One of the authors I challenged was Eberhard Knobloch (who, incidentally, was just awarded the most prestigious prize in the field). He replied with indignation in a letter to the editors, calling my paper “completely unacceptable.” I have submitted a reply which will hopefully appear soon.

In the meantime, a review of my paper just appeared in the Zentralblatt. The review is written by Paolo Bussotti, who spent three months as a guest researcher hosted by Knobloch in 2014. This so-called review is not really a review of my paper at all, but rather a regurgitation of Knobloch’s letter to the editors, which Bussotti follows slavishly.

Bussotti cites Knobloch’s letter parenthetically, but in no way indicates that “his” critique is in fact nothing but a point-by-point regurgitation of everything Knobloch said in his letter. Bussotti’s phraseology will lead readers to think that he is offering an independent judgement, when in reality he is parroting Knobloch’s letter. For instance, Bussotti writes:

> [Blåsjö’s] main theses can be summarized in two items: 1) … 2) … let us start from what I have indicated as item 2)

This summary of my view in terms of these two theses is due to Knobloch, who even explicitly labelled them (1) and (2). But those who do not have Knobloch’s letter in front of them will surely be mislead by this kind of phrasing into believing that Bussotti has carried out his own independent analysis, rather than simply transcribed almost literally the exact view of his friend, who is one of the parties in the conflict.

Although Bussotti obediently follows Knobloch on every single point of substance, he does manage to introduce some absurd misunderstandings of his own. For instance, he writes:

> The whole question turns around the interpretation of the sentence translated by Knobloch as “It serves, however, to lay the foundations of the whole method of indivisibles in the soundest way possible” and by Blåsjö as “Whence it will be permissible to use the method of indivisibles proceeding by spaces formed by steps or by sums of ordinates as strictly demonstrated”. The two translations are not significantly different and the whole question concerns the interpretation of that “it”.

It is difficult to fathom how Bussotti could have gotten it into his head that these two quotes are “two translations” of the same passage an even that they are “not significantly different.” They are of course completely different quotes and obviously do not refer to the same passage in Leibniz. The relevant quote in my paper is on a different page altogether (137), with a translation that follows Knobloch virtually verbatim.

In any case, the notion that “the whole question” comes down to this one sentence (as Bussotti claims twice) is absurd. It does, however, square well with Knobloch’s letter, which opens with a critique of my reading of this passage.

As for the substantive point at stake, it concerns whether the “it” in question refers to Proposition 6 (as the standard view has it) or to the idea of its proof (as I claim). Bussotti regurgitates (without saying so) Knobloch’s argument that it must be the former, for reasons of Latin grammar. I do not deny that the “it” is Proposition 6 grammatically speaking. But this proves nothing. As seen in my paper, in the very same passage Leibniz uses the very same “it” as follows: “In it, it is demonstrated in fastidious detail that …” Thus Leibniz is obviously using “it” (i.e., “Proposition 6”) quite loosely as a way of referring to the whole passage of text (somewhat like a chapter heading, say), rather than to the propositional statement per se (which is what Knobloch’s interpretation needs). Thus my interpretation is not at all inconsistent with the text.

Knobloch also raised a quibble about whether Leibniz’s proposition should be called a foundation of infinitesimal geometry or of infinitesimal calculus. Knobloch tried to allege that he spoke only of the former and that the latter is an anachronistic misnomer introduced by me. Bussotti duly parrots the same point:

> infinitesimal geometry [is] partially different from infinitesimal calculus, a difference which [Blåsjö] seems, at best, to underestimate, as he uses indifferently both expressions.

But the insinuation that I somehow introduced this false equivocation is absurd. The notion that Leibniz’s proposition provides a foundation for the calculus is clearly and explicitly present in the works I criticise. In fact, later, when it suits his purposes, Bussotti himself goes on to reaffirm exactly this:

> Leibniz’s proposition 6 offers a general foundation to integral calculus …, no doubt about this.

Why, then, is he bitching that I spoke of calculus instead of infinitesimal geometry, if he himself thinks there is “no doubt” that my terminology is accurate? Bussotti’s critique is not even coherent, let alone sound.

Bussotti’s review ends with an accurate and revealing observation:

> The approach of [Blåsjö] does not seem favourable to edify new and collaborative researches in the line traced by Knobloch and by the other scholars who have studied Leibniz [for] many years. … My conviction is that new insights as to the concept of rigour in Leibniz can be achieved taking into account that the general picture traced by these authors is basically correct.

This seems to me an accurate description of a kind of implicit axiom of modern historiography, namely that scholarship should be collaborative rather than critical. You should pat your friends on the back, not question them. This may be a sound policy if we want academia to be a feel-good social club. But as a recipe for intellectual progress I think it is fundamentally misconceived.

