Chop, Chop, Chop

At any given time, that which is highly respectable  is already dead, work on the advancing fronts of a field is mixed (thus the soaring observer in the sky discerns some merit in it) and that which is below the salt and off the reservation is the future. The last category might be described more mildly, as is done by Aaron Preston in an article on Analytic Philosophy concerning metaphysical system builders. That activity is not countenanced by today’s philosophers, he says; not, he adds, “as a respected professional activity” (link).

My own honored mentors on the nature of culture insisted that absolutely everything is saturated with the feelings of the time. But they hoped to teach that cultures change; they were satisfied if only that was understood; therefore they did not carefully describe the coexistence of ossification, transformation, and emergence.

In a field like mathematics which is inaccessible until its extreme abstraction is penetrated enough to reveal some of its meaning (or lack thereof), the cultural influence is difficult to discern. But sure enough it’s there. I only briefly ventured into analytic philosophy in the first place in order to confirm my impression, prompted by the fact that at least three big names in math had played roles there (Friedrich Frege, Bertrand Russell, and Willard Quine). Thus I once more had to enter that unfortunate slaughterhouse where all your hear is chop, chop, chop. Life is stopped at the door and isn’t permitted in; inside blood and guts and shanks of meat. It pleased me to discover that this form of philosophy, while evidently absolutely dominant in the English-speaking world, and spreading to other parts, is already showing advanced decay—and is dominant because of that. Here is a field that attempted to materialize meaning, thus to make it fit for scientific study. This was achieved by turning philosophy into linguistics, semantics, and grammar and forcing its statements to be expressed in formal reductive logic. In the process it caused meaning to vanish, which is the life of that cattle, retaining only its grammar: cattle made meat.

I find this fascinating. Mathematics emerged as a distinct language by means of which additional layers of meaning in reality could once be made accessible. And like any other language—ordinary, philosophical, poetic—so also math retained ambiguities and marvels. But when it had been reduced to its pure grammar by analytic chop, chop, chop it ended up with a vast immensity of tiny marbles in fixed categories the endless rearrangement of which into meaningless patterns is now the only “respected professional activity” in the realms of higher math. Or so at least Morris Klein observes in his worthy exposition of math—although he does so with a certain amount of nostalgia. (Speaking of language, the right word here is really the German Wehmut.)

Gödel’s Proof—Or Was it a Spoof?

When it comes to higher math, it’s a good policy to stay cheerful—and to resist the waves and waves of frustration that well up. Take one of the more prominent figures in that field, Kurt Gödel (1906-1978); he is famed for his incompleteness theorems. I managed to find a partial translation of his first paper (link). It was written when Gödel was but 25. He does the job—the job is to humble mathematicians forever—in just 15 pages (the paper is longer, but it begins and ends with the translator’s notes). It’s good to stay cheerful because with effort the essence will emerge—not from fully penetrating the actual originals, mind you, but because with the help of others one can get there. By essence I mean, enough to satisfy me that there is something worthwhile present here.

Gödel labored at a time when mathematicians were endeavoring to prove that various systems of mathematics were both complete and consistent. If they were neither—or one but not the other—the foundations of mathematics were in trouble. Those engaged in such labors were the really big names in twentieth century math: Abraham Fraenkel, Friedrich Frege, David Hilbert, Giuseppe Peano, Bertrand Russel, and Ernst Zermelo. What Gödel proved, and thus upset the apple cart was:

1. If the system is consistent, it cannot be complete.

2. The consistency of axioms cannot be proven within the system.

The reason for good cheer is that Gödel proved the liar’s paradox mathematically.  That paradox originates with Epimenides, an ancient Cretan philosopher-wit who asserted “All Cretans are liars.” If taken as a true statement, it is a lie; if as a lie, it contradicts itself. The modern way is to ponder the truth-value of  “This sentence is false.” Gödel substituted “not provable” for lie or falsity. He showed that such a statement can be formulated mathematically so that it is equally contradictory: if proved it is false, if disproved it is true.

Now completeness asserts that every proposition framed by a formal system can be proved. But consistency demands that the outcome of any process must result either in truth or falsehood, never both. Gödel therefore showed that mathematical systems are either one or the other: if they are consistent, they are incomplete, if they are complete, they are inconsistent. The consistent system must exclude the formula Gödel framed using the rules of the system and thus be incomplete. The complete system will include the Gödel number but, producing at least one paradoxical result, will be inconsistent.

But what Gödel actually proved, it seems to me, is that Man is not God—although we kind of knew that already. He himself was not only a true believer, he bought the traditional package with all of its frills and was dead certain that he would survive his own death. Good for you, Kurt Gödel. I bet you are still chuckling over your life’s achievements somewhere out there in the transcending ether.

