9 ms·

What can we gain by losing infinity?

127 points by Tomte 6mo ago

zkmon 6mo ago
Mathematicians sound like priests. They vehemently defend they regime. They enjoy exclusivity by ensuring complexity and extreme formalism.
deleted 6mo ago
[deleted]
jcgrillo 6mo ago
In school I developed a strong hunch that continuity and infinity are "convenient delusions" we have that allow us to process the otherwise horrific complexity of the world. Experiencing time, sound, or visual motion as continuous, rather than discrete signal inputs is so much simpler. Similarly, the mathematical tricks and shortcuts we can use on well behaved continuous functions are both "unreasonably effective" and... probably not grounded in actual reality[1]? But damn are they convenient. [1] EDIT: the reasoning is simple, if naive: the largest quantities we can measure are not, in fact, infinitely large, and the smallest ones we can measure are not, in fact, infinitesimally small. So until you show me an infinitesimal or an infinity, you're just making them up!
- drpixie 6mo ago
  I've always felt that to treat infinity as number is to commit a category error (aka type conflict), to confuse the process with the outcome of the process. Infinity has proven to be very useful, but usefulness doesn't make it always valid.
- magicalhippo 6mo ago
  All math is made up. We've specifically made up math in a way which is useful to us. You can make up math using different rules[1][2], and get different possibilities. [1]: https://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic https://en.wikipedia.org/wiki/Non-standard_model_of_arithmet... [2]: https://en.wikipedia.org/wiki/Internal_set_theory https://en.wikipedia.org/wiki/Internal_set_theory
- analog31 6mo ago
  "All models are wrong but some are useful." -- George Box
- rdevilla 6mo ago
  > Experiencing time, sound, or visual motion as continuous, rather than discrete signal inputs is so much simpler. Some practice with Mahasi Sayadaw style "noting" can train you into seeing your phenomenological experience as a stream of point-events between which we weave the illusion of continuity.
rambojohnson 6mo ago
finite moments. cherish them.
wizardforhire 6mo ago
Normally amps only go up to ten… but this one goes to eleven. …it’s one louder ain’t it!?!
freetime2 6mo ago
> 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. I'm hoping this is just bad writing from Quanta rather than something "ultrafinitists" truly believe. I really don't think it's that complicated. Even pre-schoolers, competing to see who can say the highest number, quickly learn the concept of infinity. Or elementary school students trying to write 1/3 as a decimal. Of course you need to be careful mapping infinity onto the physical world. But as a mathematical concept, there is absolutely nothing wrong with it. > Mathematicians can construct a form of calculus without infinity, for instance, cutting infinitesimal limits out of the picture entirely. This seems like a useful concept that also doesn't require denying the very obvious concept of infinity.
- nkrisc 6mo ago
  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?
  - jcgrillo 6mo ago
    > Yes, they could on indefinitely Only if they live forever, which they won't. They can only count so fast, and there are only so many of them. Even if every atom in the observable universe was counting at, idk, 1GHz, that's still a finite number. The universe is not (as far as we know for certain) infinitely old. Time may extend infinitely into the future, or it may not. We don't know. So far as we know for sure everything is in fact finite.
    - nkrisc 6mo ago
      Correct. Will they? No, they won’t, because they will die some day.
      - jcgrillo 6mo ago
cyberax 6mo ago
It's not a new idea, and it's a challenging one to investigate. Without real numbers (that are infinitely long) most of the calculus stops working. And everything that depends on it. Perhaps we can recover some of it by treating the infinitely variable values as approximations of the more discrete values and then somehow proving that the errors from them stay bounded, for at least some interesting problems.
bmenrigh 6mo ago
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.
- jcgrillo 6mo ago
  All indications seem to be that things are only lost, not gained. But that doesn't mean it doesn't hew closer to how things actually are. But if that's how reality actually is, then developing a rigorous understanding of it can only be a good thing, right?
  - bmenrigh 6mo ago
    Rejecting infinity is a purely philosophical stance that doesn’t teach us anything about reality. There is a big difference between “infinity doesn’t exist” and “infinity doesn’t exist physically”. I should also add that the resolution of zeno’s paradox in the form of calculus where and infinite set of steps can occur in a finite time (or infinite set of distance can span a finite total distance) is conceptually very simple and useful. Rejecting it as unphysical, or saying it must imply time or space come in discrete chunks, is not contributing to an understanding of reality unless the rejection also comes with a set of testable (in principle) predictions.
