TITLE: Does mathematics require axioms?
QUESTION [83 upvotes]: I just read this whole article:
http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf
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.

REPLY [4 votes]: Back to the original question: Does Mathematics Require Axioms?
The best answer I can think of is: Not at all - until they do.
In actual practice, working mathematicians go about developing new mathematics using tools of ordinary human thinking and speaking - they model abstract objects as pictures (in the head or on the board); they 'look at the objects' to 'see' what features they have; they use previous results in arguments very informally; they 'handwave' arguments; etc.  When they appeal to some property to justify an inference in chats with colleagues, they don't bother justifying the appeal as long as the colleague accepts it.  Even when they finally write up their results and have to be more technically precise, they still use a lot of informal language and almost never refer back to specific axioms for justification, because they expect their readers will know what they mean.
And it is a historical fact that all of the major mathematics that has been created was well-developed quite a bit before anyone felt a need to introduce axioms.  The creation and explosive develoment of Calculus/Analysis proceeded for over two hundred years before people felt a need to axiomatize it (or rather the Real Numbers on which it is based).  The basic results of Geometry were well known before Euclid wrote the Elements.  Heck, people were doing arithmetic, and later number theory, thousands of years before anyone thought to create axioms for the natural numbers.
Judging from history, it seems mathematicians resort to axiomatization in two circumstances: (a) They need to teach a subject to ordinary students rather than dedicated mathematicians, and the old 'handwaving' won't do; (b) The old 'handwaving' unexpectedly leads to contradictions or other bogus results.  Geometry is the prototype for (a) - Euclid was a teacher, and needed a textbook to organize the subject for his pupils.  Set Theory is a classic case of (b) - Cantor's own reasoning produced overt paradoxes, which were eliminated by axiomatization [Zermelo, Russell, etc].  Calculus was a combination of both - axiomatization began because mathematicians like Bolzano and Weirstrauss had to teach it to ordinary students, but found all the usual arguments to be both logical gibberish [infinitesimals??] and pedagogical disasters.