Is mathematics a science?

This is being much discussed in Which came first: the young earth, or the inerrant scripture?, where it is way off topic. I suggest a new thread, perhaps in Is It Science? for continuing the discussion of mathematics.I will take the position that mathematics is not a science, since the word "science" has come to mean an empirical study.The role of mathematics within science can be explored in this thread. That would allow addressing the question of whether mathematics is a language, as suggested in Message 108.

I would say you are about right there.There are times when mathematicians are thinking a bit like scientists, doing experimentation (with mathematical objects). So it isn't all deduction from axioms. However, what is observed from experimentation is never considered sufficient to constitute a proof. For that one needs deduction.

Several people have said that math is a language. But I think mathematicians might disagree with that.Of course, mathematics includes its own notational language. But mathematics is very much about methods and procedures, and some of those methods do prove useful in science.

There is a lot more to mathematics than symbol manipulation. Many of the most important mathematical research papers consist mainly of informal prose with relatively little use of symbols. Roger Penrose wrote two books "The Emperor's New Mind" (1989) and "Shadows of the Mind" (1993), in which he claims to disprove the possibility of AI. The basis of his argument is that computers cannot do math.The general view is that Penrose's arguments don't work. The critics point to flaws in his claimed proof. However, I think it fair to say that nobody has proved that computers can do math.My point -- there is a large difference between doing computation, and doing mathematics. Your claim that math is a language is far from proven.

I disagree with that. The more usual statement is that mathematical proofs can be translated into formal proofs. However mathematics papers can contain a lot more than proofs. It is a widely held view that Goedel's incompleteness theorem showed that this goal was (and is) unachievable. An online review of Goedel's theorem can be found here. Quine argued against that view in his highly respected paper "Truth by Convention".

My main area happens to be functional analysis. That's correct. But many people see it as showing that there are mathematical statements which cannot be proved from the axioms, but which we can see as being true (and prove true by means of a meta-theory).I'm not quite sure I agree with that conclusion, but it is what mathematical platonists tend to conclude. Sorry about that. It was the best link I managed to find, but it wasn't a very good one. Quine's paper is, quite explicitely, about mathematical truth. The link tries to describe it in a broader context, and that tends to confuse the issue. Incidently, the paper is probably not worth reading unless you are into the philosophy of mathematics. But why should we start with Zermelo-Frankel? Surely those are not encoded in our genes.From my perspective, the development of axiom systems such as ZF is an important part of mathematics. That is where much of the creativity and inventiveness is needed. By looking at only proofs derived from axioms, and ignoring the origin of the axiom systems, you omit some of what I consider to be the most important parts of mathematics. You cannot get from ZFC to, say, the Hahn-Banach theorem (of functional analysis) merely by tedious applications of logical operations. You first have to introduce new concepts, new definitions, new axioms (such as the defining axioms of a Banach space). And introducing such new concepts is, in my opinion, an important part of mathematics.

Topological groups and function spaces over such groups. However, I've been mainly doing computer science for quite a while now.

Ph.D. was related to fixed point properies on topological groups.

Sure, I agree with you. But you cannot do it "through relatively simple, yet very tedious, applications of logical operations" as was suggested in Message 18. For example, you would first need to introduce the notions from point set topology. That doesn't look too hard when you have at your disposal a rich natural language, but how are you going to do that when all you have is first order predicate calculus and the axioms of ZFC?

That's a strange way of putting it.No, I'm not saying that. But I am saying that microevolution cannot lead to macroevolution unless there is selection.

Selection is part of the mechanism, not part of the definition.Presumably one could have micro-evolution as a result of neutral drift (as in the neutral theory).

Sure. It's artificial selection. Not being a platonist, I would agree.I suspect platonists would also agree. I guess they might say that the proof exists in some platonist sense, but needs to be discovered.