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I came across this quote by Eric Weinstein that "when things got supposedly more advanced, they actually got simpler because mathematicians started revealing what was powering all the things that you previously learn." I was wondering if anyone came to the same conclusion, and if so, care to give an example of such a revelation.

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  • $\begingroup$ Example: When Euclid proposed his Elements. More about him can be found below: math.twsu.edu/history/men/euclid.html $\endgroup$ – Mr Pie Nov 22 '17 at 3:19
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    $\begingroup$ most of the essential things i learned in undergrad boil down to simple consequences of properties of linear operators over an appropriate vector space, and certainly the latter theory is relatively more modern and "advanced". on the other hand, while the objects you look at in functional analysis are often indeed formally "simpler" (less equipment is involved in constructing them) i would hesitate to say they are simpler to think about... $\endgroup$ – enthdegree Nov 22 '17 at 10:10
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    $\begingroup$ A lot of advanced things generalize more rudimentary things. This is by design. $\endgroup$ – immibis Nov 22 '17 at 21:26
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    $\begingroup$ One of my revelations was that all the rules for derivatives - chain rule, multivariate chain rule, product rule, etc., all boil down to $d(f\circ g) = df \circ dg$. $\endgroup$ – Paul Sinclair Nov 23 '17 at 4:46
  • $\begingroup$ Depends what you consider proofs. Usually theory is considered advanced math. $\endgroup$ – Andrew Scott Evans Nov 24 '17 at 1:13
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Terry Tao once talked about a concept I had devised too: Symmetrization. For this an example is: "A man saw a circle, another saw a rectangle: the two were seeing a cylinder". This means that whenever you have two dissimile structures, what you like to do as a mathematician is finding a "bigger picture", this is not to say that everything is compatible, just the opposite: this means to know exactly what constraints and what liberties are in the problem we are understanding, this means not everything is possible. So then, here comes an important element.

"Advanced math makes math simpler, not more difficult" which is like killing bugs like the irrationality of $\sqrt{2}$ with atomic bombs like Fermats Last Theorem means nothings because simply the latter knowledge goes first. Theres a clear ordering of the value of knowledge created by its logical interdependences.

I wouldnt say advanced math "powers" basic math, its the opposite (remember the previous paragraph) but its definitely great when you can have a bigger picture and retrieving also the details in your mind which makes computation easier. It would be some sort of putting some results at the same plateau, etc ,etc.

[Its important to note here that knowledge is stable, and so the brain can remember better when there are logical connections between facts, this way, computation, os power usage becomes more efficient. Seeing the bigger picture (a collection of principles) and not only sparse results does exactly that.]

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    $\begingroup$ I believe Fermat's Last Theorem is not powerful enough to prove $\sqrt{2}$ irrational. It can show irrationality for $\sqrt[3]{2}$ and higher roots of course. $\endgroup$ – Tavian Barnes Nov 23 '17 at 14:48
  • $\begingroup$ @TavianBarnes yes, because $p^2+p^2=q^2$ can be solved over integers $\endgroup$ – Michael Freimann Nov 23 '17 at 17:37
  • $\begingroup$ you are both right, my bad $\endgroup$ – user499752 Nov 23 '17 at 19:12
  • $\begingroup$ @MichaelFreimann your comment somehow seems off... $\endgroup$ – Servaes Nov 24 '17 at 2:14
  • $\begingroup$ @Servaes no, I point out why the proof fails $\endgroup$ – Michael Freimann Nov 24 '17 at 5:16
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This is true whenever you learn a "powerful tool" in high school maths. The two main examples that come to mind are trigonometry and calculus.

In trigonometry, we had to memorise a large number of trig identities. We would learn that $\tan$ is basically "defined by" $\tan(x) = \sin(x)/\cos(x)$, but apart from this it was all just memorising the double angle formulae and so on. Early on in university maths, I learned that $e^{iz} = \cos(z) + i\sin(z)$, and all of the trig identities become obvious. But even then, I did not really understand what $e^{iz} = \cos(z) + i\sin(z)$ meant. Eventually, I studied complex analysis, and the notion of complex exponentiation made sense. At this point, everything that flowed on from it became clearer. (As an added bonus, understanding the proof behind the residue theorem meant I was also finally able to see why various integrals I memorised for my physics courses worked.)

