoc (# 3216) on :

quote:
Hedgehog: But there are "larger" infinities out there, as Chapelhead mentioned: the transfinite numbers.

Yes there are.

quote:
Hedgehog: IIRC, the infinite number of decimals between 0 and 1 is a larger infinity than the infinite number of fractions.

And there is a mathematical proof of that, too.

Yes there is. The proof of that is a bit more complex.

quote:
Hedgehog: But I don't remember it because it was 30+ years ago and I had no use for the knowledge until right now.

See? Your maths teacher told you that it would come in handy some day [Smile]

[ 15. September 2015, 21:15: Message edited by: LeRoc ]

Posted by Jack o' the Green (# 11091) on :

Is that the same or similar to 'countable' and 'uncountable' infinities (e.g. the infinite of integers, and the infinite set of real numbers?)

Posted by Sober Preacher's Kid (# 12699) on :

quote:
Originally posted by mr cheesy:
Problem is, Alan, you're probably the only person here who could follow beyond the first sentence of your post.

[I really wish I had brought up a different example other than imaginary numbers. I hate imaginary numbers..]

Speak for yourself. [Devil]

Posted by Dafyd (# 5549) on :

quote:
Originally posted by Jack o' the Green:
Is that the same or similar to 'countable' and 'uncountable' infinities (e.g. the infinite of integers, and the infinite set of real numbers?)

That's the same. The fractions (and indeed all the algebraic numbers - that is the fractions, plus square roots, cube roots, etc.) called a countable infinity because you can put them into a list that you can count your way through.
The real numbers are an uncountable infinity because you can't put them into a list to count your way through them.

Posted by Adeodatus (# 4992) on :

To begin with, I'd like to suggest that while i isn't a "real" number (in the mathematical definition of "real"), it's every bit as real or unreal as any other number. One interpretation of i is that on a piece of 2-dimensional graph paper, if 1 is 1 unit along the x-axis, i is 1 unit along the y-axis. That's pretty real.

quote:
Originally posted by Alan Cresswell:
But, as many of you will know, i turns out to be incredibly useful.

Is this like saying that if God didn't exist, it would be necessary to invent him? That whether he exists or not, "God" is necessary because of his usefulness?

It's an interesting idea, and my private personal creed has for a long time been like that of C.S.Lewis' Puddleglum, whom I'm going to quote here at greater length than he's usually given, and hope it isn't too long a quote for the Hosts -

quote:
"Suppose we have only dreamed, or made up, all those things - trees and grass and sun and moon and stars and Aslan himself. Suppose we have. Then all I can say is that, in that case, the made-up things seem a good deal more important than the real ones. Suppose this black pit of a kingdom of yours is the only world. Well, it strikes me as a pretty poor one. And that's a funny thing, when you come to think of it. We're just babies making up a game, if you're right. But four babies playing a game can make a play-world which licks your real world hollow. That's why I'm going to stand by the play world. I'm on Aslan's side even if there isn't any Aslan to lead it. I'm going to live as like a Narnian as I can even if there isn't any Narnia."

My faith is a devotion to an idea that perhaps someone one had, of something better and brighter. I suppose you could accuse my faith of being mere utopianism.

But utopianism isn't the real problem. The real problem with faith is that it either has to be real, or it has to be better than real. And we find ourselves in a world where people of faith so often give their lives to making real life worse for others, not better. This is my problem with faith - that it can be as ugly as it is beautiful, and God as harmful as he is useful.

Posted by Alan Cresswell (# 31) on :

quote:
Originally posted by Crœsos:

quote:
Originally posted by Alan Cresswell:
It makes you wonder, if something is that useful in such a wide range of situations ... could it be a real, rather than imaginary, entity.
I think you're getting caught up in the semantics. Imaginary numbers aren't "imaginary" in the sense of having no real existence any more than irrational numbers are unamenable to human reason. They're called these things largely for historic, contingent reasons and because they have to be called something.

