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What can we gain by losing infinity?

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Re: What can we gain by losing infinity?

#101

Post by amelius »

computably wrote: post #59
amelius wrote: post #55

Earlier quoted context omitted.

The problem with infinity is that it's a hack. It is basically the NULL pointer of mathematicians. An instance of a number that has a special meaning that breaks the abstraction of numbers. If you want to do things with infinity, fine, but then do it properly and write things like lim x->inf (your expression with x here)

> An instance of a number that has a special meaning. Not really. There are infinitely many infinities. Infinite numbers are not particularly more special than real numbers, complex numbers, matrices, functions/operators, etc.

Infinite numbers break the abstraction of numbers in important ways.

Complex numbers do too (e.g. there is no ordering). But at least that doesn't cause as much confusion as some of the problems with infinite numbers.

Re: What can we gain by losing infinity?

#102

Post by mcontrac »

gumgumpost wrote: post #99
mcontrac wrote: post #58

Earlier quoted context omitted.

What people might not be understanding is that mathematics is inherently built... ZFC was pored over for years and eventually the community concluded it was a good system to (a) preserve most, if not all, of the mathematics that had already been done and (b) build more mathematics. You can have gripes over whether or not pure math is compatible with the physical world but we're not exactly close to solving that probl…

Don't know much about the field, but isn't he implying it could make math more compatible with the physical world? Math as a field seems like a deep rabbit hole that sometimes describes our reality.

The trouble is that if you want math to be standardized/able to be described in a sort of "objective" manner (what mathematicians call a proof), you'd like to start out with a set of axioms that are not themselves provable in the traditional sense but from which everything else can be proven. If you leave out infinity, it turns out that your set of axioms isn't really powerful enough to do anything, much less the kind of math that physicists require to describe the universe. If you keep it in, your set of axioms is SO powerful that you end up proving things that don't seem compatible with the physical world. There is no objective solution to this problem. The mathematical community chose the latter because, well, it helped us prove cooler and more sophisticated stuff. Some of that stuff is a beautiful way to describe the physical world (try googling something like representation theory), while other things make us question our intuition about it (i.e. string theory).

Re: What can we gain by losing infinity?

#103

Post by kenferry »

Ifkaluva wrote: post #18

I don’t understand, and I hope it’s just bad writing. Certainly you can build a branch of mathematics without an axiom of infinity, and that’s fine, it’s math over finite sets. However, an axiom of infinity is independent, it doesn’t contradict anything in standard formalizations, and so it doesn’t make sense to say “infinity is wrong”. He may think the axiom of infinity isn’t satisfied by our real physical world, bu…

I think you can reframe this and better understand the point these mathematicians are making.

The vast, vast majority of mathematics DOES use infinities. That's the standard perspective. The question is whether there is good, interesting, useful mathematics to be explored by disallowing that concept.

The way I see it, Gödel's, Turing's work and complexity theory come out of this line of thinking about _effective_ computation. This is an argument for exploring the mathematics that arises when you don't think of actual computer math as an imperfect approximation of the real numbers, but rather as a mathematical object in its own right.

I would guess (?) it's more interesting for floating point math and related than for integer math, because for integer math it's already well explored in group theory.

Re: What can we gain by losing infinity?

#104

Post by scrubs »

getnormality wrote: post #56

> To Zeilberger, believing in infinity is like believing in God. It’s an alluring idea that flatters our intuitions and helps us make sense of all sorts of phenomena. But the problem is that we cannot truly observe infinity, and so we cannot truly say what it is. When the author says we cannot truly observe infinity, what does that mean? Infinity is a mathematical symbol we can observe. We can't observe infinitely ma…

You cannot observe infinity operationally. Take 0 and add 1 repeatedly. For what n does n+1 become infinite? Never. Since you can't construct infinity you can only believe in it like God.

Hence the jargon "completed infinity". Semantically --- but not in the symbols themselves 0+1+1..." one can pass from finite to infinite by arguing since every n has a successor define Z to be the set of all successors "completing into infinity".

Not having infinity is the real reason a+b=b+a can't be proved in ultra finitism. Induction which depends on the idea of completed infinity is what is otherwise is used.

Re: What can we gain by losing infinity?

#105

Post by nextaccountic »

IsTom wrote: post #92
nextaccountic wrote: post #89

Earlier quoted context omitted.

