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!
> 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.
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.
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.
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)
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
> 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.
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.
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.
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.
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.
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
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 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.
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.
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."
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.
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.
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.
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
> 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.
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
>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.”
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 :)
> 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.
54 comments
[ 3.0 ms ] story [ 66.0 ms ] thread[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!
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.
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 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.
Paradoxes comes from contradictions, a mathematical system that contains contradictions is a failed mathematical system.
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.
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)
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
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.
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.
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.
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.
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.
https://en.wikipedia.org/wiki/Axiom_of_determinacy
You can idealise it like many things in mathematics, but implementation details fail compared to the abstract ideals.
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
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?
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.
Also, all of the components of an atom, electrons and nucleus, have mass.
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.
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.
BTW, the article is really badly written.
Hn loves to dismiss him as a crank, which I think is overly harsh.
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
Zeilberger is intellectually honest in a way that Wildberger is not.
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.
> computers handle math just fine
strong disagree tbh
[0]: https://en.wikipedia.org/wiki/Projectively_extended_real_lin...
>“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?
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/
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.