Of course, and the word "imaginary" was first used in a derogatory sense - because, as Sipech said in correcting a point in my OP, they were thought to be fallacious, and a useless mathematical curiosity. Sometimes derogatory names stick (in the other thread I also mentioned "Big Bang" as a derogatory term that endured).

Posted by Alan Cresswell (# 31) on :

quote:
Originally posted by Schroedinger's cat:
Concepts like i do not have a sensory-world-analogue, so we treat these with more suspicion. We instinctively struggle with comprehending this, even if we can accept it as something that is useful, and an idea that exists within a conceptual paradigm (like, the world of Maths). The thing is, it is no more or less extant that those things we can visualise. They are just less intuitive.

In the same way, I think, God is not something/one we can relate to the physical, sensory world, by his nature. That doesn't mean he is any less real, just that his "reality" is not one that is limited by our sensory perception.

Which is a great way of expressing what I was grasping for in my speculative OP. [Overused]

Posted by orfeo (# 13878) on :

quote:
Originally posted by Schroedinger's cat:
I think the point about mathematical concepts like i is that they do not directly map onto something in the real world. The integers do - we count things using them, we can count 1 cushion, for example - it doesn't make them any more extant, but they relate to something we can see and identify. 1 cushion is a concept that we can relate to, and so we tend to accept that "1" is something extant.

Concepts like i do not have a sensory-world-analogue, so we treat these with more suspicion. We instinctively struggle with comprehending this, even if we can accept it as something that is useful, and an idea that exists within a conceptual paradigm (like, the world of Maths). The thing is, it is no more or less extant that those things we can visualise. They are just less intuitive.

In the same way, I think, God is not something/one we can relate to the physical, sensory world, by his nature. That doesn't mean he is any less real, just that his "reality" is not one that is limited by our sensory perception.

This is true of lots of things. We have abstract nouns for emotions, for example. You can't use your senses to directly detect love, anger, hope, glory or justice.

Posted by Schroedinger's cat (# 64) on :

orfeo - maybe (and if those help you grasp God then fine). I do think that our sensory world can "see" these things in action - our physical bodies know love, hate; we see acts of justice and mercy. We use words that convey these ideas that are within our sensory remit.

Even with completely abstract concepts like "an idea forming", we "see" this in the expressions and actions of a person. i is more like that actual biological and neurological processes of "an idea forming".

And Alan likes my explanation of his idea? wow. I win.

Posted by Full of Chips (# 13669) on :

Of course there are two square roots of -1.

If i^2 = 1, then (-i)^2 = -1 x i x -1 x i = 1 x i^2 = -1

So (-i) is also a square root of -1.

The question is, which of the two square roots of -1 did we label i (or j) in the first place?

The mathematics of complex numbers is based on an unknowable choice which creates two identical systems pi out of phase with each other.

I expect we make unknowable choices in our understanding of God, resulting in belief systems out of phase with each other too.

[ 16. September 2015, 18:57: Message edited by: Full of Chips ]

Posted by LeRoc (# 3216) on :

quote:
Full of Chips: The question is, which of the two square roots of -1 did we label i (or j) in the first place?

(It doesn't matter.)

quote:
Full of Chips: I expect we make unknowable choices in our understanding of God, resulting in belief systems out of phase with each other too.

Nice one!

Posted by Full of Chips (# 13669) on :

Quote:
Le Roc:
quote:
Full of Chips: The question is, which of the two square roots of -1 did we label i (or j) in the first place?
(It doesn't matter.)

Well I agree it doesn't matter if you just want to use complex numbers. It does matter however if you care about whether mathematical systems you are constructing are natural in the sense of being categorically unique as opposed to merely one of several isomorphic objects that could result dependent on some choice.

I freely admit though that even in advanced mathematical texts of that nature, it generally results only in a little footnote that says "*subject to a choice of square root of -1".

It also matters (though perhaps only to pedants) if you want to say, as Alan did, that "i is the square root of -1".

