Total Posts:8|Showing Posts:1-8
Jump to topic:

Axiom of Choice

tarkovsky
Posts: 212
Add as Friend
Challenge to a Debate
Send a Message
7/6/2015 3:40:26 PM
Posted: 1 year ago
So, as we know, the consistency of the Axiom of Choice (AOC) isn't up for debate as Godel put that discussion to bed some time ago. However, whether or not we should adopt AOC seems to be a touchy subject for some. On what basis should we refuse AOC? I find that most arguments repudiating AOC are made on the basis of unwanted results in the topics of mainly analysis and (modern) geometry. (Non-continuous solutions to Cauchy-Equations, bizarre decompositions, etc.). For me, this is no problem at all as these are essentially two of my least favorite topics and already replete with ugliness in my mind. That is, it doesn't make much of a difference to me if they get uglier as they were ruined to begin with. If we take a much more general and appealing topic like Topology (which is related but, in my mind, superior to the aforementioned) we find that AOC makes everything much simpler and more beautiful.

With that in mind, I say the 'ugliness' begotten by AOC in the topics of analysis and geometry is, in fact, a reason to adopt (rather than repudiate) AOC. It serves as a criterion for acceptance; that we should be suspicious of those topics which become ugly with respect to this very natural and intuitive principle. I find that it is usually constructivists (mathematicians who are secretly just engineers/scientists) who take issue with this axiom (perhaps with the exception of Cohen who i'm not even sure rejects AOC).
RuvDraba
Posts: 6,033
Add as Friend
Challenge to a Debate
Send a Message
7/6/2015 11:57:51 PM
Posted: 1 year ago
At 7/6/2015 3:40:26 PM, tarkovsky wrote:
On what basis should we refuse AOC?
I understand that nowadays in math, nobody really does, Tarkovsky, except certain kinds of constructive logicians. Some Pure Mathematicians still get anxious about it, but that might be for psychological or historical reasons.

For members who aren't familiar with the Axiom of Choice, it comes up in Set Theory, a branch of Pure Mathematics dealing with collections of things. You may remember Venn diagrams from primary school -- that's set theory.

This uncontroversial-sounding axiom that says if you have any number of boxes, each with one or more white ping-pong balls inside, you can pick exactly one ping-pong ball from each box and paint it red, so that each box has exactly one red ping-pong ball.

That's obviously true when you have a finite number of boxes, each with some finite number of ping-pong balls, but should mathematicians assume it to be true when you have an infinite number of boxes, or some boxes have an infinite number of ping-pong balls?

Why does that matter, you might ask? We never really have an infinite number of anything in real life. Well, mathematicians care because if it's true it makes some proofs much easier, while if it's false it makes some proofs harder and longer, and mathematicians were worried that it might make some proofs impossible.

They also care, because (for example), if it's not true then it might make navigating around the universe very hard, if the universe has infinitely many coordinates and they're hard to tell apart. (Yes, mathematicians worry about such things.)

So can it be proven? Because if it could then that would be convenient.

It turns out, though, it can't. If you assume reasonable things about boxes and ping-pong balls, you can't prove that you can choose a ping-pong ball from a box of infinite ping-pong balls. (You can if they all have numbers on, say, and you know what the smallest number will be, but not if not.)

So is it false? Because if it's false then that would be both weird and disturbing.

Irritatingly, it's not provably false either.

So what is it?

It turns out, it's an axiom: you can choose to believe it or not.

So which should mathematicians believe?

In practice, they weren't willing to trust it until they knew it wasn't going to lead them into inconsistencies. But as our Original Poster mentioned above, it doesn't. However it does produce some weirdness.

For example, if you believe the Axiom of Choice, it turns out that you also believe there's a way of cutting up a ping-pong ball and rearranging the pieces so it makes two identical ping-pong balls of exactly the same volume.

This is called the Banach-Tarski Paradox [https://en.wikipedia.org...], and nobody really wants to believe that.

On the other hand, if you believe the Axiom of Choice, most of the time things are easier, and not very weird -- they just work like finite boxes and finite ping-pong balls. So a lot of mathematicians use the Axiom of Choice when convenient, and then just check that they didn't accidentally create two ping-pong balls when they really meant to create only one.

