Digestion and Absorption scroll down to see diagrams An Analogy Suppose you are interested in purchasing a Pizza store and wish to investigate how productive the store is without the present owner knowing because, you fear the owner will raise the price. So, instead of going into the store and watching what happens and asking to examine the books that record expenses and profits, you decide to watch the store from outside.
We may take advantage of libraries in programming but even then they may not work on all of the physical platformsbut it's not that well defined, what is "the fundamental language" of mathematics; at least not as of today.
Although I find it cool that tools like Coq are getting increasingly popular, so we will eventually get there.
It forced me to pay attention and as a bonus I have a consistent set of notes for most undergraduate mathematics: Seemed like deaf people missed a good opportunity, but then I realized that how we talk and think is shaped very much by our language.
Similarly scholars from different backgrounds have different ways of expressing a precise mathematical concept. This one paper I read implicitly treated complex scalars as real vectors, took me some time to clear up the confusion. The glib answer is that math is a superset of programming and there are non-computable arithmetic functions, so programming is not rich enough to describe what you need to describe when you are talking about math.
I think a more intuitive answer is that in math you are generally talking about relations between infinite sets of objects, whereas in programming you are living in a in practice finite space and are evaluating everything into an integer.
Programming is about calculating integers, mathematics is about proving theorems. The language has to be very different because what is crucial about evaluating integers unambiguously is peripheral to proving theorems and vice versa.
You can ask, why is it so painful to give a formal proof of correctness of your program? It's impossible to do for all but the most trivial programs. You know that a brilliant person, given years of work, might be able to come up with a proof of correctness, but you also know that the language they would use to do that would be very different from the language that you as programmer would use to write your program.
So it goes in both directions. Programs evaluate numbers, they are essentially adding machines, and central to that is the erasure of state via addition. Math is about symbolic relations and erasure of state is via relations on infinite sets.
For example, you want to prove that if a group has a prime number of elements, then it must be cyclic. So pick a generator, raise it to powers, and get a subgroup. Then by Cauchy's theorem, the order of the subgroup has to divide the order of the main group, and that order is prime, so therefore the subgroup has to be the whole group.
There are many things to unpack in that statement. For example, Cauchy's theorem, which says that if a A is a subgroup of B, then the order of A divides the order of B.
So it's bit like adding numbers, in that you forget state, but the rules are much more complicated involving quantifiers and a huge universe of sets. To build a computer that could reason that way, you'd need an infinite set of registers, rules for quantifiers, etc.
You can try to simulate special cases of that in software, say with an SMT solver. If you look at the DSL of writing code for SMT solvers, it looks different from the code that a programmer is used to writing, and the best SMT solver can't really do anything too interesting from a math point of view, with very rare exceptions.
Moreover in math as practiced, no one except some logicians writes in formal logic. It would be too cumbersome. Simple statements like "a harmonic function on a disk achieves its maximum and minimum value on the boundary of its domain" would require thousands of pages of formal symbols and quantifiers.
This is a very narrow view of mathematics. Mathematics is much, much deeper than that. Even the pure algebra is no longer about "symbolic relations".
In fact, today the correct view of mathematics would be much closer to that of theoretical physics: It turns out that the dependently-typed lambda calculus is sufficient.
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