[ 16. September 2015, 19:31: Message edited by: Full of Chips ]

Posted by LeRoc (# 3216) on :

quote:
Full of Chips: It does matter however if you care about whether mathematical systems you are constructing are natural in the sense of being categorically unique as opposed to merely one of several isomorphic objects that could result dependent on some choice.

No. I'm sorry, but no. There is no choice that we're making here. Try to formulate what you're saying here mathematically. You'll run into problems.

quote:
Full of Chips: I freely admit though that even in advanced mathematical texts of that nature, it generally results only in a little footnote that says "*subject to a choice of square root of -1".

I don't believe you. Show me a mathematical text that has such a footnote.

quote:
Full of Chips: It also matters (though perhaps only to pedants) if you want to say, as Alan did, that "i is the square root of -1".

This is true. He should have said something like "i is one of the square roots of -1". (Actually, a pedant could still have problems with that. The correct phrasing is "i is one of the two solutions of x²+1=0".)

But he's a physicist. What can I say? [Smile]

Posted by Full of Chips (# 13669) on :

Quote:
Le Roq

"No. I'm sorry, but no. There is no choice that we're making here.

...

I don't believe you."

There's a nice wee discussion of it here

See the section entitled "i and -i"

There are footnotes like I have described in, for example, algebraic topology texts where functorial constructions (such as homotopy groups or K Theory) with complex coefficients are being constructed. Those constructions are natural up to a choice of the square root of -1.

I don't care enough whether you believe me or not to go trawling through one to find such a footnote and, anyway, I think the complex field example given in the link is easier to get.

[ 16. September 2015, 20:49: Message edited by: Full of Chips ]

Posted by LeRoc (# 3216) on :

quote:
Full of Chips: See the section entitled "i and -i"

This section literally sais "However, no ambiguity results ..." All of this is just a matter of notation. There is no mathematical choice here, no ambiguity.

quote:
Full of Chips: There are footnotes like I have described in ...

No, there are no such footnotes.

I'm sorry, but what you are saying is absolutely bullshit.

We are talking mathematics here. This isn't a matter of opinion that people can simply disagree about on a bulletin board. The only way to go about this is to forget about sloppy definitions like "i is the square root of -1", which is not how i is defined mathematically, and move to formal definitions of complex numbers. This shows how that is done.

You want to make a mathematical statement? Talk mathematics. Define what you are talking about mathematically, and move from there.

Posted by Barnabas62 (# 9110) on :

The topics which cause temperature rises between Shipmates never cease to amaze me.

LeRoc, Full of Chips, I recommend that you de-escalate, else you may run into Commandment 3 issues. I can follow the differences between you sufficiently to see you're getting narked.

Barnabas62
Purgatory Host.

Posted by Jack o' the Green (# 11091) on :

Maybe they should have a custard π fight to get it out of their systems.

Posted by agingjb (# 16555) on :

Maths? We invent the questions and discover the answers.

BTW, I remember an Only Connect Round 2 where the first reveal was "Octonions"; it's one of those nice but slightly amazing things about maths that the answer in the fourth term is immediately deducible.

Posted by mark_in_manchester (# 15978) on :

quote:
The only way to go about this is to forget about sloppy definitions like "i is the square root of -1",

Wow, you must really have hated my '2 girls on the swings' analogy!

I don't like to think how you might have reacted in class to 'the thing in the power to the right of e^j is the phase; and the thing to the left of the 'e' is how big it is'. Or indeed to the phrase 'think of impedance as 'hard-to-move-ness'!

[Big Grin]

(Yes, the magnitude was sometimes complex too, but hey, we got there in the end. Such are engineering courses in lower ranking UK universities. Those which still attempt any formal content at all...).

Posted by LeRoc (# 3216) on :

quote:
agingjb: BTW, I remember an Only Connect Round 2 where the first reveal was "Octonions"; it's one of those nice but slightly amazing things about maths that the answer in the fourth term is immediately deducible.

