Use of infinity in mathematics from the constructivists?

[disregardthat]

How are you, if you are, responding to the critique about the use of infinity in mathematics from the constructivists?

 

[JSuarez]

First, there are no such thing as "constructivists"; there are a lot of variants of constructivism that differ widely in their assumptions.

Second, most forms of constructivism do not reject infinity altogether (to my knowledge, only the most extreme finitists, reject any form of infinity); what some constructivists object to is some uses of actual infinity (complete, infinite entities; some accept them up to enumerability, some go higher); but practically all accept infinite sets up to the enumerable ones.

Third: personally, I don't respond; I'm a Platonist.

 

[disregardthat]

First, there are no such thing as "constructivists"; there are a lot of variants of constructivism that differ widely in their assumptions.

Second, most forms of constructivism do not reject infinity altogether (to my knowledge, only the most extreme finitists, reject any form of infinity); what some constructivists object to is some uses of actual infinity (complete, infinite entities; some accept them up to enumerability, some go higher); but practically all accept infinite sets up to the enumerable ones.

Third: personally, I don't respond; I'm a Platonist.

Any constructivist is a constructivist however, and they all have in common their rejection of the use of actual infinity. I believe potential infinity is allowed in most directions.

My question is still: how do you defend the use of the actual infinity (if you can't find a logical (syntactical) isomorphism in finitistic means), like the set of real numbers, in mathematics? Being a platonist doesn't skip this problem.

For example: is the statement A: "There exist an integer n in $$\mathbb{N}$$ such that property P is satisfied" a logically sound statement if you are unable to create the algorithm to produce this integer? What would be the syntactical equivalence? Surely, the (possibly isomorphic to A) statement B: "(1 satisfies P) or (2 satisfies P) or (3 satisifies P) or... etc" is not a statement that can be stated in its entirety in finistic means, thus B must be rejected as a statement altogether (as the the argument goes..).

Suppose it was proven that there does not exist an algorithm which can produce n. Could A possibly be meaningful?

Such errors (?) is what makes the notion of an actual infinite set troublesome.

 

[Dragonfall]

Third: personally, I don't respond; I'm a Platonist.

Platonism is so... 350 BC.

 

[disregardthat]

I hope to get a response on this, so I'm bumping it.

 

[JSuarez]

Apologies for the late response, but I fell prey to deadlines.

For example: is the statement A: "There exist an integer n in such that property P is satisfied" a logically sound statement if you are unable to create the algorithm to produce this integer?

I think you want to ask if this statement is meaningful, even if you are unable to produce a witness for the quantifier ("logically sound" has a different technical meaning). Given this, my answer is yes, if the statement was deduced by correct application of classical logical principles, in which I include proofs that consider the natural number set as a completed, actual, totality. Reducing matters of truth to constructive proof is, in my view, too restricted and, more importantly, unjustified. Nevertheless, I'll happily concede that constructive proofs yield, in general, more information.

What would be the syntactical equivalence?

Here, I'm afraid I don't understand what you mean. Syntax is the set of rules that specify the well-formed expressions in a language; the above natural language statement can be translated in, for example, first-order logic, by:

$$\exists nP\left(n\right)$$

Surely, the (possibly isomorphic to A) statement B: "(1 satisfies P) or (2 satisfies P) or (3 satisifies P) or... etc" is not a statement that can be stated in its entirety in finistic means, thus B must be rejected as a statement altogether (as the the argument goes..).

No, but it can be stated meaningfully in infinitary logic and, as I am yet to see a convincing argument that non-finitistic methods are harmful, I have no problem with it.

Suppose it was proven that there does not exist an algorithm which can produce n. Could A possibly be meaningful?

As far as I am concerned, yes. Look, you can't even name (or describe by finitistic means) an non-denumerable infinity of real numbers; but they work just fine, and I'm not going to stop using them because of merely philosophical position, given Philosophy's track record of stability (and even sanity). I do not concede that there is a "first philosophy"; my personal stance is naturalistic and I'm all for the Indispensability Arguments: if it works in the actual world, then philosophy must justify that first, and propose alternatives (especially more restrictive ones) later.

My question is still: how do you defend the use of the actual infinity (if you can't find a logical (syntactical) isomorphism in finitistic means), like the set of real numbers, in mathematics? Being a platonist doesn't skip this problem.

As I said, it's not up to me, nor I feel the need, to defend infinitistic methods; they yield valuable results. It's up to the finitists to defend why we should abandon them.

Platonism is so... 350 BC.

Yup, but it's still alive and kicking; many new & shiny things are not.

 

[disregardthat]

Constructivism is not necessarily finitism, and I'm not referring to finitism specifically.

No, but it can be stated meaningfully in infinitary logic and, as I am yet to see a convincing argument that non-finitistic methods are harmful, I have no problem with it.

That depends on the definition of "harmful". The axiom of choice leads to a wide array of non-intuitive results which may be regarded as "harmful", at least it wreaks havoc with intuition. Basically, the axiom of choice implies the existence of some inherently non-constructive mathematical objects. This contrast with intuition can hardly be recognized as "working" in the actual world.

 

[JSuarez]

Intuition is, at best, a psychological construct that may serve as an aid in science, but it's not something that can dictate what is scientifically valid or not. The strong form of the axiom of choice (there are several forms) has, indeed, non-intuitive consequences but they are also valid non-intuitive consequences (that is, they correctly follow from it). That's all there is to it; if intuition doesn't like them, too bad for it.

Most scientific theories have non-intuitive consequences; I will not even go as far as Quantum mechanics, because classical physical theories also have them, but physicists will hardly let go of them for that.

