## Monday, January 20, 2014

### Math vs Logic

Math provides a series of games, which can be usefully applied to reality when their rules closely mirror those of some real system.

Originally, it would seem that math started out as just another part of language. Language itself has been described as a game: a useful set of rules which we follow in order to get things done.

Eventually, math developed into a sort of sub-language or sub-game with a clearly independent set of rules. The objects of mathematical language were much different from the typical objects of everyday language, being more abstract while simultaneously being atypically precise, with very definite behavior. This "definite behavior" constitutes the rules of the game. The Pythagorean cult nurtured the idea of formal derivations from axioms or postulates.

(Around this time, Plato's idea of a separate pure mathematical reality started to seem plausible to some folks.)

Notice, however, that math still seemed like a single game. Euclid's Elements provided a wonderfully unified world of mathematics, in which number theory was considered as a geometric topic.

I think it wasn't until the 1800s that it started to become clear that we want to view math as a rich set of different possible games, instead. Non-euclidean geometry was discovered, and mathematicians started to explore variant geometries. The use of imaginary numbers became widely accepted due to the work of Gauss (and Euler in the previous century), and the quaternions were discovered. Group theory was being applied and developed.

Once we have this view, math becomes an exploration of all the possible systems we can define.

Within the same century, logic was gaining teeth and starting to look like a plausible foundation for all mathematical reasoning. It would be overstating things to claim that logic had not developed within the past two thousand years; however, developments in that century would overshadow all previous.

In using the number two thousand, I refer to Aristotle, who laid out the first formal system of logic around 350 BC. Aristotle provided a deep theory, but it was far from enough to account for all correct arguments. Euclid's Elements (written just 50 years later) may have used exceedingly rigorous arguments (a bright light in the history of mathematics), but they were still informal in the sense that there was no formal system of logic justifying every step. Instead, the reader must see that the steps are justified by force of reason. Aristotle's logic was simply too weak to do the job.

It was not until Frege, publishing in the late 1800s, that this became a possibility.

Frege attempted to set out a system of logic in which all of math could be formalized, and he went a long way toward achieving this. He invented modern logic. In the hands of others, it would become the language in which all of mathematics could be set out.

So, we see that logic steps in to provide a sort of super-game in which all the various games of mathematics can be played. It does so just as a unified picture of mathematics as a single game is crumbling.

My point is to answer a question which comes up now and then: what is the dividing line between math and logic? Is logic a sub-field of math, or is it the other way around? The reality is complex. Logic and math are clearly distinct: logic is an attempt to characterize justified arguments, whereas math merely relies heavily on these sorts of arguments. A system of logic can be viewed as just another mathematical system (just another game), but it must be admitted that logic has a different "flavor" than mathematics. I think the difference is in how we are trying to capture a very large space of possibilities within a logical system (even when we are studying very restricted systems of logic, such as logic with bounded quantification).

Ultimately, the difference is a historical one.

All facts cited here can be easily verified on Wikipedia. Thanks for reading!