Thursday, October 19, 2006

Math, GKC, and an Ignatian Asylum

Today the "North American Martyrs" are recalled, which brings up some very interesting mathematics.

Now, the connection between these two must seem very strange, but - as Bunny Watson explains in "Desk Set", a hilarious Tracey/Hepburn movie, "I associate many things with many things". The Chesterton translation of this exalted dictum of computer and also library science is simply "I never can really feel that there is such a thing as a different subject." Perhaps I ought to give the next verses, which will help you understand a bit more:
There is no such thing as an irrelevant thing in the universe; for all things in the universe are at least relevant to the universe. It is my psychological disease (since one must have a psychological disease of some sort nowadays, and this is the best I can do), it is my psychological disease that I never can see disconnected things without connecting them together in a train of thought.
[GKC, ILN Feb 7, 1906 CW27:126]
So - as I was saying, today is the feast of the North American Martyrs, one of whom was St. Isaac Jogues, a Jesuit priest.

Now, I live in southeastern Pennsylvania, in a little town which was once a larger city. And to its west is an even smaller town called... but let me reserve that for a moment. You see, when I say the name of that town, or hear it said, I have a completely different sense of its meaning than just about anyone else from this area of Pennsylvania. For whenever I hear that town named, I think of the (former) Jesuit Novitiate of St. Isaac Jogues, where one can find the grave of Father Walter Ciszek, who had been a prisoner in Siberia for many years, and whose Cause is now being considered in Rome... So this is what I think whenever I hear someone mention "Wernersville". But for everyone else it has a very different meaning.

For New Yorkers, it would be like saying "Bellevue" (remember in "Miracle on 34th Street"?) - or for Londoners, "Bedlam" which degenerated from "the hospital of St. Mary of Bethlehem"; GKC often speaks of "Hanwell" in the same fashion. You know: asylums, or mental wards... Remember, in Dickens' A Christmas Carol:
"There's another fellow," muttered Scrooge; who overheard him: "my clerk, with fifteen shillings a week, and a wife and family, talking about a merry Christmas. I'll retire to Bedlam."
Even our Lord took advantage of such word-play when He used the old name for the Jerusalem "town dump" - Gehenna - to suggest a more horrifying terminus for ontological waste.

Jesus, Dickens, and Chesterton often took advantage of such extended terms, in which the words mean something more than their literal sense - we call such things figures of speech:
The phrases of the street are not only forcible but subtle: for a figure of speech can often get into a crack too small for a definition. Phrases like "put out" or "off colour" might have been coined by Mr. Henry James in an agony of verbal precision. And there is no more subtle truth than that of the everyday phrase about a man having "his heart in the right place." It involves the idea of normal proportion; not only does a certain function exist, but it is rightly related to other functions. Indeed, the negation of this phrase would describe with peculiar accuracy the somewhat morbid mercy and perverse tenderness of the most representative moderns. If, for instance, I had to describe with fairness the character of Mr. Bernard Shaw, I could not express myself more exactly than by saying that he has a heroically large and generous heart; but not a heart in the right place. And this is so of the typical society of our time.
[GKC Orthodoxy CW1:233]
Quoting this so soon after a mention of Scrooge cannot help but recall the Grinch, upon whose conversion his heart "grew three sizes"... but let me proceed.

I have, on my "ready reference" shelves, a book called Classical Rhetoric for the Modern Student by Edward P.J. Corbett, which spends some 20 pages defining and exemplifying some figures of speech: the simile, metaphor, oxymoron, hyperbole, periphrasis, synecdoche, metonymy, and (horror!) the pun. Perhaps it seems strange to find that puns are formal things, and have their place - just as jokes and good humor do! (If you don't believe me, you can look it up in the Summa, it's II-II Q168 A4 Resp.)

The pun for us today, you will find, is technically an antanaclesis, that is a "repetition of a word in two different senses".

