1-2a2 Logical reasoning is no less a physical process than scientific reasoning is, and it is inherently fallible
1-2f1a3c A central flaw in Kantian philosophy = the doctrine that certain truths about the physical world could be ‘known a priori’ - that is to say, without doing science
* Replace mathematics with physics and logic with mathematics: 1-1a4b0b According to Peirce, mathematics precedes all other fields of study, and only studies imbued with a strong mathematical foundation were worthy of the label ‘science’. Mathematics cannot be derived from logic.
Deutsch: “Our knowledge of whether a proposition is true or false always depends on knowledge about how physical objects behave … Mathematical truth is absolutely necessary and transcendent, but all knowledge is generated by physical processes, and its scope and limitations are conditioned by the laws of nature … So, a computation or a proof is a physical process in which objects such as computers or brains physically model or instantiate abstract entities like numbers or equations, and mimic their properties. It is our window on the abstract … Consequently, the reliability of our knowledge of mathematics remains for ever subsidiary to that of our knowledge of physical reality. Every mathematical proof depends absolutely for its validity on our being right about the rules that govern the behaviour of some physical objects, like computers, or ink and paper, or brains. So, contrary to what Hilbert thought, and contrary to what most mathematicians since antiquity have believed and believe to this day, proof theory can never be made into a branch of mathematics. Proof theory is a science: specifically, it is computer science … The object of mathematics is to understand – to explain – abstract entities. Proof is primarily a means of ruling out false explanations; and sometimes it also provides mathematical truths that need to be explained. But, like all fields in which progress is possible, mathematics seeks not random truths but good explanations.”
- 1-2f1b4 The content of a theory is in what it rules out (and how!)
- 1-2b In the quest for truth, what matters is the explanation and not where it came from, because even if gods reveal something to you it’s just as fallible as ever (the quest for truth ≠ a quest for certainty-justification-necessity)
- 1-2g2s7b5 Proof theory is computer science and not mathematics
Why there are such things as mathematics lectures rather than just lists of proofs