Quote of the Week, 2014-11-12

Between philosophy and pure mathematics there is a certain affinity, in the fact that both are general and a priori. Neither of them asserts propositions which, like those of history and geography, depend upon the actual concrete facts being just what they are. We may illustrate this characteristic by means of Leibniz's conception of many possible worlds, of which only one is actual. In all the many possible worlds, philosophy and mathematics will be the same; the differences will only be in respect of those particular facts which are chronicled by the descriptive sciences. Any quality, therefore, by which our actual world is distinguished from other abstractly possible worlds, must be ignored by mathematics and philosophy alike. Mathematics and philosophy differ, however, in in their manner of treating the general properties in which all possible worlds agree; for while mathematics starting from comparatively simple propositions, seeks to build up more and more complex results by deductive synthesis, philosophy, starting from data which are common to all knowledge, seeks to purify them into the the simplest statements of abstract form that can be obtained from them by logical analysis.

Bertrand Russell, Our Knowledge of the External World, Lecture 7

Retrieved from Project Gutenberg

I have areas of agreement and areas of disagreement with this post. I would put both mathematics and philosophy, as well as fields of study like constitutional law, largely in a class of knowledge referred to as formal knowledge. For me, this is knowledge derived from systems we set up, such as logic, uses propositions we assert to be true. In this sense, it is true mathematics, philosophy, or the law would be the same in any alternate world, as long as you hold the assumptions that they make to be unchangeable.

On the other hand, Russell wrote these lectures in 1914. long before the Kurt Gödel's incompleteness theorems and Paul Cohen had proved the independence of the Continuum Hypothesis in set theory. We can certainly talk about one possible world where the Continuum Hypothesis is true, and another where it is not true. In that case, we can't say mathematics will be identical in these two possible worlds. The would hold true for any branch of philosophy (or any other formal system). We will always come across unprovable statements, which may be true or false, and discuss possible worlds for each case. The assumptions of mathematics are not unchangeable, but instead, up to the decision of the mathematician.