I just read this whole article:
which is also discussed over here: Infinite sets don't exist!?
However, the paragraph which I found most interesting is not really discussed there. I think this paragraph illustrates where most (read: close to all) mathematicians fundementally disagree with Professor NJ Wildberger. I must admit that I'm a first year student mathematics, and I really don't know close to enough to take sides here. Could somebody explain me here why his arguments are/aren't correct?
These edits are made after the answer from Asaf Karagila.
Edit $\;$ I've shortened the quote a bit, I hope this question can be reopened ! The full paragraph can be read at the link above.
Edit $\;$ I've listed the quotes from his article, I find most intresting:
- The job [of a pure mathematician] is to investigate the mathematical reality of the world in which we live.
- To Euclid, an Axiom was a fact that was sufficiently obvious to not require a proof.
And from a discussion with the author on the internet:
You are sharing with us the common modern assumption that mathematics is built up from "axioms". It is not a position that Newton, Euler or Gauss would have had a lot of sympathy with, in my opinion. In this course we will slowly come to appreciate that clear and careful definitions are a much preferable beginning to the study of mathematics.
Which leads me to the following question: Is it true that with modern mathematics it is becoming less important for an axiom to be self-evident? It sounds to me that ancient mathematics was much more closely related to physics then it is today. Is this true ?
Does mathematics require axioms?
Mathematics does not require "Axioms". The job of a pure mathematician is not to build some elaborate castle in the sky, and to proclaim that it stands up on the strength of some arbitrarily chosen assumptions. The job is to investigate the mathematical reality of the world in which we live. For this, no assumptions are necessary. Careful observation is necessary, clear definitions are necessary, and correct use of language and logic are necessary. But at no point does one need to start invoking the existence of objects or procedures that we cannot see, specify, or implement.
People use the term "Axiom" when often they really mean definition. Thus the "axioms" of group theory are in fact just definitions. We say exactly what we mean by a group, that's all. There are no assumptions anywhere. At no point do we or should we say, "Now that we have defined an abstract group, let's assume they exist".
Euclid may have called certain of his initial statements Axioms, but he had something else in mind. Euclid had a lot of geometrical facts which he wanted to organize as best as he could into a logical framework. Many decisions had to be made as to a convenient order of presentation. He rightfully decided that simpler and more basic facts should appear before complicated and difficult ones. So he contrived to organize things in a linear way, with most Propositions following from previous ones by logical reasoning alone, with the exception of certain initial statements that were taken to be self-evident. To Euclid, an Axiom was a fact that was sufficiently obvious to not require a proof. This is a quite different meaning to the use of the term today. Those formalists who claim that they are following in Euclid's illustrious footsteps by casting mathematics as a game played with symbols which are not given meaning are misrepresenting the situation.
And yes, all right, the Continuum hypothesis doesn't really need to be true or false, but is allowed to hover in some no-man's land, falling one way or the other depending on what you believe. Cohen's proof of the independence of the Continuum hypothesis from the "Axioms" should have been the long overdue wake-up call.
Whenever discussions about the foundations of mathematics arise, we pay lip service to the "Axioms" of Zermelo-Fraenkel, but do we ever use them? Hardly ever. With the notable exception of the "Axiom of Choice", I bet that fewer than 5% of mathematicians have ever employed even one of these "Axioms" explicitly in their published work. The average mathematician probably can't even remember the "Axioms". I think I am typical-in two weeks time I'll have retired them to their usual spot in some distant ballpark of my memory, mostly beyond recall.
In practise, working mathematicians are quite aware of the lurking contradictions with "infinite set theory". We have learnt to keep the demons at bay, not by relying on "Axioms" but rather by developing conventions and intuition that allow us to seemingly avoid the most obvious traps. Whenever it smells like there may be an "infinite set" around that is problematic, we quickly use the term "class". For example: A topology is an "equivalence class of atlases". Of course most of us could not spell out exactly what does and what does not constitute a "class", and we learn to not bring up such questions in company.