| > Proof: Starting at A (or later down the chain/network of theory you built up), you can just retrace your way back to W. This perfectly explains why W is true. How does that differ from "showing that it is true?" > why you think your claim is true "Why you think your claim is true" is not the same as "why it is true". I contrasted "showing that it is true" versus "showing why it is true" as a shortcut to show the distinction; since "why" something is true can also be interpreted as showing something is true, i.e. how we know it is true. Let me try to elaborate what I mean by "showing why it is true": You can trace through a sequence of proof steps, and confirm each one is true and therefore the conclusion is true, without understanding the whole. For example, even an automated prover can do this, but, like a Searle's Chinese room, it ia without understanding. In the natural sciences, the distinction is clearer, because you show that something is true by empirical observation and experiment: what is the colour of starlight? what is the trajectory of the moon? But there are also theories to explain these facts: the life-cycle of stars; red-shifting due to acceleration; inverse square law of gravity. Though we do bottom out to "it just is". --- Even for highschool elementary aithmetic algebraic, that seem general, there are (sort of runtime) exceptions: divide by zero (and multiply by zero). You must learn notation, variations amd abuses of notation. You have to learn definitions, but I think that's fair enough. They are the rules of the game, and with even slightly different definitions, you're simply playing a different game. But I think elementary geometry doesn't suffer from this: even though it's not taught rigorously these days (from Eulcid), it still makes sense from first principles/axioms, and you can see it is true. It may simply be that modern mathematics has become so divorced from the directly graspable (and yet so powerful and useful), that you just have to teach it without understanding. "Young man, in mathematics you don't understand things. You just get used to them." -
John Von Neumann (criticism here: https://math.stackexchange.com/questions/11267/what-are-some...) |