F comes before M

My subject is the claim I’ve often seen that mathematics is the source of science and thus the father of modern technologies. A list then usually follows ending with radio, television, and of late the Internet. Just last night I read this paragraph in a distinguished book on Mathematics:

However, the Kantian explanation that we see in nature what our minds predetermine for us to see does not fully answer the question of why mathematics works. Developments since Kant’s time such as electromagnetic theory can hardly be endowments of the human mind or the mind’s organized sensation. Radio and television do not exist because the mind organized some sensation in accordance with some internal structure then enabled us to experience radio and television as consequences of the mind’s conception of how nature must behave.

                                                                                                                                            [Morris Kline, Mathematics, p. 342.]

Sure enough. But the thought here goes astray. It suggests that mathematics lies behind electromagnetism—and radio and television. No. Faraday discovered electromagnetism by experiment—before young Maxwell came around to translate it into elegant mathematical concepts. Thus F came before M. But it wasn’t Maxwell’s equations that led to radio but more unruly inventiveness by the likes of Edison, Tesla, Marconi, Morse, and the like; you look in vain for paeans to math in their histories. Television got itself going in 1884 when a twenty-three-year old German student named Paul Nipkow punched holes into a disk; he spun the disk above an illuminated picture and sliced, diced, and subdivided it into many tiny images that we now call rasters. That was the beginning.

No. Mathematics is the immensely helpful servant of science—and technology belongs to the inveterate tinkerers. Later, when inventions come to be commercialized and engineers get going in rationalizing the processes, each of them, of course, has had to master calculus and so forth and be handy with equations—although the results of the most useful of these are in the handbooks already.

My image is that of two horses harnessed to the same cart. One is strong and unruly, the other is elegant and spirited. The strong one does most of the pulling, but when it comes time to take photographs of the team, people take the picture from the side of the spirited, elegant horse. Oh, just look at it snort!

There is also a hierarchy here. Math belongs to the upper classes. Faraday came from a poor working class family and was self-educated (as an apprentice in a bookshop); he knew very little math; when he rose in stature and worked as an assistant to Sir Humphry Davy at the Royal Institute, people there did not consider him a gentleman. Maxwell came from the nobility, his father a lawyer and financially secure. The inventors like to grub about with matter, the mathematicians are more at home in the airy realms of concepts. Can we do without them? No. But let’s not forget where science really starts.

Finally, mathematics works because, when successful, it models reality. And reality works.

Not a Lot of Popularizers

1975 The Tao of Physics

1979 The Dance of the Wu Li Masters

1984 The Looking Glass Universe

1984 In Search of Schrödinger’s Cat

1988 The Symbiotic Universe

1988 A Brief History of Time

1989 Coming of Age in the Milky Way

The 1970s and 1980s produced a rash of popular books on physics. In 1994 came Michio Kaku’s Hyperspace, another book I bought along the way, but the curious thing is that string theory does not lend itself to popularization quite so much—either that or the hot air has cooled in this balloon: we don’t have a string of books on string theory; it is too evidently a theory based on pure mathematics. When one of those twin brothers goes off on a decades-long trip to outer space at speeds close to the speed of light—and returns to find the other twin an old man while he is still full of testosterone—why that’s a worthy plot. Trips into Hilbert space, a mathematical dimension, just don’t have the same sort of impact.

The less accessible a subject, the less it will be known to the public—and the more so, if it is deemed important, will it be wrapped in awe. Mathematics wins that prize hands down. I’ve been reading Morris Kline’s book, Mathematics: The Loss of Certainty, a Christmas gift from Brigitte—she who knows what I need. It is not an attempt at popularization, to be sure, but the closest thing we’re likely to get. It was published in 1980 by Oxford University Press and tells the (I’m not kidding) nail-bitingly suspenseful story of the history of math. As Brigitte will testify, I’ve read many, many books of which, at first, I’ve understood at most, say, twenty percent of the content. I have some of the characteristics of the junk yard dog. This book is one of them. It is my conviction that anything made by humans is accessible—if only one makes the effort to penetrate the subject. Eventually, as John von Neumann said of math, you get used to it. And after years, one fine day, we find out that it’s true. The grand old patterns of human nature appear quite clearly again, and what felt like impenetrable fog becomes the same-old. The mild reward is that, at that point, you can eventually feel the problems the great but largely unknown names (who’s ever heard of Kronecker, Borel, Lebesgue , and Baire, for instance) actually felt as real. In my own case, alas, once I’ve penetrated the actual pattern of the thing, I tend to lose interest. I’m interested in the shape of things. For me it’s all about orientation. I appreciate the work of popularizers, and almost-popularizers like Morris Kline, because they let me get there faster.

Sometimes it does take decades to get anywhere at all. It’s been a long time since I’ve first started looking into physics—a subject entirely inaccessible until one has managed at least a certain level of comfort with mathematics, which, these days, is physics. Until then a vast complex field that throws huge shadows over everything, from practical life to cosmology, has the aspect of watching an elaborate thirteenth century Japanese drama unfold, told entirely in Japanese, and all you get is the emotional toning of the harsh shouts of the samurai engaged in its battles.