    - jcgrillo 6mo ago
      > There is a big difference between “infinity doesn’t exist” and “infinity doesn’t exist physically”. Is there? I think one could make a decent case for "nothing exists which doesn't exist physically[1]". [1] https://plato.stanford.edu/entries/physicalism/ https://plato.stanford.edu/entries/physicalism/ EDIT: you could even probably claim "nothing exists which isn't physically measureable" which may or may not be a stronger claim depending on your point of view. EDIT AGAIN: rate limited by this dogshit website :D but I'll respond to this comment here: > Which is exactly why I mentioned rejection of zero, negative numbers, etc. You can reject them, but doing so just throws away useful tools without gaining anything in return. Yeah! I fully agree. I can see no obvious benefit to rejecting these powerful tools. However, important discoveries often happen in non-obvious directions, and exploring unexplored territory is generally worthwhile. So the fact that it doesn't seem immediately useful doesn't mean it's not worth trying!
Ifkaluva 6mo ago
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, but that’s not a math question! There’s nothing logically inconsistent about infinite sets nor their axiomatizations.
- marcus_holmes 6mo ago
  > But in the late 1800s, Georg Cantor and other mathematicians showed that the infinite really can exist. I think, as I understand it, the objection is this. The proposition that infinity is "real", and there are actually infinite (not just very many) things.
  - SideburnsOfDoom 6mo ago
    > The proposition that infinity is "real" As far as I can tell, numbers aren't real either. "Twelve" isn't a thing that exists in itself in the physical universe, it's an abstraction over some features of reality. "Infinity" is another abstraction, but it's not the same kind of abstraction as "Twelve". It's a further step. All mathematics is abstractions, layered on each other. See also "God created the integers, all else is the work of man"
    - marcus_holmes 6mo ago
      Agreed, but I can say "there are 102 keys on this keyboard" or "there is 105ml of coffee in this cup" and that meaningfully describes reality to some extent. "there are infinite stars" does not meaningfully describe reality. For a number of reasons, not least of which would be how we would verify that - we cannot count the stars if they are infinite because we would need an infinite amount of time to do it. We can't shortcut this process, either, any method of counting stars that allowed us to count more stars more quickly would still run into the problem that it would take infinite time to count them. We can never know if something "real" is actually infinite, or if it's just very large. But disallowing infinities from our concept of the universe does do cool things; we no longer get the endless "in an infinite universe, everything possible must happen an infinite number of times, so therefore <this ridiculous thing> must be happening" arguments. The universe is not infinite, or if it is infinite we cannot measure that or confirm it, so <ridiculous thing> isn't happening unless probability says it will happen in a merely very large universe.
gmuslera 6mo ago
The first thing that came to mind reading the article is that you need only 60ish digits of pi to calculate the circumference of the universe with a resolution of a Planck length, or something like that. You can have all the digits you want, but at some point you are beyond what is possible in reality, and giving back wrong answers for what you are trying to achieve.
alex_young 6mo ago
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 imagine that's the smallest you can go. We can still talk about half of a down quark, or half of that, etc. Say, uh, infinitely so. There you go, everything is infinite. That wasn't so hard was it?
- marcus_holmes 6mo ago
  I think you missed the point. So, firstly, you have split the particle 5 times. That's not infinite times. You can split it more, so that would be 6 times. And more. Even if you could split it 1000 times, that's not infinity. The standard argument for infinity is that "you can always add 1 to any number, so there must be an infinity of them", and the refutation is that no matter how many times you add 1 to a number, all you've done is create a larger number. You never reach the point of actual infinity, no matter how long you keep doing this. You need to have infinite time in order to create an infinity by adding 1 to each number, so you're starting with the axiom that infinity exists (because you need an infinite number of operations to actually create an infinity). If you don't start with that axiom, then you can never reach infinity by addition (or any operation).
  - alex_young 6mo ago
    Time has nothing to do with it. There are an infinite number of ways to divide anything. You don’t need time to prove that. Whatever number you think of you can divide by a larger number.
