To me, ultrafinitism is an intellectual curiosity. Proofs which depend upon the axiom of choice, or a notion of infinity, do not always correspond to algorithms which terminate in finite time. Such results are "useless" to a computational mathematician. The goal of determining how much of mathematics can translate to an finite universe is, in my opinion, a noble one.
A hill I would die on, for sport: "the logarithm of the biggest number I see reason to believe in is the combined size of my RAM and HDD."
The key word being "believe." Mathematicians do not tend to operate on faith. Proofs invoking unrealistic axioms are perfectly acceptable, valid proofs. One needn't "believe" anything.
- Infinitism, people who believe in the existence of completed infinite sets, and transfinite numbers more generally.
- Finitism, people who believe that "infinite" means "arbitrarily large" or "unbounded", but nothing more.
- Ultrafinitism, people who think every quantity has an upper bound due to mathematics not being separate from the physical world.
Zeilberger identifies as an ultrafinitist. I personally think finitism is the most plausible one. It accepts an intuitive notion of potential infinity, but it avoids basically all the mathematical paradoxes of infinity and the philosophically extravagant mathematics of transfinite set theory.
So, um, does that mean there's a "biggest number"? One you can't +1?
My sense is that infinitary axioms like the existence of N, successor, induction, etc., basically allow mathematicians to write shorter and more straightforward proofs. They are likely to be sound until the cows come home, and moreover they represent what mathematics is essentially all about - finding some shortcut through the calculations we need to make, understanding what part of some structure or procedure is irrelevant.
And that probably makes as much sense (if not more) as the alternative for all I know. There is a way to think of mathematics as corresponding to computation in a way where anything that cannot be computed cannot be proven. If you cannot "construct" or "compute" infinity, then you cannot say that it exists.
I think what I am describing corresponds to the "constructivist" view of mathematics, but tbh I don't know enough about this to say whether the framework allows for infinities.
In a computer, this text is stored in binary format. That's a base2 number. Arbitrary concepts can be encoded into a form that can then be represented and worked with by mathematical operations.
If you want to get really cafe philosophical about it, the universe can be broken down into its smallest elements, quantified, and represented by numerical values. And thus the universe is a number. Of course, there's all sort of "gotchas" and "whatifs" and this is a wildly over-simplified version.
But even with that arbitrary-size integer arithmetic, your mathematical universe is still bounded by your computational environment's resources.
-Do you believe in 1?
-Yes.
-Do you believe in 2?
-Yes.
-Do you believe in 4?
-[slight pause] Yes.
-Do you believe in 8?
-[slightly longer pause] Yes.
Eventually you realize each question takes him twice as long to answer as the last.
And poorly-founded, I would add. Cantor's diagonal proof is, quite literally, question begging. Sure, you can construct a number not in the infinite set of natural numbers via that diagonalization method, but only if you're at infinity, or if you make very liberal use of "..." -- otherwise, you cannot.
And Skolem was quick to discredit Cantor's extravagances, on grounds that tend to support finitism generally.
Cantor's diagonalization argument doesn't rely on the contents of an infinite list of numbers, or on numbers themselves. You don't need to know anything about what "..." represents and you don't need to be "at" anywhere. There is no need to compute anything with concrete numbers.
The only thing the argument relies upon is whether the fundamental structure of numbers allows for defining a bijection. The proof/computation is an abstract relation, not a concrete one with actual number objects.
That there is no bijection must be the conclusion, not an assumption, otherwise the argument would simply be question begging. If you can't define a new diagonal number in the first place, you don't have an argument why you can't have a bijection.
With finitism it doesn't seem like you can define a diagonal number, because then Cantor's infinite table is merely potentially infinite in width and height. For the diagonal to be finished, the table has to be at least as wide as it is tall. But if you list numbers by their lexicographic order, the list grows much faster in height than in width. For base 10, for n columns (digits), you get 10^n rows (numbers). Then you can't ever finish a diagonal that isn't on the list. The number of digits doesn't grow fast enough to cover the entire list. You don't get the diagonal in the limit.
Now if you assume infinitism, you can simply declare that the height and width of the list is the same (or even that the width is larger), and therefore the diagonal can be constructed. But that's not a convincing assumption for a finitist (or ultrafinitist).
