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When students ask me for the use of the formal and abstract theory, I often would like to give answers they wouldn't understand. For instance, one application of abstract vector spaces and the banach fixed-point theorem is solving differential equations by applying the theory to function spaces. That's too much for a beginner.

A nice one is what is called the "football theorem" in german (Satz vom Fußball). Compare a football lying on the center spot before the game starts to the same football lying on the center spot before the 2nd half starts. There will be (at least) 2 point which are exactly in the same position, how ever much the ball was turned during 1st half. The proof is given by the fact that the composition of rotations still is a rotation having an eigenvector in three-dimensional space.

Is there more like this?

Edit: I think of first-year undergraduate students which are confronted with formal definition and proof for their first time and ask "why?".

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    $\begingroup$ I think that this is a fairly wide question without a specific subject and level stated. For example Brouwer Fixed Point Theorem or the Four Color Theorem would yield interesting examples, dependent upon the class. $\endgroup$
    – Chris C
    Mar 21, 2014 at 15:55
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    $\begingroup$ @ChrisC, the four color theorem is something everybody can grasp, the proof is really understood by nobody. $\endgroup$
    – vonbrand
    Mar 21, 2014 at 16:36
  • $\begingroup$ @ChrisC thank you, I have updated my question. $\endgroup$
    – Anschewski
    Mar 21, 2014 at 17:35
  • $\begingroup$ Once you leave the "immediately visible" for more abstract stuff, you soon reach areas where there the only way of "seeing" a result is through it's proof. Comes with the territory. $\endgroup$
    – vonbrand
    Mar 23, 2014 at 1:08
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    $\begingroup$ I disagree. And I think part of the intent of this question is that we, as practiced mathematicians, have managed to push that "immediately visible" barrier further along via "seeing" thru proofs. How can we help students push their own barrier further? $\endgroup$
    – Brendan W. Sullivan
    Mar 23, 2014 at 17:42

4 Answers 4

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To give a concrete example, consider the following task in geometry:

Let $A,B,C,D$ be four distinct intersection points of two parabolae with perpendicular axes of symmetry. Prove that $A,B,C,D$ are concyclic.

There is an almost trivial formal proof using analytic geometry. Any other approach I know is close to nightmare. You can find more about it at math.se: intersection of two parabolae.

If you dislike this particular example, there are combinatorial identities which are quite easy to prove formally (e.g. by induction or some manipulation), but rather hard via counting arguments. There were even some open problems (I don't know if those are open anymore) about identities for which there was no combinatorial interpretation known.

Another counter-intuitive example is about vector spaces (see here):

$$\dim(U +V + W) \neq \dim U + \dim V + \dim W - \dim (U \cap V) - \dim (U \cap W) - \dim (V \cap W) + \dim(U \cap V \cap W)$$

despite quite obvious

$$\dim (U + V) = \dim U + \dim V - \dim (U \cap V).$$

Although it is possible to describe it intuitively, it's much better to just give a counterexample (which constitutes a formal proof).

I hope this helps $\ddot\smile$

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Here's one that just occurred to me: differentiation. First-year undergraduates are likely to have been told that the derivative of $x^n$ is $nx^{n-1}$, and that this is the gradient of the graph $y=x^n$, but I can't think how you would explain why that is true without writing out the definition and seeing what happens.

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    $\begingroup$ I'm a few years late, but here's a possible "physicist's approach" to differentiating $x^n$: first by considering the units of $\frac{\mathrm{d}}{\mathrm{d}x} x^n$ it is clear that the answer must be proportional to $x^{n-1}$. To find the proportionality constant, consider a unit $n$-cube: increasing its side length by $\mathrm{d}x$ amounts to gluing a thin slab of volume $\mathrm{d}x/2$ to each of its $2n$ sides, increasing its total volume by $n\,\mathrm{d}x$, hence $\frac{\mathrm{d}}{\mathrm{d}x} x^n|_{x=1}=n$. $\endgroup$
    – David Zhang
    Mar 4, 2018 at 23:30
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    $\begingroup$ @DavidZhang Sounds dubious to me, especially considering that derivatives don't have units (along with all other mathematical functions). $\endgroup$
    – Jessica B
    Mar 5, 2018 at 22:22
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Edit: I think of first-year undergraduate students which are confronted with formal definition and proof for their first time and ask "why?".

Responding to your edited side-question: I'm always happy to have this question asked, because it cuts to the heart of the mathematical discipline. It's worth spending a few minutes on anytime it comes up. My current answer ticks off several bullet points:

  • To guarantee that our theorem/shortcut is actually correct.
  • To explain exactly why the theorem works in simpler steps.
  • To come to a communal agreement before moving on.
  • To practice old skills and methods that will be used later.
  • As an aid for remembering the structure of the theorem or formula.

In many ways the mathematics discipline is wonderful because it's the only field where we can accomplish all of this, essentially for free, by just talking and pencil and paper. Only math can actually prove things for certain (other sciences at best develop theories from inductive evidence). Only math can establish the principles by pure logical thought, without resorting to expensive or complicated experimental apparatus. Only math allows everyone to comment and participate, without trusting some authority that outside evidence was acquired correctly.

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There are many existence proofs that are formally not difficult but I think could be useful examples .E.g., Euclid's proof that there is no largest prime.Rolle's theorem (calculus) and its main corollary,the Mean Value theorem. There are also some puzzles that should only be solved analytically rather than constructively, such as "Can a chessboard, with 2 diagonally opposite corner squares removed, be tiled by non-overlapping 1x2 dominoes?" (Black/white square parity problem.)

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  • $\begingroup$ Could you expand on one or more of your answers (which run the gamut from number theory to calculus to recreational mathematics!) and how they answer the question originally posed? In their stated form, I am not sure why/how they would serve as good examples for the OP's purpose. $\endgroup$
    – Benjamin Dickman
    Nov 21, 2015 at 9:29
  • $\begingroup$ I was thinking about useful results that can be proven without any specific construction, The formal theory doesn't merely catalog the rules but allows us to consider it from the top down, so to speak, which leads to different , more general questions with useful answers. Instead of differentiating specific functions, finding tangents, etc.It also helps to clarify what we are actually talking about. "A vector (is something that ) has position and direction.". To which I once replied "So a bus is a vector?" $\endgroup$
    – DanielWainfleet
    Nov 21, 2015 at 19:17

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