A second related concept was calculus. When I first "learned" that $\int \frac{\mathrm{d}x}{x} = \ln(x)$, I had no idea why. Most of integration was just memorisation in high school. In my first maths course at university, I was finally properly taught what functions were, and what some of their properties are - and about inverse functions. Now, together with the chain rule, and knowing that $x\mapsto \ln(x)$ is the inverse of $x\mapsto e^x$, the lecturer showed us a simple reason why the derivative of $\ln(x)$ is $1/x$ (assuming it is in fact differentiable).

$$1 = x' = [e^{\ln(x)}]' = e^{\ln(x)}\cdot \ln'(x) = x\ln'(x)$$

Thus $\ln'(x) = 1/x$, and so an antiderivative of $1/x$ is $\ln(x)$. Seeing the proofs of the fundamental theorem of calculus was fantastic as well. In high school I asked why integration was antidifferentiation, and I was told "of course it is", but it was never obvious at all. They seemed like entirely different concepts so I couldn't understand how everyone was so okay with the close way they're tied together. Studying real analysis really helped to understand why calculus works.

As for the maths I've learned at university, all of my courses were well designed so they didn't use any tools that weren't proven in the course/earlier courses. I can't give any higher-level examples for your question.

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    $\begingroup$ I take issue with your simple 'reason' for the derivative of $\ln$. The chain rule is valid only if the derivatives of the two functions exist. If you do not show that, then you cannot apply the chain rule. So it is not so 'simple' as your answer makes it seem to be. $\endgroup$ – user21820 Nov 22 '17 at 8:51
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    $\begingroup$ @HuiWang: This answer is also wrong that integration is 'of course' the same as anti-differentiation. It is not. Let $f$ be the function on reals such that $f(0) = 0$ and $f(x) = x^2·cos(1/x^2)$ for every real $x \ne 0$. Then $f$ is differentiable, and hence $f'$ has an anti-derivative. But $f'$ cannot be integrated (Riemann or Lebesgue) on any interval around $0$. $\endgroup$ – user21820 Nov 22 '17 at 8:57
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    $\begingroup$ I would like to add to this answer on how Complex Mathematics made this... well... less complex for me in electrical engineering. Doing the math on alternating currents involve lots and lots of vector operations, especially when inductors and capacitors are involved. While vector operations are easy to grasp on a conceptual level, it makes for some pretty messy math. But if you instead of vectors switch to a complex representation of currents, voltages and impedance, then everything just drops back to Ohm's Law and becomes stupefyingly easy. $\endgroup$ – MichaelK Nov 22 '17 at 9:06
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    $\begingroup$ I would like to nit-pick about where you say "the antiderivative". There is no "the" antiderivative, there never is. There is only an antiderivative. $\endgroup$ – Arthur Nov 22 '17 at 10:06
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    $\begingroup$ @user21820 But this answer isn't claiming that "integration is 'of course' the same as anti-differentiation". This answer is simply saying that's how it was presented in high school. $\endgroup$ – David Z Nov 23 '17 at 1:15
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The view that advanced mathematics is "what is really going on" in elementary mathematics is generally a fallacy in my opinion.

It is easy when learning (or even when teaching) to fall into the trap of thinking that, for example, trigonometric identities are because of complex numbers. But this amounts, I believe, to a large extent to confusing cause for effect.

It would be truer to what is really going on to say that we care about complex analysis because it happens to simplify (among plenty of other things) reasoning about trigonometry.

There is an infinity of possible "advanced" structures that we could reason about mathematically -- but the ones that get any effort spent on them are the ones that can (or are hoped to) lead to answers to questions that we can already ask without the advanced theory. Often they do this by unifying and generalizing concepts we already have.

An advanced theory earns its way by answering questions that can be asked in more elementary terms but are too hard to be answered by elementary techniques. Along the way, as a side effects, it is common for a lot of questions that can be answered with elementary tools to become simpler and easier to answer using the advanced theory -- and as a matter of research direction this is often used as a touchstone for whether we're getting closer to beating the currently-too-hard problems.

And, of course, once we have the advanced theory, that gives us a chance to ask even harder questions that we couldn't even have thought of before, which it will become the task of the next advance to solve.