I know what octonions are and I think I know what Only Connect is (through the Circus thread), but I don't understand what you're saying here? It sounds like fun.

Posted by mark_in_manchester (# 15978) on :

I know I may be talking to myself, but I'm enjoying a trip down memory lane.

So continuing my tour of dodgy pedagogical techniques, here's a class call-and-response along the lines of Bruce Forsyth's (70s UK game show host - you say the words in bold type) 'Nice to see you' - 'To see you - nice' .

Celebrant: 'Why do we use complex exponentials as trial solutions for second order differential equations?'

Congregation: 'Becuase they're a piece of piss to differentiate and integrate'.

Posted by Paul. (# 37) on :

Only Connect the TV show has a round where they reveal the connections one by one and you have to predict the fourth one - so there's a connection and an order. Obviously more points the less has been revealed so far.

Posted by LeRoc (# 3216) on :

quote:
Paul.: Only Connect the TV show has a round where they reveal the connections one by one and you have to predict the fourth one - so there's a connection and an order. Obviously more points the less has been revealed so far.

I get that, but I admit that I'm still puzzled what the sequence would be in the case where the first reveal is octonions.

Octonions → quaternions → complex numbers → real numbers?

Posted by Sipech (# 16870) on :

*2nd attempt at posting* [Mad]

What is particularly interesting in the "higher-dimensional" numbers is that you lose a helpful property with each new set you use.

With a move from the real to the complex numbers, you lose the property that a square root is necessarily real.

When you move to the quaternions, you lose the property of commutitivty under multiplication. We are used to the idea that a times b is the same as b times a; the order doesn't matter. But now, while i.j = k, j.i = -k

Some find this incredibly confusing, yet neither division nor subtraction are commutative binary operations, even when acting on the set of real numbers.

With octonions, you go even further and lose the property of associativity. This is the property that a.(b.c) = (a.b).c Then it starts to get weird.

And when you go to the 16-dimensional version, sedonions, your head is likely to be spinning like a character in The Exorcist!

Posted by Paul. (# 37) on :

quote:
Originally posted by LeRoc:

quote:
Paul.: Only Connect the TV show has a round where they reveal the connections one by one and you have to predict the fourth one - so there's a connection and an order. Obviously more points the less has been revealed so far.
I get that, but I admit that I'm still puzzled what the sequence would be in the case where the first reveal is octonions.

Octonions → quaternions → complex numbers → real numbers?

I assume so, but I didn't see the episode.

Posted by agingjb (# 16555) on :

Yes, division algebras, which have dimensions 1, 2, 4 , and 8.

Sipech has covered the details better than I can.

So people had invented the question about in what systems can you have arithmetic (including division) with more or less usual rules. Hamilton tried to construct a system with three basic elements, and discovered you need four. Cayley and Graves quickly (and separately) discovered an eight dimensional system, and it turned out that there weren't any more.

You can read more on wiki etc., my point is to illustrate invention (of questions) and discovery (of answers) in maths.

Posted by LeRoc (# 3216) on :

quote:
Sipech: What is particularly interesting in the "higher-dimensional" numbers is that you lose a helpful property with each new set you use.

Yes, this is true.

quote:
Sipech: And when you go to the 16-dimensional version, sedonions, your head is likely to be spinning like a character in The Exorcist!

In a sense, the buck stops at eight. With sedenions, you can multiply two of them that are not zero and get zero as a result, you can't really define a norm ("distance") for them ... You lose so many properties that you basically end up with strings of numbers.

Posted by Schroedinger's cat (# 64) on :

There was an episode of Only Connect, where the first clue was "Intake", and the team correctly guessed the fourth as being "Exhaust" (I had just about got that this might be the solution, but would not have pressed!).

