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.
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.
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.
I don't think it's bad writing. These people actually get angry at the idea that other people do math that might not connect to the real world. And they specially have it out for infinity.
I say do whatever math you like. It is helpful to know what math you are doing. For instance, while I don't have a "problem" with the Axiom of Choice per se I do like clean specifications of when we are using it and when we are not, because it is another example of when we detach from reality as we know it. I don't have a problem with detaching from reality as we know it, I just like there to be awareness that we have.
But plenty of math is detached from reality. Honestly we don't observe very many "mathematical entities" at all; I've never seen a graph. I've never seen hyperbolic space. I'm aware of the many places aspects of them seem to map to reality, but I've never actually seen a literal graph in the real world.
Personally I am reminded of the way that we model our computers with Turing Complete formalisms, despite the fact they are observably not Turing Complete and are technically just finite state machines. However, the observation that they are "just" finite state machines doesn't move us closer to an understanding of how our computers work, it moves us farther away. Even though computers are completely real-world phenomena, if you want to understand the issues raised by things like Turing Incompleteness and other such things in the real world, you're going to be exponentially better off using Turing Machine formalisms and simply noting that you may run out of memory or practically-available computational resources before a calculation can complete than trying to build a new set of formalisms around finite state machines. We can be in an engineering context where we are well aware of the finite nature of everything we are doing because it all comes back to real, physical machines, but it's still easier to model with infinity than without it.
In that context, the real utility of "infinity" is less "an infinite number of things" than "you will never reach for another X [byte of RAM, byte of disk, CPU cycle, incrementing counter, etc.] and be told you're out of resources". Basically we write our proofs, formal or informal, as ignoring "what if I reach for this resource and it's not there?" for every such resource and every time we reach for a resource, which is quite often. You could go through a system and add a "what if" check for every such instance, but it's way cheaper to just buy another stick of RAM or tweak the program to take fewer resources than it is to try to deal with the exponential-with-a-large-exponent explosion of states this causes mathematically.
Your response essentially assumes formalism - mathematics is a game with rules (axioms, inference rules, etc), and all rules are in themselves equally valid, it is just a question of whether the game they produce is playable (i.e. produces interesting or useful theorems). Formalism has no objection to infinities: the axiom of infinity is just another axiom, in itself as valid as any other-but one which produces a near-endless array of interesting results.
Formalism is a very common approach in the philosophy of mathematics-but it isn’t the only one, and it is not the philosophy which motivates ultrafinitism.
Another viewpoint is that mathematical objects somehow really exist; mathematics is more than just a symbol manipulation game. One variation of this is (mathematical) Platonism, which believes they exist in some timeless realm beyond this physical universe; that view has no issue with infinities either, since adherents of this view generally believe that realm to be infinite and filled with infinities.
Yet another view is conceptualism-mathematical objects really exist, but in the human mind. And this is the viewpoint that motivates ultrafinitism - the human mind is finite, so infinite mathematical objects cannot really exist in it, or at least not in the fullness of the sense that finite objects can; and that turns out to be true, not just for infinities, but also for overly large finitudes.
This idea that some mathematical objects are in a philosophical sense “more real” than others is a big motivator of mathematical constructivism-trying to find axioms which respect that philosophical distinction, and work out what the consequences of those axioms are. Ultrafinitism is just a particularly extreme form of constructivism, which adopted a stricter “criterion of reality” for mathematical objects than most constructivists do
I think you're conflating opinions about when math is useful with opinions on the nature of math itself. Formalism does not assume that "all rules are equally valid". You can be a staunch formalist and yet still believe that X set of axioms are the only useful ones and everyone who assumes different axioms is wasting their time. You could be a formalist and still believe that the concept of infinity is leading math astray from useful math. Many of the differences you lay out seem to just be in people's opinion on which axioms are useful and which aren't. That's still formalism.
Setting that aside, it's very difficult for me to take non-formalist views of mathematics seriously. I strongly suspect that anyone who subscribes to those views has some deep-seated confusion in their heads.
> Platonism, which believes [mathematical objects] exist in some timeless realm beyond this physical universe
This is equivalent to formalism, except perhaps in how the mathematician feels about it. What could any possible difference be? In what way could it ever matter in the slightest whether something "really exists", if we define that to be so weak as to include "in some timeless realm beyond this universe"? Surely pink goblins "really exist" in this sense as well. With such a weak definition, the difference between your "really exists" and my "really exists" is purely emotional.
> Yet another view is conceptualism-mathematical objects really exist, but in the human mind.
