“Do not all charms fly / At the mere touch of cold philosophy?” Keats famously lamented. Getting his cue from Keats, Poe also keened
Science! true daughter of Old Time thou art!
Who alterest all things with thy peering eyes.
Why preyest thou thus upon the poet’s heart,
Vulture, whose wings are dull realities?
One of the most memorable expressions that have originated from Keats’ poem above is the accusation of Unweaving the Rainbow, which is also the title of a famous book by Dawkins. In the first part of the following short video, Feynman addresses this accusation beautifully.
In most scientists’ frame of mind, understanding only adds to the wonder. On reflection it becomes evident that the same frame of mind is also shared by a huge population of non-scientists. In researching NY-Times most e-mailed articles of the year, challenging cerebral science articles have surpassed expectations. The researchers concluded that readers crave and share articles that inspire awe. Far from robbing the poet of inspiration, as Poe accused, it seems Science indeed only adds to it.
Closer to home, a similar argument can be made by noting the popularity of The Masked Magician and other magic exposés. If these are any indication, most people crave more than what the simple magic show offers. By saying that they want in on the trick, these people are trading the wonder of mystery for the wonder of understanding.
Like with flow, when it comes to providing insights Mathematics is up there at the top of the ladder. Indeed, it can safely be argued that insights are math’s raisons d’être. One of the reasons the Haken and Apple proof of the Four Colour Theorem has been so controversial is precisely because, as Paul Halmos puts it, “we don’t learn anything from the proof.” Halmos was convinced that someday in the future a more elegant solution will be published that will truly explain the theorem. An interesting historical survey of the 4CT and its philosophical implications briefly quotes Halmos and continues:
The proof gives no indications as to why it is that we only need four colours to colour maps; why not thirteen? What is so special about the number four? As good as an “oracle” gets, it still can’t provide an explanation for that one! Stewart agrees with Halmos and Hersch joins them in their disappointment about there being no “enlightenment” brought out through the proof. Some go as far to claim that mathematics is not about finding answers, but about the methods used to obtain them.
Mathematicians tinker with concepts and every so often realize that many of the problems they’re dealing with share some common features. This is a clear indication that there is an underlying common structure, which they then proceed to abstract, name and define. This process may take a very long time to accomplish. The concept of open sets, so fundamental to topology is such an example. In fact, it took well over a century to put calculus on firm ground (which we now call analysis). Reading some of the history of mathematics, I am often surprised at how recent are the concepts that we happily teach to high-schoolers and undergrads. These concepts hide an astounding level of sophistication and a mountain of historical development.
Once commonalities are delineated and clearly defined, we then regard them as “chunks” and use them as building blocks to go even higher, even deeper. This is the reason that the most advanced research-level mathematics of previous decades is now regularly taught to undergraduates. This is exactly how math progresses.
Mathematics is the language with which we capture the most basic and simple patterns and concepts. We then idealize and expand them. Thus, mathematical insights are insights into the most fundamental patterns we’ve been able to uncover so far. Any progress in mathematical understanding is also a progress in our ability to capture even more fundamental patterns. This explains Wigner’s “unreasonable effectiveness of mathematics.” It also explains why mathematical insights are felt to be so much more penetrating, so much more insightful in comparison to other insights (excepting maybe psychological-philosophical insights, which are of a different kind).
Insight is so fundamental to mathematics that it is inextricably tied to its self-conception. It is a measuring rod against which mathematical achievements are judged. This can be seen in mathematicians’ aesthetic appreciation, an expression of their mathematical value judgments. A theorem is beautiful when it provides insight. A theory is deep when it highlights unexpected connections (think of Galois theory, for example). Euphoria in mathematics is understanding; moments of realization and insight are always awash with joy.
Math enlightens…
[Imported from my now defunct blog 10,000 Hours of Mathematics.]