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This is probably a bit of a boring and ridiculously noob question, so apologies in advance, but:

Is everything in maths essentially built on adding, subtracting, multiplying and dividing? Are they the first principles on which everything else is built?

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The quote from Eichler seems relevant: "There are five elementary arithmetical operations: addition, subtraction, multiplication, division, and ... modular forms." –  Timothy Wagner Nov 27 '10 at 0:25
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I don't understand the downvote. IMO, this is a pretty reasonable question, especially when looking at it from a noob viewpoint. –  Aryabhata Nov 27 '10 at 0:37
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That depends on what you mean by "everything," as well as "built." –  Qiaochu Yuan Nov 27 '10 at 0:52
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I don't understand the upvotes. The question is very vague E.g. You could argue that everything in the universe is essentially built on adding, etc. You could also argue that almost nothing in mathematics is essentially built one adding, etc. -- There's no explanation of why the long list of topics on the mathematics page on wikipedia are considered to be advanced forms of "adding, subtracting, multiplying and dividing". This would help discern what it means to be "essentially built on". –  Douglas S. Stones Nov 27 '10 at 1:07
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@Bill: I think that’s what I’m asking. (i.e. is integer arithmetic powerful enough to represent all of mathematics?) –  Paul D. Waite Nov 29 '10 at 12:11

2 Answers 2

up vote 45 down vote accepted

Not really. For one thing, before you can "add", "subtract", "multiply", and "divide", you have to have something that you are "adding", "subtracting", "multiplying", or "dividing." So that "something" has to precede these operations. And for another, there are many notions that cannot really be expressed in these terms. To take a familiar one if you've taken Calculus, the idea of "limit" is not one that can be expressed in terms of adding, subtracting, multiplying, or dividing, but rather is expressed in terms of "approaching", or of being "arbitrarily close to". (This is one reason why Newton and Leibnitz are so rightly hailed as revolutionaries: the ideas of Calculus were very much unlike what mathematics had been up to that point.)

However, one has to start somewhere. Historically, mathematics started from different starting points in different cultures; counting was certainly one of the firsts things that comes up. But some cultures then realized that they need some more general ideas that went beyond simply counting (with its addition, multiplication, etc). Greeks settled on geometry as their underlying structure, and in fact translated everything about numbers into statements about geometry: lines, points, circles, squares, etc. Other cultures did not.

These days we usually take the route of axiomatic theories, where we have some "primitive notions" that are not defined, and some rules about how we can talk about them and their properties, and then we try to build everything on top of that. One can start from the natural numbers (the "counting numbers") using, for instance, the Peano axioms, and start building from there. Then you are actually starting before you get to "addition", "multiplication", "subtraction", and "division", with only the notion of "number", "next one", and induction. Or we can have other starting points.

Now, today, there is usually one of two starting points: Set Theory, and Category Theory. Both presume that we agree on the basics of logic (the meaning of things like "and", "or", "if... then...", and "not", the meaning of "is equal to", and things like that); and the rules of inference, which are rules that let you go from some statements others; for instance, the rule that says that if you know that A implies B, and you also know that A is true, then you know that B is true.

In Set Theory, you start with the idea of "set" and "element of"; these ideas are not actually defined, you just give some axioms that describe things you can say about them. From these axioms one can define numbers, addition, multiplication, etc. And also build other notions that are orthogonal to natural numbers and their usual operations.

In Category Theory, you start with the idea of "objects" and "maps" between objects (again, not defined; you only get some axioms that tell you how you can talk about them); then you can use these notions to construct everything else in terms of them.

So, rather, in modern mathematics everything is either "sets all the way down", or everything is "categories all the way down". If there is one common set of presuppositions, it's just basic logic.

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Fantastic - that sort of overview is exactly what I was hoping for. (The thank-you comment is long overdue - maybe I didn't have enough rep to comment when I accepted.) –  Paul D. Waite Aug 15 '13 at 16:12

This is an interesting rhetorical question. On the one hand, many theoretical mathematical concepts transcend arithmetic. However, computer algebra systems (such as Mathematica) are able to express answers to those sorts of questions using a CPU that performs a limited range of simple computations. So the answer is a matter of opinion. Or there is no answer?

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