Below, I will discuss some few of the epiphanies that I have had over the years.
For now, I include three of particular importance to my own thinking, that truly altered how I approach some types of thinking. Others might follow later, but most epiphanies of true magnitude have usually been of a more personal nature, which (a) makes them less interesting to the public at large, (b) makes me more cautious of sharing them.
I stress the difference between an epiphany and an insight. Arguably, all epiphanies are insights, but far from all insights are sufficiently unexpected, change my worldview sufficiently, whatnot, that I would apply the word “epiphany”. (And one man’s epiphany might be another man’s insight and vice versa.) Indeed, most insights tend either to be the result of a longer process of accumulated thinking, reading, and observation, or to come in a more “expected” manner, as when I read a book on a topic and expect to gain new insights into the topic.
The headings used are somewhat limited and do not necessarily catch the full range of the epiphany.
A second year course in mechanics made heavy use of Lagrangians and Hamiltonians. I was soon enamored with the ease and generality that problems could be solved, and proceeded to use them for every problem that I encountered—magic solutions!
This applied to the final written examination too, which I aced. I felt quite happy with myself—until the professor went through his solutions to various problems on the blackboard: One of the problems involved a falling ladder on frictionless wheels, and we were supposed to calculate some particular value (maybe, the time it took the ladder to hit the ground after some event). I had needed a page-or-so to write down my (correct) solution, without bothering to understand the problem. The professor began by noting that there was no side-ways force, applied regular, Newton-style, mechanics, and was done in four-or-so lines. He had actually understood the problem; I had merely solved it.
A secondary lesson, if not an epiphany to me, is the importance of occasionally solving a single problem in multiple ways. Had I done so through the course of the semester, I would likely have developed a more nuanced and less enthusiastic view of “magic” solutions at a much earlier stage.
More generally, I have always had a liking for “meta-solutions”, where a more general problem can be solved once and the solution easily applied to more specific versions of the problem.
(And the current text notwithstanding, I still see a very great value in them. Looking specifically at Langrangian/Hamiltonian solutions, they certainly provide a very different perspective, which is valuable in its own right, and can be generalized to a wider set of contexts than Newtonian mechanics. Note that I speak of “meta”, however, and not “magic”.)
A much more trivial, but easier to understand, example is solving a single-variable, second-degree equation by completing the square, or even jumping directly to the standard formula for the roots, instead of finding an ad-hoc solution to the equation.
In a twist, partially echoing the epiphany at hand, long-after-school encounters with such equations, e.g. in the context of the history of mathematics or with equations restricted to rational roots, have made me spend time on such ad-hoc solutions, and I have gained far more insight than by merely completing the square. Indeed, those who merely memorize the standard formula will gain no insight at all. (And, no, I have not memorized it. This also raises some concerns of whether formula-based school math brings enough return on invested time.)
The impulse to write this text was the reading of James Gleick’s “Genius: The Life and Science of Richard Feynmann”, where I encountered a reversal of my own original take. Feynmann appears to have had a considerable early scepticism where I had an early enthusiasm—and exactly due to issues of understanding. To quote the book (with reservations for transcription errors):
Feynmann refused to apply [the Langrangian technique]. He said he would not feel he had understood the real physics of a system until he had painstakingly isolated and calculated all the forces.
A famous problem involves three switches and three light-bulbs. A typical formulation:
You are in the cellar of a house. In front of you are three on-off switches, each controlling a light-bulb upstairs, each currently in the “off” position. You may flip the switches as often as you please. What is the least number of trips upstairs that you need in order to determine what switch controls what light-bulb?
(A more stringent formulation would include additional restrictions, e.g. that no helper is available, but would do little to further this text, as the point is not to pose a problem, merely to use the problem as illustration.)
On my own first encounter, I almost immediately arrived at the solution “two”, e.g. in that I flip two of the switches, go upstairs (1), check which light-bulb is still “off”, go downstairs, flip one of two switches now “on” back to “off”, go upstairs (2), and check which of the previously “on” light-bulbs is now “off”. This gives a trivial mapping between the switches and the light-bulbs.
The alleged answer, however, was “one”.
I dug down, cut out everything that seemed irrelevant, and abstracted by converting the switches and the light-bulbs to binary numbers, and going through all combinations manually, applying binary trees, whatnot. I found no way to go below “two”. Math does not lie—and I concluded that something was amiss with the official answer.
I consulted the provided solution—and found that abstracting had been exactly the wrong thing to do. Taking advantage of the fact that light-bulbs were used, not e.g. binary numbers or diodes, it is possible to solve the problem with only one visit upstairs: Proceed as before, but replace that first visit upstairs with a sufficient wait that the “on” light-bulbs gain enough in temperature. During the one visit (which, in the original scheme, would have been the second), now note which light-bulb is respectively “on”, “off and warm”, and “off and cool”.
Such examples are another good reason why “word problems” in math are best avoided: How do we know what characteristics of the problem actually match reality? Certainly, if confronted with this particular problem in real life, my chances of solving the problem would have been better, because I could actually draw on reality—not just words. (But I make no statement of how large those chances would have been, beyond that “better”.)
