[eimrine]

Experimental mathematics is not a field of mathematics, but an approach. In the link you have given I found two approaches called experimental mathematics: proofing theorems by computer contrary to proofing with a pensil, and searching new problems contrary to solving some existing ones.

[yesenadam]

I was responding to your apparent claim that experimentation is not possible in mathematics. Maybe I misunderstood what you meant.

From Halmos' quote from the wiki page:

"Mathematics is not a deductive science—that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork. You want to find out what the facts are, and what you do is in that respect similar to what a laboratory technician does"

- it sure sounds like he thinks experiment/experimentation a key part of maths.

[Burner7799]

What evidence would convince you that mathematics is empirical?

If your argument was a tautology then what’s the point?

Would you consider interactive theorem proving to be empirical or not? The sort of experiment you perform is you try to see if the computer accepts your proof or not.

[sudosysgen]

Interactive theorem proving is not an experiment on nature because the theorem priver is itself a mathematical construct.

[ukj]

It sure seems like the goal posts are being moved.

Every experiment is an experiment of nature.

Humans are part of nature. Human constructs are part of nature.

Every interaction that results in the testing or falsification of some hypothesis is a valid experiment. Irrespective of the object being experimented/interacted with.

I am experimenting with (testing/falsifying hypotheses against) your linguistic constructs right now.

[sudosysgen]

I think you're missing the point. Mathematics isn't a natural science, and that's what I was trying to point out. There is a term for doing science on human constructs - it's called social science. Is your thesis that mathematics is a social science?

> Humans are part of nature. Human constructs are part of nature.

Humans physically are part of nature, sure. Human constructs are not considered to be as far as classification in the social and natural sciences.

[ukj]

I am not sure why you are using the phrase "natural science", is there any other kind? Lets revisit with the wikipedia definition:

  Science (from Latin scientia 'knowledge') is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe.

That seems pretty general and all-inclusive to me. Any knowledge about humans; or societies; or the stuff human societies invent is knowledge about the universe. Because we are part of the universe.

Now, you seem to be trying to differentiate the different sciences from one another - tell us why. What's the purpose of drawing a distinction between science and (what you call) "natural science"?

[sudosysgen]

I am not doing so. I am saying that if you want to classify mathematics as a science on the basis of the existence of theorem validation software on which someone may experiment, then it is a science on human constructs, ie, a social science, and not a natural science.

The argument here for math to be a science is very tortured and contrived, but if it has any even technical validity then it makes the argument for math as a social science.