Month: September 2016

The Joker’s Stars and Stripes

The other day I was at the library reading this article about how unexpected the demise of the USSR really was, and how to this day it is still not really understood. Then I looked up, watching people walking around the carrels and clicking away at computer terminals. It was as innocuous a scene as one could hope to find, and yet in that moment there was a subtle transition that I cannot describe except to say that, for the first time, I really felt that fascism is somewhere we could go in this country.

The outward patterns of life remain largely the same. Yet a hunger and a rage has slipped in somewhere, and at the same time a supreme lassitude that would never lift a finger in protest; it is in the movements, the expressions of people everywhere. One feels it in every quarter.

There is fatigue with normalcy, fatigue with order, fatigue even perhaps with reason itself. Fairness, equality and justice are on everyone’s lips, and the concept of rights has never been more widely invoked. But underneath, for more and more citizens, these terms have come to be seen as objects of cynical manipulation or weapons for tantrum-throwing. The names of great ideals have become mere phantoms or memes–either objects of derision, or sound patterns no more meaningful or idealistic than the pounding of a fist on the dinner table or the honking of horns in traffic.

On the eve of the first debate, Donald Trump–a casual bigot who romanticizes violence against opponents and exhibits an alarming power to cast “truth” as whatever he says often and loudly enough–stands once again at the top of the polls, or at least in a virtual tie. On the other side of the aisle stands the establishment’s lackluster champion: a terminally uncharismatic former first-lady with a record of committing major errors in judgment, compromising herself to corporate and elite interests across the board, boosting for unnecessary military interventions and now possibly, of failing health.

In anything like a functioning democratic culture, I cannot easily believe either of these people would be in a serious position to attain the highest office in the land; but if they somehow were, there would, I think, be mass mobilizations and civil disobedience aimed at delegitimizing their candidacy and setting up a recall process of some kind.

Instead, a Joker-like desire to “just watch the world burn” has quietly taken the place of ideals of any recognizable sort, much abetted by the mass retreat in recent years into an isolating digital realm of virtual consumer comforts–a realm that seems to systematically un-train just the kind of sustained attentiveness towards wider issues that is needed to protect or even articulate political ideals. (I will never forget one young adult at a recent philosophical discussion group, who insistently and in full seriousness argued that his “philosophical ideals” consisted of the right to enjoy a nice cup of Starbuck’s once a day. To my astonishment, the other millennials soon chimed in with agreement.)

Against the hopelessness and humiliation that has quietly built up across so many walks of life  in this country–plus the strangely listless inability to formulate a critique or find enough common cause with one’s own neighbors to coalesce into movements–it is as though all hope has been invested in the catharsis of one sublime outburst of revenge against the power structure that (largely correctly) is deemed the source of their humiliation.

It seems to matter little (or is left altogether unthought) whether such revenge would justify ushering in, as it surely would, an age of even greater dysfunction and misery than came before. At first, fascism works by seduction, not by intimidation–and certainly not by far-sightedness.

In that moment in the library, I realized there is an incredible tension that runs through the center of human life–a tension between, on the one hand, the sense of overwhelming permanence and universality that comes from having been immersed in a certain social or political arrangement all your life; and on the other hand, the facts of history that tell us all such arrangements are inherently unstable.

There is nothing holding us up: the unthinkable can happen if we let it. And moreover, when or if it does, there is no guarantee that it will make any sense at all, even in retrospect.

Having lived in the US all my life, I have known it as a remarkably free country–one with a sclerotic ruling class and a limited civic-democratic culture, admittedly, but nevertheless rule of law, due process, a strong Bill of Rights, an occasional ballot measure, and some other basic checks against total statist lunacy. None of that has to last if no one will fight for it–and certainly not if they secretly crave its opposite.

In a way, Trump, with his searing militarism, nativism, and denial of climate change, is the demonic inverse of Obama’s soaring promises of dramatic “hope” and “change”. Let’s hope that Trump’s promises go just as undelivered as Obama’s–or better yet, do more than hope. Find people of like mind and raise your voices against this sham, or find a third party candidate. Do something. Protest, dammit, protest.

Three Tiny Essays, #2: Madness, Mining and Magnitude

I.

A great many of the most influential scientists and mathematicians were outsiders, odd characters or holders of odd beliefs; Cavendish, Einstein, Dirac, Newton, Wallace, Crookes, and many others. The most correct way to describe this situation is that very many people who have been right have been odd, but being odd does not make one likely to be right.

Modern academic science, with its essentially bureaucratic structure, tends to focus on the truth of the second part of this statement and put the first part out of its mind, gravitating towards uniformity, patronage, and respectability. Cranks and crackpots follow the opposite pattern, naturally gravitating to the first part of the statement and forgetting the second, and so they wear their odd beliefs, intractable personalities, or outsider status on their sleeves as somehow evidence that whatever they believe or say is a suppressed or overlooked gem.