You can be formalist and still argue about whether humans invented or discovered math. Beyond that, this is again just relying on the weakest possible definition of "really exists", with some added human-centric arrogance added in. Crows can count to 5; it's patently absurd to claim they are using something that is "not mathematics" or some completely alien form of mathematics that humans cannot access, because it's crow-brain math rather than human-brain math. This sounds like the Copenhangen Interpretation but for math: humans brains are magic! What are we doing? What are we talking about?
> This idea that some mathematical objects are in a philosophical sense “more real” than others is a big motivator of mathematical constructivism
Yet again, this is still formalism. Up until here, you've used the word "real" in such a weak tautological sense as to have no connection to our (or any possible) universe. But here, you've switched back to "real" meaning "having any bearing on our universe". So you're saying "constructivists consider different axioms useful than ZFC mathematicians do." More often they don't even really think about usefuless at all, it's just something that caught their interest and they decided to explore it.
> I think you're conflating opinions about when math is useful with opinions on the nature of math itself. Formalism does not assume that "all rules are equally valid"
I think you're misinterpreting what I was saying. Of course, a formalist will say that some rules are "more valid" in the sense that they produce more interesting or useful theorems. My point was, to a formalist, there is nothing more to be said about the validity of axioms than the value of the theorems they produce. Whereas, from certain other perspectives in the philosophy of mathematics, that is not the only grounds on which axioms can be judged.
> This is equivalent to formalism, except perhaps in how the mathematician feels about it. What could any possible difference be? In what way could it ever matter in the slightest whether something "really exists", if we define that to be so weak as to include "in some timeless realm beyond this universe"? Surely pink goblins "really exist" in this sense as well. With such a weak definition, the difference between your "really exists" and my "really exists" is purely emotional.
You sound like a logical positivist. And that's the issue – if your philosophical assumptions are positivist, then non-positivist philosophies of mathematics (and of anything else) simply aren't going to be intelligible to you. They can only make sense if you are at least willing to doubt for a moment your positivist assumptions.
> Crows can count to 5; it's patently absurd to claim they are using something that is "not mathematics" or some completely alien form of mathematics that humans cannot access, because it's crow-brain math rather than human-brain math. This sounds like the Copenhangen Interpretation but for math: humans brains are magic! What are we doing? What are we talking about?
Conceptualism claims that mathematics exists in the mind–but it doesn't claim necessarily only human minds. If animals have minds too, then mathematics can exist in animal minds as well, even if in a much more rudimentary form. I doubt any conceptualist would say, that if intelligent extraterrestrial life were discovered to exist, that their minds wouldn't contain mathematics simply because they are a different species from homo sapiens.
> So you're saying "constructivists consider different axioms useful than ZFC mathematicians do." More often they don't even really think about usefuless at all, it's just something that caught their interest and they decided to explore it.
There are different types of constructivists: (a) those who have a philosophical commitment to constructivism; (b) those who are interested in constructivism for practical reasons (related to computer science); (c) those who are just interested in it as an interesting mathematical system to explore. You can be (b) or (c) without needing any philosophical commitments at all, and they are completely compatible with a formalist philosophy of mathematics. And, quite possibly, the majority working in constructive mathematics today are (b) or (c) not (a). But, historically, the founders of constructive mathematics (e.g. Brouwer) were very much (a) not (b) or (c).
> There simply is no "non-formalist" mathematics.
I think you are conflating mathematics with the philosophy of mathematics – they are two distinct disciplines. Disagreements about the philosophy of mathematics make no direct difference to mathematics itself; at the margins, they can influence judgements about which problems are interesting – although, even there, a person can find ultrafinitist mathematics interesting without needing any philosophical commitment to an ultrafinitist philosophy of mathematics.
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.
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 problem... if we were, then physicists would have a much easier time lol
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).
> 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”.
Suppose we start with ZFC - Infinity as our base system. Then the negation of Infinity is consistent with this system. But adding Infinity itself makes the system strictly stronger, since ZFC proves the consistency of ZFC - Inf: in particular, in ZFC, we cannot prove that Infinity is consistent with ZFC - Inf.
In other words, in principle, it might be the case that ZFC - Inf is consistent, yet ZFC itself has a contradiction. In practice, most people believe that ZFC is also consistent, but we have no way to prove it a priori without accepting even more new axioms.
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.
> 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.
> 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.
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"
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.
> but I can say "there are 102 keys on this keyboard"
You can, but are there? Is the space bar grouped with them? You can decide either way, and either choice is defensible, so the count is dependant on our human categories.