Let’s Hear it for the Minus

Negative numbers deserve respect such as they rarely ever get.

In Math they have been dubbed absurd, an adjective that really hurt

When the nasty appellation saw its earliest application

In quite ancient, hoary times. Too long have nasty crimes

Like that stained Abstraction. Our times now call for action!

Let us rise now and defend numbers we can’t apprehend

Wearing those humbling minus signs they’re forced to show in lines

Doing subtraction which adds or division which just pads

Positive numbers’ sums. Let’s join and clear those slums

Below the Zero’s sway. Arise from Berkley to Bombay.

Negatives have a true domain in which they ought to reign

Supreme rather than merely be used when needed—cavalierly.

Wine, Loaf, Thou—and Math

Omar Khayyám (1048–1131), known to most by means of Edward FitzGerald’s translation (using a loose meaning for that word) of the Rubaiyyat, was also and perhaps predominantly a mathematician and a philosopher of the school of Avicenna. As I noted earlier (here), I am now reading a wonderful book on mathematics—one of whose early themes is Euclid’s fifth postulate about parallel lines. Khayyám played in that game too, and it delighted me to discover the page of a Persian manuscript of his reproduced by Wikipedia (link). Looks odd that, doesn’t it. FitzGerald was a wonderful poet but apparently challenged in finding original material. His Rubaiyyat is a kind of free, rearranged, and often reinterpreted rewrite. Robert Graves offered a more authentic version, but got shouted down. Meanwhile I note that people of a certain stripe—those who, like Khayyám, believed in the priority of intuition in knowledge—nonetheless often spend huge chunks of their time on mathematics, and the higher the individuals’ rank and fame, the more likely that turns out to be. Avicenna (Ibn Sina), whose thought Khayyám followed, was of that persuasion too.

Math Note

The book under consideration is Morris Kline’s Mathematics: The Loss of Certainty. Wonderful book—and a nice companion to one I’d recently read on Hellenistic Science (link on this blog). For as long back as I am able to remember, I’ve always thought of mathematics as a language—natural, perhaps, for someone like me who had to master three languages beyond my mother tongue before I was sixteen—and someone who got deeply into computers more or less by playing around with their insides. This work tells me that math was long and traditionally viewed as something else—the code deeply embedded in Nature and revealing the secrets of God’s design. The loss of certainty, therefore, is a modern phenomenon, another great disillusionment—but one I had been spared. Math viewed as language explains more. It suggests that math has two aspects: its rules of application, thus its grammar, and the meaning assigned to perfectly legal equations and functions—which may be quite defective.

‘Twas brillig, and the slithy toves

Did gyre and gimble in the wabe:

All mimsy were the borogoves,

And the mome raths outgrabe.

     Lewis Carroll, Jabberwocky

Nothing wrong with the grammar here. Now when it comes to reality, some of the most familiar aspects of which quite escape genuine physical grasp—such as the workings of gravity—giving explanations for them using the pristine grammar of math but meaningless concepts is good fun for toves and the mome.

Pierre de Fermat: Happy 410th

French mathematician Pierre de Fermat was born August 17, 1601. Google dedicates its search logo to the event today, the reason why I know. As in the case of all the great mathematicians, I greatly admire Fermat without grasping anything he tackled—much less why he’d bother. But I’ve looked into his life—as also the lives of several others over the years—because they are in some ways kin. I’ve spent my life pondering great puzzles, but in my case none is solvable whereas, in theirs, many were, if not by ordinary humans.

Fermat is most famed for his Last Theorem. The theorem is that there are no whole number (integer) solutions for this equation:

xⁿ + yⁿ = zⁿ if n is greater than two

…which is what Google’s logo reproduces. Fermat jotted into the margins of Dophantus’ Arithmetica the following famous phrase. “I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.”

The issue goes back to Pythagoras’ theorem, namely: “In a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides,” which makes:

x² + y² = z²

Finding numbers that satisfy this relationship became a preoccupation in mathematics; there are endlessly many, among them 3, 4, and 5. What Fermat claimed was that no such triplets were possible if the power is greater than two.

Fermat wrote his intriguing teaser in 1637. He never published his “marvelous proof.” It took 358 years before Andrew John Wiles, a British mathematician, published a proof in 1995. The story of that proof is told in Fermat’s Enigma, by Simon Singh, Walker and Company, New York, 1997. I got my copy from Brigitte as a birthday gift, my 62nd, in 1998. It’s a truly fascinating but, in the end, a not very satisfactory story. Wiles’ proof takes more than 100 pages to present and makes use of the most modern techniques of numbers theory—which, believe me, are not at all accessible to mere mortals. The margin of Arithmetica was not enough for Fermat, but two or three pages of parchment presumably would have been…