The creative and flexible middle, those who dared to think strange disturbing things (for all real strangeness is disturbing) but nevertheless retained just enough coherence and honesty to engage successfully with institutions and with the rest of humanity, seems today quite lost in that divide.

II.

Something seems to be underway in biology that mimics what has been underway in particle physics, namely a period of disillusionment brought on by diminishing returns on old brute-force experimental thinking.

The physicists felt convinced that, if smaller accelerators had shown them so much, building an even bigger accelerator would give them more leads, open up spectacular new theories. The LHC would be the key to a new understanding of the universe and a new epoch of theories (supersymmetry and strings).

The LHC was indeed built, ran successfully at full power–and found nothing beyond the Higgs, a particle predicted by existing theory back in the 1960s. This is the so-called “Nightmare Scenario”: no new physics. With this summer’s release of data from the LHC’s 2016 runs, which confirm these negative results to 6 standard deviations, the problem has become undeniable. As there will probably be no new accelerators larger and more powerful than the $10 billion LHC in the foreseeable future, more than ever theoretical physics finds its faith in expansion and intensification dashed.

Roughly analogously, biology has placed great hope in the continual expansion of throughput, particularly of sequencing power and data analytics. If only one can sequence more millions of base-pairs per hour, can sequence more species simultaneously, can store them by the terabyte in ever growing, better curated data banks and comb them with enough types of algorithms and statistical tests using ever larger amounts of processing power, we surely must eventually find ourselves the masters (and understanders) of life. Moreover, we surely must at last become able to calculate the character of an individual directly from their genomic makeup, just as we make any other calculation. Not only medical problems, but ultimately social and psychological problems as well, could thereby become controllable and tractable.

Yet it seems likely that what biologists will find from such approaches, if they find much of anything reproducible at all, is not hard principles of life but effects that are squishy–either so subject to vast change over the history of the organism and so dependent on the organism’s own choices that stable definitions become impossible, or so hugely multivariate that the curse of dimensionality makes predictive distinctions completely impractical at any level of precision that would be useful. Much as string theory and the faith in ever-larger accelerators has led physics into a seeming blind alley, the belief in sheer throughput and processing as the solution to life and human individuality seems likely to turn out, by and large, to have brought us little closer to understanding ourselves (if not farther from it). Size and complexity cannot forever compensate the limitations of an underlying conception.

III.

Perhaps it is best to look at discovery and invention in terms of the metaphors that preside during their appearance. When discoveries are abundant and expected, a “mining” mentality obtains. When they are rare, a “miracle” mentality is more typical. Put another way, the sheer concentration of fundamental scientific and technological discoveries and upheavals in the 19th and 20th centuries kept us from appreciating their singular, miraculous quality; we allowed ourselves to think they were being mined, hence that all that was needed was to keep digging diligently and new ones would continue appearing indefinitely.

On the other hand, even mining cannot go on forever in the real world; therefore, in a curious irony, the modern determination to see these shattering discoveries as merely the first products of a mining operation had to be shored up by smuggling in a new form of the miraculous, to wit: the limitless insight and innovative power of the human mind. Thanks to its unique and now scientifically-mediated power to discern the absolute essence of reality, we assured ourselves that humanity would be able to mine the vein of fundamentally transformational discoveries indefinitely, priming further expansion by artfully changing tack (or frack?) whenever scarcity or limitation reared their heads.

Now we begin to realize that the mining process has slowed, that even our scientifically-adjusted vision is not without serious glares and aberrations, and above all, that we have staked our future as a civilization on achieving an ongoing succession of ever-greater miracles. “It’s a near miracle that disaster has been avoided this far”, observes Chomsky, “and miracles do not go on forever”.

Reading List Roundup II: “A Brief History of Progress”

A few months ago, I’d been looking for a good concise extension and filling-in of the history of the Idea of Progress, to support some of what I had read by others on this enormous subject–particularly Heidegger’s “The Question Concerning Technology” and John Michael Greer’s trenchant series of expository essays from 2013.

Judging from the title, Ronald Wright’s “A Brief History of Progress” looked like just the thing. The reader reviews only were quite glowing, and though the book dates from 2004, I thought, what does such piddling a span matter in the scheme of global history and the evolution of grand ideas? So I snapped it up and sat down with it the moment it arrived in the mail.

Unfortunately, despite the title, Wright’s book has relatively little to say about the idea of Progress, or its intellectual origins and development. While providing a reasonably well-written rundown of various historical episodes that certainly do suggest potential repercussions and follies of short-sighted human expansion and overshoot (the destruction of habitat by farming and hunting; the fall of Rome, Easter Island, the Mayan city-states, etc.), Wright’s treatment even of these episodes was mostly re-hash, and I suspect will seem so for most readers with even a moderate grasp of history.

