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.”

and:

“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,

and therefore

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.

Submit this article to tons of worthy magazines or online forums (but not “fora”.)

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I’ll put in a plug for platonism. You didn’t dismiss platonism, only weighed against it in tone, motivating my comment. Your primary aim, as I understand it, was to look at mathematics in the context of human life, including how mathematical thought feels (mathematics has that aspect, and intensely, for all who come near it). One important psychic role of mathematics has been to posit an eternal realm beyond human strife, as does religion, with related benefits for the sense of meaning. Your post describes how Russell initially saw math in those quasi-religious terms , but backed away later on. I hadn’t known of this. Head platonist quits!

The issue remains: is mathematical platonism the correct metaphysical viewpoint, independent of the feelings, religious or otherwise, that it might evoke?

Are mathematical objects internal to the human world? “Yes”, is the implication that I read from, eg

“Mathematics has ceased to seem to me non-human in its subject-matter.”-Russell

“It is a form of language (or perhaps even a mood)” – you

What about the quadrillionth digit of pi, or the trillionth prime, or the membership status in the Mandelbrot set of a point that nobody has happened to look at yet (and nearly all points are like that)? Each of these has a definite value – a value that all humans and computers (assuming competence) will agree on if they carry out the observation. These observations resemble peeks into the oceans of Ganymede in that no such peek has happened yet, but if it does happen, its result is well-determined independent of human creativity or desire. A device could be designed that would blow up a dam or kill a cat (Godel’s cat) dependent on outcome, with no human in the loop. Saying that such objects are internal to the human world seems to require extension of the meaning of “human world” beyond all doings and thoughts of humans that have occurred until now, and into the realm of all possible thoughts and all physical realizations based on those thoughts. A realm which the universe held in waiting until our arrival?

Maybe no one other than us humans have had the concept of pi or of a prime, but now pi and the primes have been exported into the non-human realm, and this has been required for their observation (beyond the tiny samples that fit in human minds).

You may not want to say that pi and friends “exist”, but their properties are well-determined prior to observation. What do you call such things? My platonist language game allows the phrase “existant, in the mathematical sense”

Is this way of talking erroneous? If so, as our friend Paul always asks, what is at issue? What hangs upon the answer?

I am sympathetic to the project of thawing the ice of platonism, at investigating how mathematics fits into human life, and how it evolved in the context of the human situation. But will the the puddle evaporate? I don’t think so.

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Chris,

I don’t see it as possible to refute (or for that matter guarantee) Platonism, nor is that my goal. Indeed, “refutation” is itself a logico-mathematical notion. Rather, I tend to see Platonism as a branch of human ego or even human self-obsession disguised as transcendence–the more so when its proponents selectively invoke it just for mathematical concepts. (Claiming it for EVERY sort of concept at least feels less like special pleading.)

I think it’s telling that you pose the question as whether “mathematical Platonism is the correct metaphysical viewpoint”, for this carries many assumptions: that there **already exists** one viewpoint, and also **already exists** one discrete property of “correctness”, such that these things pair together in one and only one possible configuration, that itself **already exists** and merely awaits our finding it. In effect, by your very way of phrasing the issue you have already assumed Platonism.

Why should our bookkeeping methods, like mathematics or chess moves or legal proceedings, have no human content whatsoever, and somehow instead reach across and beyond all space and time into an unsullied realm of absolute Trueness? How would anyone ever underwrite such a massive claim, except that we cannot imagine it another way? Why, for instance, should the fact that Ptolemy’s Theorem expresses in a nifty form something about cyclic quadrilaterals mean that it isn’t still tautological–that it actually tells me anything true outside of what’s already given by such human bookkeeping?

Mathematics is a useful tool and a sometimes fascinating way of examining certain kinds of conceptual structure. For me claims for anything far beyond that just seem unlikely, uncompelling, unnecessary. They may even be counterproductive–enshrining a sort of mental idol that holds itself supreme over, and stands in the way of developing, other kinds of seeing and thinking.

As for computers agreeing, they agree because they are designed to agree: this is part of their function and utility; it’s the nature of the game. Saying computers following the right steps will agree no more requires a pre-existent transcendental Answer Key than saying dancers following the right steps won’t fall over each other.

You also say that “now pi and the primes have been exported into the non-human realm”. Even assuming that “pi” and “primeness” are somehow able to have a distinct existence “in themselves” outside any mathematician’s or engineer’s mind–in effect again sneaking in Platonism before you argue for it–in what sense do you mean for this “exporting” to have happened? How is a transcendent, mind-independent, eternal entity supposed to be “exported” into the universe by mortal human minds, having never been there before? Tricky.

Maybe the “puddle”, as you put it, doesn’t have to evaporate; it’s sufficient that we remind ourselves that whatever we like or tend to believe, it’s almost certainly not “ultimate truth”–a tremendous, potentially calamitous pairing of words. To horribly over-stretch the metaphor, a humbled Platonistic puddle may be as good as a dried-up one.

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“Why should our bookkeeping methods, like mathematics or chess moves or legal proceedings, have no human content whatsoever, and somehow instead reach across and beyond all space and time into an unsullied realm of absolute Trueness?”

I don’t think that math has no human content. Did my comment indicate otherwise?

