What can we gain by losing infinity? • Hacker News Board

Re: What can we gain by losing infinity?

#91

Post by eggn00dles » Thu, Apr 30, 2026, 6:42 PM UTC

ianburrell wrote: post #84

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.

top quarks exist as lone quarks and decay before they can pair

Re: What can we gain by losing infinity?

#92

Post by IsTom » Thu, Apr 30, 2026, 7:10 PM UTC

nextaccountic wrote: post #89

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.

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.

Re: What can we gain by losing infinity?

#93

Post by jballanc » Thu, Apr 30, 2026, 7:23 PM UTC

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…

I think Douglas Adams had one of the best quotes regarding observing infinity:

"Infinity itself looks flat and uninteresting. Looking up into the night sky is looking into infinity – distance is incomprehensible and therefore meaningless."

Re: What can we gain by losing infinity?

#94

Post by reliablereason » Thu, Apr 30, 2026, 7:40 PM UTC

bmenrigh wrote: post #8

The article doesn’t really tell us what is gained by rejecting infinity. And in general, why not also reject zero, negative numbers, irrational numbers, complex numbers, uncomputable numbers, etc.? Seems like an article about quacks that can’t even agree on what the bounds and rules of their quackery are.

Removes paradoxical stuff like claims that there are bigger and smaller infinities.

Paradoxes comes from contradictions, a mathematical system that contains contradictions is a failed mathematical system.

Re: What can we gain by losing infinity?

#95

Post by injidup » Thu, Apr 30, 2026, 7:43 PM UTC

Stopped reading at.

> computers handle math just fine with a finite allowance of digits.

Go try and write yourself a robust algorithm to do booleans on polygons or calculate a voronoi diagram. The finite nature of floating point is the mother of all leaky abstractions and bites you in the arse any time you think you are smart enough to roll your own algorithms.

Re: What can we gain by losing infinity?

#96

Post by lostmsu » Thu, Apr 30, 2026, 7:45 PM UTC

alex_young wrote: post #85

lostmsu wrote: post #53

Earlier quoted context omitted.

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.

Half of something has a value, and that value is not infinity. You need to be more specific about how exactly do you get infinity from the fact that half of something has a value.

Re: What can we gain by losing infinity?

#97

Post by looneysquash » Thu, Apr 30, 2026, 7:57 PM UTC

nextaccountic wrote: post #89

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.

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 there is a symbol, and there's an actual hill.

It is an adjective and not a noun. It's not real unless it is describing something else.

My point being that regular finite numbers are not real either. So what's wrong with infinity? Or the square root of 2, or pi?

Re: What can we gain by losing infinity?

#98

Post by _alternator_ » Thu, Apr 30, 2026, 8:14 PM UTC

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 rarely understood that infinity isn't something mathematicians made up to make things more complex, it's an abstraction that makes a lot of ideas vastly simpler.

This is alluded to in the article; it's challenging to prove a+b=b+a without infinity (though if you do modular/wraparound arithmetic it becomes straightforward).

It seems to me (not an expert in this area by a long stretch) that ultrafinite mathematics could basically be a branch of theoretical computer science in the sense that people seem interested in procedures to generate the numbers. In this regard, it's a bit surprising that TCS wasn't mentioned in the article.

Re: What can we gain by losing infinity?

#99

Post by gumgumpost » Thu, Apr 30, 2026, 8:48 PM UTC

mcontrac wrote: post #58

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…

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.

Re: What can we gain by losing infinity?

#100

Post by montjoy » Thu, Apr 30, 2026, 8:59 PM UTC

I wonder how this affects that combining General Relativity with Quantum Mechanics leads to mathematical infinities “that render calculations meaningless.”