Just as Wernersville means to me both the Jesuits and a mental ward, "figure" means both a structure of words, but also a structure of numbers. And few will think of Chesterton as being a math wizard, but don't forget my first quote! Hence I will conclude with some examples of powerful Chestertonian mathematics:

If two sides of a triangle are always greater than the third side (and all this I steadfastly believe) it can be proved from three-cornered hats or three-cornered tarts. I object to that fastidious mathematician who refuses to prove it except from the two secret triangles of the pentacle. [ILN Sep 17, 1910 CW28:60]

People talk about priest-craft, but there is no proof that the most priest-ridden people believe what a priest wrote on a parchment more than what a priest said with his own lips. Many people argue nowadays about whether education itself is not too arrogant an assumption of superiority by one generation over another. They suggest that it is an abuse of strength to teach a child anything so controversial as the multiplication table, or to prejudice and poison his mind with anything so narrow and sectarian as the A B C. But there is no proof that any children in the past could disbelieve what a schoolmaster stated in class viva voce [spoken aloud], but were bound to believe whatever he wrote on the blackboard. This strange idea of the infallibility of the written or printed word will have rather remarkable results in the immediate future, unless I am very much mistaken... [ILN Sep 18 1926 CW34:166-7]

Life (according to the faith) is very like a serial story in a magazine: life ends with the promise (or menace) "to be continued in our next." Also, with a noble vulgarity, life imitates the serial and leaves off at the exciting moment. For death is distinctly an exciting moment. But the point is that a story is exciting because it has in it so strong an element of will, of what theology calls free will. You cannot finish a sum how you like. But you can finish a story how you like. When somebody [Newton] discovered the Differential Calculus there was only one Differential Calculus he could discover. But when Shakespeare killed Romeo he might have married him to Juliet's old nurse if he had felt inclined. And Christendom has excelled in the narrative romance exactly because it has insisted on the theological free will. [Orthodoxy CW1:341-2]

For [the imagination] has laws of its own, which man has never been able to turn into a code. Only anybody who understands poetry knows when poetry has fulfilled those laws; as certainly as a mathematician knows when a mathematical calculation is correct. Only the mathematician can explain, more or less, why the answer is exactly right; and the lover of poetry can never explain why the word or the image is exactly right. [ILN Aug 4 1934, collected in As I Was Saying]

I do not set myself up here as a judge of the judgment or taste that Mr. Wells shows in these curious, intermittent outcries against the Christian mysteries. But I am quite certain that I should not like to talk in that way about the Buddhist mysteries or the Moslem mysteries. I should hold myself free to reason respectfully against the negative or quietist quality expressed in the image of Buddha in a trance. But it would give me no particular satisfaction to say that a fly might settle on his nose. I should hold myself free to make any fair and decent case against a fanatical simplification in the mind of the Moslem fakir when he rushes on the knives or flings himself on the sand. But I should not think it adequate to say that, after lying prone in the desert dust, with his face towards Mecca, his face would probably want washing. It is not appropriate, because it is not commensurate; it is not on the scale of the things with which his spirit is concerned. It consists, as I have said, in winking the other eye and not seeing the other half of the picture; or even the other half of the equation. All philosophers, sceptical or mystical or both, are working at that immense algebraic equation, and trying to find the exact relation indicated by saying that x = y. For one school, x may be only the unknown quantity; and y, by a sort of pun, may appear as a sort of question. For another, there may be an answer as well as a question, and the x may have a meaning, as it has in the shorter form of Xmas. But both mathematicians are bound to deal with both signs. Neither has found even a negative solution if it does not cover both sides of the equation. And to think about the relation of life and lice by thinking about the lice and not about the life is really to refuse to think about either. [ILN Nov 10, 1934]


  1. It is not quite true that “When somebody [Newton] discovered the Differential Calculus there was only one Differential Calculus he could discover.”

    Leibniz also discovered Differential Calculus, in a different form, arguably, a more durable and suitable calculus:

    “The infinitesimal calculus can be expressed either in the notation of fluxions or in that of differentials. “

    “Because Newton had been approaching calculus primarily in regards to its applications to physics, he purported curves to be the creation of the motion of points while perceiving velocity to be the primary derivative. Conversely, the calculus of Leibniz was applied more to discoveries in geometry made by scholars such as Descartes and Pascal. Since "Leibniz' approach was geometrical," the notation of the differential calculus and many of the general rules for calculating derivatives are still used today, while Newton's approach, which has in many aspects, fallen by the wayside, was "primarily cinematical" (Struik, 1948).”

    Wild Goose

  2. Wow, Nancy, this is a first - a GKC debate about math!

    But of course we should have expected this, for as GKC said, though there is a "common idea that mathematics is a dull subject, whereas the testimony of all those who have any dealings with it shows that it is one of the most thrilling and tantalising and enchanting subjects in the world." [GKC, "A Defence of Bores" in Lunacy and Letters]

    So. With all due respect, lest our discussion turn into another famous debate, I would like to offer an additional correction.

