Learning is ineffective if attempted in a linear fashion. Fine details are best learnt, appreciated and remembered by those that can spontaneously describe and answer questions about the coarser details of the subject at hand. Therefore, it can be valuable for more advanced books to revisit “elementary” concepts because rarely is anything sufficiently elementary that nothing more remains to be known. The following quote is apt; the emphasis is my own. (All quotes below are taken from the preface of C. Lanczos (1966) Discourse on Fourier Series.)
By the nature of things it was necessary to develop the subject from its early beginnings and this explains the fact that even so-called “elementary” concepts, such as the idea of a function, the meaning of limit, uniform convergence and similar “well-known” subjects of analysis were included in the discussion. Far from being bored, the students found this procedure highly appropriate, because very often exactly the apparently “elementary” ideas of mathematics — which are in fact “elementary” only because they are relegated to the undergraduate level of instruction, although their true significance cannot be properly grasped on that level — cause great difficulties in proceeding to the more advanced subjects.
There are three interwoven aspects of mathematical knowledge:
- Intuition — the “pictures” one forms (consciously or subconsciously) in one’s head when reasoning about a problem or endeavouring to generalise a concept.
- Rigour — the formalisation and verification of definitions and proofs.
- Communication — the transfer of mathematical knowledge from one person to another.
Pictures, formulae and discussions are generally how mathematical knowledge is communicated from one person to another. It is important though to isolate this from the actual understanding of mathematics itself. The formula is not in itself what is “understood” by someone reading it. Rather, seeing the formula conjures up a wealth of images in one’s subconscious mind which can be then refined further and reasoned with; seeing the formula primes relevant areas of the cortex facilitating subsequent thought. One “understands” because one can reason with it and answer questions about it, for instance, one can graph it, differentiate it, find its zeros, write down its Taylor series expansion, draw a relevant right-angled triangle and so forth. [Understanding is therefore relative to the questions one has asked oneself or otherwise encountered to date.] Memorising a result does not immediately lead to understanding. Understanding occurs only after one’s mind has formed associations that link the result with other stored knowledge. The degree of understanding is related to the scope and complexity of such associations.
To distinguish rigour from intuition, consider reading the proof of a theorem. It is possible to check a proof is correct without having any sense of actually “understanding” the proof, or even of “understanding” why the theorem should be true. In fact, it is possible to come up with a proof without “understanding” it! That is to say, by trial-and-error and (subconscious) pattern recognition (e.g., making substitutions and transformations one has seen before without quite being sure one is heading in the right direction), one can write down an algebraic proof of a theorem in convex analysis, say, without being able to offer any geometric picture or other explanation to justify how the proof was found. It is often worth the extra effort to develop a sense of intuition about theorems and their proofs. Intuition and rigour together provide a sense of understanding and increase one’s fluency in mathematics.
In some cases, intuition and rigour go hand in hand; one can translate directly one’s intuition into a proof. That this is not always the case is perhaps the only reason why teaching and learning mathematics is non-trivial: It is easy to convey rigour (at least, no harder than programming a computer), and all too easy for an author to convey only rigour and leave it to the readers’ mathematical maturity and ingenuity to deduce the intuition for themselves. Conveying intuition is not necessarily more difficult, but for an author, there are apparently drawbacks.
The most obvious drawback is verbosity. Stating and proving a theorem in its full level of generality (à la Bourbaki) takes considerably less space than does proving a basic theorem using one technique, then motivating the generalisation of the theorem, then pointing out why the proof of the basic theorem does not generalise, then motivating a new proof technique and finally proving the general theorem. Another drawback is imprecision and inaccuracy; intuition need not be precise nor even accurate for it to be valuable, yet some authors may be uncomfortable committing to paper anything even remotely inaccurate.
Tracing the historical development of a subject can provide a wealth of intuition. Here is what Lanczos has to say on the matter.
… a close tie with the historical development seemed appropriate, although the author is well aware that this exposes him to the charge of datedness. We have to be “modern” and there are those who believe that before the advent of our own blessed era the pursuers of mathematics lived in a kind of no-man’s-land, bumping against each other in the gloomy haze that pervaded everything (“Euclid must go!”). But there are others (and the author belongs to the latter group), who believe that the great masters of the eighteenth and nineteenth centuries, Lagrange, Euler, Gauss, Cauchy, Riemann, Fourier, Dirichlet, and many others, were not necessarily lacking mathematical intelligence and some of them might even be comparable to the geniuses of today.
One wonders if occasionally intuition is purposely withheld, the false reasoning being that the merit of an idea is judged by how complicated it is to understand. Merit should be judged by originality, usefulness and simplicity rather than complexity. A thing is understood when it appears to be simple.
To display formal fireworks, which are so much in the centre of many mathematical treatises — perhaps as a status-symbol by which one gains admission to the august guild of mathematicians — was not the primary aim of the book.
In conclusion, when teaching or learning mathematics, keep in mind that intuition and rigour are distinct aspects whose synergy forms mathematical knowledge. Intuition and rigour should be learnt together because rigour without intuition is like owning a car without the key; you can admire the car for its beauty but you cannot get very far with it.