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Ishaan comments on Open thread, Jan. 26 - Feb. 1, 2015 - Less Wrong Discussion
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Comments (431)
What's wrong with treating axioms as assumptions?
I don't think it's "wrong" in the sense of "incorrect"... it's just that if you don't also realize that axioms are arbitrarily constructed "universes" and that all math takes place in the context of said fictional "universes", you kind of miss the deeper point. Thinking of them as assumptions is a simple way to teach them to beginners, but that's a set of training wheels that aught to be removed sooner rather than later, especially if you are using axioms for math.
And , handy side effect, your intuition for epistemology gets better when you realize that. (In my opinion).
Well, they are a set of assumptions on the basis of which you proceed forward. Starting with a different set will land you in a different world built on different assumptions. But I see it as a characteristic of assumptions in general, I still don't see what's so special about axioms.
When you assume the parallel postulate, for example, you are restricting your attention to the class of models of geometry in which the parallel postulate holds. I don't think that's a useful way of thinking about other kinds of assumptions such as "the sun will rise tomorrow" or "the intended audience for this comment will be able to understand written English".
(At least for me, I think that the critical axiom-related insight was the difference between a set of axioms and a model of those axioms.)
What is useful depends on your goals. The difference is still not clear to me -- e.g. by assuming that "the intended audience for this comment will be able to understand written English" you are restricting your attention to the class of situations in which people to whom you address your comment can understand English.
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.
I think I see what you mean. I would probably describe it not as a difference in the properties of axioms/assumptions themselves, but rather a difference in the way they are used and manipulated, a difference in the context.
I do not recall a realization similar to yours, however, perhaps because thinking in counterfactuals and following the chain of consequences comes easy to me. "Sure, let's assume A, it will lead to B, B will cause C, C is likely to trigger D which, in turn, will force F. Now you have F and is that what you expected when you wanted A?" -- this kind of structure is typical for my arguments.
But yes, I understand what you mean by blocks in empty space.
I don't think this is really the same skill as following counterfactuals and logical chains and judging internal consistency. Maybe the "parallel postulate" counterfactual was a bad example.
It's more the difference between
"Logic allows you to determine what the implications of assumptions are, and that's useful when you want to figure out which arguments and suppositions are valid" (This is where your example about counterfactuals and logical chains comes in) [1]
and
"Axioms construct / pin down universes. Our own universe is (hopefully) describable as a set of axioms". (This is where my example about building blocks comes in) [2]
And that's a good way of bridging [1] and [2].
I am not too happy with the word "universe" here because it conflates the map and the territory. I don't think the territory -- "our own universe", aka the reality -- is describable as a set of axioms.
I'll accept that you can start with a set of axioms and build a coherent, internally consistent map, but the question of whether that map corresponds to anything in reality is open.
I very strongly do. I think the universe is describable by math. I think there exist one or more sets of statements that can describe the observable universe in its entirety. I can't imagine the alternative, actually. What would that even be like?
That's actually the only fundamental and unprovable point that I take on faith, from which my entire philosophy and epistemology blossoms. ("Unprovable" and "faith" because it relies on you to buy into the idea of "proof" and "logic" in the first place, and that's circular)
I don't necessarily think we can find such a set of axioms. mind you. I can't guarantee that there are a finite number of statements required, or that the human mind is necessarily is capable of producing/comprehending said statements, or even that any mind stuck within the constraints of the universe itself is capable. (I suppose you can take issue with the use of the word "describable" at this point). But I do think the statements exist, in some platonic sense, and that if we buy into logic we can at least know that they exist even if we can't know them directly. (In the same sense that we can often know whether or not a solution exists even if it's impossible to find)
No "universally compelling arguments in math and science" applies here: I can't really prove it to you, but I think anyone who believes in a lawful, logical universe will come around to agree after thinking about it long enough.
Well, I was thinking that in those other cases, you consider the other possibility (e.g., that nobody who reads my comment will understand it) and dismiss it as unlikely or unimportant. It doesn't even make sense to ask "but what if it turns out that the parallel postulate doesn't actually hold after all?"
Am I explaining myself any better?
Is my reply to Ishaan helpful?