barbarianhorde wrote:It is not a mathematical quibble but a philosophical one which means it could end up telling mathematics what to do, after all philosophy still trumps mathematics by being the arbiter of what constitutes true correspondence.

I don't necessarily agree, but I will stipulate for sake of discussion that philosophy trumps math. BUT philosophy has perfectly well accepted infinitary math. Wasn't Russell a philosopher? The philosopher of math Penelope Maddy has written

Believing the Axioms (parts I and II. I linked to part I). She walks through each axiom of ZFC (Zermelo-Fraenkel set theory with Choice), places it in historical and philosophical context, and describes the principles by which we accept it today.

So it is NOT true that philosophy says one thing and math another. Some philosophers argue for finitary math, I suppose, just as some mathematicians study finitary math. It's interesting to study! But mainstream math accepts infinity and the mainstream philosophers of math do too.

So I don't agree with your conclusion even if I accept your premise. There's no dispute between philosophy and math when it comes to the mathematical infinite.

barbarianhorde wrote:So what is actually in play here is what a "set" means, not what infinity means.

Yes that is very true and also insightful. In elementary education we tell people that a set is a "collection of objects." But of course that was Frege's idea, demolished by Russell. A set is, in fact, a highly technical gadget with

no definition at all. A set, in mathematics, is any object that obeys the axioms of set theory. And what axioms are those?

Any axioms you like, subject only to consistency and interestingness. And we don't even know for sure if our axiom systems ARE consistent.

If anyone wants to argue that mathematics is based on a pile of sand, you will get no argument from me. It's the job of philosophy to explain why all this obvious

nonsense is so damn

useful.

barbarianhorde wrote:I can now see how Serendipper considers this as strictly speaking an unwarranted shortcut. But I don't contest that it is useful. Shortcuts are very often useful, look at the Panama Canal.

Great example! And I don't disagree with Serendipper on this point either. There is no logical or moral reason why we should prefer one assumption over another, when it comes to allowing the actual infinite into math. All we have is a centuries of experience that when it comes to understanding the physical world, mathematics is

indispensable. SEP has an article on this

indispensability argument.

barbarianhorde wrote:I personally don't deny the mathematical set. I just deny the philosophical set.

Ok. But will you stipulate at least that most philosophers accept modern math? The axiom of infinity dates back to Frege and Russell and Zermelo and all those other ancients.

barbarianhorde wrote: I mean I deny that this set of infinitely many numbers has any meaning outside of how the set is being made useful.

Ah. Ok. Meaning outside of utility. Well, tell me this. A Martian physicist comes to earth and sees a traffic light. She can tell red light from green light by the wavelength. But she cannot tell you which is stop and which is go. That's a socially constructed fact that has

meaning only because it's useful. We could make green mean stop and red mean go, and that would be just as valid a choice. There's no inherent meaning in the colors.

So I would say that it's true there's no "meaning" to set theory outside of how we use it. But so what? Most of reality is that way. Civilization is one abstraction piled on another. None of it has any meaning outside of how we as humans use it. Your criticism of set theory is a criticism of the foundation of civilization: namely, the human power of abstraction.

Our ability to make the abstract real.

barbarianhorde wrote:I know that. So the question is how far we want to allow mathematics to operate in defiance of physics.

But it's not. A lot of math comes directly from physics. Physics finds modern infinitary math indispensable. Even though the universe might be discrete, the math used by the physicists is infinitary. That may be a puzzle; but it is also a FACT.

Your beef is with the physicists, not the mathematicians! The mathematicians invented this crazy non-Euclidean geometry, but it was the physicists who decided it was the best way to understand the world. I hope you see my point!

barbarianhorde wrote:Because philosophy is about reliability and not about speed. The power gained from seeing sets as potentially having infinite size may come with a drawback of making it doubtful if sets can be trusted, if they can still logically correspond to another set.

Well maybe set theory can't be trusted. Make your case. What does that mean? What if it can't?

barbarianhorde wrote:I guess what I mean is all questions like, how does the set of integers correspond to the set of real numbers?

The integers are a proper subset of the reals. Additionally, the reals can be set-theoretically constructed from the integers. That is, if all we had was the integers, we would first create the rationals as certain equivalence classes of integers; then we'd create the reals as certain subsets of the rationals.

barbarianhorde wrote: Does the fact that the second is infinity squared make the former into the root of infinity?

I would not say the reals are infinity squared and in fact that's wrong. What is true is that the cardinality of the reals is the same as the cardinality of the set of subsets of the integers. Is that what you meant?

barbarianhorde wrote:If this can't be addressed there is a logical problem with the infinite set, even if it can still do mathematical work, creating a subprime mathematics bubble.

I hope I addressed it. There is no sense in which "the reals are infinity squared" is meaningful. The reals are in fact essentially the same set as the collection of subsets of the integers. You can encode each as the other.

barbarianhorde wrote:

I mean in the sense that it identifies infinity. Thats where philosophy begins to not be bored.

All the philosophers of math I know accept mathematical infinity. I must be reading the wrong philosophers. What does "believe in" mean? Just that we accept it for being useful; and we rely on experience that the history of math is the history of weird stuff that someone realized was actually useful. Negative numbers, complex numbers, irrational numbers, non-Euclidean geometry. So ... what is the meaning of the mathematics of infinity? Perhaps we'll know in a hundred years.