    I should not have added [Newton] to GKC's words. It was in the 1660s (or so) that both Newton and Leibniz "discovered" calculus; yes, almost simultaneously, though Newton seems to have priority.

    Alas, "a quarrel arose between the followers of Newton and the followers of Leibniz, and unhappily it grew into a quarrel between the great men themselves..." [The World of Mathematics 143, 286 et seq; Purcell, Calculus 156, 278]

    Here, one might hear GKC add: "...what men have before now done for the abstractions of theology I have little doubt that they would, if necessary, do for the abstractions of mathematics. If human history and human variety teach us anything at all, it is supremely probable that there are men who would be stabbed in battle or burnt at the stake rather than admit that three angles of a triangle could be together greater than two right angles." [GKC, ibid.]

    Yes, there are two representations, but there is only one differential calculus. (You see, mathematics is NOT a liberal art, but a science.) Newton and Leibniz gave two ways of describing the one single differential calculus.

    Curiously, if there were indeed two kinds, today we would have both a "Newtonian calculus" and also a "Leibnizian calculus" - as we have both the differential and also the integral calculus, and as we have both Euclidean and non-Euclidean geometries (which is a far more debatable topic, given the GKC quote above about triangles!)

    Sure, there are different representations or ways of expressing the idea of differential calculus - but there is only one idea being expressed by the different notations. Lest there be some doubt, "as early as 1665" (VNR Concise Encyclopedia of Mathematics, 406) Newton recognized his fluxions as being the inverse of integration; that inverse is none other than differential calculus, which Leibniz wrote as dy/dx.

    Indeed, differential calculus is a paradox GKC would delight to write poems about: the "slope of a curve"!

    Remember, it's not the slope of THIS curve, or THAT curve, but the slope of a curve:

    f'(x) = the limit as delta x approaches zero of (f(x+ delta x) - f(x))/(delta x)

    However, and this is really the important point: both of these brilliant mathematicians (polymaths, in fact) did not actually complete the calculus, for it was not until Cauchy and others in the 1800s brought rigor to the scheme.

    Newton comes from the west, climbs a mountain and draws a map showing the way to the top. Leibniz comes from the east, climbs the same mountain and writes a detailed narrative of the journey. But it is still only one mountain and both have reached its summit! GKC's point is that differential calculus could not possibly be a "story" which you or I or Tolkien or Rowling (or GKC!) could end as we liked. It is a singular idea, regardless of its notation - just as five is 5 or cinco or fünf or quinque or (3+2) or 20/4 or V or (0000 0101) or "one hand"...

  3. Hello Dr. Thursday,

    You have brought up an excellent topic. The way I see it, the main point of your post was that math and science have their own fixed rules, while poetry has its own, due to the free will of the author or the creator. (“But when Shakespeare killed Romeo he might have married him to Juliet's old nurse if he had felt inclined.”) You are saying that there is only one calculus, while there may be a virtually infinite number of plays or plots along the lines of Romeo and Juliet, limited only by the author’s imagination. But I think that would be like comparing apples and oranges.

    By pointing out that there have been two theories or models of calculus, I have raised an objection to that premise. The fact that calculus needed other mathematicians to be “completed” (and is it complete today?), points to the fact that neither Newton’s, nor Leibniz’ model was adequate, (although Lebnitz’ was more intuitive and longer lasting), and therefore Newton could not have dicsovered the “only one Differential Calculus he could discover”, precisely because he only discovered a part of it, and the lesser part of it compared to Leibnitz!

    No scientist or mathematician can discover the whole of the reality, we can only get windows, or pictures limited by frames, when we look at it. And, curiously, we each see a somewhat different picture. (One of the early Greeks, Heraclitus, or Democritues?, proposed this vision of science and reality, the different views we each get, which was a rather insightful idea.)

    Besides, not even the accumulation or intergration of knowledge (be it Calculus or Science or Theology), can completely reveal the whole mystery of reality. Calculus itself is only one of the models of mathematics, suitable for some things, less suitable for others. Yet, not even the whole of Mathematics with all its disciplines contains the whole of reality. (Those who try to climb the higher mountan of the Infity Calculus may easily end up in the lunatic asylum, like its founder Georg Cantor.)