Each key is unique, with the printed letter and with the precise arrangement of atoms differing. None of them are interchangeable. So you could decide that each key is in their own category. Or you could throw them in a bag with some dice and pebbles too and count the objects. Numbers are regarded as interchangeable, one "102" is the same as any other "102". Physical objects are not.
That's what I mean by saying that counting is an abstraction, and a number isn't a thing that exists in itself in the physical universe - it only occurs not "in itself", but once we can subjectively group things. There are "twelve keys" if you decide to group them as the same thing though they're actually not really. There's no physically existent "twelve".
I fully agree that "infinity" is a different kind of concept, less physically relatable even than "102". It's a further abstraction.
> 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.
> 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
The problem to me seems to be that we are trying to map everyday language onto the mathematics. Even though we have a symbol for infinity, infinity is not necessarily a "thing" that the symbol points to.
In analysis, when we write "the limit as x goes to infinity" this translates into a logical statement like "for all x, there exists some y > x such that ..." I don't really see anything conceptually difficult or contradictory here.
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."
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.
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?"
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.
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.
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.
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 :)
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.
2^512 exists in binary notation, but not in unary notation (tally marks, successor function). We conflate these ideas of "number", trying to forget the practical differences. Quite frustrating! SHA-512 depends on the fact that computers cannot feasibly increment to 2^512. A loop cannot feasibly run 2^512 times. Strict finitists emphasize those distinctions when they say 2^512 doesn't exist.
> 2^512 exists in binary notation, but not in unary notation (tally marks, successor function).
Sorry, but this is incoherent nonsense. The existence of a number doesn't depend on its representation ... but we can in fact represent any integer 0 <= n < 2^512 with 512 bits.
> SHA-512 depends on the fact that computers cannot feasibly increment to 2^512.
non sequitur
> A loop cannot feasibly run 2^512 times.
non sequitur
> Strict finitists emphasize those distinctions when they say 2^512 doesn't exist.
I don't mean to come off as a crank. I'd like to clarify the strict finitist position in a sensible way (what they mean is that 2^512 unary notation doesn't fit in this universe, and similarly 2^(2^512) in binary notation doesn't fit in this universe and thus "doesn't exist"), but clearly I fail at meeting your requirements, sorry.
Sigh. There was no mention of 2^(2^512),only the set [0, 2^512), each value of which can be represented with a different configuration of 512 bits in any computer (or storage medium) "in this universe". That these values cannot be enumerated within a small amount of time is completely irrelevant (other than that SHA-512 cannot be reversed in practice, which as I said is a non sequitur) ... 2^512 is quite finite, even for finitists.
And "doesn't fit in this universe" is unformalizable crank nonsense.
A few formalizations exist: bounded arithmetic, internal set theory, Nelson's predicative arithmetic.
From Buss's thesis "Bounded Arithmetic": "However, a recursive function may be computable only in a theoretical sense: the time required to compute it may be far larger than the lifespan of the universe. We are more interested in feasibly computable functions, which can be calculated by today's (or tomorrow's) computers."
And then he goes and defines an arithmetic hierarchy for polynomial functions.
> 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.
> 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.
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 explanation, everyone more or less gets it.
When I was about ten, a math teacher once asked me whether the number 0.9999... (infinitely repeating) was different than 1. I said, with my child's intuition, that of course it was. He then challenged me to write down a number that was in between them, because if they were not the same number then there would be many (in fact, infinitely many) numbers between them. I couldn't, of course: the best I could do was to write 0.9999...5, which falls into the same category error as "infinity plus one / infinity plus two".
Now, decades later, I get it better. The number 0.99999... is 9/10 + 9/100 + 9/1000 + 9/10000 + ..., which approaches 1 asymptotically the same way that 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... approaches 1. Under many circumstances, you can treat that number as if it was 1, which neatly answers Zeno's Paradox. (Though beware of the limitations of that analysis: 1/n approaches infinity as n approaches 0, but 1/0 is not equal to infinity. Because 1/n approaches infinity only as n approaches 0 from the positive direction. If you look at the sequence 1/-0.1, 1/-0.01, 1/-0.001, etc. where n approaches 0 from the negative direction, that approaches negative infinity. A function that has two different limits as you approach the same number from two different directions cannot have its limit substituted like that).
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.
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 its limit if it exists.
> 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.
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.
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.
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.
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.
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.
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).
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 :)
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?
> 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 that understanding.
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.
At the bottom end we have the Planck length. How many cubic Planck lengths in the visible universe ? Anyone ? To paraphrase Bill Gates (allegedly), "(PlanckLengths/widthOfUniverse)*3 ought to be enough for anybody."
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.
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.
I wonder how this affects that combining General Relativity with Quantum Mechanics leads to mathematical infinities “that render calculations meaningless.”