    - marcus_holmes 6mo ago
      Yes, and that gets you to another number. Not infinity. You need an infinity of operations to create an infinity.
novia 6mo ago
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 of the universe in terms of the width of an up-quark or whatever, feel free to tack on some modifier to my proposed number system.
- vlovich123 6mo ago
  I want to count the number of possible permutations of the particles. We’ve now got a “larger” number than Ω will ever be able to represent by definition (even Ω² is minuscule by comparison).
  - novia 6mo ago
    yeah that seems fine. there's like no good reason to do that. are you trying to simulate reality or something? but my point still stands, choose whichever calculation you think is important to be able to do with Ω, defined as f(Ω), square it for good measure, and set that as the max, the min, and the number of numbers in between each integer. The total number of possible numbers will be ~2*f(Ω)⁴ which should be more than enough numbers :)
    - vlovich123 6mo ago
      AES256 already has more possible keys than exist atoms in the visible universe and that’s a pretty mundane thing. If you wanted to store all those keys, that’s even large. # of atoms in the universe turns into a very small very quickly when talking about permutations and permutations come up all the time (mathematical simulations, probability computations, etc). I really don’t understand what point you’re trying to make saying “pick the largest possible number relevant” as that number varies. Also, that’s just the rational numbers. There’s plenty of digits of precision needed for trajectories over galactic distances and the more precision you try to give irrational numbers, the larger your magical “largest number” needs to grow again. Also, we don’t know how big the “non observable universe” is and it’s beyond the scope of science. It very well could be an infinite number of atoms and then what?
staticshock 6mo ago
Contrarian thinking can be great because it taps into the intuition that the masses are mostly followers who can be led anywhere, not critical thinkers who've deeply examined what they believe. Being contrarian, then, is akin to staking out a new leadership position. The space of contrarian ideas is vast, and most of them are probably bad, but, nevertheless, the willingness to hold unconventional, internally consistent views should be celebrated, because it increases diversity of thought. Our collective hive mind grows stronger through heresy. However, I like my heresy with a splash of axiomatic precision, which is sadly lacking in this article.
deleted 6mo ago
[deleted]
micheles 6mo ago
I have always maintained that real mathematics starts when you address the infinite. I don't see how you can get anything interesting (like analysis, differential geometry, topology) without the assumption that the infinite exists. BTW, the article is really badly written.
krackers 6mo ago
Sad that the article doesn't mention wildberger (coincidentally similar last name), an (in)famous math youtuber that's been mentioned on HN several times before. He has a "rational trigonometry series" an approachable way to see how math would work in an ultrafinite setting.
- potsandpans 6mo ago
  I was also sad to see him not mentioned. But happy to see his name here. Hn loves to dismiss him as a crank, which I think is overly harsh.
don-bright 6mo ago
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 character is too far from origin point. Or why a spreadsheet shows rounding issues
- BigTTYGothGF 6mo ago
  > Surprised Wildberger’s youtube channel wasnt in here. Zeilberger is intellectually honest in a way that Wildberger is not.
getnormality 6mo ago
> 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 many objects, but even if we could, it wouldn't be the same as observing infinity. You can't observe the number one by observing one stone. I think there is some confusion in this article between symbols and what they can stand for, and I can't help but wonder if that same confusion is at the root of ideas like ultrafinitism.
- gbnwl 6mo ago
  The symbol is not the thing. The map is not the territory. Ceci n'est pas une pipe.
- rbtprograms 6mo ago
  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.
  - ndsipa_pomu 6mo ago
    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.
  - alestainer 6mo ago
    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.
m3kw9 6mo ago
Just thinking about when "things" began breaks the brain as there is always something before the "beginning" that you've concluded.
m3kw9 6mo ago
From your point of view, infinity stops to matter once you cease to exist in reality. So there is something to consider
12_throw_away 6mo ago
As someone who has done a whole lot of IEEE 754 floating point math: > computers handle math just fine strong disagree tbh
rikroots 6mo ago
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