I think it is relevant that the last century has blown the idea that our sensory engagement with the world is all there is, even without a spiritual dimension. The idea that "If I can't see it, it doesn't exist" should be gone (Although Dawkins does seem to be wanting a form of it back). The "real" world is much less "real" than we ever could have imagined. For me - and I know this doesn't apply to everyone - I feel that there is somewhere in that a place for God. Or rather, there is the possibility of a non-empirical entity.

Posted by Sipech (# 16870) on :

A very interesting voice on the subject is Roger Penrose. Here's a short essay he wrote which echoes the introduction of his work, The Road to Reality.

Contrary to Schroedinger's cat, the argument Penrose advocates is Platonist than modern, albeit just expressed in a modern fashion.

I sometimes wonder if that mathematical reasoning is why the church leaders I have always got on best with began by studying mathematics and then moved over to theology.

Posted by Jack o' the Green (# 11091) on :

quote:
Originally posted by Sipech:
A very interesting voice on the subject is Roger Penrose. Here's a short essay he wrote which echoes the introduction of his work, The Road to Reality.

Contrary to Schroedinger's cat, the argument Penrose advocates is Platonist than modern, albeit just expressed in a modern fashion.

I sometimes wonder if that mathematical reasoning is why the church leaders I have always got on best with began by studying mathematics and then moved over to theology.

I couldn't help thinking that the obvious answer to Penrose's implied question as to how you link the Platonic, mental and physical realms was 'God'. In other words, God creates the material universe and human minds within that, and conceives the Platonic realm of mathematical truths via his intellect.

Posted by TomM (# 4618) on :

quote:
Originally posted by Full of Chips:
Quote:
Le Roc:
quote:
Full of Chips: The question is, which of the two square roots of -1 did we label i (or j) in the first place?
(It doesn't matter.)

Well I agree it doesn't matter if you just want to use complex numbers. It does matter however if you care about whether mathematical systems you are constructing are natural in the sense of being categorically unique as opposed to merely one of several isomorphic objects that could result dependent on some choice.

I freely admit though that even in advanced mathematical texts of that nature, it generally results only in a little footnote that says "*subject to a choice of square root of -1".

It also matters (though perhaps only to pedants) if you want to say, as Alan did, that "i is the square root of -1".

But there are no properties (at least that I'm aware of) that anything is different because of which square root you have chosen.

And the description as <i>the</i> square root is perfectly normal language - we would say that the square root of 9 is 3 in normal discourse, even though we know we mean ±3.

Posted by HughWillRidmee (# 15614) on :

quote:
Originally posted by Alan Cresswell:
So, as I continue to speculate, if the atheists are right to describe God as a logically fallacious, impossible entity just as i appears what does our experience of i tell us about whether a logically fallacious and impossible entity may, nonetheless, be real?

I think you are confusing "real" with "a concept that has use".

Posted by Dafyd (# 5549) on :

quote:
Originally posted by HughWillRidmee:
I think you are confusing "real" with "a concept that has use".

What other criterion can we have for "real"?

Posted by Jack o' the Green (# 11091) on :

quote:
Originally posted by TomM:

And the description as <i>the</i> square root is perfectly normal language - we would say that the square root of 9 is 3 in normal discourse, even though we know we mean ±3.

People who know that a negative number multiplied by a negative number equals a positive number will be aware of that, but I'm not sure everyone engaging in normal discourse will be.

[ 20. September 2015, 22:13: Message edited by: Jack o' the Green ]

Posted by Palimpsest (# 16772) on :

The imaginary numbers are a historical term. Next step is to say irrational numbers are bad and make a law that pi is a simple fraction that the godly can understand.

If you're looking for mathematical analogies; how about non-Euclidean geometries. There are different systems that can be constructed depending on some unstated axioms. If you want to do a theological extrapolation, maybe there are some systems with one or more gods, and some with none. I'm not going to think about a system with an imaginary or complex god...

Posted by Jack o' the Green (# 11091) on :

quote:
Originally posted by Palimpsest:
The imaginary numbers are a historical term. Next step is to say irrational numbers are bad and make a law that pi is a simple fraction that the godly can understand.

