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I don't know if this is the right place to post this or not, but I will go ahead anyway (sorry if it ain't the right place)

Yesterday I was discussing a particular theorem of geometry with my brother which he just learnt in the school. I had asked him if he knew the proof for it, he replied saying his teacher has said that wouldn't be necessary.

Then, I asked him to sit and try to prove the theorem. He said that knowing the theorem counts for more than knowing the proof. How do I explain to him that knowing the proof is more important and how it can even help expand his thinking?

I know this question doesn't have a single pointed answer as is pre-requisite for questions posted here, but I would appreciate any replies

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Knowing both is important. Proofs can present widely-applicable techniques and sharpens your thinking skills, while theorems themselves also provide tools for tackling problems and serve as building blocks for all theory. That being said, it is nice that you are trying to get your brother to stop taking things on faith! –  Fantini Dec 10 '13 at 2:30
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I wouldn't like for a student of math to be just a trove of statements he can remember. Knowing why the statements are what they are and how they can be proved is also equally important is what I feel. Anyway I would like to hear more about this from this community –  Bach Dec 10 '13 at 2:33
    
In the real world, very few problems can be solved using a memorized theorem. You may have to calculate probabilities, but you'd better bet that it isn't as simple as looking up Permutation on wikipedia and punching the numbers in to the formula. Understanding why things work is far more valuable than what the simplest way to calculate a textbook problem. At least in my experience. –  jmac Dec 10 '13 at 5:53
    
However, sometimes it is more practical to just know the statement of the theorem... and leave the proof for when you have the time! –  becko Dec 12 '13 at 0:50
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6 Answers 6

An idea on why learning proofs (not just theorems) is important: One could say that it is unimportant to know how to prepare food because there is a McDonald's down the street. But, if a person becomes strictly reliant on McDonald's for preparing food, then we can be assured that (s)he will never be able to produce a (worthwhile) dish of their own creation.

Likewise with proofs--one could say it is unimportant to know how to "prepare" the "food" of a theorem via proof because there is the "McDonalds" of the math book nearby. But, after years of just relying on memorizing theorems, a person will never be able to come up with a sound theorem of their own.

Being able to prove something makes it much more solidified in one's mind, and gives you a tool that is applicable to many circumstances, not just a single instance. For example, my double angle formula may not be useful when I need a triple angle formula, but I could use the proof/derivation of the double angle formula to find my own triple angle! This is where proof is much more powerful than memorization.

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One approach would be to give an example of a theorem that sounds so absurd that noone in his right mind would accept it without proof. For example, mention the Banach-Tarski result to the effect that the unit ball can be decomposed into 5 parts which can then be reassembled by rigid motions into... two unit balls.

Another example would be the equality of $0.999\ldots$ to $1$. Of course, this one he won't believe even after you give him a proof :-)

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I'm sure I could write more about why proofs are important, but I believe it all boils down to this. Proofs are important because some very intuitive results turn out to be false, and if we just accept results with no proof because something 'sounds correct', then we could be led down false pathways. Who knows what horrors that could reap?

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So true. This happens all the time in higher mathematics; heaps of published proofs are wrong, and every so often a false theorem ends up being believed for many years. However, the question was more about theorems that have been independently proved by many people, which are therefore almost certainly correct. –  goblin Dec 11 '13 at 13:50
    
I see. Still, I believe the principle applies. Developing that questioning and skeptical attitude for later, when the theorems are not so clear cut. –  tylerc0816 Dec 11 '13 at 15:59
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I think the dislike for proofs, and the belief that they must not be important, comes from two misconceptions:

  1. That proofs are all two-column proofs.
  2. We prove things to find out if they are true.

The first is blatantly false. Most proofs are paragraph proofs, and don't have to show every tiny little step. No one will be displeased with you if you commute a few variables here and there without explicitly saying so. Often, proofs are taught in high school with a very specific format, and straying from it is penalized. This makes it seem pretty arbitrary and dumb.