- BigTTYGothGF 6mo ago
  > 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".
- gucci-on-fleek 6mo ago
  As the other reply alluded to, this is actually a real thing [0]. [0]: https://en.wikipedia.org/wiki/Projectively_extended_real_line https://en.wikipedia.org/wiki/Projectively_extended_real_lin...
flumes_whims_ 6mo ago
Related work: https://steve-patterson.com/logic-and-infinity/ https://steve-patterson.com/logic-and-infinity/
moralestapia 6mo ago
>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. >“Infinity may or may not exist; God may or may not exist,” he said. “But in mathematics, there should not be any place, neither for infinity nor God.” >much as, Zeilberger might say, science brought doubt to God’s doorstep. >But one day, he added, mathematicians will look back and see that this crackpot, like those of yore who questioned gods and superstitions, was right. “Luckily, heretics are no longer burned at the stake.” LOL. What is this guy's problem?
dwedge 6mo ago
It's always annoyed me that maths has loopholes around infinity to keep it consistent
amavect 6mo ago
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/ https://jeanpaulvanbendegem.be/home/papers/strict-finitism/
- rini17 6mo ago
  You know about busy beavers? These programs do fit in few bits, yet the number of states they can reach does not.
  - amavect 6mo ago
    Of course, I follow the bbchallenge project. That's like the distinction between 2^32, 4294967296, and a string of tally marks that cannot fit in this comment. In a pithy way, strict finitists prefer numbers as tally marks. I find value in that perspective.
- jibal 6mo ago
  2^512 is the number of distinct values of 512 bits ... it sure seems to succeed in existing. SHA-512 is a cryptographic hash that depends on that.
  - 6mo ago
whimblepop 6mo ago
> 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.
- amavect 6mo ago
  The quote comes from his paper "Mathematics and Faith"! https://web.math.princeton.edu/~nelson/papers/faith.pdf https://web.math.princeton.edu/~nelson/papers/faith.pdf You can find many of his papers here. https://web.math.princeton.edu/~nelson/papers/ https://web.math.princeton.edu/~nelson/papers/
injidup 6mo ago
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.
montjoy 6mo ago
I wonder how this affects that combining General Relativity with Quantum Mechanics leads to mathematical infinities “that render calculations meaningless.”
effed3 6mo ago
One thing is try to do a new(?) math in a finite way, good luck, we will see the results, and check agains is still proved. (I think others contraddiction will arise, will be funny..) Another thing is to start a holy war against a concept, to delete it, only to make disappear some problems and not work to resolve it -introducing- new concepts, the way science advances. Unless proving something -wrong- in science, to ban in the name of some beliefs, seems not a good idea to me.
jibal 6mo ago
From the comments: The notion of "believing in" axioms is absurd ... as absurd as believing in the rules of chess and disbelieving the rules of checkers. Each set of rules or axioms forms a system (possibly degenerate if the axioms are inconsistent). The rules, axioms, and systems aren't "true" or "false" -- that's a category mistake. Studying the systems resulting from the Peano axioms or ZFC is a worthwhile endeavor. Studying the systems resulting from finitist axioms may well be too, but the nonexistence of infinities in the latter doesn't mean that they don't exist in the former--that's crackpottery. Mathematics has room for both sorts of systems. Which axiomatic systems best model the world is a different matter. Now we're in an empirical realm, where there are observations, evidence, facts. And observational reports are necessarily finite, so even if there are "real" infinities they can't be demonstrated. But "all models are wrong", so both infinite and finitist axiom systems might serve as good approximations. Likewise with computer systems--all actual computer systems are finite state machines, but it's convenient and useful to model them as Turing Machines that allow for both infinite non-halting systems and finite halting systems. And since both this medium and I are finite, I will stop there.
skyberrys 6mo ago
It's easy to think of infinite counting upwards as having to come to an end eventually, but what about dividing a thing down? Can we have smaller and smaller fractions until we have an infinity of tiny bits, or does it go on for so long there is a point when all of the bits are of size 0? Anyways, I enjoyed reading the perspective of a mathematician on this.
Nevermark 6mo ago
Reality is an infinite expanse of finite things. Finite unbounded admits all the natural numbers. Allowing infinity to represent no-finite-limit, as apposed to trying to shoehorn actualized infinities (general real numbers inclusive of “unconstructible” “non-uniquely specifiable” numbers, and “concrete” higher-order Cantor cardinalities, whatever that could mean) into local structures.