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
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.
> The article doesn’t really tell us what is gained by rejecting infinity.
Decidability. The issues around undecidability all involve the lack of an upper bound. In a finite deterministic space, everything is decidable, although some things may be too costly computationally to decide.
There are several ways to go for decidability. The brute force way is computer arithmetic - there is no number larger than 2^64-1. That's how we get things done on computers, but proofs about numbers with finite upper bounds need lots of special cases. Mathematicians hate that.
I used to work on this sort of thing, using Boyer-Moore theory. That's a lot like the Peano axioms. There is (ZERO), and (ADD1 (ZERO)), and (ADD1 (ADD1 (ZERO))), etc. Everything is constructive and has an unambiguous representation in a LISP-like form.
You can have recursive functions. But they must be proven to terminate, by having a nonnegative value which decreases on each recursive call. There is a distinction between "infinite" and "arbitrarily large". You can talk about arbitrarily large numbers, but you cannot get to 1/2 + 1/4 + 1/8 ... = 1. You can have integers and rational numbers of arbitrary size, but not reals.
Set theory was interesting. Rather than axiomatic set theory, I was using lists as sets, with the constraints that no value could be duplicated and the list must be ordered. Equality is strict - two things are equal only if the elements are all equal, compared element by element. It's possible to prove the usual axioms of set theory via this route. The ordered criterion requires proving things about ordered list insertion to get there. It's ugly and needs machine proofs.
I was doing this back in the early 1980s, when machine proofs were frowned upon. Mathematicians were still upset about the four-color theorem proof. It's all case analysis, with thousands of cases. That's more acceptable today.
Looked at in this light, infinity is a labor-saving device to eliminate special cases, at a potential cost in soundness.
> Looked at in this light, infinity is a labor-saving device to eliminate special cases, at a potential cost in soundness.
Or it is something that clearly conceptually exists, and makes simplistic reductionist viewpoints impossible to prove, which frustrates those who attempt to extend them into twisted metaphysical conjectures.
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?
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.
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!
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!
> 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.
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.
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.
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.
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
> 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.
>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.”
>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.”
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.
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.
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?
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.
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.
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).
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.
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.
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.
Not from “that half of something had a value”, but from “that half of any thing has a value”.
If you accept that every natural number has a successor which is a natural number, and no two natural numbers have the same successor, and that there’s no loops (e.g. by saying that there’s a total order on natural numbers and that any natural number is less than its successor), then there can’t be a finite collection which is all the natural numbers.
You could say “there’s no collection which has all the natural numbers”, which, ok, how do you want to talk about things true of all natural numbers then?
Formulating descriptions of physics without the axiom of infinity (or, without something to play the role of the real numbers) is super icky. You, in practice, can’t do any significant mathematical physics in an ultrafinitistic approach.
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.
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.
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.
I don't think it's bad writing. These people actually get angry at the idea that other people do math that might not connect to the real world. And they specially have it out for infinity.
I say do whatever math you like. It is helpful to know what math you are doing. For instance, while I don't have a "problem" with the Axiom of Choice per se I do like clean specifications of when we are using it and when we are not, because it is another example of when we detach from reality as we know it. I don't have a problem with detaching from reality as we know it, I just like there to be awareness that we have.
But plenty of math is detached from reality. Honestly we don't observe very many "mathematical entities" at all; I've never seen a graph. I've never seen hyperbolic space. I'm aware of the many places aspects of them seem to map to reality, but I've never actually seen a literal graph in the real world.
Personally I am reminded of the way that we model our computers with Turing Complete formalisms, despite the fact they are observably not Turing Complete and are technically just finite state machines. However, the observation that they are "just" finite state machines doesn't move us closer to an understanding of how our computers work, it moves us farther away. Even though computers are completely real-world phenomena, if you want to understand the issues raised by things like Turing Incompleteness and other such things in the real world, you're going to be exponentially better off using Turing Machine formalisms and simply noting that you may run out of memory or practically-available computational resources before a calculation can complete than trying to build a new set of formalisms around finite state machines. We can be in an engineering context where we are well aware of the finite nature of everything we are doing because it all comes back to real, physical machines, but it's still easier to model with infinity than without it.
In that context, the real utility of "infinity" is less "an infinite number of things" than "you will never reach for another X [byte of RAM, byte of disk, CPU cycle, incrementing counter, etc.] and be told you're out of resources". Basically we write our proofs, formal or informal, as ignoring "what if I reach for this resource and it's not there?" for every such resource and every time we reach for a resource, which is quite often. You could go through a system and add a "what if" check for every such instance, but it's way cheaper to just buy another stick of RAM or tweak the program to take fewer resources than it is to try to deal with the exponential-with-a-large-exponent explosion of states this causes mathematically.