]]>It would be sad indeed if the leading scientists in history committed such elementary blunders and couldn’t avoid even the crudest forms of social bias. If so, we should have to accept the postmodern historiography that takes science to be a social construct determined by the powers that be rather than by objective truth.

Let’s see if this is the right conclusion on Darwin. Saini bases her point on this quote:

“The chief distinction in the intellectual powers of the two sexes is shewn by man’s attaining to a higher eminence, in whatever he takes up, than can woman.” (Descent of Man, 361)

It is indeed hard to dispute, as Darwin observes, that “lists … of the most eminent men and women in poetry, painting, sculpture, music …, history, science, and philosophy” have mostly men in them.

But is this due to biology? Saini evidently thinks it would be a naive fallacy to assume as much. So what is the alternative? That these inequalities are arbitrary social constructs with no biological basis? Then how to explain that they have persisted across cultures and millennia? Did men just happen to obtain the upper hand once upon a time thousands of years ago and then doggedly managed to maintain their arbitrarily constructed advantage without interruptions across countless revolutions, bloody wars, religious upheavals, and the rise and fall of empires? And also the same chance occurrence took place many times over in one geographically isolated civilisation after another?

I don’t think you have to be “blinded by bias” to infer that there are biological factors at play here. Of course this does not mean that men are more intelligent than women. Maybe they just have more muscles and maintain their advantage by force, for example. So is Darwin’s mistake that he assumed “intellectual powers” to be the explanation? His view is rather more nuanced. He in fact explicitly denies that there is an innate difference in this regard:

“It is, indeed, fortunate that the law of the equal transmission of characters to both sexes prevails with mammals; *otherwise*, it is probable that man would have become as superior in mental endowment to woman, as the peacock is in ornamental plumage to the peahen.”

Instead, he attributes the advantage of men to “higher energy, perseverance, and courage.” If there is any innate gender difference, says Darwin, it is this: “Man is the rival of other men; he delights in competition, and this leads to ambition which passes too easily into selfishness. These latter qualities seem to be his natural and unfortunate birthright.”

Today, “although men do not now fight for their wives, and this form of selection has passed away, yet during manhood, they generally undergo a severe struggle in order to maintain themselves and their families; and this will tend to keep up or even increase their mental powers, and, as a consequence, the present inequality between the sexes.” Therefore, “in order that woman should reach the same standard as man, she ought, when nearly adult, to be trained to energy and perseverance.”

In sum, Darwin denies that men are inherently more intelligent than women. Instead he attributes their higher prominence in intellectual pursuits to differences in attitude, and notes that present societal circumstances play a large part in this. Altogether, I do not think this warrants the conclusion that Darwin’s reflections are self-serving “Victorian male” make-believe rather than science.

]]>Some day I will write a follow-up paper, but for now let us consider Ragep’s most overarching argument. He thinks independent discovery by Copernicus is implausible because:

> Perhaps most importantly, why would someone seek to start from scratch when it was certainly known in the fifteenth and sixteenth centuries that Islamic astronomers still had much to teach their European counterparts? (194)

Is this true? Did Europeans at the time consider Islamic astronomers way more advanced than themselves? Is there even a shred of evidence that Copernicus ever held such an opinion? No.

Ragep opts to back up his claim with one single, peculiar reference: the chapter by Feingold in Ragep (ed.), Tradition, Transmission, Transformation. Here’s what Feingold has to say:

> Most of those who sought access to Arabic science were animated by … “reductionist” motives: They viewed “the achievement of Islamic scientists … merely [as] a reflection, sometimes faded, sometimes bright, or more or less altered, of earlier (mostly Greek) examples.” Certainly they recognized the existence of a considerable body of scientific knowledge available in Arabic, but it was usually adjudged either as derivative of the Greeks or, at best, the fruit of sheer drudgery. (445)

A few had “great hopes” to find “most precious stones for the adornment and enriching of my syntaxis mathematike” “in that happy Arabia” (447), and set out to learn Arabic for the purpose. But this was soon followed by a “rapid decline of such studies” (448).

> Some were simply disillusioned by what they viewed as the small return on their investment. John Greaves, for example, griped … that the drudgery he had put himself through editing Abulfeda’s Geography was simply not worth it: “to speak the truth, those maps, which shall be made out of Abulfeda, will not be so exact, as I did expect; as I have found by comparing some of them with our modern and best charts. In his description of the Red sea, which was not far from him, he is most grossely mistaken; what may we think of places remoter?” (448)

Others too lamented “how greate the losse of time was to study much the Eastern languages” and no longer “much care for to trouble myself about the keys [to oriental learning] when there was no treasure of things to be come at.” (449) Francis Bacon agreed:

> “The sciences which we possess come for the most part from the Greeks. … Neither the Arabians nor the schoolmen need be mentioned; who in the intermediate time rather crushed the sciences with a multitude of treatises, than increased their weight.” (443-444)