127 points by Tomte 6mo ago

zkmon 6mo ago

Mathematicians sound like priests. They vehemently defend they regime. They enjoy exclusivity by ensuring complexity and extreme formalism.

deleted 6mo ago

[deleted]

jcgrillo 6mo ago

In school I developed a strong hunch that continuity and infinity are "convenient delusions" we have that allow us to process the otherwise horrific complexity of the world. Experiencing time, sound, or visual motion as continuous, rather than discrete signal inputs is so much simpler. Similarly, the mathematical tricks and shortcuts we can use on well behaved continuous functions are both "unreasonably effective" and... probably not grounded in actual reality[1]? But damn are they convenient. [1] EDIT: the reasoning is simple, if naive: the largest quantities we can measure are not, in fact, infinitely large, and the smallest ones we can measure are not, in fact, infinitesimally small. So until you show me an infinitesimal or an infinity, you're just making them up!

drpixie 6mo ago
I've always felt that to treat infinity as number is to commit a category error (aka type conflict), to confuse the process with the outcome of the process. Infinity has proven to be very useful, but usefulness doesn't make it always valid.
magicalhippo 6mo ago
All math is made up. We've specifically made up math in a way which is useful to us. You can make up math using different rules[1][2], and get different possibilities. [1]: https://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic https://en.wikipedia.org/wiki/Non-standard_model_of_arithmet... [2]: https://en.wikipedia.org/wiki/Internal_set_theory https://en.wikipedia.org/wiki/Internal_set_theory
analog31 6mo ago
"All models are wrong but some are useful." -- George Box
rdevilla 6mo ago
> Experiencing time, sound, or visual motion as continuous, rather than discrete signal inputs is so much simpler. Some practice with Mahasi Sayadaw style "noting" can train you into seeing your phenomenological experience as a stream of point-events between which we weave the illusion of continuity.

rambojohnson 6mo ago

finite moments. cherish them.

wizardforhire 6mo ago

Normally amps only go up to ten… but this one goes to eleven. …it’s one louder ain’t it!?!

freetime2 6mo ago

> 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. I'm hoping this is just bad writing from Quanta rather than something "ultrafinitists" truly believe. I really don't think it's that complicated. Even pre-schoolers, competing to see who can say the highest number, quickly learn the concept of infinity. Or elementary school students trying to write 1/3 as a decimal. Of course you need to be careful mapping infinity onto the physical world. But as a mathematical concept, there is absolutely nothing wrong with it. > Mathematicians can construct a form of calculus without infinity, for instance, cutting infinitesimal limits out of the picture entirely. This seems like a useful concept that also doesn't require denying the very obvious concept of infinity.

nkrisc 6mo ago
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?
- jcgrillo 6mo ago
  > Yes, they could on indefinitely Only if they live forever, which they won't. They can only count so fast, and there are only so many of them. Even if every atom in the observable universe was counting at, idk, 1GHz, that's still a finite number. The universe is not (as far as we know for certain) infinitely old. Time may extend infinitely into the future, or it may not. We don't know. So far as we know for sure everything is in fact finite.
  - nkrisc 6mo ago
    Correct. Will they? No, they won’t, because they will die some day.
    - jcgrillo 6mo ago

cyberax 6mo ago

It's not a new idea, and it's a challenging one to investigate. Without real numbers (that are infinitely long) most of the calculus stops working. And everything that depends on it. Perhaps we can recover some of it by treating the infinitely variable values as approximations of the more discrete values and then somehow proving that the errors from them stay bounded, for at least some interesting problems.

bmenrigh 6mo ago

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.

