The Book of Uncertain Truth

September 25, 2010 § 7 Comments

Mathematics: The Loss of Certainty (Kline, Morris)

Logic creating Mathematics, or Mathematics creating Logic, or just a clever game on paper? (MC Escher, ‘Drawing Hands’, courtesy

“The supreme triumph of reason is to cast doubts on its own validity.”
Miguel de Unamuno

The mathematics that I learnt in high school, over two decades ago, was a formidable fortress of certainty, through the sturdy bulwarks of which not a single wisp of doubt or flaw could ever hope to sneak. We knew things for a fact back then, some of which were the following:

  1. 1 + 1 = 2. Always.   It was so before the first amoeba emerged in the primordial soup and it will be so when the last star boils away into nothingness, unwitnessed by sentient beings.
  2. People discovered mathematics the same way they discovered a great Chinese restaurant in the neighborhood, a boil in their armpit, a field of wild daffodils, or North America. It was there all along, we just happened to stumble upon it, but once we knew of its existence, we put it to very good use (except for the boil, of course, and quite possibly North America)
  3. Human affairs – love, religion, politics, biology, poetry – are non-mathematical and therefore, essentially unpredictable. This, we were taught, was a shame, and a critical shortcoming of these subjects.
  4. The physical world, on the other hand, runs on mathematical principles. This is simply so. We didn’t think to ask why. Unexplained physical phenomena remain unexplained only until the right mathematical model is found for them. It is merely a matter of time, and several ridiculously bright people are already working hard on these problems.

It is worth noting that the most complex mathematical concepts I mastered in high school – integral calculus, analytic geometry and probability – were all products of the 17th century, “discovered” over three hundred years before I learnt them. And while I was dimly aware of some unpleasant results in 19th and 20th century physics (Relativity and Uncertainty, of course, and am I the only person to have found the Second Law of Thermodynamics strangely disturbing?), I was not remotely aware of any limitations to the power of mathematics itself.  I was not even aware of the profound 18th century doubts raised by Berkeley and Hume.

Things got slightly complicated in college, with Gauss, Fourier, LaplaceJacobi, and Hamilton , but the most recent of these techniques, the analytical one developed eponymously by Messrs. Kutta, M. W., and Runge, C., came into being around 1900 – nearly a century before I studied it. I still was not taught about any schism in mathematics – it was still a perfect science, admitting of no defect and confident in its correctness. Physical science had, by then, lost its sheen of mathematical perfectness for me. To the extent that I could see and feel it – in laboratories and engineering applications – science wasn’t perfect, with measurement and parallax errors to be accounted for, and more often than not, an additional percentage crudely slapped on to the mathematical answers to handle deviations caused by material and human imperfection. And to the extent that the science couldn’t be seen, the models were incredibly abstract, and used mathematics of a bewildering level of sophistication without physical justification (I still can’t understand the use of complex numbers in electrical circuit analysis).

But through all this, my faith in mathematical purity remained unshaken. Reality itself might be defective, or humans might be too imperfect to know a perfect reality, or maybe I was not smart enough to know even as much as others (with higher grade point scores) did. But surely mathematics was blameless; mathematics was certainty itself.

Or was it? A friend (Sury) told me wild stories of a man named Godel, stories that I did not fully pay attention to at the time. A couple of years later, a flatmate (Ajith) loaned me Douglas Hofstadter‘s Godel Escher Bach, which I thoroughly enjoyed for its breezy style  and general cleverness. But it was only in the year 2000, when I re-read the same book after borrowing it again (this time from Ravi, who owns the current book under review as well) that the full import of Kurt Godel’s Incompleteness Theorem hit me.