 

[disregardthat]

Intuition is, at best, a psychological construct that may serve as an aid in science, but it's not something that can dictate what is scientifically valid or not. The strong form of the axiom of choice (there are several forms) has, indeed, non-intuitive consequences but they are also valid non-intuitive consequences (that is, they correctly follow from it). That's all there is to it; if intuition doesn't like them, too bad for it.

Yes, I agree, but you provided the argument that since the results "works" in the real world we should not deny the axiom. Since it clearly opposes our notions, why would it serve as a foundation for something which we consider as an extension of human logic? Validity is in any case a priori provided whenever we treat it as an axiom, however.

 

[Hurkyl]

As I understand it, in many forms of constructive mathematics, the axiom of choice is a theorem. The intuitive point being that it's easy to well-order "constructions".


The claim that there exist things like real numbers that you cannot write down is somewhat sketchy -- it's confusing internal and external. See Skolem's paradox for a related example.

 

[CRGreathouse]

Any constructivist is a constructivist however, and they all have in common their rejection of the use of actual infinity. I believe potential infinity is allowed in most directions.

This is quite wrong. I would say that almost all constructivists accept (actual, completed) infinities. Even an extreme finitist might accept the existence of ω = {0, 1, 2, ...}; this is a theorem of Zermelo set theory, minus the axiom of infinity, plus the negation of the axiom of infinity!
http://us.metamath.org/mpeuni/omon.html

More typical constructivists would accept the axiom of infinity and thus the set-hood of ω, while rejecting all forms of choice (beyond those provable in ZF, of course).

Less strict constructivists might accept weakened forms of choice as well as the axiom of infinity...

Extreme constructivists, of course, simply accept V = L and have AC and GCH for free. :PI think the term you're looking for isn't "constructivist" nor even "finitist", but "ultrafinitist".

 

[JSuarez]

Yes, I agree, but you provided the argument that since the results "works" in the real world we should not deny the axiom. Since it clearly opposes our notions, why would it serve as a foundation for something which we consider as an extension of human logic?

As long as any axiom gives us interesting results, and does not lead to an obvious inconsistency (it may well lead to a non-obvious one, but as almost all consistency proofs must be relative to the consistency of more powerful and less understood theories), I'll use it; and not only the usual suspect AC, CH, etc. I'm convinced that we will have to accept far less intuitive axioms, like the large cardinal ones, particularly if some ongoing research projects, like Woodin's $$\Omega$$-Logic are succseful.
I am not saying that we should appeal to it in situations where it's not needed, or even use in its full force when weaker and more intuitive versions will do, but if it is needed for an interesting result then, by all means, use it. I don't see it as a "foundation" as well; usually, people looking too hard for foundations are also looking for absolute certainties, and I don't think that's a feasible (or even useful) goal. Personally, I stick to Stewart Shapiro's position of "Foundationalism without Foundations", meaning that foundational questions are interesting and worth pursuing, but skeptical about any definitive foundations.

As I understand it, in many forms of constructive mathematics, the axiom of choice is a theorem. The intuitive point being that it's easy to well-order "constructions".

Up to enumerable constructions, yes. And many of them accept the weaker forms, like enumerable choice and dependent choice.

The claim that there exist things like real numbers that you cannot write down is somewhat sketchy -- it's confusing internal and external. See Skolem's paradox for a related example.

I'm not following you here. I didn't intend to present it as an argument. I was simply stating that we cannot describe most real numbers by a finite sentence. I was not even thinking in a context where the Lowenheim-Skolem theorems are applicable: I was referring to the second-order formalization of the (ordered and complete) real field.

This is quite wrong. I would say that almost all constructivists accept (actual, completed) infinities. Even an extreme finitist might accept the existence of ω = {0, 1, 2, ...}; this is a theorem of Zermelo set theory, minus the axiom of infinity, plus the negation of the axiom of infinity!
http://us.metamath.org/mpeuni/omon.html

You are right, and I said it in an earlier post: only the most extreme finitists reject enumerable sets.

 

[CRGreathouse]

You are right, and I said it in an earlier post: only the most extreme finitists reject enumerable sets.

I missed your earlier post.

 

[disregardthat]

Great replies!

This is quite wrong. I would say that almost all constructivists accept (actual, completed) infinities.

I think the term you're looking for isn't "constructivist" nor even "finitist", but "ultrafinitist".

How would you define an actual and completed infinity in that case?

 

[CRGreathouse]

How would you define an actual and completed infinity in that case?

$$\aleph_0$$, $$\aleph_\omega$$, $$\beth_2$$, etc. on the cardinal side; $$\omega$$, $$\epsilon_0$$, etc. on the ordinal side.

 

[JSuarez]

(A supplement to CRGreathouse post)

How would you define an actual and completed infinity in that case?

The discussion has, so far, revolved around the Axiom of Choice but, as far as I know, very few people questions the Axiom of Infinity, which is also an existencial statement, that explicitly introduces an (actual) infinite set in the hierarchy. More, without this axiom, it's impossible to have infinite sets.

 

[CRGreathouse]

The discussion has, so far, revolved around the Axiom of Choice but, as far as I know, very few people questions the Axiom of Infinity, which is also an existencial statement, that explicitly introduces an (actual) infinite set in the hierarchy. More, without this axiom, it's impossible to have infinite sets.

I agree -- the Axiom of Infinity is well-accepted. I, for example, have certain finitist/constructivist tendencies but accept the axiom.

There are toy model of set theory -- most famously Boolos' General Set Theory -- that do not include the Axiom of Infinity or equivalent, but essentially every set theorist assumes it; non-set theorists don't even consider the possibility of leaving it off.