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Ishaan comments on Open thread, Jan. 26 - Feb. 1, 2015 - Less Wrong
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1) The idea of constructing things out of axioms. This is probably old hat to everyone here, but I was clumsily groping towards how to describe a bunch of philosophical intuitions I had, and then I was learning math proofs and understood that any "universe" can be described in terms of a set of statements, and suddenly I understood what finally lay at the end of every chain of why?s and had the words to talk about a bunch of philosophical ideas...not to mention finally understanding what math is, why it's not mysterious if physics is counterintuitive, and so on. (Previously I had thought of "axioms" as"assumptions", rather than building blocks.). Afterwards, I felt a little cheated, because it is a concept much simpler than algebra and it ought to have been taught in grade school.
2) Something more specialized: I managed to get a B.S. in neuroscience without knowing about the thalamus. I mean, knew the word and I knew approximately where it was and what it did, but I did not know that it was the hub for everything. (By which I mean, nearly every connection is either cortico-cortico or cortico-thalamic). After graduation, I was involved in a project where I had to map out the circuitry of the hippocampus, and suddenly... Oh! This is clearly one of the single most important organizational principles of the brain and I had no idea. After that, a whole bunch of other previously arbitrary facts gradually began to made sense...Why did no one simply show us a picture of a connectome before and point out that big spot right in the middle where it all converges?
3) We learned all this minutia of history, but no one really talked about the hunter-gatherer <--> agriculture transition and its causes. Suddenly, historical trends in religion, the demographic transition, nutrition, exercise, cultural differences, and a bunch of other things start clicking together.
I think what all these 3 things have in common, is that they really aught to have been among the very first lessons on their respective subjects...but somehow they were not.
This is a pet peeve of mine, Axioms as assumptions (or self-evident truths) seem to be a very prevalent mode of thinking in educated people not exposed to much formal maths.
What's wrong with treating axioms as assumptions?
Well, it's hard to articulate. There's of course nothing wrong with assumptions per se, since axioms indeed are assumptions, my peeve is with the baggage that comes with it. People say things like "what if the assumptions are wrong?", or "I don't think that axiom is clearly true", or "In the end you can't prove that your axioms are true".
These questions would be legitimate if the goal were physical truth, or a self-justifying absolute system of knowledge or whatever, but in the context of mathematics, we're not so interested in the content of the assumptions as we are in the structure we can get out of them.
In my experience, this kind of thing happens most often when philosophically inclined people talk about things like the Peano axioms, where it's possible to think we're discussing some ideal entity that exists independently of thought, and disappears when people are exposed to, say, the vector space axioms, or some set of axioms of set theory, where it becomes clear that axioms aren't descriptions but definitions.
Actually, you can ignore everything I've said above, I've figured out precisely what I have a problem with. It's the popular conception of axioms as descriptive rather than prescriptive. Which, I suppose OP was also talking about when they mentioned building blocks as opposed to assumptions.
That's a valid question in a slightly different formulation: "what if we pick a different set of assumptions?"
But that, on the other hand, is pretty stupid.
Well, normally you want your axioms to be descriptive. If you're interested in reality, you would really prefer your assumptions/axioms to match reality in some useful way.
I'll grant that math is not particularly interested in reality and so tends to go off on exploratory expeditions where reality is seen as irrelevant. Usually it turns out to be true, but sometimes the mathematicians find a new (and useful) way of looking at reality and so the expedition does loop back to the real.
But that's a peculiarity of math. Outside of that (as well as some other things like philosophy and literary criticism :-D) I will argue that you do want axioms to be descriptive.
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].
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?
It's not very amenable to teaching.
The grandparent said "prevalent mode of thinking in educated people" -- what's convenient for teaching is not very relevant here.
About that, how would you evaluate the state of the typical undergrad neuroscience curriculum today and how relevant it is to modern knowledge about the organization and workings of the brain?
Hmm
I think the undergraduate curriculum is good enough to get the average college student up to a level where they are comfortable reading and understanding a scientific paper in biology even if they start out with only a very rudimentary knowledge of science coming in.
You spend the first 3 years kind of learning the basic fundamentals of biology, like how evolution and cells and hormones and neurons work, and I think for Lesswrong's demographic that sort of thing is general knowledge so most of y'all could skip it without any major issues. I found these courses challenging because of the amount of stuff to memorize, but I am not sure I found them useful. I kind of wish I could have replaced some of those introductory courses with work in computer science or a stronger biochem/chem foundation, because I already knew about evolution and mitochondria and that sort of thing.
