I’m reading through some of Wittgenstein’s lectures on the foundations of mathematics. I’m not real sure what to expect. I thought I’d write up some notes on it as I went along. This is, in my opinion, the only way to read Wittgenstein. I figured I’d post them in case anyone can help shed some light on what is happening or is interested.

Wittgenstein’s approach to understanding in the first few lectures (I assume throughout) seems to be that to understand a concept is to be able to use it. Suppose that someone says that they understand the same concept I do and we use it in the same way in several cases but then diverge. Wittgenstein seems to think that this means that we understand it differently, because our uses of the concept diverge. At the start of lecture two, Wittgenstein distinguishes two criteria for understanding. One is where you respond to q question of understanding by responding, “Of course.” If asked whether one understands “house”,the response will be an affirmative. The other criteria is how the word is used, indicating houses, etc. Wittgenstein seems to cast some doubt on the first criterion of understanding. He seems to be unsure what justifies it except the second criterion.

This is probably related. He thinks that part of understanding what a mathematical discovery is consists in seeing the proof of it. He gives a sample exchange:”Suppose Professor Hardy came to me and said “Wittgenstein,I’ve made a great discovery. I’ve found that …” I would say, “I am not a mathematician, and therefore I won’t be surprised at what you say. For I cannot know what you mean until I know how you’ve found it.” We have no right to be surprised at what he tells us. For although he speaks English, yet the meaning of what he says depends upon the calculations he has made.” The proof of a mathematical claim isn’t an application of the claim. It does demonstrate a use of the concepts involved, which use, I suppose, gives the meaning of the proposition. I should note that it is a little weird for the exchange to talk about a discovery, at least from Wittgenstein’s perspective. Near the end of the first lecture he says that he will try to convince us that mathematical discoveries are better described as mathematical inventions.

Why would we need to see the proof of a statement to understand it? The proof provides an illustration of the use of the concepts involved. This begins to shed some light on why different proofs of one proposition are interesting. The proposition has been show to be true once demonstrated, assuming the proof is good, so additional confirmations of this aren’t that exciting. What is useful is seeing different ways in which the concepts can be used together. This seems to result in understanding the proposition in different ways. Does it result in different meanings for the proposition? Not sure. It isn’t clear what role, if any, the idea of meaning plays in these lectures.

In lecture four, Wittgenstein talks about people who use things that look like mathematical propositions without them being integrated into a wider mathematical context. The example is kind of weird. It is a group of people that measure things with rulers then measure to figure out how much something will weigh. The question is whether they are working with physical or mathematical propositions. Wittgenstein thinks “both” might be a reasonable answer, but he gives a follow-up suggestion that might indicate that this isn’t his view. He says that there is a view that math consists of propositions and there is another view that it consists of calculations. It seems like the latter will be his view.

The first few lectures have been a bit difficult to pull together, although going forward a bit, I think I’m getting a better sense of some of the issues. More notes to follow I expect.