I’ve been reading Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem, by Simon Singh. Normally, I’d shy away from a book like this — after all, it’s about math! — but it was required reading for my book club, and it’s proven to be delightful. To the extent there is math in it, Singh masterfully simplifies complex ideas so that even math illiterates like myself can understand them. Indeed, I suspect that, if I’d had a teach like Singh when I was in school, one who teaches why something matters, or how it came to be, rather than just demanding that one memorize meaningless formulas, I might not be the math illiterate (and math phobe) that I am today.
But my ruminations about books and math aren’t actually why I’m writing right now. Instead, I wanted to comment on the different types of thinking in the sciences. I’m ashamed to admit that I never really sat down and analyzed the different intellectual approaches people on the “science side” use. To me, the world was binary: science mind (including math) and not science mind (including me). Sure I knew that engineers could be a bit obsessive compulsive, but it was a trait I admired, so I never thought more about it.
What never occurred to me, however, is that specific branches of science demand different approaches to finality — or, as it’s called in math, “absolute proof.” Let me have Singh describe this concept. I’ll quote at some length from his text at pages 20-22 (in the hard copy version 0f his book):
The story of Fermat’s Last Theorem revolves around the search for a missing proof. Mathematical proof is far more powerful and rigorous than the concept of proof we casually use in our everyday language, or even the concept of proof as understood by physicists or chemists. The difference between scientific and mathematical proof is both subtle and profound, and is crucial to understanding the work of every mathematician since Pythagoras. The idea of a classic mathematical proof is to begin with a series of axioms, statements that can be assumed to be true or that are self-evidently true. Then by arguing logically, step by step, it is possible to arrive at a conclusion. If the axioms are correct and the logic is flawless, then the conclusion will be undeniable. This conclusion is the theorem.
Mathematical theorems rely on this logical process and once proven are true until the end of time. Mathematical proofs are absolute. To appreciate the value of such proofs they should be compared with their poor relation, the scientific proof. In science a hypothesis is put forward to explain a physical phenomenon. If observations of the phenomenon compare well with the hypothesis, this becomes evidence in favor of it. Furthermore, the hypothesis should not merely describe a known phenomenon, but predict the results of other phenomena. Experiments may be performed to test the predictive power of the hypothesis, and if it continues to be successful then this is even more evidence to back the hypothesis. Eventually the amount of evidence may be overwhelming and the hypothesis becomes accepted as a scientific theory.
However, the scientific theory can never be proved to the same absolute level of a mathematical theorem: It is merely considered highly likely based on the evidence available. So-called scientific proof relies on observation and perception, both of which are fallible and provide only approximations to the truth. As Bertrand Russell pointed out: “Although this may seem a paradox, all exact science is dominated by the idea of approximation.” Even the most widely accepted scientific “proofs” always have a small element of doubt in them. Sometimes this doubt diminishes, although it never disappears completely, while on other occasions the proof is ultimately shown to be wrong. This weakness in scientific proof leads to scientific revolutions in which one theory that was assumed to be correct is replaced with another theory, which may be merely a refinement of the original theory, or which may be a complete contradiction.
I know that, having read that, you’re thinking exactly what I’m thinking: Global Warming. You’re thinking of falsified data, of non-vanishing glaciers, of robust polar bear populations, and of the other cascade of data showing wrong-headed theories supported by bad, careless, or out-and-out fraudulent “science.” Credulous people, ideologically driven people, and people who confuse scientific theory with the absolute proof of a mathematical theorem were willing to accept that “the science is settled.” But unlike math, which can see a theorem being finally and definitively proved, real science is never settled, and anyone who claims that must be a liar.
Certainly, we know that some scientific theories are more stable than others, and we’ve built large parts of our world on that. But when people purport to take the dynamics of the sun, the moon, the earth and predict the climate outcome years or even decades in advance, and then it turns out that they’ve done so entirely without regard to the sun, the moon, and the earth, you know you’ve got mysticism and faith, and nothing remotely approaching science, let alone the sureties of math.
I’ll leave you with a joke, also from Singh’s book, although it originally comes from Ian Stewart, in his book Concepts of Modern Mathematics:
An astronomer, a physicist, and a mathematician (it is said) were holidaying in Scotland. Glancing from a train window, they observed a black sheep in the middle of a field. “How interesting,” observed the astronomer, “all Scottish sheep are black!” To which the physicist responded, “No, no! Some Scottish sheep are black!” The mathematician gazed heavenward in supplication, and then intoned, “In Scotland there exists at least one field, containing at least one sheep, at least one side of which is black.”
Since you’re all much cleverer than I at jokes and bon mots, I’ll leave you to imagine what the AGW “scientist” would have said upon seeing that sheep in that field.