The second is a little more subtle. Proofs do, in fact, prove that things are true. One benefit of that is that we can be absolutely certain we're working "on solid ground", so to speak. But if you aren't sure if something is true, you don't go trying to prove it first. The more important aspect of a proof is that it is a justification as to why something is true. Proving the "why" of something gives you

  • the potential for more generalization
  • a larger toolbox for proving other things

For the first point, if you notice Möbius inversion works for the totient function, you could just prove that and call it a day. But if you look at the proof, you might notice that it doesn't actually involve computing the totient function at all, and it just uses the fact that it's multiplicative. (Well, more likely it'd take a bit of needling to give up that fact, but it's still possible to pry it out) This gives you a far more general theorem, one that would be much harder to find if you only knew the special cases, without proof.

For the second point, anorton's answer has a very good example involving double and triple angles.

Ultimately, proofs are for verifying results that you're pretty sure are true. Because everything you're taught has been proved, this seems useless for quite some time. But when you actually do need to find new things, proofs are indispensable.


The other comparison I've heard is to science labs. Yeah, why go and test that $F = ma$ is true if we can just check the physics book? Because that's not how science is done out in the real world. You use it to support hypotheses that you have. (Although in math, you get to prove them!)

On the other hand, this can be a dangerous analogy, because this is exactly the opposite of how math works. You don't conjecture things on paper and prove them in the lab, you conjecture with numerical examples (the lab) and then you prove things on paper.

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Proofs are important because they tell you why the theorem is true: no amount of practical experience with the use of some helpful formula can ever explain its existence, and this renders mathematics just a bland narrative of assertions to be judged on the merit of their applicability to one's occasional interests. To take a geometric example, I don't think it's at all relevant to my life that the incircle of a triangle is centered on the intersection of the three angle bisectors, but after a moment's thought, I found a cute proof by symmetry; you wouldn't think that would work, given that the triangle likely is completely asymmetric. So now I like the theorem, and ten minutes ago, I didn't care. I still have no use for it.

Edit: The proof, by the way, is to focus on the circle rather than the triangle. Pick any two tangent lines and let $P$ be their point of intersection. The line from $P$ to the circle's center is an axis of symmetry for this picture, which you can prove by dropping radii to the points of tangency and using similar right triangles. Therefore it bisects their angle, and that means that the center lies on the angle bisector between any two sides of a triangle to which it is tangent. If that's not Euclid's proof, it certainly could be.

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I particularly like this answer because it emphasizes the multiplicity of proofs. Practically speaking, proving the same result in different ways leads to different generalizations. Aesthetically speaking, as pointed out in this answer, different proofs appeal to different readers. –  Karl Kronenfeld Dec 10 '13 at 3:33
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In my opinion, they're not important. But let me qualify that in a massive way.

Suppose we're interested in solving some particular real-life problem. We're going to make some "let" statements. Like maybe we've got $(x,y) \in \mathbb{R} \times \mathbb{R}$. And to solve the problem, we're going to let $(r,\theta)$ satisfy $x = r\cos \theta$ and $y = r\sin \theta$. Why are we allowed to do this? We're actually leveraging a theorem:

For all $x,y \in \mathbb{R}$, there exists $r \in \mathbb{R}_+$ and $\theta \in \mathbb{R}$ such that $x = r\cos \theta$ and $y = r\sin \theta$.

This legitimizes the "Let" statement. So theorems tell us the rules of the game; and therefore, the range of problems the human race is capable of solving is hard-limited by the theorems we know. So theorems are important.

Does that mean you personally ought to know the proof of a theorem? That we're not allowed to use it 'til we know the proof? Of course not.

That being said, if you want the ability to prove your own theorems and thereby move the human race forward, expanding the range of problems that we're capable of solving, well you'd better start going beyond the question: "What can I do with this theorem?"

You'd better start asking: "How do we even know its true?"

By the way, most theorems in mathematics aren't proved because of their direct applications to real world problems. This begs that we ask: "Why should we care about theorems that don't have direct applications to real-life problems?" You should post another question if this topic interests you.

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