Your response essentially assumes formalism - mathematics is a game with rules (axioms, inference rules, etc), and all rules are in themselves equally valid, it is just a question of whether the game they produce is playable (i.e. produces interesting or useful theorems). Formalism has no objection to infinities: the axiom of infinity is just another axiom, in itself as valid as any other-but one which produces a near-endless array of interesting results.
Formalism is a very common approach in the philosophy of mathematics-but it isn’t the only one, and it is not the philosophy which motivates ultrafinitism.
Another viewpoint is that mathematical objects somehow really exist; mathematics is more than just a symbol manipulation game. One variation of this is (mathematical) Platonism, which believes they exist in some timeless realm beyond this physical universe; that view has no issue with infinities either, since adherents of this view generally believe that realm to be infinite and filled with infinities.
Yet another view is conceptualism-mathematical objects really exist, but in the human mind. And this is the viewpoint that motivates ultrafinitism - the human mind is finite, so infinite mathematical objects cannot really exist in it, or at least not in the fullness of the sense that finite objects can; and that turns out to be true, not just for infinities, but also for overly large finitudes.
This idea that some mathematical objects are in a philosophical sense “more real” than others is a big motivator of mathematical constructivism-trying to find axioms which respect that philosophical distinction, and work out what the consequences of those axioms are. Ultrafinitism is just a particularly extreme form of constructivism, which adopted a stricter “criterion of reality” for mathematical objects than most constructivists do
I think you're conflating opinions about when math is useful with opinions on the nature of math itself. Formalism does not assume that "all rules are equally valid". You can be a staunch formalist and yet still believe that X set of axioms are the only useful ones and everyone who assumes different axioms is wasting their time. You could be a formalist and still believe that the concept of infinity is leading math astray from useful math. Many of the differences you lay out seem to just be in people's opinion on which axioms are useful and which aren't. That's still formalism.
Setting that aside, it's very difficult for me to take non-formalist views of mathematics seriously. I strongly suspect that anyone who subscribes to those views has some deep-seated confusion in their heads.
> Platonism, which believes [mathematical objects] exist in some timeless realm beyond this physical universe
This is equivalent to formalism, except perhaps in how the mathematician feels about it. What could any possible difference be? In what way could it ever matter in the slightest whether something "really exists", if we define that to be so weak as to include "in some timeless realm beyond this universe"? Surely pink goblins "really exist" in this sense as well. With such a weak definition, the difference between your "really exists" and my "really exists" is purely emotional.
> Yet another view is conceptualism-mathematical objects really exist, but in the human mind.
You can be formalist and still argue about whether humans invented or discovered math. Beyond that, this is again just relying on the weakest possible definition of "really exists", with some added human-centric arrogance added in. Crows can count to 5; it's patently absurd to claim they are using something that is "not mathematics" or some completely alien form of mathematics that humans cannot access, because it's crow-brain math rather than human-brain math. This sounds like the Copenhangen Interpretation but for math: humans brains are magic! What are we doing? What are we talking about?
> This idea that some mathematical objects are in a philosophical sense “more real” than others is a big motivator of mathematical constructivism
Yet again, this is still formalism. Up until here, you've used the word "real" in such a weak tautological sense as to have no connection to our (or any possible) universe. But here, you've switched back to "real" meaning "having any bearing on our universe". So you're saying "constructivists consider different axioms useful than ZFC mathematicians do." More often they don't even really think about usefuless at all, it's just something that caught their interest and they decided to explore it.
There simply is no "non-formalist" mathematics.
> I think you're conflating opinions about when math is useful with opinions on the nature of math itself. Formalism does not assume that "all rules are equally valid"
I think you're misinterpreting what I was saying. Of course, a formalist will say that some rules are "more valid" in the sense that they produce more interesting or useful theorems. My point was, to a formalist, there is nothing more to be said about the validity of axioms than the value of the theorems they produce. Whereas, from certain other perspectives in the philosophy of mathematics, that is not the only grounds on which axioms can be judged.
> This is equivalent to formalism, except perhaps in how the mathematician feels about it. What could any possible difference be? In what way could it ever matter in the slightest whether something "really exists", if we define that to be so weak as to include "in some timeless realm beyond this universe"? Surely pink goblins "really exist" in this sense as well. With such a weak definition, the difference between your "really exists" and my "really exists" is purely emotional.