Godel’s First Theorem of Incompleteness, in very brief terms, proved that a rigorously consistent mathematical system can never be complete – it will always have unprovable statements.  In short, there are things – mathematical things – that will never ever be known for sure. The significance of Godel’s Theorem is immense within, and practically impossible to appreciate outside, the context of the history and philosophy of mathematics, and I have no intention of attempting to explain its significance here. To do so would take a book – the very book, perhaps, that is the topic of this review. There is a very satisfying circularity involved in that statement, and circularity, as we know, leads but to paradox, a place where there is much weeping and gnashing of teeth.

Hofstadter’s book, written in 1980, was a hymn of praise to Godel’s Theorem, celebrating in slow motion every loop and twist of the intellectual roller-coaster ride that the theorem truly is. Kline‘s book was written a year before that, and it involves more history and philosophy than worship and wordgames. He traces the evolution of mathematics from Pythagoras to the Bourbakists, placing each philosophical thought and theory in its appropriate context, and discusses these thoughts with intelligence and passion along the way.

Do numbers really exist, independent of  human thought and experience? Is mathematics a collection of abstract truths? Is there a parallel world of abstract, perfect ideas, of which we are imperfect copies, as Plato believed, or is the rough-edged universe we see around us the only one there is, as Aristotle did? Is all of mathematics only a branch of logic? Or is logic a branch of mathematics? Is there a logical basis to why we think Mathematical axioms are true, or are they intuitively known to us as true? Or are the truths of mathematics not ‘true’ at all, but mere conventions, flavors of the month, fervently believed in until their contradictions become popular? Should we not believe in anything that hasn’t been rigorously proved yet? But what exactly is a proof?  Is (1+1= 2) a statement about the real world, or about the way our brain perceives the world, or is it merely a game played with symbols on paper, signifying nothing? And above all else, why does mathematics explain the real world so well? Is it because God is the ultimate geometer and arithmetician, as Leibniz believed, or is it that we perceive the world through our senses, which have mathematical filters, and thus we perceive only what passes through, while actual reality will remain forever unknown, as Kant said?

These are fascinating topics for debate, that thinkers have been preoccupied with for millennia. Matters came to a head in the first half of the 20th century, in science and art, in mathematics and music, in poetry and politics, and there were events and discoveries that fundamentally altered our perception of truth, beauty and justice.  Amazingly, none of this is taught in schools even today – I mean not just the mathematical and scientific discoveries of the 20th century, but the overall philosophy of the post-modern world, with its rejection of a single truth, its entirely different ways of looking at the world. Around the world, children in schools are taught only what mankind already knew over a century ago, and taught to think only how people thought a century ago, when the world was a more confident and optimistic place (mistakenly so,  for the most part) – and many of us live our entire lives without ever learning that anything changed in the 20th century in poetry and philosophy, mathematics and art, history and science.

Kline wants to educate us, however, at least with regard to one subject. He wants to wake us up, and inform us, like Nietsche‘s Zarathustra, that what we were taught in school as our one true god, the purest science and fount of truth, mathematics, is indeed dead, and has been so for the last few generations. And what’s more, he knows what killed the god: the tendency of number theorists to ignore the soundness of the real world as a basis for their work and their subsequent embarcation upon flights of introverted fantasy, which resulted in acrimonious failure and utter confusion, somewhat like an aging rock star under the influence of recreational drugs, believing he can break the land speed record, and ending up crashing his mini-Cooper into a shopfront.

While the book is a fantastic read, I find myself not wholeheartedly in agreement with Kline’s overarching agenda of extolling the virtues of applied mathematics while decrying “pure” mathematics. If mathematics is completely separated from contexts where it could be meaningfully applied, it is, after all, the pure mathematicians who would suffer, but it is a choice they should be allowed to make. Theoretical physicists and perhaps even the occasional engineer would carry the torch of mathematical innovation forward. But is that so bad? Does it really matter if it is a mathematically-minded physicist who springs to the rescue or a physics-savvy mathematician? To ask, as Kline does, that all mathematicians turn their back on pure mathematics, that all mathematics be applied mathematics, is unreasonable. Man contemplates because he can, and his mind takes him wherever it can conceivably rove. He can dictate direction to his mind no more successfully than he can sit in a corner and try hard not to think of a polar bear for an extended period of time.

