Does mathematics require axioms?

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.

Answer

Is it true that with modern mathematics it is becoming less important for an axiom to be self-evident?

Yes and no.

Yes

in the sense that we now realize that all proofs, in the end, come down to the axioms and logical deduction rules that were assumed in writing the proof. For every statement, there are systems in which the statement is provable, including specifically the systems that assume the statement as an axiom. Thus no statement is “unprovable” in the broadest sense – it can only be unprovable relative to a specific set of axioms.

When we look at things in complete generality, in this way, there is no reason to think that the “axioms” for every system will be self-evident. There has been a parallel shift in the study of logic away from the traditional viewpoint that there should be a single “correct” logic, towards the modern viewpoint that there are multiple logics which, though incompatible, are each of interest in certain situations.

No

in the sense that mathematicians spend their time where it interests them, and few people are interested in studying systems which they feel have implausible or meaningless axioms. Thus some motivation is needed to interest others. The fact that an axiom seems self-evident is one form that motivation can take.

In the case of ZFC, there is a well-known argument that purports to show how the axioms are, in fact, self evident (with the exception of the axiom of replacement), by showing that the axioms all hold in a pre-formal conception of the cumulative hierarchy. This argument is presented, for example, in the article by Shoenfield in the Handbook of Mathematical Logic.

Another in-depth analysis of the state of axiomatics in contemporary foundations of mathematics is “Does Mathematics Need New Axioms?” by Solomon Feferman, Harvey M. Friedman, Penelope Maddy and John R. Steel, Bulletin of Symbolic Logic, 2000.

Attribution
Source : Link , Question Author : Kasper , Answer Author : Community

Leave a Comment