> Mathematics in the broadest sense is nothing but logic applied to axioms.
mmmm. Lots of people would disagree with this :-)
> Until computers came along, math was entirely a theoretical pursuit
Not at all; mathematics always had applications. Not all mathematics, but it's certainly not that case that math was "all theory" and without serious application before computing.
> because for the first time we can take a mathematical idea from inside our heads and express it in a physical computer which executes it according to the logic we've defined.
We've had calculation assistants in various forms for centuries. Also, ballistics and construction based on mathematics are more-or-less "taking a mathematical idea and expressing it in the physical world".
Well, "computer" originally referred to a person performing the task of computation. In that sense, "Until computers came along, math was entirely a theoretical pursuit" may be accurate - it is hard to make math practical without computing anything. It's certainly not the case that it was purely theoretical until we had mechanical or electronic computers.
I disagree. The term "computer" was typically reserved for people who exclusively did often pre-defined computations, and didn't contribute much other than the computation to the mathematics at hand. That is, someone else had come up with the novel method of computation, and someone else had formulated the problem. The human computer was applying the computation to problem that someone else handed them.
There was plenty of mathematics before mathematics got to the point where we needed dedicated humans who spend their whole day computing things in a prescribed manner.
But regardless, for some reason I think the original post was referring to electrical/mechanical computers :)
I think it's the same question as whether someone cleaning their own toilet is "a cleaner" for the duration, on which I could go either way.
"There was plenty of mathematics before mathematics got to the point where we needed dedicated humans who spend their whole day computing things in a prescribed manner."
More to the point, there was plenty of useful application of mathematics before then. Which I certainly agree with. My point was that most (and possibly all?) early application of mathematics required computation.
My comment, though, was mostly agreeing with you - just picking apart a technicality to get at some tangential interesting questions.
So, I tried to think of counter-examples to "most (and possibly all?) early application of mathematics required computation" just for the sake of discussion.
I think I have only one, which is the establishment of axioms, both philosophically (as a method) and specifically (e.g. in Elements).
A revisionist history might say that choosing axioms doesn't require any computation, just a keen sense of style and close observation of the world.
But actually, I'm sure that the choice of axioms was a long and drawn-out process informed mostly by computation and checks that the computed values/proven theorems matched with physical intuition. After all, that's kind-of how it's done today, even by people who have lots of experience with formal systems.
Now I really want to read pre-Euclidean mathematical philosophy to see if I'm correct :-)
Yeah, I thought it was interesting space for speculation. 'S why I tried nodding toward it. Don't really know enough of the history to get terribly concrete, but that roughly corresponds to my understanding. Both the actual history and what might be theoretically possible are fascinating.
mmmm. Lots of people would disagree with this :-)
> Until computers came along, math was entirely a theoretical pursuit
Not at all; mathematics always had applications. Not all mathematics, but it's certainly not that case that math was "all theory" and without serious application before computing.
> because for the first time we can take a mathematical idea from inside our heads and express it in a physical computer which executes it according to the logic we've defined.
We've had calculation assistants in various forms for centuries. Also, ballistics and construction based on mathematics are more-or-less "taking a mathematical idea and expressing it in the physical world".