A quest for beauty and clear thinking: interviewing John Baez
mathisintheair.com
mathisintheair.com
I used to hope for this, but the more I go into mathematics, the more complex it becomes, the explanations more complex than the explained. I suspect "mathematical maturity" is the absorption of all this information. It takes years and years.
Of course once you know it it's easy - like fluency in a natural language, you aren't aware of all you've learnt.
Probably, if I had John Baez's experience, I wouldn't feel this way...
> everything has a reason
Proofs show that it is true, not why it is true.
At that point you can ask questions like “are these two reasons why theorem W is true the same or different?” which leads into things like homotopy type theory.
Proofs tell you exactly why something is true as they are nothing more than a water-tight explanation of why you think your claim is true. Oftentimes proofs actually precede the claims they are a proof of. For example, you can start with a set of axioms A from which you derive a packet of theorems and definitions B. In turn, combining (or separately) you derive more statements from A, B which we call, say, C. You can continue this process for however far and long you want. Suppose, at some point you end up at a statement W. Then you can do this:
Theorem: W.
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.
Where I agree with you is that the role of memorization is downplayed. Sometimes, it's helpful to memorize some algorithms and definitions. It clears up your mind for actual thinking. Partly why they make you memorize a lot of elementary algebra rules (without proof) in high school so that when doing Calculus you don't have to think about "lower level stuff". Often, if I produce a proof that's different from that in my book or someone else, I try to memorize their proof because likely this way of thinking will recur down the road in the book and someone else's ideas might produce insight that would be unavailable to me had I decided to stick to what I know.
edit: I'd add induction proofs usually won't tell you something you didn't know beforehand. I never read induction proofs unless the author alludes to some clever trick that's worth remembering.
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".
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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...)
An induction proof shows the recursive internal structure of a mathematical object.
https://johncarlosbaez.wordpress.com https://twitter.com/johncarlosbaez
The recent breakthrough in finding the smallest Lebesgue cover [1] was triggered by one of his blog posts about it a few years back.
1 -- https://www.quantamagazine.org/amateur-mathematician-finds-s...
I do love that dig at the insane US medical education system.
The human body is a beautifully complex system, and it's really quite fun studying it and stretching it to its limits. My biggest regret is having started lifting at a rather older age.
TBF there is some regularity to the names e.g. the finger bones (phalanges) are named in a 2D grid of thumb..litte X inner..tip:
I..V x proximal,middle,distal - Imiddle
Names simplify communication, both from educator to learner in teaching and practitioner to practitioner in use. Learning the names used for things in a field, is for that reason, usually part of learning about the things (and the field.)
(Of course, names aren't always words, they can be visual symbols, and it's not like people learning math don't have to learn a whole lot of domain-specific visual notation alongside learning the concepts that notation is used to communicate about, for the exact same reason as medical students learn the names of bones and other body parts.)