# vi21maobk9vp comments on Which fields of learning have clarified your thinking? How and why? - Less Wrong

12 11 November 2011 01:04AM

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Comment author: 11 November 2011 05:51:27AM *  0 points [-]

Set theory also helps one see mathematics as a whole and see how different areas connect to each other.

For what it's worth, I strongly disagree. For a new student too much emphasis on foundations can be a major mental block when getting used to a new idea and especially a new circle of ideas. Set theory is used very informally in most of mathematics, as a notation. To learn more than this notation is mostly unnecessary for pure math, completely unnecessary for applications of math to other areas.

Well, the difference of mathematics from natural sciences that you need not only to build a good model of something, but also to describe it using a limited set of axioms (using proofs from there on).

For some people set theory is the area of mathematics which quickly reaches proofs that are accessible to our reasoning, but transcend our intuition. Sometimes even the notions used are easy to define formally but put strain on your imagination. For people inclined to mathematics this can be a powerful experience.

Comment author: 11 November 2011 06:09:04AM 2 points [-]

But that's nothing special about set theory. I prefer to think that the role of mathematics (at least the best kinds of mathematics) is to correct and extend our intuition, not to "transcend" it. But the kind of powerful experiences you describe were available two thousand years before the invention of set theory, and they're available all over modern math in areas that have nothing to do with set theory.

Comment author: 11 November 2011 07:44:20AM 1 point [-]

Not every student would benefit from learning set theory early beyond the universally needed understanding of injective/bijective mappings, but some would. It does depend on personality. It has some relation to cultural things.

"Role of mathematics" implies relatively long run; experience is felt in a very short run.

If you want to extend your intuition into an area nobody understands well, you often need to combine quite weak analogies and formal methods - because you need to do something to get any useful intuition.

There are many branches of science where you can get an amplified feeling of understanding somthing in the area where you don't have working intuition. There are three culture-related questions, though. First, how much (true or fake) understanding of facts you get from the culture before you know the truth? Second, how much do you need to learn before you can understand a result surprising to you? Third, is it customary to show the easiest-to-understand surprising result early in the course?

Of course, for different people in different cultural environments different areas of maths or natural sciences will be best. But it does seem that for some people the easiest way to get an example of reasoning in intuitively incomprehensible (yet) area is to learn set theory from easily accessible sources.