    Mathematics, the Queen of Sciences, is thus the main coordinating science, but even in its knowable human “completeness” it is rather incomplete, and more or less just a “calculating device.” (Was it cardinal Bellarmine, or one the Jesuit mathematicians, who launched the similar criticism of Galileo’s proposition that it was only another “calculating device”?) This is where many modern scientists have gone wrong - we will never know the whole mystery of reality - and that was the point of Chesterton’s famous Chapter IV in his Orthodoxy. (It has something to do with the problem of the Tower of Babel.)

    The point is that if there is a mountain to climb, as soon as we climb it, a new vista opens up with many more and higher mountains, and we cannot be even sure that we have climbed to the very top of our mountain. (I think this metaphore is an improvement on Einstein’s doors, and Rawling’s secret underground chambers.) All that is keft now are the deadly mountains like K2 and Mount Everest.

    Another interesting thing is that the more complex our model of reality becomes, (be it math or any other science), the less intuitive it becomes. (Chesterton knew that there were only about 20 people in the world who could really understand Einstein’s ideas.) And this is where the kicker is - at this point the “rigorous” science or math has to become “intuitive”, and intuition has its deadly pitfalls and traps.

    As you are perhaps aware, mathematics itself got divided into formal and intuitive schools, and more and more mathematicians adopted the intuitive and imaginary approach to problem solving. Netwon’s science had a significant influence on the development of the world, and we still have “Kant” (that second rate philosopher, according to Chesterton), to deal with.

    The final blow came when Gödel’s proof exploded the wishful thinking of all formalists like Hilbert and Russell. (“We must know. We will know.” and they realized they will never know!) In a desperate attempt to know more, what was left, was mostly misguided fantasy trips of scientists like Fritjof Capra (who in his Tao of Physics advocated taking hallucinogenic drugs to discover new things), and “detached from reality” or “out of this world” fantasies like Harry Potter. Perhaps you have heard about the End of Science proposed by John Horgan, but it is rather in this sense that science and reason (and it seems also art) is “at the end of the tether”, as Chesterton was well aware of.

    As far as the the rules of the free will of artists, I think Leibniz may have found a better “logical” solution, just as he found an easier and safer climbing route to the top of Mt. Calculus, see:

    Wild Goose


    Chesterton was correct if he said- "I never can really feel that there is such a thing as a different subject." -- any problem can be “logically” boiled down to the same subject, and we are really discussing the same problem in a variety of ways, since the same reality is covered with different masks - Ain’t that so Dr. Thursday?

  4. To simplify things a bit. Godel proved the essential intuitive nature of mathematics. But Wild Goose, you seem to be implying that intuition and imagination are one in the same. They are in fact quite different. Without seeing a proof I may intuitively think that there are infinitely many primes or that the Reimann hypothesis is true or the Axiom of Choice. However, this is because I would think, for instance, as I do, that the Axiom of Choice is in fact the truth. Clearly, this is quite different from writing a different ending of Romeo and Juliet (or preferably some other story I rather like the ending of Romeo and Juliet). Mathematicians do not intuitively think something because that's the way it sounds nice or because they think it's "cool." It's because they think it is TRUE. That is the ultimate goal. They may and often do choose to explore an idea because it is pretty or beautiful, but not because of those goals in mind but because they know from experience that the truth most often is pretty and beautiful.

    This reminds me also of the line from V for Vendetta that says something of Artists telling the truth with lies. This is true in the sense that they are trying to express some inner truth through any means neccesary and we all understand what they're doing. Mathematicians are seeking rather than expressing truth when using intuition. Whereas, artists are expressing a truth already experienced when using imagination.

  5. Newton and Leibniz both did discover the same differential Calculus ... but that doesn't mean that they knew all of the logical consequences of it or even fully explained and described it. To use the mountain analogy used by Dr. Thursday, assume that Leibniz and Newton each reached the summit of Mt. Diffcalc and chronicled the journey and everything they saw ... this does not mean that we know everything about the mountain ... or even everything about the parts of the mountain they traversed ... and it doesn't even guarantee that everything they said about the mountain was perfectly precise. The "rigorization" of a branch of mathematics makes it more precise; it doesn't correct something previously thought to be true. Newtonian (and for that matter, Leibnizian) Calculus remain valid ... they have just been made more precise through the efforts of mathematicians such as Cauchy.

    Oh, and to the original poster, I'm glad you posted this. I'm a mathematician and a lover of Chesterton and it's nice to read some of the things he says about it. Thank you,


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