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Learning Abstract Math from First Principles?

by Teach1 min read1st Dec 20192 comments

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I find that I am usually quite good at applied math, and enjoy it. I am taking a course currently that is split into two parts, Vector Calculus and Complex Analysis. The vector calculus makes sense to me and I can see how and why it works, and I find it interesting and enjoyable to learn.

On the other hand, I spend quite a bit of mental energy wrapping my head around the hows and whys of the more abstract complex analysis. I am not sure if I enjoy abstract math or not in general because I do not understand it as well. So, my question: Does anyone have any recommended resources for learning (any) abstract mathematical topic from first principles, that explains reasonably well what's going on with the math, rather than just how to do it?

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For your specific situation, may I recommend curling up with Visual Complex Analysis for a few hours? 😊 http://pipad.org/tmp/Needham.visual-complex-analysis.pdf

On a more general note, I find that anyone who says they "learned it from first principles" is usually putting on airs. It's an odd intellectual purity norm that I think is unfortunately very common among the mathematically- and philosophically-minded.

As evolved chimpanzees, we are excellent at seeing a few examples of something and then understanding the more general abstractions that guide it on a gut level; we have an amazing ability to arrest form from thing, but our ability to go the other way around is a lot more limited.

I think most of your intellectual idols would agree that while eventually being able to build up "from first principles" is a great goal to shoot for, it's actually not the pedagogy you want. It's okay to start concrete and just practice and grind until the more abstract stuff becomes obvious!

Take it from a guy who leapt off the deep end this quarter into abstract algebra, real analysis, signal processing and probability theory at the same time -- there is no way I would be performing at the level I am in these classes if I didn't force my abstraction-loving ass down to ground level and actually just crank out problem sets until the abstractions finally started to make sense.

I don't really understand what you mean by "from first principles" here. Do you mean in a way that's intuitive to you? Or in a way that includes all the proofs?

Any field of Math is typically more general than any one intuition allows, so it's a little dangerous to think in terms of what it's "really" doing. I find the way most people learn best is by starting with a small number of concrete intuitions – e.g., groups of symmetries for group theory, or posets for category theory – and gradually expanding.

In the case of Complex Analysis, I find the intuition of the Riemann Sphere to be particularly useful, though I don't have a good book recommendation.

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6Pattern1yWithout examples of someone who did it, I'm guessing Learning from first principles is a myth.

The treatment of this subject in Novum Organum might be illuminating.

From The Baconian Method:

104. But the intellect mustn’t be allowed •to jump—to fly—from particulars a long way up to axioms that are of almost the highest generality (such as the so-called ‘first principles’ of arts and of things) and then on the basis of them (taken as unshakable truths) •to ‘prove’ and thus secure middle axioms. That has been the practice up to now, because the intellect has a natural impetus to do that
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1Pattern1yThere are small things someone might notice or come up with on their own*. Learning, the first time around, seems like it's usually not done that way. *Like how to take a sequence of numbers (1, 4, 9), and come up with a polynomial equation that fits (f=x^2). Working out that there's a faster/algebra way to add consecutive numbers together is fairly straightforward (and probably anything else Gauss [https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss] did is more impressive).
1Pattern1yMaybe Euclid [https://en.wikipedia.org/wiki/Euclid%27s_Elements] did? There's also some relevant philosophy of science from Einstein [https://plato.stanford.edu/entries/einstein-philscience/#PriTheTheDis] (a la plato.stanford.edu [https://plato.stanford.edu/]):