We also learn very little about why these cultures grew dysfunctional and collapsed, or how it has any bearing on us or our own notion of Progress, other than through the nebulous implication that if bad things could happen to others, maybe they could also happen to us. To make that connection anything other than nebulous, there would need to be not only a history of deeds or outcomes–however scary and ominous, moai toppling and so forth–but a complementary history of the thoughts and ways of seeing that made these collapses possible, and then inevitable. That thought-history would then seek to place our current ideas about Progress in a wider context, to see whether they truly deserve the world-pivoting singularity with which we imbue them.

Instead, on many of the issues of greatest concern with respect to Progress–its philosophical underpinnings; its inevitability versus its contingency; and above all, the potential alternatives that might yet allow a fulfilling and sustainable tenure for this curious hominid species on this curious sapphire planet–on these things, this light little book is light to non-existent. In fact we find only passing glances at the subject, which either did not much interest the author, or ultimately proved too thorny, too overwhelming, or too blinding for him. For Wright, it almost seems, Progress is much like the Ark of the Covenant: you can mention it all you want, even track it down to put in a museum; but whatever you do, you must not look directly into it.

This is all a pity, because the study of history and particularly of the attitudes and customs of various collapsed and collapsing civilizations regarding their future–the “shape of time”, as Greer calls it in another essay–surely is indispensable to understanding the real underpinnings of our belief in progress, and to our search for alternatives.

Wright’s discussions of progress itself can almost be counted on one hand. There is an early mention of Sidney Pollard’s definition of the concept of progress, from his 1968 work The Idea of Progress: History and Society: for Pollard, progress is “the assumption that a pattern of change exists in the history of mankind […] that it consists of irreversible changes in one direction only, and that this direction is towards improvement.”

Yet even this presents a relatively bare definition, one that scarcely touches on the psychological, mythic, and religious aspects of our collective belief in progress, especially of the technological kind. Wright does briefly mention that faith in progress

“…has ramified and hardened into an ideology–a secular religion that, like the religions that progress has challenged, is blind to certain flaws in its credentials. Progress, therefore, has become “myth” in the anthropological sense” [4].

This is surely tantalizing and cries out for the kind of scrutiny described above; what are the “flaws in the credentials”? In what way is progress similar to a “myth” or a “religion”? Yet for another 150 pages, Wright simply seems to drop these burning questions altogether, settling for what might be described as a hushed editorial griping on the sideline. We know that Wright doesn’t like what’s in the slideshow he’s giving us, but we know little else.

Later on, much as with the historical treatments, we tend to find mostly subjects and observations that, while correct and worthy in themselves, will neither surprise us much nor offer much of a new perspective. The comments about global warming and environmental degradation, for example, though alarming for 2004 standards (the date of publication), seem almost demure today, as warming rapidly accelerates and global all-time temperature records fall by the month.

One gets the impression that “A Short History” is a torso, an abortive attempt at a much more comprehensive work which midway through was scuttled and re-worked into more of an extended essay. Certainly the massive notes and bibliography section, which take up approximately 70 pages of this slim 200-page volume, point to a research process that was extraordinarily extensive considering the fairly modest text that resulted from it, and it’s doubtful that Wright truly manages to make use of this many sources in such a small space. (From my own experience in both reading and writing, I find that 4-5 good sources can provide ample fuel for dozens of pages of analysis, whereas it’s rare to work in more than 3-4 citations per page without either seriously disrupting the continuity of a text, or giving away that that author didn’t really read the cited works.)

This, too, just adds to the “Ark of the Covenant”-like impression: it’s as though in the course of his research, Wright found himself on the verge of conclusions that he could not tolerate, or at least could not articulate to his satisfaction, and settled instead for a relatively tepid historical run-down with echoes of ineffectual (therefore acceptable) hand-wringing over environmental issues and man’s folly. Despite this, “A Brief History of Progress” could well have been kept brief, I feel, while still being far more illuminating than it turned out. What was needed was the courage to entertain wide conceptual vistas and changes of view, however dangerous and disorienting–and it is in this, I suspect, more than any scholarly shortcoming, that the effort ultimately fell short.

***

Finally, I also have been meaning to review E. O. Wilson’s “The Meaning of Human Existence”, so I’ll post it separately in a little bit–it seems to be running into more detail than I expected (what else is new). In the meantime, I hope the Einstein equations and the history of Progress will provide my esteemed readers sufficient light reading.

Reading List Roundup: “A Most Incomprensible Thing”

While I’m working out some other things to ramble about, thought I’d finally do a run-through of a few books I’ve read lately.

***

First off, partly out of my ongoing fascination and uncertainty regarding the recent LIGO announcements, I began a number of months ago to feel very insecure regarding my knowledge of general relativity (GR). Who doesn’t, right?

