The Art of Logic

• Contrary to how it might seem, maths isn't about right and wrong, and nor are most arguments. They're about the sense in which something is right and wrong, depending on world views. If people disagree, it's often a result of different points of view stemming from different fundamental beliefs, not that one is right and the other is wrong.

I: The Power of Logic

1. Why Logic?

• The idea of logic is to have clear rules so that conclusions can be dran unambiguously and consistently by different people.
• The rules of scientific discovery involve experiments, evidence and replicability. The rules of mathematical discovery involve logical proof. Mathematical truth is established by constructing logical arguments, and that is all.
• Maths is the study of how logical things work: it's the logical study of how logical things work.
• A powerful aspect of abstraction is that many different situations become the same when you forget some details.
• Mathematics is a framework for finding similarities between different parts of science, and my research field, category theory, is a framework for finding similarities between different parts of maths.
• When we look for similarities between things we often have to discard more and more layers of outer details, until we get to the deep structures that are holding things together.
• Making analogies is the essence of mathematical thinking, where we focus on important features of a situation to clarify it, and to make connections with other situations. In fact mathematics as a whole can be thought of as the theory of analogies.
• In normal language people judge things not only by context but also relative to their own experiences; logical explanations need to be independent of personal experiences.