Mathematics and logic are, firstly, systems whose original purpose for existence is the same: to provide a system for more effectively operating within our world. Both, as they began, did so through a codification of common sense. They quantified our experience and defined what operations we could apply to those quantities so that we could systematically find facts of which we were previously unaware. The distinction between the two was in what they attempted to quantify: mathematics was primarily concerned with objects in the world, such as money, land, and later the laws of physics, whereas logic was primarily concerned with concepts in the mind, such as propositions and categories (although those concepts often related to objects).
However, despite their different foci, logic and mathematics were based on a common method: deduction, that is, a system of rules which you can apply in such a way as to arrive at a conclusion which is both new and necessarily true (pg. 99). As such, it was almost inevitable that the two would meet. One of the earliest examples of this meeting is probably Euclid’s “Elements” (pg. 324), but the true synthesis came with the work of Boole and Frege, where it was shown that logic could be dealt with mathematically, and that one could attempt to build mathematics on a logical foundation (pg. 329-330).
The question now raises itself: is mathematics a branch of logic or logic a branch of mathematics? The logicists such as Frege and Russell believed the first, intuitionists believed the latter (pg. 328). I believe both are wrong. Just as philosophy is not a branch of logic, but instead logic is the method by which we do philosophy, so too is mathematics not a branch of logic—logic is the method by which we do mathematics, in the construction of theorems and proofs. Because of this, though, logic cannot be considered a branch of mathematics: while it is a mathematical system that works in a way particular unto itself and distinct from other areas of mathematics, it nevertheless permeates the entire structure of the enterprise. If mathematics is a tree with branches, logic is how the tree grows.
However, it is here that we run into our difficulty. Because while logic is how the tree grows, logic won’t necessarily make it grow the way we want. Logic and mathematics both have the same problem that the way that they allow us to systematically analyze the world by turning that world into symbols, which are then manipulated according to a strict set of rules. This is formalism, which attempts to avoid errors due to flaws in human intuition by making logic and math completely devoid of meaning. But as long as the symbols, statements, and rules aren’t mutually inconsistent, you can come up with whatever rules you like and make whatever statements you like. Yet, only some of these rules and statements will give you a system that provides an accurate description of the world—which is, after all, the original purpose of mathematics and logic. The test of a logical or mathematical system’s truth would seem to be, then, concurrence with the actual world, with experience. But if this is the case, then why should we try so hard to logically prove that, for example, 1+1=2, as was done in 300 pages by Russell and Whitehead? To do so is to use a system whose truth is based on experience to prove something that according to experience we already know to be true!
I must admit that I don’t know the answer, and furthermore don’t know enough about mathematics to even know whether I’m asking the right questions. I can say, however, that I have a very strong conviction that we should attempt to prove as much as we can by starting with the least and most simple and obvious assumptions. I suppose this really is the maxim of all work in philosophy, mathematics, and science, which has allowed those fields to flourish: to not take anything for granted, unless one absolutely has to.
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