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I'm currently studying precalculus in high school and have no hands-on experience with advanced mathematics (calculus and beyond). Every time I learn something new, I feel the need to connect it with other branches that I've learned. I try to do this by coming up with a unified representation of mathematics; however, it always fails every time. Should I be looking for deeper connections in mathematics at my level, or should I be content with just learning/understanding each topic? Does the abstraction of mathematics come with mathematical maturity? Insightful answers (preferably from experienced mathematicians) would be greatly appreciated.

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closed as primarily opinion-based by user223391, Moya, achille hui, Shailesh, Daniel W. Farlow Jun 15 '16 at 1:05

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Yes, learn all you can, leave time for your mind to run free, and you’ll find that some things connect to other things in ways that you didn’t expect. Above all, try to find many ways of looking at one thing. You want the fourfold vision that Blake imagined Newton didn’t have. $\endgroup$ – Lubin Jun 14 '16 at 23:01
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    $\begingroup$ I should warn that, while I like this question, it risks being closed as too opinion-based and/or too broad. $\endgroup$ – Semiclassical Jun 14 '16 at 23:07
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    $\begingroup$ I suspect that, in general, spending time thinking about mathematics or trying to do things with mathematics is useful to learning mathematics more deeply. $\endgroup$ – Milo Brandt Jun 14 '16 at 23:26
  • $\begingroup$ Perhaps abstraction is making connections with new problems, while connections with previous material seems more "concrete". Both are essential aspects of the mathematical journey, at every level from counting to calculus and beyond. $\endgroup$ – hardmath Jun 15 '16 at 0:47
  • $\begingroup$ Very brief thought: Your urge to search for connections is a good one, and if you respect it, it will take you far in this subject. However, you should also learn to be satisfied with the answer "there does not seem to be a connection here". Sometimes you will be wrong: there is a connection that is just hard to see. But sometimes, the reason a connection is hard to see is because it doesn't exist. $\endgroup$ – Eric Stucky Jun 15 '16 at 1:27
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I would absolutely disagree with those who would discourage such thinking. Mathematics was founded by people who asked "why?", who asked how things fundamentally work. New questions in mathematics are all tied to this way of thinking. It is true that you lack much of the fundamentals necessary to abstract much, but do whatever you can.

At every level of mathematics there are connections you can find... let me point out Pascal's Triangle as an easy example. The triangle shows a beautiful connection between probability and polynomial expansion (i.e. Algebraic manipulation) and is not at all "expected" in the conventional sense.

Note here that I am not a formal mathematician... while I have studied partway through Complex Analysis on my own I remain a High School student myself, so my views may be biased. Nevertheless, just browse through some of the popular questions and answers on this site. While some are popular for no good reason, many are highly voted because of the deep insights involved - again, deep thinking is the core of mathematics.

I would further add that you are not too young to do mathematics... while abstraction definitely comes with mathematical maturity, mathematics is often actually considered a young man's game. While you may, again, be limited in how far you can abstract at this point, do as much as you want. There is no harm in it, and all it could lead to is a more inquisitive mind, another key to being a mathematician or having appreciation for mathematics in general.

As a closing remark, let me make a comparison I often make... that of mathematics to artwork. Both have their applications, and their abstractions. What good has abstract art done for the world? One could argue very little, but I would say otherwise... it has inspired many men and likely led to many great things. Abstract mathematics is much the same. If you get nothing else but enjoyment for the beauty of mathematics from your ponderings, so be it.

Addendum
I am curious what kinds of mathematics you have considered generalizing? I can try to help point you in the right direction if you wouldn't mind sharing the aspect of pre-calculus you have desired to abstract.

Here is a bit of abstraction from my days taking a Pre-Calculus course: $$\sum_{k=0}^n \binom n k (-1)^k (x+k)^n = (-1)^n n!$$
While I wasn't able to prove this statement until much later, I discovered that identity by just subtracting adjacent integers in polynomial expansions, which had long bored me in my Algebra class due to their "perceived uselessness" (at the time that is.. my views have clearly changed!). If you desire to see a proof of this you can see my question here. (Note that the easiest elementary proof would be to first read the comments and then the accepted answer and combine the two, though this may be above your level)

I would cite an example such as this as reasoning for why abstraction is so useful, even at lower levels of mathematical maturity... that formula above is what really sparked my love of mathematics, along with a curious few other identities I discovered. "Why is that sum independent of $x$?!?!" I thought to myself at the time, and thoughts such as that deepened my understanding of mathematics as a whole

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  • $\begingroup$ May I ask why the downvote? I am curious what someone would like to me to improve on $\endgroup$ – Brevan Ellefsen Jun 14 '16 at 23:13
  • $\begingroup$ I have considered abstraction in the complex plane, specifically as shown towards the end of this video: youtube.com/watch?v=F_0yfvm0UoU . He describes concepts such as basic arithmetic as movements in the complex plane, there is also a follow up video about derivatives that elaborates on this visualization: youtube.com/… . After looking at more advanced math concepts, I have become more and more unsure about the accuracy of this visualization. $\endgroup$ – guest Jun 14 '16 at 23:33
  • $\begingroup$ @guest ah, 3Blue1Brown... a fantastic channel. The visualization is actually very accurate! The basic idea here is that $e^{ix} = \cos x + i\sin x$, a formula used often in mathematics. There is actually a way to connect this to those rotations and movements shown to Pre-Calculus! Remember your unit circle? It has $\cos \theta$ as the $x$ coordinate and $\sin \theta$ as the $y$ coordinate. This also holds in the complex plane! In this case we have $\cos x$ being the horizontal component again (our real part, the horizontal axis) and $\sin x$ being the imaginary part (the vertical axis once... $\endgroup$ – Brevan Ellefsen Jun 14 '16 at 23:37
  • $\begingroup$ again, as noted by the "$i$" out in front.) Because of this connection we see that $e^{ix}$ represents a rotation in the complex plane, which can then be extended to explain the concepts shown in the video! $\endgroup$ – Brevan Ellefsen Jun 14 '16 at 23:39
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    $\begingroup$ Your explanation of the concepts in the video and your experience with abstraction during your precalculus course encourages me to continue exploring mathematics- even if I can't connect all the dots just yet. Thanks for the advice! $\endgroup$ – guest Jun 14 '16 at 23:53
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Sadly there are alot of topics in pre-calc that are pretty much useless and far too biased toward geometric intuition rather than abstract math. You'll find that a good many mathematicians taught themselves a variety of topics at an early (around what I presume is your) age. You are short changing yourself if you only learn what is put in front of you. Go to the library and grab a book on linear algebra. Despite the fact that it often has calculus as a pre-requisite, it is a very good place to start learning abstract mathematics. Try to find a text that emphasizes the abstract perspective, rather than the geometric.

Also, you have to understand that your early mathematics (moreso in high school but most college calculus classes are just as guilty) classes will try very hard to convince you that mathematics is a disjoint set of unrelated algorithms and concepts. Resist this. Always try to put concepts in context, and the fact that you are already thinking about this is a good sign for you.

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  • $\begingroup$ I feel the need to point out that "abstract" and "geometric" can go together quite well. As, for example, in the work of Sophus Lie. Although there's no need to favor one over the other, I prefer geometry! $\endgroup$ – pjs36 Jun 14 '16 at 23:38
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    $\begingroup$ I mean more in the sense that one appeals to a visual argument or "eye test" as a primary means of understanding or proving something. I think geometric intuition is better served to supplement abstract mathematics rather than supplant the latter entirely in the some of the most formative years of a students career. $\endgroup$ – Wavelet Jun 14 '16 at 23:52

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