When Alexis de Tocqueville visited the United States of America in 1831, he noted, with some amusement,  the state of science education in the country.  He said, rather sweepingly:

In America there are few rich people; therefore all Americans have to learn the skills of a profession which demands a period of apprenticeship. Thus America can devote to general learning only the early years of life. At fifteen they begin a career; their education ends most often when ours begins. If education is pursued beyond that point, it is directed only toward specialist subjects with a profitable return in mind. Science is studied as if it were a job and only those branches are taken up which have a recognized and immediate usefulness…There is no class, then, in America, which passes on to its descendents the love of intellectual pleasures along with its wealth or which holds the labours of the intellect in high esteem.

Well, circumstances have changed, and there are several rich people in America these days, and not everyone needs to begin a career at 15. Reading Kline, I realize that the spirit of America has not changed in some important ways; perhaps the disdain of the kind of science that is not readily applicable to practical and commercial betterment of life is right up there with gun rights and the taxation of tea.

Besides, there is undeniable intellectual beauty in pure mathematical contemplation, and that, to me, is justification enough. Like poetry, or rock music, or philosophy, or sport, or art, things that are of no direct use or utility do have a place in our lives. Some of our most glorious triumphs as a civilization, our most immortal contributions as a species, our greatest moments of unadulterated joy as individuals, are attributable to our appreciation of some of these “useless” things.

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§ 7 Responses to The Book of Uncertain Truth

  • ravi says:

    I must confess I do not remember what Kline’s prescriptive or “ideological” bits are, but I agree wholeheartedly with you on pure vs applied mathematics. More in fact… the history leading up to Gödel incompleteness, as you know, is one (as Kline recounts) of attempts (particularly by Russell) to shore up the foundations in the face of flights of ill-understood application.

    As you sort of write, Gödel’s result (with overdue nods to Skølem and Church) is sublime and it’s significance is either poorly understood or misappropriated (as some — though not I — would accuse Penrose of doing).

  • ravi says:

    One thing that I found a bit irritating in Kline’s book was his casual repetition of the old idea that the Arabs and Hindus restricted their mathematical investigations to the merely practical, never stopping to wonder or abstract further. Per Kline, it took the Greeks to really get into the conceptual and abstract stuff. Similar claims are made about democracy, rationalism, so on.

    This I have always found strange not because of any misplaced nationalism, but simply because I have found all exceptionalist or singularist ideas unsustainable, and almost always unsubstantiated.

    On that front, this is an amusing read:

    • psriblog says:

      Yes that piece did raise my hackles as well but I did think it balanced out a tiny bit that irritating and insistent spam mail that I still get once every few months that says, among other things, that Indians invented mathematics 5,000 years ago. Or some such. Do you know which one I’m talking about? That is as exceptionalist, unsubstantiated and historically wrong as claims go – but am sure by now millions of Indians believe it.

      More on the topic when we meet next – but I also think on reflection that Kline was alluding to the fact that Hindus and Arabs didn’t base their math on a solid axiomatic footing (as the Greeks did with their geometry) and preferred stating results to deriving proofs. Which is fair criticism, surely.

  • SKP says:

    don’t know why reading this article reminded me of the above…it took me sometime to track this visualization, well worth a look, if you have not seen it…would guess that it was this gathering of reference to pure math…complex analysis…what one might not see might be simpler when looked from a higher dimension…how does one come up with a higher dimensional structure unless unmoored from practical considerations for a short while, at least?

    hope to read this book, but your summary/review did not make me want to prioritize it.

  • What a perfect style of writing.

  • […] I recognize identical processes at play, for instance in the cases of non-Euclidean geometry and number theory. Most compelling are Kuhn’s descriptions of the period of crisis, when rival paradigms clash, and […]

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