> Thomas Sprat, the official historian of the [Royal] Society, was willing to admit that the Arabs were “men of deep, and subtile Wit,” but he also felt it unnecessary to discuss them in surveying the progress of knowledge because their studies “were principally bent, upon expounding Aristotle, and the Greek Physitians.” Besides, “they injoy’d not the light long enough. … It mainly consisted, in understanding the Antients; and what they would have done, when they had been weary of them, we cannot tell.” (454)

> More disparaging was Joseph Glanvill who faulted the Arabs principally for their blind devotion to Aristotle. … “These Successors of the Greeks did not advance their Learning beyond the imperfect Stature in which it was delievered to them.” (454)

> William Wotton [held that the Arabs] “translated the Grecian Learning into their own Language [but] had very little of their own, which was not taken from those Fountains.” … “There is little to be found amongst them, which any Body might not have understood as well as they, if he had carefully studied the Writings of their Grecian Masters. … There are vast Quantities of their Astronomical Observations in the Bodleian Library, and yet Mr. Greaves and Dr. Edward Bernard, two very able Jugges, have given the World no Account of any Thing in them, which those Arabian Astronomers did not, or might have not learnt from Ptolemee’s Almagest, if we set aside their Observations which their Grecian Masters taught them to make.” (455)

> Theophilus Gale … [argued that] it is not Aristotle … who should be blamed for breeding that “Sophistic kind of Disputation, which now reigns in the Scholes.” This was the doing of his Arab commentators, Averroes and Avicenna in particular, “who, being wholly unacquainted with the Greek Tongue, were fain to depend upon the versions of Aristotle, which being very imperfect, left them under great darknesse and ignorance touching Aristotle’s mind and sense; whence there sprang a world of unintelligible Termes and Distinctions, with as many Sophistic Disputes and Controversies. These the Scholemen (more barbarous than the Arabians) greedily picked up … and incorporated with their Theologie.” (456)

All of this is quoted from the one article Ragep himself singled out as support for his claim that it made no sense for people like Copernicus to think for themselves since they had so much to learn from the much wiser Arabic sources. If this is the evidence in favour of his claim, you can imagine for yourself what evidence against it would look like.

]]>If the truth is simple, and the right interpretation is half a page long, you can’t publish it. But if you “problematise” the question and bring in an assortment of irrelevant material, chances are that you can put together twenty pages of subtleties and footnotes. And you can certainly publish that, because everybody knows that’s what scholarship is supposed to look like.

Repeat this for a few generations and the papers with the erroneous view have now become forty pages apiece since they have to include baroque analyses of each other in addition to the misconceived primary evidence the mistaken view was based on in the first place.

The further this goes on, the more naive you will look if you speak the simple truth. “But there’s an enormous literature on that!” people will exclaim with indignation. Experts upon experts have piled on the footnotes and devoted entire careers to the issue. Surely so many eminent scholars cannot be wrong. Meanwhile, the simplistic view you espouse has not been expressed by anyone with the proper titles and credentials since practically the age of the dinosaurs (i.e., more than half a century ago).

But the fact that a certain view dominates the latest papers in the latest journals doesn’t mean it has won the day by merit, only that academic evolution is bound to produce organisms that thrive on the excrement of another. The law of the academic jungle is not survival of the fittest; it’s survival of the most publishable.

It is only natural that bottom feeders become more hostile to outsiders with every passing generation. The more established they become, the greater their stake in insisting that quantity of footnotes is a proxy for expertise. Then those pesky people who speak the simple truth are simpletons by definition, and no one needs to face the unpleasant prospect that they’ve been living in the wrong ditch for generations.

These forces make it natural and predictable that historical research will take us further and further from the truth. In time, as academics invest more and more in their erroneous interpretations and build entire schools upon them, they even develop an instinctive hostility toward the truth, since, at that point, accepting the truth is tantamount to challenging the territorial hegemony and survival of their entire tribe.

]]>This definition is clearly meaningless drivel. How can such a masterful work, which is clearly written by a top-quality mathematician, to open with such junk?

Russo proposed a compelling answer to this conundrum. It goes as follows.

Euclid didn’t define “straight line” at all. His focus was on the overall deductive structure of geometry, and for this purpose the definition of “straight line” is essentially irrelevant, as indeed shown by the fact that the utterly useless Definition 4 is never actually used anywhere in the Elements.

Archimedes agreed with Euclid as far as the Elements were concerned, but in the course of his further researches he found himself needing the assumption that among all lines or curves with the same endpoints the straight line has the minimum length. He therefore stated this property of a straight line as a postulate in the context where this was needed.

The Hellenistic era, which included Euclid and Archimedes, was one of superb intellectual quality. Unfortunately it did not last forever.