You sound like a logical positivist. And that's the issue – if your philosophical assumptions are positivist, then non-positivist philosophies of mathematics (and of anything else) simply aren't going to be intelligible to you. They can only make sense if you are at least willing to doubt for a moment your positivist assumptions.
> Crows can count to 5; it's patently absurd to claim they are using something that is "not mathematics" or some completely alien form of mathematics that humans cannot access, because it's crow-brain math rather than human-brain math. This sounds like the Copenhangen Interpretation but for math: humans brains are magic! What are we doing? What are we talking about?
Conceptualism claims that mathematics exists in the mind–but it doesn't claim necessarily only human minds. If animals have minds too, then mathematics can exist in animal minds as well, even if in a much more rudimentary form. I doubt any conceptualist would say, that if intelligent extraterrestrial life were discovered to exist, that their minds wouldn't contain mathematics simply because they are a different species from homo sapiens.
> So you're saying "constructivists consider different axioms useful than ZFC mathematicians do." More often they don't even really think about usefuless at all, it's just something that caught their interest and they decided to explore it.
There are different types of constructivists: (a) those who have a philosophical commitment to constructivism; (b) those who are interested in constructivism for practical reasons (related to computer science); (c) those who are just interested in it as an interesting mathematical system to explore. You can be (b) or (c) without needing any philosophical commitments at all, and they are completely compatible with a formalist philosophy of mathematics. And, quite possibly, the majority working in constructive mathematics today are (b) or (c) not (a). But, historically, the founders of constructive mathematics (e.g. Brouwer) were very much (a) not (b) or (c).
> There simply is no "non-formalist" mathematics.
I think you are conflating mathematics with the philosophy of mathematics – they are two distinct disciplines. Disagreements about the philosophy of mathematics make no direct difference to mathematics itself; at the margins, they can influence judgements about which problems are interesting – although, even there, a person can find ultrafinitist mathematics interesting without needing any philosophical commitment to an ultrafinitist philosophy of mathematics.
Okay that might be true for the axiom of infinity, but it's 100% not true for the axiom of choice.
https://en.wikipedia.org/wiki/Axiom_of_determinacy
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.
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 problem... if we were, then physicists would have a much easier time lol
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).
Infinity isn't a destination, it's an iterative ongoing approach.
You can idealise it like many things in mathematics, but implementation details fail compared to the abstract ideals.
> 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”.
Suppose we start with ZFC - Infinity as our base system. Then the negation of Infinity is consistent with this system. But adding Infinity itself makes the system strictly stronger, since ZFC proves the consistency of ZFC - Inf: in particular, in ZFC, we cannot prove that Infinity is consistent with ZFC - Inf.
In other words, in principle, it might be the case that ZFC - Inf is consistent, yet ZFC itself has a contradiction. In practice, most people believe that ZFC is also consistent, but we have no way to prove it a priori without accepting even more new axioms.
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.
> 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.
> 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.
> 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"
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.
> but I can say "there are 102 keys on this keyboard"
You can, but are there? Is the space bar grouped with them? You can decide either way, and either choice is defensible, so the count is dependant on our human categories.
Each key is unique, with the printed letter and with the precise arrangement of atoms differing. None of them are interchangeable. So you could decide that each key is in their own category. Or you could throw them in a bag with some dice and pebbles too and count the objects. Numbers are regarded as interchangeable, one "102" is the same as any other "102". Physical objects are not.
That's what I mean by saying that counting is an abstraction, and a number isn't a thing that exists in itself in the physical universe - it only occurs not "in itself", but once we can subjectively group things. There are "twelve keys" if you decide to group them as the same thing though they're actually not really. There's no physically existent "twelve".
I fully agree that "infinity" is a different kind of concept, less physically relatable even than "102". It's a further abstraction.
> 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.
> 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.
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?
Well importantly like scrubs points out in a sibling comment
42+1 = 43, 42 + 1 ≠ 42, ∞ + 1 = ∞
Infinity plays by very different rules than numbers.
I actually agree, there's nothing wrong with infinity. I think finitists are silly and ultrafinitists are ultra silly.
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.
The problem to me seems to be that we are trying to map everyday language onto the mathematics. Even though we have a symbol for infinity, infinity is not necessarily a "thing" that the symbol points to.
In analysis, when we write "the limit as x goes to infinity" this translates into a logical statement like "for all x, there exists some y > x such that ..." I don't really see anything conceptually difficult or contradictory here.
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."
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.
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?"
You say that as if we knew the number of atoms in the universe, or its size, age, and "duration".
But none of this can be observed either, which in my book makes your argument a bit weak.
Your "universal reality" is a construction relying in big parts on the mathematics relying on infinity as a concept.
