Mathematicians who are finitists.

[polymath257]

All uncomputable reals, which is most of them according to classical mathematics. We might argue that such things are not "constructions", in which case, swap "mathematical object" for "mathematical construction" in my previous posts.

My point is that Cantor, using a finite number of symbols, proved that some infinite sets (the Reals) are infinite and uncountable:

Wikipedia -- Cantor's diagonal argument

The rational numbers are, however, countable:

Wikipedia -- Countable set

And, so, what's the issue here?  Is finitism any different than Creationism?

For finitistic mathematics, such proofs are irrelevant because no infinite sets (like the set of positive integers or the set of reals) even exists in the system.

Yes finitism is different than creationism because it attempts to thrash out what can be known if no infinite sets are postulated to exist. This gives, for example, information about the independence of the axioms we use. It is also closely related to things like recursion theory, questions of computability, etc .

As such it is an acknowledged are of modern mathematics, even if it is a bit off the beaten track.

Creationism, however, isn't even a part of biology. It is pure superstition.

[A Toy Windmill]

Do finistists accept the existence of irrational numbers?

I don't think so, at least not in the sense that everyone else does. The classic proof that a square's diagonal is incommensurable with its side looks finitistic to me, but that's not what a modern mathematician means by "the square root of 2 exists."

Still, it's not too shameful if the finitist position on the nature of incommensurable magnitudes is on par with the pre-moderns.

Just curious. I know that you may be playing the DA here.

I'm not. I'm not a finitist, but I see value in research in finitistic mathematics.

Modern finitism was started by David Hilbert who, as far as mathematicians goes, wasn't exactly a wally. So I don't think it is fairly compared to creationism. Hilbert regarded Cantor's mathematics as a "paradise", and insisted we would not be expelled from it, but he was also aware that it contained serious contradictions, as have the many other attempts to naively treat of infinite sets and higher order logic. Some might say that this business is inherently precarious and we should aim to build our mathematics on much more secure footing. Some say we should start with a world that is logically finite, and if some finitist shows that huge swathes of existing mathematics can be done this way, that's an awesome thing to know.

[Jehanne]

The proof of the irrationality of the square root of 2 by contradiction is what I had in mind, anything but finitism in my opinion.

[A Toy Windmill]

The proof of the irrationality of the square root of 2 by contradiction is what I had in mind, anything but finitism in my opinion.

That proof looks straightforwardly finitistic to me. Where do you think it requires quantification over all the natural numbers?

[Jehanne]

The proof of the irrationality of the square root of 2 by contradiction is what I had in mind, anything but finitism in my opinion.

That proof looks straightforwardly finitistic to me. Where do you think it requires quantification over all the natural numbers?

[A Toy Windmill]

I'm familiar with the proof. It looks easy enough to formalize in PRA.

Where do you think it requires unbounded quantification over the naturals?

[Jehanne]

I'm familiar with the proof. It looks easy enough to formalize in PRA.

Where do you think it requires unbounded quantification over the naturals?

What's the alternative?  I suppose that you, as the DA, are trying to make a distinction between a potential infinite versus an actual one.  But, here's the Axiom of Infinity:



Wikipedia -- Axiom of Infinity

Now, if one rejects the above axiom, then, one, I suppose is a finitist.  But, consider the harmonic series:



It goes on forever, but, as can be proved, it is unbounded (goes to infinity), even though though the terms of the series (its limit) goes to zero.  The sum of a p-series, where p > 1



is bounded, even though it, like the harmonic series, goes on forever.

I am not sure how finitists deal with the above conundrums, but, of course, if you tire in your role as the DA here, I understand completely!

[A Toy Windmill]

What's the alternative?  I suppose that you, as the DA, are trying to make a distinction between a potential infinite versus an actual one.

I'm not playing Devil's Advocate. This is not a debate. Either the classical proof of the irrationality of the square root of 2 is a theorem of PRA or it isn't. I'm confident that it is, and the matter can be settled definitively by providing the formalized proof. It could even be checked by a machine.

But, here's the Axiom of Infinity:



Wikipedia -- Axiom of Infinity

Now, if one rejects the above axiom, then, one, I suppose is a finitist.

Finitism has stronger constraints. ZF set theory without the axiom of infinity is as expressive as Peano Arithmetic, which is strictly more expressive than primitive recursive arithmetic.

But, consider the harmonic series:

Why are we moving away from the proof that there is no rational square root of 2? I know there are classical theorems that are not provable by finitistic methods. But you seem to be under the mistaken impression that finitism is far weaker than it is. The article I linked early discusses a paper which claims that the majority of contemporary mathematics is formally provable by finitist methods.

I am not sure how finitists deal with the above conundrums, but, of course, if you tire in your role as the DA here, I understand completely!

Statements about convergence can be finitistic. Even classically, a convergence statement about a series that converges as n goes to infinity is formalized as:

for any ε > 0, I can produce N such that f(n) - l < ε for all n > N.

Many such statements can be proven finitistically. You just need a primitive recursive procedure that sends the ε to the N. The "for any" and "for all" can both be moved outermost, so the quantification here is one that a finitist accepts.

[LastPoet]

One thing to consider is that irrational numbers are not infinite. They are precise numbers, despite the need  to represent them as an infinite convergence.

The problem is that to represent them as we usually do, as decimals, binary, etc, we would need an infinite string of symbols. The square root of 2 is a number and quite finite.

And I don't consider finitists alike creationists. They might be wrong, but at least they can pull some algebra out. Creationists can only use yhe favourite holy book.

[Jehanne]

One thing to consider is that irrational numbers are not infinite. They are precise numbers, despite the need  to represent them as an infinite convergence.

The problem is that to represent them as we usually do, as decimals, binary, etc, we would need an infinite string of symbols. The square root of 2 is a number and quite finite.

And I don't consider finitists alike creationists. They might be wrong, but at least they can pull some algebra out. Creationists can only use yhe favourite holy book.

How about transcendental numbers, such as pi? Are those finite?