I find that it is usually constructivists (mathematicians who are secretly just engineers/scientists) who take issue with this axiom

Yes. Constructively it matters. I started out researching in pure math, but now think much more like 'just' an engineer/scientist, and in that respect, it still niggles at me. I like maths to serve engineering and the sciences. When it doesn't, it gets more like philosophy, and I want to send it off to the Arts faculty for wine and cheese afternoons and late morning starts. :)

So the niggles matter, but I find more cause to object to AoC in philosophy (for example, Anselm's ontological argument for God uses it blithely -- which few philosophers seem to realise), and in certain kinds of Information Management problems where there's no canonical way to establish ordinality.

I hope that may be interesting and/or useful. But if not, here's an xkcd joke about pumpkin-carving that picks up on the Banach-Tarski paradox: https://xkcd.com...
tarkovsky
Posts: 212
Add as Friend
Challenge to a Debate
Send a Message
7/7/2015 8:55:11 PM
Posted: 1 year ago
At 7/6/2015 11:57:51 PM, RuvDraba wrote:
Yes. Constructively it matters. I started out researching in pure math, but now think much more like 'just' an engineer/scientist, and in that respect, it still niggles at me. I like maths to serve engineering and the sciences. When it doesn't, it gets more like philosophy, and I want to send it off to the Arts faculty for wine and cheese afternoons and late morning starts. :)

Sounds like your Arts department knows what's up. Wine and cheese afternoons with a little pure mathematics to settle the stomach? What's not to love about that? Nah, I say the more useless the math is the better. There's something suspicious about useful mathematics. (That's a joke but, like any joke, there's some truth to it).

So the niggles matter, but I find more cause to object to AoC in philosophy (for example, Anselm's ontological argument for God uses it blithely -- which few philosophers seem to realise), and in certain kinds of Information Management problems where there's no canonical way to establish ordinality.

I hope that may be interesting and/or useful. But if not, here's an xkcd joke about pumpkin-carving that picks up on the Banach-Tarski paradox: https://xkcd.com...

Never seen how AOC can be used in an ontological argument. Then again I've never really seen a 'formalized' (clearly not in the sense of formal systems, but I mean rigorous and symbolic) ontological argument. I know there's something about Godel and one of his own ontological arguments but meh can't be bothered with it, it's just not interesting to me.
RuvDraba
Posts: 6,033
Add as Friend
Challenge to a Debate
Send a Message
7/7/2015 10:36:29 PM
Posted: 1 year ago
At 7/7/2015 8:55:11 PM, tarkovsky wrote:
At 7/6/2015 11:57:51 PM, RuvDraba wrote:
I like maths to serve engineering and the sciences. When it doesn't, it gets more like philosophy, and I want to send it off to the Arts faculty for wine and cheese afternoons and late morning starts. :)

There's something suspicious about useful mathematics. (That's a joke but, like any joke, there's some truth to it).

There is certainly something suspicious about useful mathematics, T. Since utility is defined in terms of what we know, and maths doesn't differentiate between articulating what we know and what we don't, when we have a mathematics that's entirely useful, how do we know we have all of it? :)

I find more cause to object to AoC in philosophy (for example, Anselm's ontological argument for God uses it blithely -- which few philosophers seem to realise), and in certain kinds of Information Management problems where there's no canonical way to establish ordinality.
Never seen how AOC can be used in an ontological argument.

As you may know, AOC has hundreds of equivalent formulations. But a famous one for set theory is Zermelo's Well-Ordering Theorem, that for every set X, there's a strict, total ordering such that each nonempty subset of X has a maximum under that ordering. [https://en.wikipedia.org...] This is valuable for the ordinals and cardinals because it means that transfinite induction works.

At the risk of luring theologists out of the woodwork, Anselm's ontological argument proposes that God is the unique, maximal member of the set of everything, well-ordered by greatness -- the idea being that somehow, you could compare any pair of elements in the universe and work out which is greater, so that a unique, greatest member is guaranteed. [http://www.iep.utm.edu...]

