Loved it up until the introduction of proofs. Still can't wrap my mind around it. I'm beginning to think it's because "proofs" don't prove anything, they are just an argument that someone might find convincing.
You can evaluate the validity of a proof by checking the truthfulness of every statement which was used to argue that the proof holds
Miniature example of a proof with relaxed rigor:
Proove that 3+0=3
Proof: The above statement is a direct consequence of the additive identity axiom which states that x+0=x, if x is a real number. So the only thing we need to check is if 3 is a real number, and we know that it is. The statement holds, end of proof
So to check the validity of this proof you could check if that axiom really exists and check if 3 is a real number, if some of those is false than the proof is invalid
Edit: Or imagine that you have a DB full of axioms, theorems, prooven statements which you can use to proove a given statement. Then you could proove something by just referencing those. Eg. in the example above lets say that the neutral identity axiom has id 3, and the fact that 3 is a real number has an id 103. Then I could just say since 3 and 103 3+0=3
* There is a finite set of logical rules. For example, one rule is that if A => B and B => C are true, then A => C is true.
* We start with a finite set of given facts.
* At each step, we state a new fact and note it is true by combining previous facts with a rule.
* At the end, we have the claim we started with.
Example: prove 14 is an even number.
1. If a number equals two times an integer, it is even (given fact).
2. 7 is an integer (given fact).
3. 14 = 2 times 7 (laws of multiplication).
4. 14 = 2 times an integer (combining (2) and (3)).
5. 14 is even (combining (1) and (4)).
At a more complex level, e.g. proofs by induction are often no more than programs with for loops.
What are your favorite Math and Programming books?
Any lists for favorite Math book that you put together?