Michael Eriksson
A Swede in Germany
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Personal epiphanies

Introduction

Below, I will discuss some few of the epiphanies that I have had over the years.

For now, I begin with two 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.

The importance of understanding a problem

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.


Side-note:

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.)



Side-note:

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.


The risk of abstracting too far

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”.


Side-note:

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.