I found myself lately vaguely recalling a quotation by Bertrand Russell that I had read some years ago on the Web, but had been unable to place or verify at the time. Over the last several days it began nagging at me again, as though clawing its way up from the subconscious, and a few Google searches sufficed to locate it in “The Retreat from Pythagoras”, a short retrospective written late in Russell’s life (available in the Basic Writings). It says:
“Mathematics has ceased to seem to me non-human in its subject-matter. I have come to believe, though very reluctantly, that it consists of tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-footed animal is an animal. I think that the timelessness of mathematics has none of the sublimity that it once seemed to me to have, but consists merely in the fact that the pure mathematician is not talking about time. I cannot any longer find any mystical satisfaction in the contemplation of mathematical truth.”
In his early adulthood, during the years of his influential work on the foundations of mathematics, Russell had been an avid Platonist, thoroughly taken with the idea of a transcendent mathematical “heaven” where absolute truths about Number dwell, eternally, free from the tumult and ugliness of the real world, and ready and waiting for our minds to grasp them. Yet with time and harsh experience this excitement dimmed, leading Russell to this much more reserved, even glum attitude towards the subject.
This transformation fascinates me, for I myself (while having nothing approaching Russell’s mathematical gifts) have experienced both attitudes to some degree.
I tend to strongly suspect mathematics is not pure, not a bridge to a transcendent other realm, but is a human creation, a peculiar subset of our evolved reasoning abilities–a kind of toolbox of useful concepts, responses, and operations accumulated from our long sojourn in this universe, reaching back through the entire tree of life to the first cells. It is a form of language (or perhaps even a mood) in which our thoughts, obeying certain very rigid but deceptively simple constraints, are made to assume the role of elementary, interchangeable parts, combining with total predictability into an endless range of structures–which retroactively seem “inevitable”.
George Lakoff and Rafael Núñez, in their book Where Mathematics Comes From, (for full disclosure: I’ve only read the preface) made one of the better known attempts to broach this general view of mathematical truth as not inhuman and remote, but deeply entwined with the biological and cultural history of human beings. These authors claim that the Platonic perfection that so captivated Russell, the smooth inevitability and infinite rigor of mathematical conceptions, is a kind of deception, concealing a much more complex, ultimately physiological processes: “the intellectual content of mathematics does not lie where the mathematical rigor can be most easily seen—namely, in the symbols”, they write; “…Rather, it lies in human ideas”. In turn, “…our ideas are shaped by our bodily experiences—not in any simpleminded one-to-one way but indirectly, through the grounding of our entire conceptual system in everyday life.”
Lakoff and Núñez are not unmindful of the extraordinary seductiveness of the Platonic story of mathematics, as exemplified by Russell’s youthful experience. They dub this seductiveness the “Romance of Mathematics”, and break it down into a series of constituent creeds, for example:
“Mathematics is part of the physical universe and provides rational structure to it. There are Fibonacci series in flowers, logarithmic spirals in snails, fractals in mountain ranges, parabolas in home runs, and p in the spherical shape of stars and planets and bubbles.”
“To learn mathematics is therefore to learn the language of nature, a mode of thought that would have to be shared by any highly intelligent beings anywhere in the universe.”
Not at all surprisingly, the authors, while sympathetically admitting to the beauty of this vision, contend that every element of the Romance is false, an intellectual myth formed out of an ignorance of our true, metaphorical way of thinking. “”
However, there are imaginable explanations for the Romance of Mathematics besides the cognitive-metaphorical one sketched out by Lakoff and Núñez, or the map-of-tautologies one expressed by the aging Russell. We can say that the metaphors themselves are ad hoc, or that they are themselves far too complex to be the “basis” of anything. We can make recourse to the fact that no two mathematicians, working independently, have thus far ever arrived at contradictory answers to the same problem using correct logic–whereas many have, in fact, independently arrived at the same answer by widely different though correct means–and then thrill at the thought that this surely can be no coincidence. To all this, we can add to this that Lakoff and Núñez are far from having the kind of experimental warrant needed to back up their framework; “cognitive science” remains severely limited in terms of predictive power, while attempts to localize complex concepts or behavior to specific regions of the brain so far have, at best, a checkered track record.
Probably the most compelling argument, though, is simply that most mathematicians themselves tend to be more or less overt Platonists regarding mathematical entities, while remaining no more likely than anyone else to attribute eternal “essences” to everyday objects or notions derived from common parlance. Whereas endocrinologists tend not to think of an eternal “essence” encompassing the hypothalamus, and lawyers tend not to posit a boundless ream of patent laws beyond the world of phenomena, mathematicians the world over, throughout history, have been drawn to the very conceptions that Lakoff and Núñez, and the older (and wiser?) Russell so vividly dismiss.
I’d like to share a small personal example of the Romance of Mathematics, in the form of a fairly basic geometric theorem that has been known for well over a millennium: Ptolemy’s Theorem.