Onwards to pi. Consider again the computer programmed to cruise out into the far reaches of pi (or whatever mathematical structure might be of interest). Yes, of course, we know how it will go about its work, but not what it will find. Nor can we influence the outcome with thoughts or wishes. I am not claiming any importance for the answer (unless hooked to poison gas that will kill Godel’s cat). Still, the answer will come from somewhere, some process of the world which is not internal to the thinking of any human mind. Note that there are many algorithms for computing pi, so we are talking about pi itself not the output of any particular algorithm, let alone a particular run of a particular algorithm. The world somehow has all those digits lying in wait for inspection, and will reveal exactly the same sequence, whatever the method chosen. If you hate the word existence for such mathematical structures, maybe you will accept Wittgenstein’s phrase: “not a something, but not a nothing either” (which he applied to the sensation of pain). This is what I mean by “exporting”. We can place pi out there in the world beyond the touch of human hands. Not only can we, but it has been done many times.

I have the sense that the difference between our thoughts here involves philosophical idealism on your side, as opposed to realism on mine. (The realism of my scenario is intended to be strictly physical, but with mathematical consequences). For you, perhaps, (just speculating), pi has no meaning outside the human mind, so a computer churning out its digits can have no significance of any kind beyond electrons whirring about (cat or no cat). I admit, I can’t quite grasp your standpoint, and you can’t seem to grasp mine (or perhaps your grasp is complete, and if I had it, I’d dismiss platonist thoughts too).

In the few years since I’ve started talking philosophy on a regular basis, I’ve noticed that the effect has been as often to demonstrate the impermeable isolation of my own mind, as to justify optimism about possibilities of mutual understanding. Still, I find it to be the topic of conversation to which I am most attracted.

“a humbled Platonistic puddle may be as good as a dried-up one.”

Pretty harsh on us puddle-dwellers!

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Chris,

I didn’t intend to be “harsh”–the puddle imagery was your own after all–but I do think it is wise to hold one’s philosophical views with a certain humbleness, and not less so when it comes to asserting vast detailed knowledge of an entire, but at best quite indirectly observable, realm of existence.

As for your thinking math has “no human content”, I did infer that. The term “mathematical Platonism” does usually indicate that math, or whatever else you count as part of the Ideas, is human-independent; it existed forever before us, and will exist forever after we are gone, and we discover it by doing proofs or contemplating Forms, etc. So when you then talk of investigating how math “evolved” or how it got “exported” into the universe by human invention, I have to say you appear to be proposing something radically–albeit intriguingly–different from mathematical Platonism. It reminds me a bit of Smolin’s talk about how the laws of physics “evolve”. (Mathematical Goadism?)

To torture the puddle metaphor and your own earlier comment a little more, your question seems to be, can the “ice of Platonism” be “melted”, without evaporating it? Well, if by “melt” we mean to dissolve the fixed relations and shapes that constitute the solid, while somehow retaining the bulk, I think this is bound to nullify anything resembling the traditional Platonic view–whose very focus is on the inevitability and fixity of relationships between elements as much as on the elements themselves. So evaporated or not, I suspect the “Platonic puddle” is not really very Platonic any more.

I also don’t think the idealism-versus-realism dichotomy helps much here either. What I’ve said doubting Platonism has almost no truck with idealism–at least in the “world is my representation” sense. (I’m not sure we even “represent” the world.) By contrast, Platonism is usually considered an example of Idealism–hence the term, “Platonic Idealism”.

Contrary to the claims of idealism, I don’t reject that there is something “outside” our selves; rather I suspect that we are unable to correctly or definitively represent (or otherwise catch hold) of it, and I tend to doubt that it takes the form of an infinite, insubstantial Answer Key holding in advance all the answers to any question anyone might ask. It might be so, and the “outside” does seem to have certain regularities which are quite useful; but there is no warrant for insisting that those regularities must be absolute and unchanging, or somehow ineffably pegged to mathematical concepts, or much less dictated by Forms in a Platonic heaven. Again, believe it if you want, but believe it humbly/cautiously.

With regard to pi, you say “The world somehow has all those digits lying in wait for inspection, and will reveal exactly the same sequence, whatever the method chosen.” There is a certain uncanny feeling there, perhaps. But you are again neglecting the fact that these different algorithms were specifically designed, by people, to converge, in this case to the ratio of a circumference to a diameter–none of which needs exist in any pure form “out there”.

I’d also note that Wittgenstein, in your quote, was referring not to universal mathematical entities, but to sensations. That is a huge difference. At any rate, I’ll see your Wittgenstein and raise you two, as these seem to bear closely on your standpoint:

‘The general form of propositions is: This is how things are.’ —That is the kind of proposition that one repeats to oneself countless times. One thinks that one is tracing the outline of the thing’s nature over and over again, and one is merely tracing round the frame through which we look at it. [PI, #114]

and also:

Here I should first of all like to say: your idea was that that act of meaning the order had in its own way already traversed all those steps: that when you meant it your mind as it were flew ahead and took all the steps before you physically arrived at this or that one. Thus you were inclined to use such expressions as: “The steps are really already taken, even before I take them in writing or orally or in thought.” And it seemed as if they were in some unique way pre-determined, anticipated–as only the act of meaning can anticipate reality. [PI, #188]

It might be we are talking past each other. One thing I will agree with you on, from your first comment, was: “One important psychic role of mathematics has been to posit an eternal realm beyond human strife, as does religion”. Certainly harboring doubts about Platonism can mean the loss of a certain spiritual consolation. But, to me, with or without that overarching superstructure of Ideas, the world somehow seems just as astounding.

Cheers–

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