But those numbers that require the totality of my computer's N bits of storage to represent? I can, without reservation, believe that every integer between zero and 2^N-1 exists. But incrementing that biggest number will be challenging. I hear you scoff; surely there are bigger numbers, you can just buy more hard drive. And I can! But there are only finitely many hard drives available for purchase, many more than I can afford, so I'd rather save my money and believe in less.
IMO they're neither "faith" nor "hypotheses", although "whether a set of axioms is self-consistent" is probably a hypothesis.
Axioms are just building blocks. They don't even need to be "true" in any real sense. They're just arbitrary determinations or rules that we make up, and then we figure out what the consequences of these choices are.
In practice we often try to come up with axioms that have some practical value in the real world, but this is not necessarily or always how we go about doing maths.
Fun fact: For certain kinds of results, there is an explicit procedure for transforming any given ZFC-proof into an IZF-proof. By ZFC, I mean classical "uncomputable" mathematics with the axiom of choice and with the law of excluded middle. By IZF, I mean (a specific flavor of) constructive mathematics without those two contested axioms.
In particular, this metatheorem applies to termination results. So feel free to use the axiom of choice and the law of excluded middle to your heart's content when proving that a program terminates; you can trust that at the end of the day, your proof can be transformed to a constructive proof.
Slides which I hope are also accessible to non-specialists in proof theory: https://www.speicherleck.de/iblech/stuff/37c3-axiom-of-choic...
More pointers to the literature: https://math.stanford.edu/~feferman/papers/reductive.pdf
"Everyone believes that the Empire State Building is real, because it is possible for almost anyone to go and see it for himself. By the same token, anyone who takes the trouble to learn some mathematics can “see” the set of natural numbers for himself. So, Gödel reasoned, it must be that the set of natural numbers has an independent existence, an existence as a certain abstract possibility of thought.
I asked him how best to perceive pure abstract possibility. He said three things, i) First one must close off the other senses, for instance, by lying down in a quiet place. It is not enough, however, to perform this negative action, one must actively seek with the mind, ii) It is a mistake to let everyday reality condition possibility, and only to imagine the combinings and permutations of physical objects — the mind is capable of directly perceiving infinite sets, iii) The ultimate goal of such thought, and of all philosophy, is the perception of the Absolute. Gödel rounded off these comments with a remark on Plato: “When Plautus could fully perceive the Good, his philosophy ended.”"
[https://medium.com/@rudyrucker/memories-of-kurt-godel-84f66e...]
Also far from mainstream, but in an opposite direction
Maybe Zeilberger would respond that that machine might need to use unbounded amounts of memory. If you fix a bound on memory, i.e. take infinity out of the question, maybe you can't construct such machines.
A Turing machine with a tape of only constant length is equivalent to a finite-state automaton. If the length of the tape is a linear function of the length of the input, on the other hand, the machine is a linear bounded automaton.
https://en.wikipedia.org/wiki/Linear_bounded_automaton
https://en.wikipedia.org/wiki/Finite-state_machine
https://en.wikipedia.org/wiki/Template:Formal_languages_and_...
So if "nonsense" breeds meaning, is it really nonsensical?
[1]: The birth of complexity theory is attributed to Hartmanis and Stearns's 1965 "On the Computational Complexity of Algorithms, which very directly builds on Turing's work: https://www.ams.org/journals/tran/1965-117-00/S0002-9947-196...
How bad would the analogy be to a calculation with imaginary numbers that still yields a real result?
Would you say that the rationals (equivalence classes of ordered pairs of integers) are "less real" than the integers?
[1]: https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathe...
> But just like Gödel, [Turing] missed the point! Neither of them proved that "there exist true yet unprovable statements". Rubbish! Every meaningful statement is either provable (if it is true) or disprovable (if it is false). What they did (meta-)prove, by a Reductio Ad Absurdum clever argument, is that many statements that were believed to be meaningful, are really utterly devoid of meaning. As I have already preached in this sermon, the problem is the fictional (and pernicious!) infinity. In particular, Turing thesis's is utter nonsense, talking about "oracles", that lead to lots of beautiful, but fictional and irrelevant work by logicians and theoretical computer scientists.