Heron of Alexandria lived about 300 years later. This was a much dumber time. Fewer people were capable of appreciating the great accomplishments of the Hellenistic era. But Heron was one of the best of his generation. He could glimpse some of the greatness of the past and tried his best to revive it.

To this end Heron tried to make Euclid’s Elements more accessible to a less sophisticated audience who didn’t have the background knowledge and understanding that Euclid’s original readers would have had. He therefore wrote commentaries on Euclid, trying to explain the meaning of the text.

To these new, dumber readers of geometry, it was necessary to explain for example what a straight line is. Heron realised that Archimedes’s postulate captured well the essence of a straight line. However, it is not itself suitable as a definition, because a line should have the property of being the shortest distance between any two of its points, not be phrased in terms of only two fixed points, as in Archimedes’s postulate.

Heron therefore explained that “a straight line is [a line] which, uniformly in respect to [all] its points, lies upright and stretched to the utmost towards the ends, such that, given two points, it is the shortest of the lines having them as ends.” The phrase “uniformly …” obviously refers to the universality of the shortest-distance property applying to any two points on the line.

Now fast-forward another 300 years. Intellectual quality has now plunged deeper still. Geometry is in the hands of rank fools. Euclid’s Elements, which was once written for connoisseurs of mathematical subtlety, is now used by schoolboys who rarely get past Book I and “learn” that only by mindless rote and memorisation. A time-tested way (still in widespread use today) to “teach” advanced material to students who do not have the capacity to actually understand anything is to have them blindly memorise a bunch of definitions of terms.

In this context, therefore, there is a need for an edition of the Elements which includes many definitions of basic terms, which must be short and memorisable, and which don’t need to make mathematical sense.

In this era of third-rate minds, some compiler set out to put together an edition of the Elements that would satisfy these conditions. Heron’s commentary on the Elements is appealing in this context since it affords opportunities to focus on trivial verbiage instead of hard mathematics. But Heron’s description of a straight line is still too complicated. It’s too long to memorise as a “soundbite” and the mathematical point it makes is moderately sophisticated.

The compiler therefore makes the decision to simply cut Heron’s description off after the bit about “uniformly in respect to [all] its points.” This solves all his problems in one fell swoop. The only drawback is that the “definition” becomes utter and complete nonsense, but since the whole purpose of it is nothing but blind memorisation anyway this doesn’t matter anymore.

This is how the ridiculous Definition 4 — a mutilated vestige of what was once a very good definition — ended up in “Euclid’s” Elements.

I have included excerpts from Russo’s paper in my History of Mathematics Reader. There you can find the above argument in his own words.

]]>I wrote these notes to supplement a conventional book, because in my opinion typical intro to proofs courses are fundamentally questionable in their very nature. The premise of such a course is rather like that of My Fair Lady: uncivilised students must be taught to “talk the talk.” It’s the dress code and table etiquette of mathematics. In their fanatical devotion to the clinical, sterilised, Bourbaki way of doing mathematics, these courses have forgotten that precision elocution is pointless unless you have something to say. These courses deem actual mathematical content “too messy” and instead feed the student only fake theorems specifically concocted for the sake of being amenable to the desired mould of what a mathematical proof should look like.

Consider these notes, then, my pauper’s rebellion. Down with haut bourgeoisie snobbery for snobbery’s sake, down with fake perfume and powdered wigs. Let us have the courage to tackle real mathematics as it occurs in nature. Let us put meaning and purpose and exciting ideas first, and let us accept the airs and graces of the mathematical aristocracy only after they have proved their worth in this enterprise.

]]>I maintain that it is crucial to keep the mathematician’s point of view alive in historical scholarship. Technical acumen and an empathic sense of how a mathematician thinks are essential tools for understanding past mathematics. And this is best done by one who lives and breathes mathematics on a daily basis in the classroom, at the colloquium, and by the coffee machine — in short, from within a department of mathematics.

Fields-medalist David Mumford put it well when relating his “personal experience reading Archimedes for the first time”: “after getting past his specific words and the idiosyncrasies of the mathematical culture he worked in, I felt an amazing certainty that I could follow his thought process. I knew how my mathematical contemporaries reasoned and his whole way of doing math fit hand-in-glove with my own experience. I was reconstructing a rich picture of Archimedes based on my prior. Here he was working out a Riemann sum for an integral, here he was making the irritating estimates needed to establish convergence. I am aware that historians would say I am not reading him for what he says but am distorting his words using my modern understanding of math. I cannot disprove this but I disagree. I take math to be a fixed set of problems and results, independent of culture just as metallurgy is a fixed set of facts that can be used to analyze ancient swords. When, in the same situation, I read in his manuscript things that people would write today (adjusting for notation), I feel justified in believing I can hear him ‘speak’.”