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.
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.
The symbol is not the thing. The map is not the territory. Ceci n'est pas une pipe.
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/
You know about busy beavers? These programs do fit in few bits, yet the number of states they can reach does not.
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.
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.
2^512 exists in binary notation, but not in unary notation (tally marks, successor function). We conflate these ideas of "number", trying to forget the practical differences. Quite frustrating! SHA-512 depends on the fact that computers cannot feasibly increment to 2^512. A loop cannot feasibly run 2^512 times. Strict finitists emphasize those distinctions when they say 2^512 doesn't exist.
> 2^512 exists in binary notation, but not in unary notation (tally marks, successor function).
Sorry, but this is incoherent nonsense. The existence of a number doesn't depend on its representation ... but we can in fact represent any integer 0 <= n < 2^512 with 512 bits.
> SHA-512 depends on the fact that computers cannot feasibly increment to 2^512.
non sequitur
> A loop cannot feasibly run 2^512 times.
non sequitur
> Strict finitists emphasize those distinctions when they say 2^512 doesn't exist.
Cranks.
I don't mean to come off as a crank. I'd like to clarify the strict finitist position in a sensible way (what they mean is that 2^512 unary notation doesn't fit in this universe, and similarly 2^(2^512) in binary notation doesn't fit in this universe and thus "doesn't exist"), but clearly I fail at meeting your requirements, sorry.
Sigh. There was no mention of 2^(2^512),only the set [0, 2^512), each value of which can be represented with a different configuration of 512 bits in any computer (or storage medium) "in this universe". That these values cannot be enumerated within a small amount of time is completely irrelevant (other than that SHA-512 cannot be reversed in practice, which as I said is a non sequitur) ... 2^512 is quite finite, even for finitists.
And "doesn't fit in this universe" is unformalizable crank nonsense.
Over and out.
A few formalizations exist: bounded arithmetic, internal set theory, Nelson's predicative arithmetic.
From Buss's thesis "Bounded Arithmetic": "However, a recursive function may be computable only in a theoretical sense: the time required to compute it may be far larger than the lifespan of the universe. We are more interested in feasibly computable functions, which can be calculated by today's (or tomorrow's) computers."
And then he goes and defines an arithmetic hierarchy for polynomial functions.
> 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.
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/
> 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.
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 explanation, everyone more or less gets it.
The obvious way to win in this game, that probably many kids discover is to define your number as "whatever number the other kid says, plus 1".
The game is to name a larger number. Is infinity a number? Some would say, “no”.
Eventually, one of the kids will name a largest number, because no one else will name another and the game ends with a largest number.
It is possible that aliens exist, so is that proof aliens exist?
Is is possible to create ever larger numbers, but is that proof that infinity exists other than as a fanciful idea in our minds?
And then the next kid says "infinity plus two", which is a perfectly acceptable progression, and the cycle starts again.
When I was about ten, a math teacher once asked me whether the number 0.9999... (infinitely repeating) was different than 1. I said, with my child's intuition, that of course it was. He then challenged me to write down a number that was in between them, because if they were not the same number then there would be many (in fact, infinitely many) numbers between them. I couldn't, of course: the best I could do was to write 0.9999...5, which falls into the same category error as "infinity plus one / infinity plus two".
Now, decades later, I get it better. The number 0.99999... is 9/10 + 9/100 + 9/1000 + 9/10000 + ..., which approaches 1 asymptotically the same way that 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... approaches 1. Under many circumstances, you can treat that number as if it was 1, which neatly answers Zeno's Paradox. (Though beware of the limitations of that analysis: 1/n approaches infinity as n approaches 0, but 1/0 is not equal to infinity. Because 1/n approaches infinity only as n approaches 0 from the positive direction. If you look at the sequence 1/-0.1, 1/-0.01, 1/-0.001, etc. where n approaches 0 from the negative direction, that approaches negative infinity. A function that has two different limits as you approach the same number from two different directions cannot have its limit substituted like that).
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 its limit if it exists.
> 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.
INFINITY PLUS 1
Uncountable infinity
24 is the highest number. Where you gonna go from there?
> 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.
Correct. Will they? No, they won’t, because they will die some day.
Maybe an exceptional one will realize the task's futility and invent infinity as a way to rationalize giving up
As someone who has done a whole lot of IEEE 754 floating point math:
> computers handle math just fine
strong disagree tbh
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.
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.
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.
Related work: https://steve-patterson.com/logic-and-infinity/
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.
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.
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).
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 :)
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?
> 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 that understanding.
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.