And as for vulgar fractions...

Posted by mousethief (# 953) on :

quote:
Originally posted by balaam:

quote:
Originally posted by HCH:
Some people have commented that God invented the counting numbers (1, 2, 3, ...) and all other numbers are human inventions.
Modern maths was impossible until the Arabs invented the concept of 0 (zero).

The Arabs didn't invent it. They borrowed it from India.

quote:
Originally posted by Schroedinger's cat:
I think the point about mathematical concepts like i is that they do not directly map onto something in the real world. The integers do - we count things using them, we can count 1 cushion, for example - it doesn't make them any more extant, but they relate to something we can see and identify. 1 cushion is a concept that we can relate to, and so we tend to accept that "1" is something extant.

But the integers include the negatives and 0, which are abstract and unreal when counting cushions. You can't touch zero cushions. And –3 cushions is flat-out nonsense.

quote:
Originally posted by Jay-Emm:
Another way 1 and -1 fundementally differ is that 1 is the identity for multiplication, and -1 isn't. (That is 5*1 is still 5, 5*-1 is no longer 5).

Plus, one is GREATER THAN zero, and the other is LESS THAN zero. Also, they are 2 apart, whereas 1 is zero apart from 1, and -1 is zero apart from -1. Also, one has a real square root and the other doesn't. They're two different numbers, no matter how you slice it. They happen to be the same distance from 0 on a one-dimensional number line. I'm the same distance from St John's, Newfoundland, as Wesley J is, but that doesn't put me in Switzerland or him in Seattle.

quote:
Originally posted by LeRoc:
The "kind of infinity" (mathematicians call it "cardinality") of the integers is actually the same as the "kind of infinity" of fractions. There is a rather beautiful proof of that.

I go back and forth on that one.

quote:
Originally posted by Dafyd:

quote:
Originally posted by HughWillRidmee:
I think you are confusing "real" with "a concept that has use".
What other criterion can we have for "real"?

"exists"?

Posted by SusanDoris (# 12618) on :

quote:
Originally posted by HughWillRidmee:

quote:
Originally posted by Alan Cresswell:
So, as I continue to speculate, if the atheists are right to describe God as a logically fallacious, impossible entity just as i appears what does our experience of i tell us about whether a logically fallacious and impossible entity may, nonetheless, be real?
I think you are confusing "real" with "a concept that has use".

Agreed, and I suppose it could be argued that the concept of a god was useful for power and control when a priest could put the fear of, say, Zeus into people.
I read part of the first post and then skipped to the second half of page 2! Although many find the subject of advanced maths extremely daunting, not to say frightening, no-one believes that the subject or any of its complications is an entity which can take power or control. They know it is the humans using it one might need to be wary of at times! [Smile]

[ 21. September 2015, 07:22: Message edited by: SusanDoris ]

Posted by Curiosity killed ... (# 11770) on :

Maths runs the algorithmic trading across stock exchanges. Has caused a few stock market disasters - the 2010 Flash Crash for one.

And then there are the computers that could own themselves Radio 4 clip here - comes from the programme Future Proofing, discussing automated cars that can bank the money book more cars being built and repairs. Then there are the drones that could do the same.

Still think maths and technology relies on humans to control what happens?

Posted by Jay-Emm (# 11411) on :

quote:
Originally posted by HughWillRidmee:

quote:
Originally posted by Alan Cresswell:
So, as I continue to speculate, if the atheists are right to describe God as a logically fallacious, impossible entity just as i appears what does our experience of i tell us about whether a logically fallacious and impossible entity may, nonetheless, be real?
I think you are confusing "real" with "a concept that has use".

I don't think so, you could probably argue that, with say classical waves (and when solving the right kind of cubics). Where the imaginary part [chosen to constantly balance each other out] 'just' acts to keep the real sines and cosines in place. It's not as pretty, but it could be done consistently with whats observed (I think).