jcgrillo 6mo ago
All indications seem to be that things are only lost, not gained. But that doesn't mean it doesn't hew closer to how things actually are. But if that's how reality actually is, then developing a rigorous understanding of it can only be a good thing, right?
- bmenrigh 6mo ago
  Rejecting infinity is a purely philosophical stance that doesn’t teach us anything about reality. There is a big difference between “infinity doesn’t exist” and “infinity doesn’t exist physically”. I should also add that the resolution of zeno’s paradox in the form of calculus where and infinite set of steps can occur in a finite time (or infinite set of distance can span a finite total distance) is conceptually very simple and useful. Rejecting it as unphysical, or saying it must imply time or space come in discrete chunks, is not contributing to an understanding of reality unless the rejection also comes with a set of testable (in principle) predictions.
  - jcgrillo 6mo ago
    > There is a big difference between “infinity doesn’t exist” and “infinity doesn’t exist physically”. Is there? I think one could make a decent case for "nothing exists which doesn't exist physically[1]". [1] https://plato.stanford.edu/entries/physicalism/ https://plato.stanford.edu/entries/physicalism/ EDIT: you could even probably claim "nothing exists which isn't physically measureable" which may or may not be a stronger claim depending on your point of view. EDIT AGAIN: rate limited by this dogshit website :D but I'll respond to this comment here: > Which is exactly why I mentioned rejection of zero, negative numbers, etc. You can reject them, but doing so just throws away useful tools without gaining anything in return. Yeah! I fully agree. I can see no obvious benefit to rejecting these powerful tools. However, important discoveries often happen in non-obvious directions, and exploring unexplored territory is generally worthwhile. So the fact that it doesn't seem immediately useful doesn't mean it's not worth trying!

Ifkaluva 6mo ago

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, but that’s not a math question! There’s nothing logically inconsistent about infinite sets nor their axiomatizations.

marcus_holmes 6mo ago
> But in the late 1800s, Georg Cantor and other mathematicians showed that the infinite really can exist. I think, as I understand it, the objection is this. The proposition that infinity is "real", and there are actually infinite (not just very many) things.
- SideburnsOfDoom 6mo ago
  > The proposition that infinity is "real" As far as I can tell, numbers aren't real either. "Twelve" isn't a thing that exists in itself in the physical universe, it's an abstraction over some features of reality. "Infinity" is another abstraction, but it's not the same kind of abstraction as "Twelve". It's a further step. All mathematics is abstractions, layered on each other. See also "God created the integers, all else is the work of man"
  - marcus_holmes 6mo ago
    Agreed, but I can say "there are 102 keys on this keyboard" or "there is 105ml of coffee in this cup" and that meaningfully describes reality to some extent. "there are infinite stars" does not meaningfully describe reality. For a number of reasons, not least of which would be how we would verify that - we cannot count the stars if they are infinite because we would need an infinite amount of time to do it. We can't shortcut this process, either, any method of counting stars that allowed us to count more stars more quickly would still run into the problem that it would take infinite time to count them. We can never know if something "real" is actually infinite, or if it's just very large. But disallowing infinities from our concept of the universe does do cool things; we no longer get the endless "in an infinite universe, everything possible must happen an infinite number of times, so therefore <this ridiculous thing> must be happening" arguments. The universe is not infinite, or if it is infinite we cannot measure that or confirm it, so <ridiculous thing> isn't happening unless probability says it will happen in a merely very large universe.

gmuslera 6mo ago

The first thing that came to mind reading the article is that you need only 60ish digits of pi to calculate the circumference of the universe with a resolution of a Planck length, or something like that. You can have all the digits you want, but at some point you are beyond what is possible in reality, and giving back wrong answers for what you are trying to achieve.

alex_young 6mo ago

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 imagine that's the smallest you can go. We can still talk about half of a down quark, or half of that, etc. Say, uh, infinitely so. There you go, everything is infinite. That wasn't so hard was it?

marcus_holmes 6mo ago
I think you missed the point. So, firstly, you have split the particle 5 times. That's not infinite times. You can split it more, so that would be 6 times. And more. Even if you could split it 1000 times, that's not infinity. The standard argument for infinity is that "you can always add 1 to any number, so there must be an infinity of them", and the refutation is that no matter how many times you add 1 to a number, all you've done is create a larger number. You never reach the point of actual infinity, no matter how long you keep doing this. You need to have infinite time in order to create an infinity by adding 1 to each number, so you're starting with the axiom that infinity exists (because you need an infinite number of operations to actually create an infinity). If you don't start with that axiom, then you can never reach infinity by addition (or any operation).
- alex_young 6mo ago
  Time has nothing to do with it. There are an infinite number of ways to divide anything. You don’t need time to prove that. Whatever number you think of you can divide by a larger number.
  - marcus_holmes 6mo ago
    Yes, and that gets you to another number. Not infinity. You need an infinity of operations to create an infinity.

