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