I guess you could even do something similar to allow you to work with FT's (though quite how I don't know).

But in Quantum Mechs how do you plan to work round that. Where the observations depend on the modulus, so while I guess you could put a different backend it needs to add/multiply in the same way as complex numbers. And those wavefunctions clearly exist.

[ 21. September 2015, 07:43: Message edited by: Jay-Emm ]

Posted by HughWillRidmee (# 15614) on :

quote:
Originally posted by Curiosity killed ...:
Maths runs the algorithmic trading across stock exchanges. Has caused a few stock market disasters - the 2010 Flash Crash for one.

And then there are the computers that could own themselves Radio 4 clip here - comes from the programme Future Proofing, discussing automated cars that can bank the money book more cars being built and repairs. Then there are the drones that could do the same.

Still think maths and technology relies on humans to control what happens?

Maths is a way of describing concepts is it not? - humans control how they describe it.

Technology is the application of scientific knowledge for practical purposes, we may delegate some of this application to tools such as computers but there is a massive way to go before this is, entirely, out of human control. There is a very long chain from finding raw materials to automatic stock exchange disasters and humans are currently involved in (almost?) every step.

Posted by HughWillRidmee (# 15614) on :

quote:
Originally posted by Jay-Emm:

quote:
Originally posted by HughWillRidmee:

quote:
Originally posted by Alan Cresswell:
So, as I continue to speculate, if the atheists are right to describe God as a logically fallacious, impossible entity just as i appears what does our experience of i tell us about whether a logically fallacious and impossible entity may, nonetheless, be real?
I think you are confusing "real" with "a concept that has use".
I don't think so, you could probably argue that, with say classical waves (and when solving the right kind of cubics). Where the imaginary part [chosen to constantly balance each other out] 'just' acts to keep the real sines and cosines in place. It's not as pretty, but it could be done consistently with whats observed (I think).

I guess you could even do something similar to allow you to work with FT's (though quite how I don't know).

But in Quantum Mechs how do you plan to work round that. Where the observations depend on the modulus, so while I guess you could put a different backend it needs to add/multiply in the same way as complex numbers. And those wavefunctions clearly exist.

If you're saying that God can be and cannot be simultaneously, that good can be bad and good at the same time, that a(the) deity can both be and not be then you are doing what religion has always done - moving the goalposts when it is demonstrated that the current position is untenable. You're also inventing a new kind of god aren't you - one that is incompatible with Christianity but enables a sort-of-deistic probably-wishful-thinking halfway house to rational atheism. [Devil]

Posted by Alan Cresswell (# 31) on :

quote:
Originally posted by Jay-Emm:
But in Quantum Mechs ... those wavefunctions clearly exist.

Clearly? They either exist, or they don't. No, wait, they exist and they don't. Something like that, go ask a cat.

The equation of a wavefunction can be said to exist, it can be written down on a piece of paper (depending on the wavefunction, it may need to be a large piece of paper). But, is it real like waves on the ocean, or like music? Or, a mathematical representation of a probability distribution?

What is existence anyway?

Posted by agingjb (# 16555) on :

Hmm. I'd have said that "imaginary" is misleading and "real" is an arbitrary term coined to make a distinction.

Existence? I'd claim that mathematical entities are answers to carefully posed questions.

Posted by Jay-Emm (# 11411) on :

quote:
Originally posted by HughWillRidmee:

quote:
Originally posted by Jay-Emm:

quote:
Originally posted by HughWillRidmee:
I think you are confusing "real" with "a concept that has use".
I don't think so,...[... snip]
it needs to add/multiply in the same way as complex numbers. And those wavefunctions clearly exist.
If you're saying that God[... snip] [/QB]

Huh, where did that come from, what has that to do with complex numbers.

Posted by Schroedinger's cat (# 64) on :

quote:
Originally posted by Alan Cresswell:
Clearly? They either exist, or they don't. No, wait, they exist and they don't. Something like that, go ask a cat.

Like I'd know.

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