Derivation vs. proof

I imagine that Ramanujan and many other gifted mathematicians achieved results by derivation, meaning they followed a trail that led to a result, then proved it later if at all. Is this so and if so, why aren't derivations emphasized in education instead of proofs? I find the statement of a theorem without context and without relevance (bearing, concern, objective) and without origin, to be incomprehensible. Is it just me? Do folks who seem to absorb these like sponges have some special facility that I lack? I seem to recall reading that when Newton was criticized for not showing how he came to his conclusions, he said that when his building is finished, he takes down the scaffolding. Doesn't the derivation of a result reveal its bearing? I rarely find that a proof does.

Thank you,

Solution 1:

I know this is an ancient question, but given that there are no answers, let me give it a shot. As a disclaimer, this is a rather philosophical (or soft, if you may) answer, which I believe is justified by that the question is a soft one.

First things first, I believe calling "following a trail that led to a result" a "derivation" is misleading: the word "derivation" somewhat has a connotation of a construction, or taking explicit steps to find some other thing from one thing. So instead let us call what you call a "derivation" "intuition". Of course a better way to go about this is not to use such problematic words and make use of a spectrum that includes philosophy, mathematics and physics (and quite possibly other disciplines as well, but this whole idea is all so cumbersome to write down for now) (In your case I would consider you closer to physics, for instance, whereas I am more towards philosophy, and I am considering the spectrum to be somewhat of linear nature, which is unjustified). Indeed as you have said, oftentimes intuition comes before concrete proofs in mathematics: one makes sure that something has to be the case, and then tries to prove it. This is more so in the frontiers of mathematical knowledge, for there are no nice roads to walk on there; only muddy (and often dark) trails.

Next, we should determine what kind of education you are talking about. For your concern to make sense, I believe we should restrict education to education given by an institution: colleges, highschools etc.. So we disregard all self-studies, where one is basically by himself. For a similar reason, also disregard post-qualification exams in graduate life. With this restriction I can now talk about my understanding of the subject.

I have taken courses from mathematicians with diverse understandings of how a course should be thought, e.g. one doing every single proof and throwing in historical context every once in a while, another giving no proofs but only sketches, or ideas of proofs at most, another mostly demonstrating theorems on examples etc.. But in any of my studies I have seen that intuition and rigor go hand in hand, and that's what makes mathematics both very strict and also very flexible: otherwise either everything is trivial, or "can be seen from the picture" (as it happens in geometry), or else you lose track of what kind of objects you are working with (as it happens in analysis). Of course these are extreme (and to a greater extent false, consequently) statements, but in a "sentimental level" they have some truth in them. As long as there is human interaction I believe intuition is never left behind in communicating mathematics, though if you provide some clarification maybe I can modify my answer.

That leaves one final thing to be accounted, namely written mathematics. Again, from my experiences, mathematics addressed to a general audience, or expository mathematics usually does emphasize intuition, so that leaves mathematics textbooks for the [candidate] mathematicians, where everything has the potential to seem like cryptic and incomprehensible: out of the blue results, without any motivation provided for the reader that it is a crucial thing to learn about them. But here again, the answer lies in realizing that written mathematics expects a lot from its reader, until the reader has internalized what has been written, that is. Oftentimes, perhaps possibly because of the concise character of a mathematical script motivation, context and intuition is packed in one footnote; or such a part may lack altogether. In any case, mathematical writing expects you to develop your own intuition (and see that the results, when ordered that particular way (which may not be unique), makes a very nice trail to follow (even though muddy and dark it may be)). A quote from Gilbert Ryle, from his book The Concept of Mind, summarizes my point quite nicely I believe:

"Euclid's Elements are neither a sealed, nor an open, book to the schoolboy." (p. 56, the University of Chicago Press)

Needless to say, as I am a person who has a lot to say about this, you are not alone. I don't know about them folks, but I certainly don't have such special faculty (though it feels so when I am reading something about what I have already internalized, which I believe happens to all). You are correct in remembering that Newton was criticized. One critic of him was (Bishop) Berkeley, and his critique was about the foundations of Newton's calculus. I have not read any of Newton's work, nor the critique of Berkeley, but even the title says a lot: "The Analyst: A DISCOURSE Addressed to an Infidel MATHEMATICIAN. WHEREIN It is examined whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith". If my memory serves right, the critique was how one could use a quantity that is very small but not zero as zero at some place and as nonzero at some other place. This issue, of course took a while to be resolved, viz. fast forward to the (precise) definition of a limit. Finally, indeed a proof may fail to provide an understanding to the reader, but without it, all we are left with are things we want to be true very, very strongly.

EDIT: I have just encountered a very explicit emphasis on intuition in Spivak's A Comprehensive Introduction to Differential Geometry, Vol. 1. On p. 84 he says:

"The tangent bundle is the true beginning of the study of differentiable manifolds, and you should not read further until you grok it.*"

And the footnote the asterisk leads to reads as follows:

"[...] [The word "grok"s] sense is nicely conveyed in The American Heritage Dictionary: "To understand profoundly through intuition or empathy"."