Primitive Notions
Posted on September 9, 2010 in Asides
(last edited on April 29, 2014 at 1:25 am)
(last edited on April 29, 2014 at 1:25 am)
From Wikipedia’s article on Axioms: A lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from the mathematical assertions (axioms, postulates, propositions, theorems) and definitions. — that might explain why nothing made sense to me in math class. They could have told me why. Apparently, it’s because if you just accept something as true, it helps streamlines the learning process. If you call that learning. I call it indoctrination.
7 Comments
I agree 100%.
Have you ever read Jung’s auto biography? As a child he had a hard time accepting the premise ‘if a=b and b=c then a=c’ (or something like that, I can’t remember exactly).
I always thought this was extremely interesting. On one hand, he could not accept simple symbolic mathematics. On the other hand, he was able to identify symbols, archetypes, and the collective unconscious. Some how the part of his intellect that rejected a=c gave him insight into psychology and archetypes.
That’s not what it means. It means that whenever you try to use an example to illustrate your point, people end up arguing the example instead of the point. No example = no stupid argument.
joyce: No, I haven’t read Jung’s auto biography, but it sounds like I might enjoy it! Thanks for the comment!
Amanda: Oh, that makes more sense, and perhaps this is why I had difficulty with it in the first place…shifting the attention from an illuminating example to an illuminating point. I never understood what the point was of mathematics as it was presented to me in school in its various forms. I would catch glimpses of it here and there, but never as a system.
The problem is, if you don’t use an example, people have a hard time really understanding the concept. If you DO use an example, some people will fixate on the example and not learn the concept. I hear the Japanese educational system has a good way to get around this (which is why they’re so much better at math than we are), but I don’t know what it is.
Math was invented/developed by people who don’t have a problem working without specifics, so we all agreed to drop the specifics. Which is great for us, but lousy for the rest of the world.
Amanda has the right idea.
I would add that axioms are way of building common terminology. Once a low-level, shared concept is is linked to an agreed upon definition, it becomes a lot more efficient to discuss higher-level ideas that build on those concepts. You get to a solution faster by not having to come to consensus on the basic concepts underpinning your argument. (No reinventing the wheel, as it were.)
I’m sorry to read that you weren’t taught in an environment that encouraged(?) questioning these axioms and the assumptions underlying them. I’m an engineer by trade and spent quite a bit of my formative years convincing myself that many of those assumptions are valid, so I agree that it can be done. Didn’t you have to write out proofs in primary school? c.f. < http://en.wikipedia.org/wiki/Mathematical_proof >
You find this sort of thing all the time in education. (Bona fides: son of two educators and married to another. I know, what’s wrong with me?) It is a pretty efficient way to get to definitions and simple operations without arguing the example, as pointed out above. Get into the “buts” and “whys” too quickly, and it can easily go off the rails, whether you’re teaching basic mathematics or the history of science or journalism or whatever. Another way of stating it is that you have to know the rules before you know when to break them. Where this approach goes off the rails is when educators try to stay within this space beyond the point in time when it’s a useful place for learners to be in, for whatever reason; this can be very difficult when learners are at radically different levels of understanding. (Bona fides: gifted kid myself and father of two gifted, but very different, kids. I hate seeing them bored with certain subjects and unable to seek help in others because so much comes to them so easily. I also have more appreciation for what my parents went through.) It also argues against using word problems in math before learners have a good grasp of the fundamentals, a corollary to my contention that advancing without those fundamentals is insane.