Comment by ypcx | Hacker News Reader

Comment by ypcx | Hacker News Reader

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Tell me if I'm wrong, but I have this theory that the whole Math could be expressed on top of a single basic operation, which is addition.

For example 2 * 3 is (2 + 2 + 2).

  Or sin(7) is (7 * (1 + -0.1666666664*(7*7) + 0.0083333315*(7*7)*(7*7)
  + -0.0001984090*(7*7)*(7*7)*(7*7) + 0.0000027526*(7*7)*(7*7)*(7*7)*(7*7)
  + -0.0000000239*(7*7)*(7*7)*(7*7)*(7*7)*(7*7)))

(the multiply operator left in for brevity, but can be easily replaced by addition)

If my assumption is true, then Math is really just an arcane and archaic form of a computer-like language.

And at the same time, it's possible that the operations meant by Math's various symbols, however arcane, can be understood much better (especially with the use of modern tech, animations, etc.) than we understand them from classic texts, but those who learnt them haven't spent enough effort to create more visual explanations of it, based on which others, and also themselves, could understand those operations better, and then be able to build on top of them and even evolve Math further. (I know that I don't know what I don't know, but I have tendency to assume that I know a lot more than I really know, thus I don't explore the known deeper/further.)

It sounds like you're thinking of (or working towards) Peano Arithmetic¹, which (to oversimplify) basically starts with "0", "1", "=" and "+", and works from there. Whether that's "enough" rather depends on what you're trying to achieve, though. There's subgenres of mathematics that don't necessarily involve numbers at all, like geometry and topology. It would be difficult to reduce them to addition.

¹: http://en.wikipedia.org/wiki/Peano_axioms

Addition is a function and so functions would be a more basic concept. But functions are sets and sets are a more basic concept. Thus we should study sets. Everything comes from sets. Replace addition with sets and your first sentence would be closer to the truth.

I strongly disagree with your last paragraph though.

And you can take sets as something built on top of categories (specifically, topoi), so category theory is more foundational than set theory.

http://plato.stanford.edu/entries/category-theory/

http://sakharov.net/foundation.html

http://en.wikipedia.org/wiki/Topos

So you think the whole of math is just numerical computation? You think there is nothing more to math than computing values of certain classes of functions? This is what you get when your only exposed to so called higher math is calculus and perhaps a few courses on differential equations, with emphasis on numerical methods.

There are areas of math that don't fit this simplistic view, things that don't have analytic in their name, like topology, non-Euclidean geometry, mathematical logic etc. Algebra in general is all about studying structure that ensues when you have bunch of objects and operations you can do with them with certain properties. This structure doesn't depend on the nature of object or operation. For example rotation group etc. Then there are things like measure theory, fractals, and things like non-commutative operator algebras that are useful for applications in quantum mechanics. All in all a huge chunk of math is not about computation. Bigger in fact than the part about actual computation, esp. computation of real numbers.

I had a similar reaction after reading the beginning of this article. I feel as though modern algebra uses symbolic manipulation as a means of qualitative discovery and understanding.

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