And: Given a solution like the one using temperature, how do we know whether a teacher/TA/professor/whatnot will consider the solution smart (full points) or smart-ass (potentially, zero points)?
Indeed, with a slightly different formulation, different generations might have had very different takes. Say that the problem spoke of “lamps”. Someone of my generation or older might have associated lamps with heat, while someone of a more modern generation might not, being used to diodes and similar solutions that waste less energy on heat. (And a real-life version of the problem might not have been compatible with the given solution today.)
For that matter, even using “light-bulb”, someone of a younger generation might lack the association of heat with light-bulbs, too, not just with lamps. While this would not invalidate the solution, it would make the problem unfair to some degree, just like it is more reasonable to expect knowledge of, say, a 1999 event from someone who was actually an adult in 1999, than someone who was a child, than someone who was not yet born.
In my mid-teens, I developed a great and sudden love for math and, of course, soon tried my hand at proving “Fermat’s last theorem” (aka “Fermat’s audacious unproved claim”).
I grabbed pen and paper, scribbled for a few minutes, and soon found a proof.
While I thought highly of myself, this seemed much too good to be true, and I began to look closer at the implications of my proof. A first observation was that I had also “proved” that there were no solutions even for n=1 and n=2, and none even when x, y, and/or z were not positive integers. However, there are infinitely many such solutions. (n=2, x=3, y=4, z=5 is likely the best known and one particularly interesting through the sequentiality.)
I now went back to check where I had gone wrong. I soon found that I had turned a "<" or ">" in the wrong direction—a beginner’s error. In a next step, I had to conclude that I could not conceivably have proved the theorem by the approach taken, which relied on inequalities among unrestricted real numbers. Moreover, this should have been obvious, because any proof would necessarily have ruled out the known solutions “even for n=1 and n=2” and “even when x, y, and/or z were not positive integers”—another beginner’s error.
Once the embarrassment had faded, I had learned the very central lesson to spend some time thinking of whether a certain approach to a problem can at all work and, more generally, whether an approach seems sufficiently likely to work (has a sufficient expected payout relative costs, etc.) as to be worth pursuing.
Notably, this is not restricted to math, but has a wide range of applications. Consider trying to improve the world by just writing a website—and, yes, I am considering various options for better leveraging and whatnot when the time is right.
However, my writings also have other reasons, including the fun, the personal satisfaction, and an approach to self-development that, to date, has been highly successful.
An important insight, if not an epiphany, is that a certain activity can serve more than one end, and that a dubious success with the one end can be outweighed by success among the other ends.
The same, to some degree, applies to my attempts to prove Fermat’s last theorem: While I certainly hoped to find a proof, finding a proof is unlikely to have been my main purpose. (Even pre-epiphany, I was well aware of how many others had failed over such a long time and what this implied for my chances.) Instead, my main purpose(s) were almost certainly some mixture of learning-something-about-math and having-fun-with-math, likely with a dash of because-its-there added.
Unfortunately, it is not always possible to make predictions about what can or cannot succeed with what likelihood, or not possible to do so without so much effort that the attempt might just as well be made. Particular problems include lack of prior experiences, domain knowledge, and, for many problems, an underestimation of the stupidity/irrationality/whatnot of fellow humans.
However, the option is often there and a rough idea can often be found by drawing on the experiences of others. (With the downside that a great domain knowledge might be required.) Many problems, and political problems especially, go back to exactly previous experience being ignored, be it out of genuine ignorance, sheer stubbornness, ideological conviction, a wish to achieve some other goal than the official, or similar. Consider e.g. the many, many failures at improving school by just throwing more money at the problem or by trying to push the umpteenth alleged silver bullet in educational reform. I am often reminded of the attributed-to-Einstein claim that the definition of insanity is doing the same thing over and over and expecting different results. (As to “if at first you don’t succeed, then try and try again”, it depends exactly on what the chances of success, the payout from success, etc., is—if sufficiently favorable, it is good advice; if not, it is not.)
During this period of love for math, I did ultimately make considerable inroads, including proofs for a few specific values of n, and for some relationships between x, y, and z. I seem to recall a proof for all even n, but I could misremember after more than three decades.
In a twist, I did find a promising approach involving inequalities for the general case, which I pursued through quite a few values, but then abandoned exactly through considerations of success rate relative the great effort necessary and the great risk of errors that could have invalidated all the effort.
(The details are lost in the fogs of time, but the general idea was to assume a smallest solution of x, y, and/or z, for a given n, and then to prove e.g. that x must either be larger than some value or not an integer, which forces y and/or z to exceed some even larger value (or not be integers), which forces x to exceed some larger value still (or not be an integer), etc. The conclusion would ideally be that no smallest solution could be found. In my decision not to pursue, I drew on the meta-argument that the underlying idea was sufficiently simple that others were bound to have tried it—at least some of which had likely been better equipped to handle the resulting tricky calculations and whatnots than even a talented beginner.)