This way of doing history has widespread resonance in the global mathematical community. All mathematicians know the feeling of struggling to understand a mathematical work until it “clicks” and one feels certain that one has experienced the same idea as the author, regardless of whether he be centuries or millennia removed from us. Those of us who approach mathematical texts in this way know not to pay too much attention to superficial aspects of the presentation: scribbles of various kinds are merely imperfect representations of the author’s thought, whereas the digested “aha” insights we reach when we understand it are its true content. Mumford is right that modern historians, by contrast, are trained to categorically reject such a “gut feeling” approach and stick slavishly to the exact written word as if it were a veritable alien communiqué for which no concordance with our own ways of thinking may be assumed.

The new historiography has greatly advanced the field by offering more specialised perspectives than mathematicians alone ever could, such as histories deeply informed by broader social context and meticulous work on sources and editions according to the highest standards of textual critical apparatus. But amidst the zeal to exploit these new frontiers the field has been left with a leadership vacuum in its traditional core dominion. The time is ripe for a resurgence of the mathematician’s perspective, whose cross-fertilisation with modern developments will bring great fruits.

Here in Utrecht we keep alive this mathematical tradition of historical scholarship. We have a legacy of generations of quality history of mathematics being done in a Mathematical Institute that commands the highest international respect. We are widely recognised as the natural heirs of this way of doing history. It is not for nothing that Jan Hogendijk was awarded the European Mathematical Society’s inaugural Otto Neugebauer Prize, epitomising our continuity with the Göttingen mathematical tradition, while Henk Bos was awarded the Kenneth O. May Prize, the highest honour of the International Commission for the History of Mathematics.

I have taken up the role of torch-bearer of this movement in both words and deeds. I offered a big-picture vision for its enduring relevance and importance in a programmatic paper on the historiography of mathematics, and my more specialised works instantiate these ideals. A notable example is my paper reviving and defending the geometrical algebra interpretation of the history of Greek geometry: an issue where the battle lines have traditionally run along departmental divisions, and the older interpretations being advanced by mathematicians like van der Waerden and Weil have been singled out for criticism as emblematic of the dangers of the mathematically-oriented approach to history. With no mathematicians forthcoming anymore to challenge them, the humanistic historians who dominate the field today had been lulled into a consensus, to the detriment of the vitality of our field. The same dynamic is at play in many other cases as well, which is why the mathematician’s point of view has much to offer, not only in terms of subject-matter insights, but also for stimulating diverse and critical thought in the field.

My first publication, on the isoperimetric problem, symbolises how my point of origin is mathematics itself. Although it is a work of history in that it gives an exhaustive survey of historical solutions of the problem, it is clearly driven by a mathematician’s delight at beautiful proofs drawn from wide-ranging fields of mathematics, such as complex analysis and integral geometry.

My conception of the history of mathematics as being first and foremost about the development of mathematical ideas gives my work educational and expository appeal for a broad mathematical audience. My work on the history of the calculus, for instance, is fertile soil in this regard, and I have made the most of this in my free calculus textbook and several papers making classroom-relevant aspects of my research accessible to a wide readership.

This grounding in the mathematician’s point of view serves me well in my more historical work, where an intuitive sense of what makes sense mathematically often leads me to different interpretations than those who are guided more by contextual considerations external to the mathematical argument itself. I have taken on historians and philosophers along such lines for instance in my Copernicus paper and my paper on Leibniz’s early work on the foundations of the calculus, to name but two prominent examples.

Thinking like a mathematician also gives me a perspective on broader issues missed by historians and philosophers who keep technical mathematics at arm’s length. For example, in my dissertation I studied Leibniz and argued that from the corpus of his technical mathematical works there emerges a clear and unmistakeable picture of his conception of the purpose and method of geometry. This has wide-ranging implications for understanding the scientific and philosophical thought of that era generally, yet this perspective has been neglected since it is expressed “only” implicitly in the mathematical works. But as Albert Einstein said: “If you wish to learn from the theoretical physicist anything about the methods he uses, I would give you the following advice: Don’t listen to his words, examine his achievements.”

Mathematicians conversant with the history of science are also much needed to analyse technical issues. My paper on Copernicus is a case in point. In this article I refute an argument due to Swerdlow that has been considered crucial for over forty years and has been widely cited as decisive by historians who had not themselves worked through its technical mathematical basis.

On the other hand, mathematical understanding is not all it takes to do history of mathematics. Mathematicians who turn to history without background and training in this field often make grave errors of their own. I expose and refute many such errors of anachronism in my dissertation and elsewhere. One example is my paper on what is often called Leibniz’s proof of the fundamental theorem of calculus but which is actually nothing of the sort. The notion that this is Leibniz’s proof of this theorem is widely repeated in numerous sources. It is a notion that seems very plausible to anachronistic eyes looking only at a short piece of Leibniz in isolation, but in reality it is simply false, as becomes clear when the work is studied in its proper context. Another example is my paper rehabilitating Jakob Steiner’s geometry from anachronistic misjudgements. To clear up these kinds of things the field desperately needs proper professional expertise in both history and mathematics.

