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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?

#81

Post by BigTTYGothGF »

rikroots wrote: post #64

Last year I made the mistake of asking ChatGPT what the world would look like if `∞ === -∞` and it took me seriously (I think) and led me on an hours-long dance where in the end it had me trying to prove, mathematically, that `2 > 1` ... and it was at that point I realised that I'm not cut out to think in numbers and maybe it was for the best that I failed my end-of-school Maths exam

> Last year I made the mistake of asking ChatGPT

That's the only way to ask it.

But in the spirit of generosity you may be interested in the "one-point compactification of the line".

Re: What can we gain by losing infinity?

#82

Post by BigTTYGothGF »

don-bright wrote: post #46

Surprised Wildberger’s youtube channel wasnt in here. People ask whats the point? For me the study of the infinitesimal vs finite has really helped me better understand issues of precision and approximation in computers. I feel like I know exactly why 1/3 plus 1/5 is not exactly 8/15 in my Calculator app. Or why points in my 3d object face are not coplanar after rotation. Or why games have weird glitches when your ch…

> Surprised Wildberger’s youtube channel wasnt in here.

Zeilberger is intellectually honest in a way that Wildberger is not.

Re: What can we gain by losing infinity?

#83

Post by freehorse »

srean wrote: post #42
ndriscoll wrote: post #26

Earlier quoted context omitted.

This is one of my life goals is to prepare my kids to troll their math teachers with the dual numbers and the claim that .999... is obviously 1-ε. Goal is to convince the teacher .999...≠1. Bonus points if they instead convince the teacher to doubt that complex numbers exist.

That would be both fun and correct. It really comes down to what semantics we attach to "=" when one of the sides is an infinite series. The "equals to" sign that we have used prior to that mental exercise was for finite terms only, we had not had to deal with infinitely many terms before that leap in thought. So now we have to extend the notion in a way that is backward compatible. A convenient one is it is equal to…

> semantics we attach to "=" when one of the sides is an infinite series

I would say that the semantics are about what an infinite series itself is, not about the equal sign. Once we have the common analytic notion of convergence of an infinite series, then the equality makes sense. The issue is that an infinite series is not an actual sum, but, formally, it is a sequence (of the partial sums). As you say, we represent the limit of the sequence of the partial sums with the same notation and only in the case that we have absolute convergence, but that's basically because we use the same notation for two different things (the sequence of the partial sums, and the limit of that). If we know we refer to the limit, I don't think there is any semantic complication with the equal sign.

Re: What can we gain by losing infinity?

#84

Post by ianburrell »

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…

You can't split a quark, partial quarks doesn't exist. In fact, singular quarks can't exist, if you try to pull quark out of nucleus, it produces another quark to pair with. Quarks can be destroyed in particle accelerators collisions but those aren't components.

Also, all of the components of an atom, electrons and nucleus, have mass.

Re: What can we gain by losing infinity?

#85

Post by alex_young »

lostmsu wrote: post #53
alex_young wrote: post #52

Earlier quoted context omitted.

Create an infinity? What does that mean? Why would you need to do that? Is there a limit to how many times something can be logically divided? If not, then there’s your infinity. It doesn’t require you to continue brute forcing it, just reason about it.

Maybe? Can you prove there's no limit? The default proof by induction requires postulate of infinity. (this statement is potentially incorrect, but takes across the point)

Does half of something have a limit? Not by its definition. Same thing with addition or multiplication. All of these only work with some concept of infinity.

We could redefine "half" to mean "half of whatever you're talking about until you get to some arbitrary limit", but doing that to all of arithmetic is going to wind up in a very odd place.

Re: What can we gain by losing infinity?

#86

Post by jcgrillo »

computably wrote: post #63
jcgrillo wrote: post #51

Earlier quoted context omitted.

> It very well could be an infinite number of atoms and then what? Where I get stuck with this is how might we measure that? Continuous measurements and infinite measurements are not something we can make. We fit continuous theories to discrete measurements--and the good ones fit really well!--but until we can measure it how can we actually know ? I concluded we just can't, and we have to be OK with that.

> We fit continuous theories to discrete measurements--and the good ones fit really well!--but until we can measure it how can we actually know? Well, physicists came up with quantum mechanics because they found a way to distinguish a genuinely discrete phenomenon. Understanding the physical universe overlaps with a subset of math. It shouldn't constrain the abstract tools which may or may not one day be useful for t…

I agree that continuity (and therefore infinity) are really useful tools. But it may also be useful to develop mathematical formalism that hews more closely to that which we can actually observe. Or not! But if nobody investigates we'll never know.

Re: What can we gain by losing infinity?

#87

Post by justonceokay »

alestainer wrote: post #80
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.

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.

> some might say

Well do you say it or are you just playing devils advocate? The post you are responding to seems very straightforward.

If you wanna go all philosophical, “real” might just be anything that is useful. In that way infinity is real because you can use it to do calculus. On the other hand, there are ways of doing calculus that do not involve thinking about infinity. But if you’re gonna count to three apples you pretty much have to go through “two” no matter what.

Re: What can we gain by losing infinity?

#88

Post by kbrkbr »

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.

Can you observe 2.34 x 10^456789 apples?

Re: What can we gain by losing infinity?

#89

Post by nextaccountic »

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…

> 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.

Re: What can we gain by losing infinity?

#90

Post by something765478 »

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.
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