Since we don’t know the number of atoms, we’d need to let omega be a function, then deal with all the edge cases, rename omega with ∞ and..
Yeah I can’t tell if op is trolling or really thinks they can just define a rational number big enough to not need infinity as a concept.
[dead]
At the bottom end we have the Planck length. How many cubic Planck lengths in the visible universe ? Anyone ? To paraphrase Bill Gates (allegedly), "(PlanckLengths/widthOfUniverse)*3 ought to be enough for anybody."
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.
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.
Mathematicians sound like priests. They vehemently defend they regime. They enjoy exclusivity by ensuring complexity and extreme formalism.
I wonder how this affects that combining General Relativity with Quantum Mechanics leads to mathematical infinities “that render calculations meaningless.”
Just thinking about when "things" began breaks the brain as there is always something before the "beginning" that you've concluded.
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".
As the other reply alluded to, this is actually a real thing [0].
[0]: https://en.wikipedia.org/wiki/Projectively_extended_real_lin...
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.
It's always annoyed me that maths has loopholes around infinity to keep it consistent
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.
> The article doesn’t really tell us what is gained by rejecting infinity.
Decidability. The issues around undecidability all involve the lack of an upper bound. In a finite deterministic space, everything is decidable, although some things may be too costly computationally to decide.
There are several ways to go for decidability. The brute force way is computer arithmetic - there is no number larger than 2^64-1. That's how we get things done on computers, but proofs about numbers with finite upper bounds need lots of special cases. Mathematicians hate that.
I used to work on this sort of thing, using Boyer-Moore theory. That's a lot like the Peano axioms. There is (ZERO), and (ADD1 (ZERO)), and (ADD1 (ADD1 (ZERO))), etc. Everything is constructive and has an unambiguous representation in a LISP-like form. You can have recursive functions. But they must be proven to terminate, by having a nonnegative value which decreases on each recursive call. There is a distinction between "infinite" and "arbitrarily large". You can talk about arbitrarily large numbers, but you cannot get to 1/2 + 1/4 + 1/8 ... = 1. You can have integers and rational numbers of arbitrary size, but not reals.
Set theory was interesting. Rather than axiomatic set theory, I was using lists as sets, with the constraints that no value could be duplicated and the list must be ordered. Equality is strict - two things are equal only if the elements are all equal, compared element by element. It's possible to prove the usual axioms of set theory via this route. The ordered criterion requires proving things about ordered list insertion to get there. It's ugly and needs machine proofs.
I was doing this back in the early 1980s, when machine proofs were frowned upon. Mathematicians were still upset about the four-color theorem proof. It's all case analysis, with thousands of cases. That's more acceptable today.
Looked at in this light, infinity is a labor-saving device to eliminate special cases, at a potential cost in soundness.
> Looked at in this light, infinity is a labor-saving device to eliminate special cases, at a potential cost in soundness.
Or it is something that clearly conceptually exists, and makes simplistic reductionist viewpoints impossible to prove, which frustrates those who attempt to extend them into twisted metaphysical conjectures.
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?
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.
> 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/
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!
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.
> thinking that there’s anything that exists
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.
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!
> 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.
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.
"All models are wrong but some are useful." -- George Box
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_arithmet...
[2]: https://en.wikipedia.org/wiki/Internal_set_theory
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.
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.
From your point of view, infinity stops to matter once you cease to exist in reality. So there is something to consider
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
> Surprised Wildberger’s youtube channel wasnt in here.
Zeilberger is intellectually honest in a way that Wildberger is not.
Normally amps only go up to ten… but this one goes to eleven. …it’s one louder ain’t it!?!
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.
>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?
finite moments. cherish them.
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.
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.
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?
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.
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
Huh? I thought color confinement prevented this?
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).
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.
Yes, and that gets you to another number. Not infinity. You need an infinity of operations to create an infinity.
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.
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.
Not from “that half of something had a value”, but from “that half of any thing has a value”.
If you accept that every natural number has a successor which is a natural number, and no two natural numbers have the same successor, and that there’s no loops (e.g. by saying that there’s a total order on natural numbers and that any natural number is less than its successor), then there can’t be a finite collection which is all the natural numbers.
You could say “there’s no collection which has all the natural numbers”, which, ok, how do you want to talk about things true of all natural numbers then?
Formulating descriptions of physics without the axiom of infinity (or, without something to play the role of the real numbers) is super icky. You, in practice, can’t do any significant mathematical physics in an ultrafinitistic approach.
> how do you want to talk about things true of all natural numbers then
There's an entire branch of math for that: https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_...
I’m aware of constructive math. You still have the type of natural numbers in that?