But saying that it is the advanced theory that "powers" or "generates" the original elementary phenomena is putting the cart before the horse.

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  • $\begingroup$ The sentence in bold is key $\endgroup$ – user499752 Nov 22 '17 at 23:20
  • $\begingroup$ I agree with your reasoning, but I'd say the conclusion is positive: it does power it in that advanced mathematics gives different and often better/simpler/shorter ways of thinking about rudimentary mathematics. (Different interpretation of the word "powers".) $\endgroup$ – dafinguzman Nov 23 '17 at 20:39
  • $\begingroup$ @AlfredfromBatman: If only it were a complete sentence. $\endgroup$ – einpoklum Nov 24 '17 at 12:27
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I think that puts it rather backwards. Simpler concepts lead to, and often mirror more complex ones. Take the concept of equality, for example. Two objects are equal if, and only if, they are the same object. From this concept, we create the concept of an equivalence relation. An equivalence relation exists between two objects if the relation R in question is reflexive, (for all a, aRa), symmetric (for all a, and b, if aRb, then bRa), and the relation is transitive (for all a, b, and c) aRb and bRc implies aRc. We can take this further, however. In topology, we say two topological spaces are equivalent if, and only if, they have exactly the same set of topological properties. This is exactly the same idea, even though in practice all we can say typically is that two spaces are not toplologically equivalent, because we can show that one has a property the other does not have. In practice, all we are concerned with is the later, because the former is unprovable. (We do not know the entire set of possible topological properties, and cannot say that two spaces have ALL properties in common.) So, we build complexity from the more simple to the less simple, and not the reverse.

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What you are asking, in some sense, is "Is abstraction useful?" Abstraction is the art of embedding easy-to-understand ideas inside of harder-to-understand ideas. As to the usefulness of this paradigm (and the answer to my paraphrased version of your question), it depends. But definitely sometimes abstraction has been helpful.

Group theory is the best example of abstraction put to good use, in my opinion, and has proven useful in an extremely wide variety of problems. Roughly speaking, the success of group theory stems from its ability to put ideas of symmetry in algebraic language. Applied to polynomial equations, it leads to Galois theory, which provided a unified way of understanding a potpourri of results like Abel's theorem, the quadratic formula, Tartaglia's formula, etc. Sophus Lie extended the abstract ideas of Galois theory to differential equations, yielding what we now know as Lie theory, which is a cornerstone of modern theoretical physics. The representation theory of groups is also extremely useful in e.g. quantum mechanics (where it clarifies the structure of angular momentum).

Category theory is an even more extreme example of useful abstraction. It has proven to be a unifying theme in topology.

On the other hand, abstraction can sometimes be distracting. Terrence Tao has stated in a blog post that in PDE theory, abstraction has rarely produced breakthroughs (although it sometimes can help produce better exposition of known ideas). To quote:

At its best, abstraction can efficiently organise and capture the key difficulties of a problem, placing the problem in a framework which allows for a direct and natural resolution of these difficulties without being distracted by irrelevant concrete details.... At its worst, abstraction conceals the difficulty within some subtle notation or concept... thus incurring the risk that the difficulty is “magically” avoided by an inconspicuous error in the abstract manipulations.

Later he states

The field of PDE has proven to be the type of mathematics where progress generally starts in the concrete and then flows to the abstract, rather than vice versa.

So the usefulness of abstraction depends on the situation. But definitely it is sometimes useful.

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  • $\begingroup$ thats why for me abstraction in mathematics is a disease. Abstraction only arises because we have already got perfecto dominion over a particular area. I dont buy the "theory builder vs problem solver mathematician diichotomy" I state as absolute truth that "theory building" is putrefact garbage and only problem solving creates real mathematics. $\endgroup$ – user499752 Feb 3 '18 at 19:58
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A yet another suggestion: Group theory.

A lot of very basic maths (like, you can switch the summands) can be generalised to group theory. The actual root of it is an (successful) attempt to marry the integers $\mathbb{Z}$ and polynomials $R[x]$.

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  • $\begingroup$ "Marrying integers and polynomials" would sound like it rather leads to (commutative) ring theory. $\endgroup$ – Henning Makholm Nov 29 '17 at 11:42

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