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What can we gain by losing infinity?

quantamagazine.org

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Re: What can we gain by losing infinity?

#71

Post by mattmanser »

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…

It seems to me that you're the one confused?

The mathematical symbol is just a representation of a concept, it's not infinity itself, you've got it backwards.

Re: What can we gain by losing infinity?

#72

Post by gbnwl »

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…

The symbol is not the thing. The map is not the territory. Ceci n'est pas une pipe.

Re: What can we gain by losing infinity?

#73

Post by amavect »

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 potential infinitism or actual infinitism. Compare this to Errett Bishop's constructive analysis that requires the calculation of bounds to real numbers, instead of classical analysis only requiring that a real number exists. Much more difficult, though more precise.

I found "On Feasible Numbers" by Vladimir Sazonov to have application for computers. In a feasible mathematics, a large number fails to exist (say, 2^512), but a proof of contradiction must exceed such a large size (perhaps larger than the universe). Likewise, we have unix time that tries to count forever, so we should pick a storage size so large that counting exceeds the heat death of the universe. 10^100 years worth of Planck seconds fits in 501 bits, so round that to 512 bits. 512 bits of time ought to be enough for anybody :)

https://jeanpaulvanbendegem.be/home/papers/strict-finitism/

Re: What can we gain by losing infinity?

#74

Post by BigTTYGothGF »

amelius wrote: post #55
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…

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

Lots of numbers have special meanings. The ancients didn't think 1 was a number, and later lots of people didn't (and some still don't) think 0 was a number.

Re: What can we gain by losing infinity?

#75

Post by whimblepop »

> One morning in 1976, the Princeton mathematician Edward Nelson (opens a new tab) woke up and experienced a crisis of faith. “I felt the momentary overwhelming presence of one who convicted me of arrogance for my belief in the real existence of an infinite world of numbers,” he reflected decades later (opens a new tab), “leaving me like an infant in my crib reduced to counting on my fingers.”

Friends don't let friends do Platonism.

For real, if you're a formalist you can ask these foundational questions without fear of this kind of dread; they become methodological rather than some kind of metaphysical mess.

Re: What can we gain by losing infinity?

#76

Post by ndsipa_pomu »

rbtprograms wrote: post #70
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…

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.

Mathematical concepts don't have to have an obviously physical analogue. I mean, you'd find it difficult to observe minus two apples and certainly tricky to observe i apples.

To my mind, maths is like a "what if?" puzzle and whether or not infinity makes sense in the physical world, there's still fun to be had by considering the consequences of it.

That also means that it can be interesting to consider limited number systems which don't have any concept of infinity.

Re: What can we gain by losing infinity?

#77

Post by ibejoeb »

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…

It's an interesting read. I don't think it's bad, but it's not rigorous or really aimed at anything in particular. Basically asking a discrete mathematician whether he needs continuity: no. It seems reasonable that we might need separate paradigms to think about different kinds of problem (e.g., is there a physical size of the universe vs. is there a biggest prime number) because we don't know yet if there is a theory of everything or if there are innate boundary layers.

It's a fun thinking prompt, and you can go down the rabbit hole of information theory and quantized spacetime. Like you suggest, it's perfectly fine to say "infinity does not exist" and also contemplate and operate on slice at a time.

Re: What can we gain by losing infinity?

#78

Post by adrian_b »

alex_young wrote: post #22

And no discussion of Zeno? Pish. The idea that nothing is demonstrative of infinity is clearly incorrect. Take the screen you're reading this on. One pixel is composed of a bunch of different atoms, and once you get down to one of them, that atom subdivides into a bunch of subatomic particles, some of which even have mass. Let's take one of those for argument's sake. Split that, and you get some quarks. Now let's ima…

The paradoxes of Zeno are caused by his lack of understanding of the symmetry between zero and infinity. It is also possible that he actually understood more than is apparent from his paradoxes, but those were intended only to troll the other philosophers.

Zeno understood things like zero multiplied by a number being zero and a number multiplied by infinity being infinity, but he did not understood that neither of zero and infinity is stronger than the other, so that the product of zero and infinity may be any finite number, i.e. the limit of a sequence of products where one factor decreases towards zero and the other increases indefinitely can be any number.

While Zeno either ignored or faked ignorance about the existence of limits of infinite sequences, other later Ancient Greek mathematicians, like Eudoxus and Archimedes, computed several limits, so they had an intuitive understanding of their behavior, even if they did not have a comprehensive theory.

Re: What can we gain by losing infinity?

#79

Post by amavect »

whimblepop wrote: post #75

> One morning in 1976, the Princeton mathematician Edward Nelson (opens a new tab) woke up and experienced a crisis of faith. “I felt the momentary overwhelming presence of one who convicted me of arrogance for my belief in the real existence of an infinite world of numbers,” he reflected decades later (opens a new tab), “leaving me like an infant in my crib reduced to counting on my fingers.” Friends don't let frien…

The quote comes from his paper "Mathematics and Faith"!

https://web.math.princeton.edu/~nelson/papers/faith.pdf

You can find many of his papers here.

https://web.math.princeton.edu/~nelson/papers/

Re: What can we gain by losing infinity?

#80

Post by alestainer »

rbtprograms wrote: post #70
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…

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.

Some might say that 2 is as made up as infinity. Let me elaborate a little - your brain together with society made an abstraction "apple", and only by not distinguishing between these "sets" of atoms you can have numbers.
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