A key theme emerging from my dissertation, which I intend to build on in future work, is the influence of classical mathematics on general scientific and philosophical thought in the early modern period, which was much more comprehensive than recognised today. It was a widespread conviction at the time that if you seek truth, you must do what the geometers did; you must replicate their method and extend it to other branches of learning and philosophy. Descartes’s Discours de la méthode (1637) is explicitly written for this very purpose; indeed this famous manifesto on the method of doubting everything clearly proclaims that “I did not doubt” that “only mathematicians” had struck upon the right way of reasoning. Likewise Hobbes writes in his Leviathan (1651) that “geometry is the only science that it hath pleased God hitherto to bestow on mankind,” and proceeds to expressly fashion his philosophy in its image. Spinoza’s Ethica (1677) declares in its very title that its is “ordine geometrico demonstrata.” Newton opens his Principia (1687) with a preface outlining what “the glory of Geometry” consists in, in order to use its example to justify his innovative scientific methodology.

But what exactly did these authors mean when they spoke of “the geometrical method”? The complexities of this question are poorly understood by scholars and historians today. The 17th-century thinkers who invoked geometry were not referring to some superficial idea of geometry as conveyed by Plato or Aristotle. They were referring to the rich picture of the geometrical method that emerges from a thorough study of technical corpus of Greek geometry, as conveyed by advanced technical writers such as Pappus. Indeed they frequently refer to this technical tradition even in works that go well beyond geometry itself: Descartes cites Pappus in his Discours; Hobbes does the same in his Elements of Philosophy; Newton cites Pappus even in the very first sentence of his Principia. These authors were thoroughly versed in the technical Greek tradition, as their mathematical works show. By citing Pappus and other technical Greek material they are signalling very clearly that when they call upon geometry they mean not some simplistic idea of axiomatic-deductive method but a rich methodology only conveyed implicitly in the finest ancient works of advanced geometry.

Unfortunately modern scholars do not share these 17th-century thinkers’ excellent technical mastery of advanced classical geometry. Consequently, current scholarship has failed to appreciate the extent to which conceptions of the geometrical method permeates 17th-century thought. One indication of this is that the crucial Book 4 of Pappus’s Collection was translated into English for the first time only in 2010. Even more deplorably, Leibniz’s published mathematical works have never been translated into English at all, even though their crucial importance is universally acknowledged. Meanwhile, any philosophical treatise of even a fraction of the importance of these works has invariably been translated multiple times, betraying the skewed and anti-mathematical emphasis of modern scholarship.

Much of the richness and impact of the mathematical perspective has therefore been missed by modern scholars since it is not spelled out in philosophical prose, neither in Greek nor in early modern times. But mathematics speak loud and clear to anyone who cares to listen, and anyone who was serious about philosophy in those eras was obviously expected to know their geometry — much in the spirit of the famous inscription above the entrance to Plato’s academy. To understand 17th-century thinkers it is time for us to start taking their appeals to geometry seriously and recognise the full scope of the rich methodological picture they drew from advanced Greek geometry.

]]>This proves that there is gender bias in academic science hiring, or so we are supposed to conclude.

My concern is this: Many faculty members want studies to prove that there is gender bias. It fits their own political and ideological beliefs. They are happy when they see studies prove this. They like to refer to such studies. I know because I follow them on Twitter.

This raises the question: Did the faculty members in the study answer truthfully, or did they “second guess” the purpose of the study and submit whatever answer would produce their own preferred outcome? They may indeed have thought to themselves: “Although I’m not biased, I am convinced that a lot of my colleagues are, so I better answer as if I was too, so that attention is drawn to this important problem and progressive measures can be taken.”

Of course the faculty members knew they were being studied and of course they had no actual stake in their replies, unlike when they’re doing actual hiring. And if we look at the prompt the faculty evaluators received, the purpose of the study is quite transparent. So they had no incentive to be truthful, but some incentive to ensure the study produced the results they favoured.

Which hypothesis is right, mine or the authors’? We could try to test it by looking at gender differences among the evaluators. If the authors are right, and there is unfair discrimination against women due to bias and prejudice, one might expect this bias to be stronger among male evaluators, since women who are themselves established scientists might be expected to be open to promising female students. If my hypothesis is the operative one, on the other hand, one might expect the opposite; that is, that female evaluators would be even more biased than men, since they arguably have a greater stake in “gaming the study” to make sure it shows gender bias. The latter is in fact what happened, though the difference is not great.