The great Alexandrian polymath Claudius Ptolemy (c. 100-170 AD) is probably best known for devising the “Ptolemaic” model of the solar system, which dominated Western astronomy till long after Copernicus’ heliocentric model debuted in the 1500s. But Ptolemy was also a geographer, music theorist, astrologer, and mathematician. The theorem which has come to bear his name states that, given a cyclic quadrilateral (a four-sided figure all of whose corners, A, B, C, and D, lie on the same circle), it must be the case that
AB•CD + BC•DA = AC•BD.
In words, this means that if you multiply the lengths of the opposite sides of the quadrilateral and then add them together, it will exactly equal the product of the lengths of the two diagonals of the quadrilateral.
Without going through the proof in detail (you can find it, like nearly everything else under the sun, on Wikipedia), it involves choosing a line segment that goes from one corner of the quadrilateral to one of the diagonals, so that you get pairs of similar triangles. This makes it possible to calculate the length of that diagonal in two different ways, which leads directly to the expected formula.
For my part, from the first time I heard of this theorem I admit to a certain pleasant amazement that it all works out so neatly and so well. Starting with idiotically obvious remarks about circles and combining them in apparently mindless ways, we come to something that is both not a mess and not completely useless–the opposite of what we expect from real life. There is a fascinating sensation, too, of generality being subsumed in simplicity. We could very easily expect, given the infinitude of ways to pick four non-identical points on the circumference of a circle and hence the infinitude of possible quadrilaterals, that the problem would be messy and unsolvable; that no relation, simple or otherwise, could be found among all these random-looking shapes, beyond the one already granted, that their vertices lie on the same circle.
The theorem works every time, and it works everywhere, which again seems remarkable: again, compare this with real life, where virtually nothing works every time, everywhere. Somehow this feeling of remarkableness itself does not get old; I still get a certain gratification from doodling the proof on a nearby sheet of paper, and finding the same conclusion awaiting, its inevitability apparently unaffected by the fact that that particular sheet had never been used to work the problem before, or that I was in a completely different mood this time around.
Note, too, what happens with the formula above if we let ABCD be a rectangle; we then have
AB=CD, BC=DA, AC=BD,
AB^2 + BC^2 = AC^2.
So we find the Pythagorean Theorem popping up, after starting out in a direction that seems to have nothing in particular to do with right triangles–it is a theorem about four-sided figures, after all. There is also a way of generalizing Ptolemy’s Theorem to all quadrilaterals, not just cyclic ones: it’s called Casey’s Theorem. Thus we see another aspect of the Romance of Mathematics: coherence. Coming at the problem from many different directions leads to surprising connections, and there are often “higher” and “lower” level interpretations of any given theorem. The Pythagorean Theorem is a special case of Ptolemy’s Theorem, and Ptolemy’s Theorem is a special case of Casey’s Theorem, and on and on.
The question is what to make of what is happening as we feel this wonderment, this almost delirious relief that something that feels like it could have been so hopeless and complicated turned out simple and clear and connected. It seems odd, almost suspicious, that a process so ostensibly cold and detachedly logical as re-proving a theorem could be accompanied by these kinds of emotional reactions.
Really, none of the options seem to explain those reactions: not the imaginability of a contrafactual world where the theorem did not work; not the presentiment of an ineffable world where the cyclic quadrilaterals and the facts about them dwell far above the world we move, eat, gossip, and evolve in; not the possibility that tautologies may come to seem profound by building them to a complexity where our minds can no longer take them in all at once.
After all this, we might be tempted to ask not whether Ptolemy’s Theorem was created or discovered–but whether it was felt. So maybe it is best to close with Russell’s own assessment of the wider changes in his life that helped loosen the grip of mathematical Platonism:
“In this change of mood, something was lost, though something was also gained. What was lost was the hope of finding perfection and finality and certainty. What was gained was a new submission to some truths which were to me repugnant. […] I no longer have the wish to thrust out human elements from regions where they belong; I have no longer the feeling that intellect is superior to sense, and that only Plato’s world of ideas gives access to the ‘real’ world.”
In the case of mathematics, our best approach may be something similar to Russell’s: to continue to accept the romance of mathematics when and if it presents itself, but without trying to assert that it is really the entire story, or the sole or best form of truth, or even a true refuge from the problems of our fleeting embodied lives. Human warmth and fallibility, however we may try otherwise, seems destined to intrude on and thaw even the most icy and insulated Platonic realms. Our attempt to vouchsafe ourselves a privileged and perfect world by indexing it to our own reasoning and then placing it infinitely beyond all other things either discloses its metaphoric and emotional underpinnings, or fails to remove the doubts it was meant to forever banish, or else becomes just one of a ironic menu of options.
One can look at this whole situation scornfully, like in Kant’s remark that “Out of the crooked timber of humanity, no straight thing was ever made”. But if we are able to trust in ourselves that chaos and freedom need not melt into one another at the mere abandoning of a definition, it may be possible to see it as a kind of blessing that the ice of Platonism can be thawed. Reaching this point, in turn, may be the best way for “humanism” to become worthy of the name.