Comment on Open thread, Jan. 26 - Feb. 1, 2015

Tue, 27 Jan 2015 05:18:41 GMT

What is useful depends on your goals

When your goal is to do good mathematics (or good epistemology, but that's a separate discussion) you really want to do that "restrict your attention" thing.

Human intuition is to treat assumptions as part of a greater sistem. "It's raining" is one assumption, but you can also implicitly assume a bunch of other things, like rain is wet., to arrive at statements like "it's raining => wet".

This gets problematic in math. If I tell you axioms "A=B" and "B=C", you might reasonably think "A=C"...but you just implicitly assumed that = followed the transitive property. This is all well and good for superficial maths, but in deeper maths you need to very carefully define "=" and its properties. You have to strip your mind bare of everything but the axioms you laid down.

It's mostly about getting in the habit of imagining the universe as completely nothing until the axioms are introduced. No implicit beliefs about how things aught to work. All must be explicitly stated. That's why it's helpful to have the psychology of "putting building blocks in an empty space" rather than "carving assumptions out of an existing space".

I mean, that's not the only way of thinking about it, of course. Some think of it as an infinite number of "universes" and then a given axiom "pins down" a subset of those, and I guess that's closer to "assumption" psychology. It's just a way of thinking, you can choose what you like.

The real important thing is to realize that it's not just about making operations that conserve truth values..,that all the mathematical statements are arbitrarily constructed. That's the thing I didn't fully grasp before...I thought it was just about "suppose this is true, then that would be true". I thought 1+1=2 was a "fact about the actual universe" rather than a "tautology" - and I didn't quite grasp the distinction between those two terms. Until I broke free of this limitation, I wasn't able to think thoughts like "how would geometry be if the parallel postulate isn't true?", because, well, "obviously (said my incorrect intuition) the parallel postulate is factual and how can you even start considering how things would look without it?"

..as I write this, I'm realizing that this is a really hard misconception to explain to one who has never suffered from it, because the misconception seems rather bizarre in hindsight once you are set right. Maybe you just intuitively get it and so aren't seeing why some people would be led astray by thinking of it as an assumption.

Reading your reply to me, you do seem to have your thoughts correct, and you seem to gravitate toward the "pin down" way of thinking, so I think for you it is perfectly okay to mentally refer to them as assumptions. But it confused me.

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