In truth, though I’d learned about special relativity in high school and had a book of Einstein’s papers sitting perpetually on my desk, I have generally been terrified of the concept of tensors–the formidable coordinate-independent mathematical creatures with which GR is written–and so my understanding of GR has been limited to the usual hand-wavy, rubber-sheet-Brian-Greene-texture-of-reality variety.

I’ve tried to remedy this over the years, but constantly found myself frustrated by a kind of stair-step division in the books and publications on the subject. On the one hand, you have the popular “rubber sheet” stuff, which tells you almost nothing about the mathematical justification or structure, just vague conceptual gosh-wow like “matter warps space and time!” and “Einstein was amazing because he saw acceleration and gravity are alike!”.

I don’t want to underplay the value of the conceptual. John Archibald Wheeler’s “A Journey Into Gravity and Spacetime” is among the finest examples of this type of GR book, and reading it did I think help put the overarching ideas in the right general places.

But to me, most physical theories are ultimately like machines built with math, and so telling just the conceptual side of GR versus knowing at least the skeleton of the math behind it is kind of like the difference between telling a story over the campfire about a wondrous hovering creature with a spinning head, and showing me the technical drawings and specs of a helicopter.

These deficiencies naturally leads one to search out the other class of GR books, which unfortunately seem mostly to be written in the manner of a congratulatory capstone, a sort of welcome-to-the-club that assumes your successful surmounting of several years worth of arduous academic arcana. Opening one of these to any page, you will likely be greeted by the smell of decades-old book glue; then, as you gaze down, you will have the strange vertiginous sensation of being about to crack your head on a huge, neverending, treadmilling cliff-face of equations.

Stuck between these extremes, I’ve looked for a book that would lay out the mathematical lineaments of GR in a non-trivial way, while still allowing someone with a bit of background in calculus and linear algebra to (mostly) see what is going on. For years, I didn’t have much luck.

But my post-LIGO perplexities led me to restart the search, leading me to “A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity”. Here, Peter Collier has put together exactly what I was hoping for: the missing link of GR pedagogy, a volume intended for those of us badly in need of, as he puts it, “something in-between, a Goldilocks volume […] neither too easy nor too difficult”, that outlines the geometric theory of gravity.

Much of the book is “backstory”, for in the interest of completeness Collier dedicates the first 100 pages or so to bringing us up to speed on the foundational mathematics, forms of notation, and also good old-fashioned Newtonian gravitation. But I found these very clearly presented and it’s a good refresher course–I was surprised how effectively years of calculus can be shoehorned into a few dozen pages of core principles and operations, without losing very much (though practice surely makes perfect).

Perhaps inevitably given the interest of tractable length and scope, the book leaves out a great many derivations. Mostly these are not missed, as making constant detours away from the main points of GR would be far more confusing than simply assuming certain needed results and letting the reader research the proofs to their taste; still, in a few cases these elisions left me frustrated. The most notable example is that Collier gives no rationale for how Einstein arrived at and justified his final field equations. (Why is that factor of 1/2 in there in front of the Ricci scalar and the metric tensor?) Collier notes that Einstein was on completely the wrong track for years before hitting on the much-loved final form, and he gives some sample problems that suggest that equation makes sense in some toy models, but stops there. Maybe this is the best one can do at this level of explanation–it’s formidable enough to be able to say what the metric tensor etc. even is–but I wish we could get just a little more of a peek into how the correct field equation comes about.

Unfortunately for the LIGO addict/skeptic, “A Most Incomprehensible Thing” also stops short of even outlining the theoretical case for gravitational radiation, except for a passing reference to them on the way to explaining the stress-energy tensor. But it lays a foundation–or at least the foundation of a foundation–for the dedicated autodidact to move on to such an understanding, and makes the LIGO papers and simulations considerably more intelligible.

There are other trends in physics that come into a new light upon reading this volume, too. For example, many of us have likely heard the accelerating expansion of the universe reported as one of the most shocking revelations in cosmology since the cosmic background radiation itself. But on page 317 of “A Most Incomprehensible Thing”, we see that acceleration is really far from an unforeseen possibility or a fundamental shock to cosmology, for there are well-established models approximating such an acceleration that date from decades before the discovery of said acceleration in the late 1990s. In particular, it seems we’re just in the phase of the universe’s life where it is best modeled by a dark-matter dominated de Sitter universe (discovered c. 1917).

Though Collier notes that the book is something of a work in progress and welcomes readers to submit corrections, the writing is crisp, providing in most cases a fine interplay with the equations and derivations. There is a punctuation error or misspelling here or there, but nothing glaring as far as I could tell. Altogether, for anyone seeking a worthy jumping-off point into a more detailed and mechanistic understanding of cosmology or fundamental physics, I’d recommend grabbing this book and clearing your reading list for a couple months: a fun but demanding ride awaits.

***

More reviews to come…

The Human Equation

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.