The other day, a high school student in our discord server made fun of Principia Mathematica and its 379-page proof that 1 + 1 = 2.
He had a point. Who needs this crap and why is it so dumb? It’s even dumber than the proof of Jordan’s Theorem which says that every closed loop divides the plane into the inside and the outside! Anyone could’ve done that when they were five with a soiled sheet of newspaper and a crayon!
In this series, I’ll try to walk us through some of the great philosophical and mathematical ideas of the late 1800s up to the mid 1900s. Along the way, we’ll walk by that proof that 1 + 1 = 2 and why it mattered. But more importantly, this will be a journey through what I call “abstract nonsense” — ideas so far removed from everyday life that they strain the limits of reason itself. We won’t be following the chronological order that these people first wrote down these ideas because sometimes it just makes more sense to go the other way.
Let’s start with a renowned philosopher named Ludwig Wittgenstein. Wittgeinstein has two books, Tractatus Logico-Philosophicus and Philosophical Investigations. The latter was published after his death, but they deal with mostly similar themes.
There’s a rich philosophical tradition around these books which we will promptly ignore. The author is dead, and I think many philosophers completely misunderstood Wittgenstein anyway.
My understanding of Tractatus goes something like this:
We are humans who have thoughts about the world. Within each of our heads is a picture of the world as we personally understand it.
Language is our tool to communicate these thoughts to other humans.
Philosophy has a lot of meaningless problems like “What is justice?” and the “Paradox of the Heap” that exists only because of the limitations of language. Words have multiple meanings. Our primitive ooga boogas have a limited capacity to transfer our personal pictures of the world to other humans.
If we accept formalization, these philosophical problems disappear. Formalization means, as in mathematics, to treat everything as a set of axioms combined with deduction rules. By applying the deduction rules to the axioms, we can end up with more knowledge that is true. And by repeatedly applying the deduction rules to the set of knowledge we have, we can get further in our search for truth.
Finally, Wittgenstein ends with “Whereof one cannot speak, thereof one must be silent.” My understanding is that this is signalling to other philosophers that with a lack of formalization, philosophy will just keep going in circles.
He expands on this idea with the posthumously published work Philosophical Investigations. He discusses formalization techniques through an analysis of what he calls Language Games.
Language Games focus on naming objects in the world and making explicit the rules of logical inference. Take the paradox of the heap, for example. It’s paradoxical because we have a name for an object that we call a heap and from common sense, we know that heaps are collections of many objects. But the paradox exists only because we named a thing without an explicit definition, just a common sense one. That is, common sense doesn’t give us a definite number or size as to what constitutes a heap. Obviously, by removing items from this heap one by one, we’ll eventually run into issues with common sense.
Wittgenstein takes issue with the paradox of the heap because he claims that the word “heap” loses meaning when it’s taken out of context. Heap means heap in the language game of common sense. In the language game of philosophy and mathematics, heap is a meaningless word.
Formalization is the way out of this predicament. By making clear the rules of the deductive game and the initial assumptions, we could push forward in understanding individual language games and maybe even on a meta-cognitive level, an understanding in the nature of language games themselves.
But as we’ll soon find out, formalization brings with it its own collection of problems. We might get rid of the paradox of the heap, but in its place we would get things like multiple infinities, the Banach-Tarski Paradox, finite-volume solids that take infinite paint to cover, and other such nonsense.
Who knows whether that’s an improvement over the status quo?