Here's some thoughts, starting from the easiest questions:

Can math be discussed in this way? Math not so much as technique and numbers, but more in terms of a conceptual discussion?

It can, and it is in many popular books (such as

*Infinity and Mind* by Rudy Rucker,

*Gödel, Escher, Bach* by Hofstadter &c), there's one trouble though: most of maths requires concrete examples and good explanation. Usually it is easiest to explain maths properly by usin "technical" presentation. If it is not used, reader can have much difficulties to keep the track at the end of discussion.

Any way, conceptual discussion and numbers &techniques do not exclude each other. Even maths is taught at schools as techniques, real maths starts from concepts, symbolic part is there just to make it easier to write, read and comprehend. Just think how much easier it is to write "1+1=2" than "one plus one equals two". When you talk about integrals or more complex things, it eventually comes impossible to remember the start of the sentence when it ends if you don't use any "technical" notation.

Is the decimal system for our own convenience, or is there something natural about ten digits?

Only natural thing with it is that we have ten fingers.

Could a number system be devised based off pi, where pi is a whole number? If so, would this make advanced mathematics any easier or less ambiguous?

You could for example define numbers A,B,C,....,J as A=pi, B=2pi, C=3pi and so on, but if you think it a while, you'll notice that natural numbers 1,2,3,... would be as hard to express as the pi is in the current system. As numbers like 1 and 2 are far more important than pi (pecause they are used for counting), it would only make advanced maths more difficult.

Natural numbers are

*natural* they are learned when we count things, pi is result of little more complex thinking, and that's why most people would have to transform numbers of this new system to the current only to think about them, and after back again to tell what was the results of their thinking.

Moreover, pi isn't such a big problem that people seem to think. Mathematicians just use the symbol and name pi like they use any number, it's nonending decimal presentation is no problem, you just use pi like any number. Actually it's a bit like using the before mentioned "new system" and the traditional system side by side: You can use numers A,B,C,...

*and* 1,2,3,... Only number of the "new system" that is useful just is A, that is pi, because B, C etc are multiples of it.

There are of course other possibilities to cunstruct a pi-based number system, but they all have the same defects as the previously mentioned: natural numbers are poorly expressible and the system is not natural.

*e* the natural number, what's so natural about that number as opposed to any other number?

As far as I know it isn't called the natural number, you're confusing it with the natural logarithm. Number

*e* is called Napier's constant.

People talk about Leibniz and Newton *inventing * calculus, but I wonder, is that really correct? Did they invent calculus, or is calculus a part of nature?

---

is this all there is to math, a symbolic expression of natural laws that in their own essence have nothing to do with math?

No, you can do maths without knowing any natural law. The difference between maths and physics is that latter tries to formulate and test natural laws, and they use maths as a tool in their task. Maths on the other hand is about abstract entities, you don't measure circle's circumference but calculate it instead, and the circles of the real world aren't even interesting when you're doing maths. Basically you have the idea of circle in your mind when you're drawing one, and think: "this isn't quite the thing I'm thinking of, but it makes thinking of it easier". Then you try to formulate the idea of circle: "It is a curve which consists of all the points at a certain distance from certain point", and try to deduce from it all the properties you want.

Same way calculus (from the standard point of view) isn't about falling bodies or weight of a stone or something like that, it's abstraction of these problems. If you operate in abstract world, you get obnly abstract results.

Officially maths isn't concerned about the validity of those abstractions, that is: once an abstraction is made, mathematicians examine only it's implications and form propositions of the form: "From these things follows these other things". They don't concern themselves with the validity of their assumptions, only about their implications.

That official interpretation of maths was made around 1900 to secure the back of maths, and it's quite ok. It's only weakness is that while it is highly justifiable, it doesn't tell what maths is to those who practice it (well to most of them at least). Most mathematicians feel they are dealing with something that has existence of it's own, and use the official interpretation about maths only when they are put back against the wall. The question of that existence is a different question, and depends on what you want to call existing.

(Sorry poor and brief formulation of those last paragraphs, the question is quite complex, and I have difficulties of expressing myself precisely in English).