Not enough that St. Anselm demands that the universe uphold the Axiom of Choice, and doesn't tell us how to define greatness so that it's well-ordered, but he also goes on to reason from possibility to necessity in a way no self-respecting modal logician would let pass unchallenged. :)

I've never really seen a 'formalized' (clearly not in the sense of formal systems, but I mean rigorous and symbolic) ontological argument.
Well, one of the problems of useless math is that it attracts philosophers, and once that happens, you know that the wine, cheese and theologians aren't far behind. :)
tarkovsky
Posts: 212
Add as Friend
Challenge to a Debate
Send a Message
7/8/2015 11:01:23 AM
Posted: 1 year ago
At 7/7/2015 10:36:29 PM, RuvDraba wrote:
At the risk of luring theologists out of the woodwork, Anselm's ontological argument proposes that God is the unique, maximal member of the set of everything, well-ordered by greatness -- the idea being that somehow, you could compare any pair of elements in the universe and work out which is greater, so that a unique, greatest member is guaranteed. [http://www.iep.utm.edu...]


Yeah I don't know how I feel about the set of everything. I know we usually begin Intro to Set theory with stuff like "the set of all chairs in this room". But I always took that to be either some abstract mental object which is supposed to represent all the chairs in the room or a particular way of looking at all the chairs in the world. In any case it's not really the material chairs in the room. I don't think the set (or class or whichever way we're going) could establish physical facts. I'm actually not too huffy about the comparison part. We could do it arbitrarily honestly and just say something is greater than something else in an arbitrary way and that by Zorns we get whatever is the greatest we'll call God. Again though, I'd put the argument down with saying the best we can do at this point is say we're setting up a God in some abstract set universe if we must have the existence of such things as sets. Don't think it matters in regard to material being.

Well, one of the problems of useless math is that it attracts philosophers, and once that happens, you know that the wine, cheese and theologians aren't far behind. :)

Haha, well I think philosophers are too used to getting nothing done to meaningfully contribute to mathematics. ( This is a joke too, but again, like any joke, there's some truth to it).
Envisage
Posts: 3,646
Add as Friend
Challenge to a Debate
Send a Message
7/8/2015 1:13:08 PM
Posted: 1 year ago
At 7/6/2015 11:57:51 PM, RuvDraba wrote:
At 7/6/2015 3:40:26 PM, tarkovsky wrote:
On what basis should we refuse AOC?
I understand that nowadays in math, nobody really does, Tarkovsky, except certain kinds of constructive logicians. Some Pure Mathematicians still get anxious about it, but that might be for psychological or historical reasons.

For members who aren't familiar with the Axiom of Choice, it comes up in Set Theory, a branch of Pure Mathematics dealing with collections of things. You may remember Venn diagrams from primary school -- that's set theory.

This uncontroversial-sounding axiom that says if you have any number of boxes, each with one or more white ping-pong balls inside, you can pick exactly one ping-pong ball from each box and paint it red, so that each box has exactly one red ping-pong ball.

That's obviously true when you have a finite number of boxes, each with some finite number of ping-pong balls, but should mathematicians assume it to be true when you have an infinite number of boxes, or some boxes have an infinite number of ping-pong balls?

Why does that matter, you might ask? We never really have an infinite number of anything in real life. Well, mathematicians care because if it's true it makes some proofs much easier, while if it's false it makes some proofs harder and longer, and mathematicians were worried that it might make some proofs impossible.

They also care, because (for example), if it's not true then it might make navigating around the universe very hard, if the universe has infinitely many coordinates and they're hard to tell apart. (Yes, mathematicians worry about such things.)

So can it be proven? Because if it could then that would be convenient.

It turns out, though, it can't. If you assume reasonable things about boxes and ping-pong balls, you can't prove that you can choose a ping-pong ball from a box of infinite ping-pong balls. (You can if they all have numbers on, say, and you know what the smallest number will be, but not if not.)

So is it false? Because if it's false then that would be both weird and disturbing.

Irritatingly, it's not provably false either.

So what is it?

It turns out, it's an axiom: you can choose to believe it or not.

So which should mathematicians believe?

In practice, they weren't willing to trust it until they knew it wasn't going to lead them into inconsistencies. But as our Original Poster mentioned above, it doesn't. However it does produce some weirdness.

For example, if you believe the Axiom of Choice, it turns out that you also believe there's a way of cutting up a ping-pong ball and rearranging the pieces so it makes two identical ping-pong balls of exactly the same volume.