The last 2 years, for me, were quite useful. In these upper level classes, professors in a given sub-specialty would select primary literature which was important to their field, and it would be discussed in depth. What you learn will largely depend on what the professor is passionate about. There are also classes where many different researchers come in and present their work, and one ends up picking up many little threads from that which can later be pursued at leisure. Despite already being comfortable with primary literature in neuroscience and psychology before joining these classes, I still found them very useful because of the specific content of the work I was exposed to. Many of these courses were technically graduate courses, but it is standard for undergraduates to attend them.
Overall, I think if you are generally comfortable reading scientiic papers in biology, bachelors-degree level neuroscience is not an extremely difficult subject for a motivated autodidact to acquire without formal coursework (assuming you have access to all the major scientific journals). The coursework is good - but there's no secret esoteric knowledge you can only acquire from the coursework. it's not necessarily better than self-study, but it's awesome when combined with self-study and is fairly decent even without any self-directed study.
Direct contact with researchers is definitely a very positive thing for keeping a pulse on stuff, knowing what is important to study and what isn't, and generally learning faster than you could on your own. You're also expected to join a lab during your undergrad, and you will inevitably learn a lot through that process.
TL:DR - As with many fields, if you want to be up to date on modern knowledge, there is absolutely no substitute for constantly skimming the new papers that come out. The undergraduate curriculum spends 2-3 years preparing you to successfully read a scientific paper in biology, and 1-2 years having you read papers which are selected to be particularly important in the field. Also, you will typically join a lab, which is certain to cause learning.
Well, algebra is also not taught in grade school. Considering Piaget's theory of cognitive development, with abstract thought only getting in place in the teens, I wonder if maybe it's not possible to teach it until middle/high school, even if its simple once the cognitive machinery is activated...
I might have misused the term - I thought up to 8th grade and perhaps even 12th grade was "grade school"? I got my sister to think algebraically with non-geometrical problems, and then apply that successfully to a novel geometrical problem with perimeters and volume when she was 10...but she wasn't able to retain it after a week. Later on when she learned it in school at an older age, she did retain it. I suspect attentional control is the limiting factor, rather than abstract thought.
But you're right, this should be tested. She's technically not a teen yet so next time she has a long holiday of free time I'll see if she can learn about basic logical properties and set theory. (It seems way easier than the graphs simultaneous linear equations she's doing right now, so I am optimistic).
I see. I'm used to it being a synonym for primary school, but according to Wikipedia, that's apparently ambiguous or incorrect.
I agree that trying to teach it and seeing what happens is the way to go. :) Although I guess there is probably a lot of individual variation, so a school curriculum based on what works for your sister might also not generalize.
This is true. It would just be a test case for whether pre-teens can learn these concepts.
If I was designing a hypothetical curriculum, I wouldn't use high-sounding words like "axiom". It would just be - "Here is a rule. Is this allowed? Is this allowed? If this is a rule, then does it mean that that must be a rule? Why? Can this and that both be rules in the same game? Why not?" Framing it as a question of consistent tautology, inconsistent contradiction as opposed to "right" and "wrong" in the sense that science or history is right or wrong.
And "breaking" the rules, just to instill the idea that they are all arbitrary rules, nothing special. "What if "+" meant this instead of what it usually means?"
And maybe classical logic, taught in the same style that we teach algebra. (I really think [A=>B <=> ~B=>~A]? is comparable to [y+x=z <=> y=z-x]? in difficulty) with just a brief introduction to one or two examples of non-classical logic in later grades, to hammer in the point about the arbitrariness of it. I'd encourage people to treat it more like a set of games and puzzles rather than a set of facts to learn.
...and after that's done, just continue teaching math as usual. I'm not proposing a radical re-haul of everything. It's not about a question of complex abstract thought- it's just a matter of casual awareness, that math is just a game we make, and sometimes we make our math games match the actual world. (Although, if I had my way entirely, it would probably be part of a general "philosophy" class which started maybe around 5th or 6th grade.)
(I'm not actually suggesting implementing this in schools yet, of course, since most teachers haven't been trained to think this way despite it not being difficult, and I haven't even tested it yet. Just sketching castles in the sky.)