Meanwhile, if one looks at real data instead of contrived experiments, “actual hiring shows female Ph.D.s are disproportionately … more likely to be hired” (source, page 5365). We see the same thing by looking at the official data from the American Mathematical Society regarding hiring and PhDs in the mathematical sciences in the United States. In the latest data, women constitute 31% of PhDs awarded and 32% of positions filled. However, women constitute only 28% among PhD recipients who are U.S. citizens. This is perhaps the more relevant ratio since, among those who do their doctorate in the U.S., those who are U.S. citizens are surely significantly more inclined to aim for a job in U.S. academia. It therefore seems that hiring institutions have a preference for women, as they indeed often state openly.

]]>That is what I and others suspect. An article in the latest issue of Isis claims to disprove us:

> This essay seeks to explain the most glaring error in Ptolemy’s geography: the greatly exaggerated longitudinal extent of the known world as shown on his map. The main focus is on a recent hypothesis that attributes all responsibility for this error to Ptolemy’s adoption of the wrong value for the circumference of the Earth. This explanation has challenging implications for our understanding of ancient geography: it presupposes that before Ptolemy there had been a tradition of high-accuracy geodesy and cartography based on Eratosthenes’ measurement of the Earth. The essay argues that this hypothesis does not stand up to scrutiny. The story proves to be much more complex than can be accounted for by a single-factor explanation. A more careful analysis of the evidence allows us to assess the individual contribution to Ptolemy’s error made by each character in this story: Eratosthenes, Ptolemy, ancient surveyors, and others. As a result, a more balanced and well-founded assessment is offered: Ptolemy’s reputation is rehabilitated in part, and the delusion of high-accuracy ancient cartography is dispelled. (687)

As an aside, here’s a pro tip for academic novices: If you have little of substance to offer, make sure to lay it on thick with self-congratulatory posturing about how your work is supposedly based on “more careful analysis” and “more balanced and well-founded assessment,” showing everything to be “much more complex” than others think. After all, who would dare disagree with someone who is so careful and balanced and ever so sensitive to complexities?

Contrary to his smug proclamations, the author’s case is flimsy. For one thing he immediately admits that, indeed, Ptolemy’s error can be solved in a single stroke by recalculating his map with Eratosthenes’s excellent value for the circumference of the earth, yielding an “uncannily accurate” map (692). But he alleges that this is a mere “coincidence” (691). What is his evidence for this?

His main argument concerns the value of the length unit “stade.” A “‘short’ stade … is implied by the high accuracy of Eratosthenes’ value for the circumference” (694), for which there is no explicit evidence in the record, unlike a “long” or “common” value for a stade that is mentioned in some sources. Of course this is not strange since our hypothesis is based on precisely the claim that many excellent sources are lost. But there is in fact implicit evidence for the short stade, as the author himself in effect admits:

> The main argument for the “short” stade is based on comparison between ancient and modern distances: those measured on a modern map are divided by their ancient counterparts in stades, giving the length of an average stade. This comparison has been undertaken repeatedly, … and invariably the average stade comes out to be much closer to the “short” value of 157.5 m than to the “common” one of 185 m. On this basis, many researchers suggest that for practical purposes ancient surveyors used a special short stade, one never directly attested in extant sources. … This stade is often termed the “itinerary stade.” This result might have been regarded as a brilliant confirmation of the “short stade hypothesis” were it not for one “but”: strangely enough, in comparing ancient and modern distances, a crucial factor has been lost sight of——namely, “measurement error.” The proponents of the “itinerary stade” proceed from a tacit assumption that ancient distances were measured almost as accurately as modern ones. However, this cannot be true, for two main reasons. First, with rare exceptions, there is no indication that distances given by ancient sources were actual measurements on the ground, rather than rough estimates deduced, for example, from the duration and the average speed of travel. Second, and most important, even when ancient distances do derive from actual and quite accurate measurements, they were certainly measured not as a crow flies but including all the twists and turns of the route. (701-702)

The author’s counterargument is very weak. It merely asserts what he is trying to prove, namely that distance measurements in the time of Eratosthenes would have been poor and naive. The obvious reply, which the author does not consider, is that it is very possible that Eratosthenes and others used much more sophisticated mathematical methods such as triangulation. Instead the author expects us to believe that the generation that gave us the geometry of Archimedes was too stupid to account for “twists and turns of the route” when estimating distances for geographical purposes. There is also some reason to think that Eratosthenes defined a new stade based on his earth measurement——another possibility ignored by the author.