This is called the Banach-Tarski Paradox [https://en.wikipedia.org...], and nobody really wants to believe that.

On the other hand, if you believe the Axiom of Choice, most of the time things are easier, and not very weird -- they just work like finite boxes and finite ping-pong balls. So a lot of mathematicians use the Axiom of Choice when convenient, and then just check that they didn't accidentally create two ping-pong balls when they really meant to create only one.

I find that it is usually constructivists (mathematicians who are secretly just engineers/scientists) who take issue with this axiom

Yes. Constructively it matters. I started out researching in pure math, but now think much more like 'just' an engineer/scientist, and in that respect, it still niggles at me. I like maths to serve engineering and the sciences. When it doesn't, it gets more like philosophy, and I want to send it off to the Arts faculty for wine and cheese afternoons and late morning starts. :)

So the niggles matter, but I find more cause to object to AoC in philosophy (for example, Anselm's ontological argument for God uses it blithely -- which few philosophers seem to realise), and in certain kinds of Information Management problems where there's no canonical way to establish ordinality.

I hope that may be interesting and/or useful. But if not, here's an xkcd joke about pumpkin-carving that picks up on the Banach-Tarski paradox: https://xkcd.com...

Thank you for the most interesting post I have ever read on DDO.
RuvDraba
Posts: 6,033
Add as Friend
Challenge to a Debate
Send a Message
7/8/2015 1:42:51 PM
Posted: 1 year ago
At 7/8/2015 11:01:23 AM, tarkovsky wrote:
At 7/7/2015 10:36:29 PM, RuvDraba wrote:
At the risk of luring theologists out of the woodwork, Anselm's ontological argument proposes that God is the unique, maximal member of the set of everything, well-ordered by greatness -- the idea being that somehow, you could compare any pair of elements in the universe and work out which is greater, so that a unique, greatest member is guaranteed. [http://www.iep.utm.edu...]

Yeah I don't know how I feel about the set of everything.

I know how I feel about it, T: thinking like 'just' a scientist/engineer, I repudiate it. :)

We know from Goedel's First Incompleteness Theorem that everything can't be enumerated in human language. (For members interested, this is unrelated to the Axiom of Choice, except in weirdness factor, but happy to explain it in a separate post.)

If it can't, then there's no decision procedure to identify it and if there's no decision procedure to even find it, then how might we hope to put a total order on it?

Even if math isn't real (contestable, for at least some math), this shows us that philosophers really need to know some before they start talking about stupidly large sets, such as the set of everything. :)

Well, one of the problems of useless math is that it attracts philosophers, and once that happens, you know that the wine, cheese and theologians aren't far behind. :)
Haha, well I think philosophers are too used to getting nothing done to meaningfully contribute to mathematics.

Yes. If philosophers can't get nothing done in a meticulously structured, fully-accountable fashion, they shouldn't try to get nothing done at all.
RuvDraba
Posts: 6,033
Add as Friend
Challenge to a Debate
Send a Message
7/8/2015 8:31:33 PM
Posted: 1 year ago
At 7/8/2015 1:13:08 PM, Envisage wrote:
At 7/6/2015 11:57:51 PM, RuvDraba wrote:
At 7/6/2015 3:40:26 PM, tarkovsky wrote:
On what basis should we refuse AOC?
I understand that nowadays in math, nobody really does, Tarkovsky, except certain kinds of constructive logicians. Some Pure Mathematicians still get anxious about it, but that might be for psychological or historical reasons.

For members who aren't familiar with the Axiom of Choice, it comes up in Set Theory, a branch of Pure Mathematics dealing with collections of things. You may remember Venn diagrams from primary school -- that's set theory.

This uncontroversial-sounding axiom that says if you have any number of boxes, each with one or more white ping-pong balls inside, you can pick exactly one ping-pong ball from each box and paint it red, so that each box has exactly one red ping-pong ball.

Thank you for the most interesting post I have ever read on DDO.

You're welcome, Envisage! Since you seem to enjoy the weird as I do, you might also be interested in this post, which I put up as an example of a scientific Intelligent Design conjecture, namely that our universe was semi-intelligently designed, as a mediocre kid's C+ grade-school project, only to be abandoned for being too boring. :)

Linky: http://www.debate.org...