> Infinity is a mathematical symbol we can observe. This is like confusing the map for the territory. Symbols live in syntax (like the syntax of programming languages), while mathematical concepts live in semantics. Infinity is not a symbol, it's not ∞. ∞ is the symbol we use to represent infinity.

There is a way to look at mathematics as just a bunch of rewrite rules for things on paper. It might not be particularly inspiring, but it's a valid way to look at things.

Indeed, there's a way to get a semantics for free, based on the syntax alone. For example, in the first order logic this is the Herbrand interpretation

https://en.wikipedia.org/wiki/Herbrand_interpretation

The point of mathematical semantics is that for any given theory, we can have other interpretations that don't just interpret symbols as themselves.

So we could conceivably imagine an interpretation where ∞ doesn't just mean literally ∞ and nothing more.

Re: What can we gain by losing infinity?

#106

Post by nextaccountic »

looneysquash wrote: post #97
nextaccountic wrote: post #89

Earlier quoted context omitted.

> Infinity is a mathematical symbol we can observe. This is like confusing the map for the territory. Symbols live in syntax (like the syntax of programming languages), while mathematical concepts live in semantics. Infinity is not a symbol, it's not ∞. ∞ is the symbol we use to represent infinity.

The number 42 is also a mathematical symbol we can observe. (Or two symbols, depending on how you want to define symbol). You can observe the symbol. You can observe 42 of some object, 42 sheep for example. You can observe a pie chart, or an actual pie, with 42% of it missing. You can observe a plank of wood that is 42 inches or centimeters long. But you can't observe 42 itself. It is not like a hill on a map, where…

I actually agree, there's nothing wrong with infinity. I think finitists are silly and ultrafinitists are ultra silly.

Re: What can we gain by losing infinity?

#107

Post by rini17 »

amavect wrote: post #73

My favorite math paper is "Is 10^10^10 a Finite Number?" by David van Dantzig. It lies more on the side of philosophy, so many can understand it easily. I first learned about it many years ago from Van Bendegem's list of strict finitism papers, and I would recommend that list for anyone interested in learning more about strict finitism. For my personal opinion, strict finitism provides a richer field of study than po…

You know about busy beavers? These programs do fit in few bits, yet the number of states they can reach does not.

Re: What can we gain by losing infinity?

#108

Post by kbelder »

kbrkbr wrote: post #88
rbtprograms wrote: post #70

Earlier quoted context omitted.

saying infinity is a mathematical symbol we can observe is simplifying it way too much, all mathematical symbols are abstractions. i can observe two apples. i cannot observe infinity apples.

Can you observe 2.34 x 10^456789 apples?

No. I believe that is more apples than there are atoms in the universe, so not only it is impossible to observe, it is a fundamental contradiction with our universal reality. No one and nothing will ever be able to observe or interact that many apples, and so a reference to that many apples is only an abstract mathematical convenience that has no direct bearing to reality.

Like infinity.

I'm not sure I actually believe that, I'm just thinking out loud. But it leads me to think the question "Does infinity exist?" should be answered with the question "An infinity of what?"

Re: What can we gain by losing infinity?

#109

Post by kbelder »

something765478 wrote: post #90
novia wrote: post #24

Take the approximate number of subatomic particles in the universe, call it Ω. Define the largest number as Ω² and the smallest number as -Ω², and define the number of decimal numbers between each integer number as Ω², evenly spaced. That should be more than enough numbers. Redefine Ω with each new discovery in physics. If this seems too conservative to you, like if for some reason you want to talk about the volume o…

This system breaks down when you start looking at permutations; there are Ω! ways to arrange your subatomic particles, and that's just in 1 dimension.

Sure. And then you can do permutations OF the permutations, until the end of the universe, and the unthinkably large result still firmly remains in the world of finite numbers.

Re: What can we gain by losing infinity?

#110

Post by tomjakubowski »

freetime2 wrote: post #12
nkrisc wrote: post #9

Earlier quoted context omitted.

I’m pretty certain a finite number of pre-schoolers can only recite a finite number of numbers. Yes, they could on indefinitely, but will they ever?

They pretty quickly realize that there is no winning because you can always just say more numbers than the last kid - there is no biggest number. Usually something like "a hundred million million million million million and two", "a hundred million million million million million and three", etc. And then someone, whose friend or older brother taught them the concept, blurts out "infinity". And after a quick explanat…

24 is the highest number. Where you gonna go from there?
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