> We can apply a simple test: the same comparative approach may be used to determine the length of the Roman mile. Since it has been firmly established as equal to 1,480 m, such comparison will yield the average accuracy of Roman distance measurements. … For this purpose I have examined more than 160 distances given in Pliny’s Natural History. … If these distances are believed to be accurate, then, by the same logic that has led us to the “itinerary stade,” we have either to conclude that Pliny’s mile was equal to circa 1,190 m, which is impossible, or to assume that Roman measurements of distances were much less accurate than Greek ones, which is hard to believe. Otherwise, we have to conclude that Pliny’s distances were overestimated by 25 percent on average——that is, by approximately the same amount that Eratosthenes’ and Strabo’s distances must have been if they were expressed in the “common” stades of 185 m. (702-703)

Apparently it is “hard to believe” that the greatest generation of geometers who ever lived were better at measuring distances than a second-rate encyclopedist from a civilisation that never contributed an iota to mathematics during its entire lifespan.

The author also maintains that “the hypothesis … contains numerous logical fallacies” (693), namely:

> The match between the recalculated Ptolemy coordinates and the modern ones, however close, does not in itself mean that the longitudinal distortion in Ptolemy’s map was due entirely to a single cause—namely, the wrong value for the Earth’s circumference. Nor does it warrant discarding two other possible causes: the exaggeration of distances and the lengthening of the stade. Another crucial point to stress is that Ptolemy’s recalculated map turns out to be accurate only in terms of spherical coordinates. This does not mean that the actual distance measurements underlying it and Eratosthenes’ value for the Earth’s circumference were equally accurate. (693)

Calling these things “logical fallacies” is just ridiculous. Obviously no one ever claimed that these things were logical implications, only that they were the likeliest explanations. The author does however commit a logical fallacy himself when he draws the non sequitur that our hypothesis is a “delusion” from the fact that Ptolemy’s map errors can be explained by other means.

Altogether, the author’s preposterously overblown claims exaggerate his case by a mile and then some. I am surprised that such wording was allowed to stand in a respectable journal. The author has not taken our hypothesis seriously, let alone given a “careful” and “well-founded” demonstration that it “proves” to be a “delusion.”

]]>Our coach wanted to make us master passers so that we could play tiki-taka. So we spent hours and hours every week on one specific drill: playing football without goals. Without the “distraction” of trying to score goals, we could focus purely on passing and possession play. That was the idea.

In reality, what happened is that we got sick and tired of this boring drill that took all the fun and excitement out of the game and turned it into a pointless drudge. Instead of making us master passers it made us disinterested slackers who didn’t see any reason to put in our best effort.

Unfortunately much curriculum planning in mathematics is based on the same hare-brained logic. Again and again we see the same pattern: in order to do B you need A, but teaching A and B together in one course would be too much, so A is detached and drilled at length in a prerequisite course.

The problem is that B is the only reason anyone is interested in A in the first place, so now you are teaching an entire course on a topic A which serves no purpose whatsoever in and of itself. By severing A from B you are guaranteeing that your course has zero intrinsic motivation. You are not helping students by giving them “a good foundation in A” before moving on to B. Rather you are obliterating the meaning and purpose of mathematics and forcing your students to approach it as an empty chore.

A notable example is the plague of “intro to proofs” courses where the pedantry and mechanics of proof writing is detached from any context where these skills serve an actual purpose. In the same way we detach integration rules from their purpose, which is solving differential equations, and we detach rings and ideals from their purpose, which is number theory, and so on.

The fallacy is one of short-sighted, non-organic thinking. “I’m a teacher of B. I find that my students are lacking in A. Let’s solve my problem by having them do tons of A in a prerequisite course.” This is B-centered thinking that zooms in on one particular issue and wreaks havoc with the cohesion and integrity of the curriculum as a whole in order to “fix” it.

The A-course is rarely a success, because it’s hard to learn something well without knowing what it’s for, and it’s hard to stay motivated and excited when there is no purpose to what one is doing. But, unfortunately, the same blinders that led to this course in the first place also means that its proponents are blind to its failures. The disastrous outcomes only leads them to double-down on their short-sighted scheme. “Now they have a whole course on A and they still don’t get it! Obviously this proves how essential it is to drill and drill and drill A before going on to B.” Thus, as with medieval bloodletting, the failure of the treatment is taken as evidence that more of it is needed.

The naiveté is the same one that led to the myth that vitamin C cures and prevents the common cold. Compare: “Vitamin C is essential to the immune system. Therefore, taking tons of vitamin C pills will keep us from ever catching a cold.” “In higher mathematics it is essential to understand concepts and techniques relating to writing proofs. Therefore, making all our students cram these skills in a dedicated course detached from any content will have them flying through later courses without impediment.”

Just as a well-rounded diet gives us all the vitamins we need in a natural way, so also a well-rounded mathematics curriculum automatically incorporates any necessary material in an organic and natural manner. Unfortunately mathematical curriculum planners do not believe in teaching mathematics they way it grows naturally and organically. Instead they would rather play the role of a hubristic doctor in a sci-fi dystopia who thinks he can “improve” on nature by replacing all organic foods with artificial capsules.

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