novia 6mo ago

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 of the universe in terms of the width of an up-quark or whatever, feel free to tack on some modifier to my proposed number system.

vlovich123 6mo ago
I want to count the number of possible permutations of the particles. We’ve now got a “larger” number than Ω will ever be able to represent by definition (even Ω² is minuscule by comparison).
- novia 6mo ago
  yeah that seems fine. there's like no good reason to do that. are you trying to simulate reality or something? but my point still stands, choose whichever calculation you think is important to be able to do with Ω, defined as f(Ω), square it for good measure, and set that as the max, the min, and the number of numbers in between each integer. The total number of possible numbers will be ~2*f(Ω)⁴ which should be more than enough numbers :)
  - vlovich123 6mo ago
    AES256 already has more possible keys than exist atoms in the visible universe and that’s a pretty mundane thing. If you wanted to store all those keys, that’s even large. # of atoms in the universe turns into a very small very quickly when talking about permutations and permutations come up all the time (mathematical simulations, probability computations, etc). I really don’t understand what point you’re trying to make saying “pick the largest possible number relevant” as that number varies. Also, that’s just the rational numbers. There’s plenty of digits of precision needed for trajectories over galactic distances and the more precision you try to give irrational numbers, the larger your magical “largest number” needs to grow again. Also, we don’t know how big the “non observable universe” is and it’s beyond the scope of science. It very well could be an infinite number of atoms and then what?

staticshock 6mo ago

Contrarian thinking can be great because it taps into the intuition that the masses are mostly followers who can be led anywhere, not critical thinkers who've deeply examined what they believe. Being contrarian, then, is akin to staking out a new leadership position. The space of contrarian ideas is vast, and most of them are probably bad, but, nevertheless, the willingness to hold unconventional, internally consistent views should be celebrated, because it increases diversity of thought. Our collective hive mind grows stronger through heresy. However, I like my heresy with a splash of axiomatic precision, which is sadly lacking in this article.

deleted 6mo ago

[deleted]

micheles 6mo ago

I have always maintained that real mathematics starts when you address the infinite. I don't see how you can get anything interesting (like analysis, differential geometry, topology) without the assumption that the infinite exists. BTW, the article is really badly written.

krackers 6mo ago

Sad that the article doesn't mention wildberger (coincidentally similar last name), an (in)famous math youtuber that's been mentioned on HN several times before. He has a "rational trigonometry series" an approachable way to see how math would work in an ultrafinite setting.

potsandpans 6mo ago
I was also sad to see him not mentioned. But happy to see his name here. Hn loves to dismiss him as a crank, which I think is overly harsh.

don-bright 6mo ago

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 character is too far from origin point. Or why a spreadsheet shows rounding issues

BigTTYGothGF 6mo ago
> Surprised Wildberger’s youtube channel wasnt in here. Zeilberger is intellectually honest in a way that Wildberger is not.

getnormality 6mo ago

> 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 many objects, but even if we could, it wouldn't be the same as observing infinity. You can't observe the number one by observing one stone. I think there is some confusion in this article between symbols and what they can stand for, and I can't help but wonder if that same confusion is at the root of ideas like ultrafinitism.

gbnwl 6mo ago
The symbol is not the thing. The map is not the territory. Ceci n'est pas une pipe.
rbtprograms 6mo ago
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.
- ndsipa_pomu 6mo ago
  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.
- alestainer 6mo ago
  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.

m3kw9 6mo ago

Just thinking about when "things" began breaks the brain as there is always something before the "beginning" that you've concluded.

m3kw9 6mo ago

From your point of view, infinity stops to matter once you cease to exist in reality. So there is something to consider

12_throw_away 6mo ago

As someone who has done a whole lot of IEEE 754 floating point math: > computers handle math just fine strong disagree tbh

rikroots 6mo ago

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

BigTTYGothGF 6mo ago
> 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".
gucci-on-fleek 6mo ago
As the other reply alluded to, this is actually a real thing [0]. [0]: https://en.wikipedia.org/wiki/Projectively_extended_real_line https://en.wikipedia.org/wiki/Projectively_extended_real_lin...

flumes_whims_ 6mo ago

Related work: https://steve-patterson.com/logic-and-infinity/ https://steve-patterson.com/logic-and-infinity/

moralestapia 6mo ago

>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. >“Infinity may or may not exist; God may or may not exist,” he said. “But in mathematics, there should not be any place, neither for infinity nor God.” >much as, Zeilberger might say, science brought doubt to God’s doorstep. >But one day, he added, mathematicians will look back and see that this crackpot, like those of yore who questioned gods and superstitions, was right. “Luckily, heretics are no longer burned at the stake.” LOL. What is this guy's problem?

dwedge 6mo ago

It's always annoyed me that maths has loopholes around infinity to keep it consistent

amavect 6mo ago

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

rini17 6mo ago
You know about busy beavers? These programs do fit in few bits, yet the number of states they can reach does not.
- amavect 6mo ago
  Of course, I follow the bbchallenge project. That's like the distinction between 2^32, 4294967296, and a string of tally marks that cannot fit in this comment. In a pithy way, strict finitists prefer numbers as tally marks. I find value in that perspective.
jibal 6mo ago
2^512 is the number of distinct values of 512 bits ... it sure seems to succeed in existing. SHA-512 is a cryptographic hash that depends on that.
- 6mo ago

whimblepop 6mo ago

> 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.

amavect 6mo ago
The quote comes from his paper "Mathematics and Faith"! https://web.math.princeton.edu/~nelson/papers/faith.pdf https://web.math.princeton.edu/~nelson/papers/faith.pdf You can find many of his papers here. https://web.math.princeton.edu/~nelson/papers/ https://web.math.princeton.edu/~nelson/papers/

injidup 6mo ago

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.

montjoy 6mo ago

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

effed3 6mo ago

One thing is try to do a new(?) math in a finite way, good luck, we will see the results, and check agains is still proved. (I think others contraddiction will arise, will be funny..) Another thing is to start a holy war against a concept, to delete it, only to make disappear some problems and not work to resolve it -introducing- new concepts, the way science advances. Unless proving something -wrong- in science, to ban in the name of some beliefs, seems not a good idea to me.

jibal 6mo ago

From the comments: The notion of "believing in" axioms is absurd ... as absurd as believing in the rules of chess and disbelieving the rules of checkers. Each set of rules or axioms forms a system (possibly degenerate if the axioms are inconsistent). The rules, axioms, and systems aren't "true" or "false" -- that's a category mistake. Studying the systems resulting from the Peano axioms or ZFC is a worthwhile endeavor. Studying the systems resulting from finitist axioms may well be too, but the nonexistence of infinities in the latter doesn't mean that they don't exist in the former--that's crackpottery. Mathematics has room for both sorts of systems. Which axiomatic systems best model the world is a different matter. Now we're in an empirical realm, where there are observations, evidence, facts. And observational reports are necessarily finite, so even if there are "real" infinities they can't be demonstrated. But "all models are wrong", so both infinite and finitist axiom systems might serve as good approximations. Likewise with computer systems--all actual computer systems are finite state machines, but it's convenient and useful to model them as Turing Machines that allow for both infinite non-halting systems and finite halting systems. And since both this medium and I are finite, I will stop there.

skyberrys 6mo ago

It's easy to think of infinite counting upwards as having to come to an end eventually, but what about dividing a thing down? Can we have smaller and smaller fractions until we have an infinity of tiny bits, or does it go on for so long there is a point when all of the bits are of size 0? Anyways, I enjoyed reading the perspective of a mathematician on this.

Nevermark 6mo ago

Reality is an infinite expanse of finite things. Finite unbounded admits all the natural numbers. Allowing infinity to represent no-finite-limit, as apposed to trying to shoehorn actualized infinities (general real numbers inclusive of “unconstructible” “non-uniquely specifiable” numbers, and “concrete” higher-order Cantor cardinalities, whatever that could mean) into local structures.