The study’s data set is impressive: 19,952 student evaluations of university faculty in courses where students were randomly allocated to instructors. Female faculty were rated lower, despite producing the same outcomes in terms of grades.

A major problem, however, is this: The evaluation forms completed by students never actually asked them to judge whether the teacher was good or bad. Here is what the students were actually asked (39):

T1: “The teacher sufficiently mastered the course content”

T2: “The teacher stimulated the transfer of what I learned in this course to other contexts”

T3: “The teacher encouraged all students to participate in the (section) group discussions”

T4: “The teacher was enthusiastic in guiding our group”

T5: “The teacher initiated evaluation of the group functioning”

When the authors say female faculty received lower evaluations, they mean lower average score on these five items. But these five items are very poorly conceived as a way of capturing teaching quality, for the following obvious reasons.

T1 is a bad measure of teaching quality since you can master the content and still be a lousy teacher.

T4 is a bad measure of teaching quality since a teacher can be enthusiastic but ineffectual, or dry but effective.

T3 is very dubious since the pedagogical strategy of calling on reluctant students is not necessarily positive.

T5 is a bad measure of teaching quality since it’s pointless if the group worked fine already. The data suggests that groups on the whole worked fine (39). If the instructor saw this and hence for this reason did not “initiate evaluation of the group functioning,” then it obviously makes no sense to punish this teacher in the course evaluations for not wasting class time on a needless group evaluation.

The instructor’s performance on T2 can by definition not be checked by controlling for course grade. It could be that female faculty were simply worse at this. The conclusions of the study follow only if we agree that the equality of grade outcomes prove that female faculty performed equally well. But T2 specifically asks for things that go beyond the course, i.e., things that do not count toward the course grade. Hence we have no way of telling whether the students’ assessment of T2 were biased or accurate.

In sum, the supposed evaluative measure of teaching quality is not a measure of teaching quality at all. The assumption—essential for the study’s conclusions—that equality of grade outcomes means equality of instructor performance on T1-T5 is unwarranted.

There are some grounds to nevertheless maintain the authors’ interpretation. One is that the bias seems to cut somewhat uniformly across T1-T5, suggesting that the students harbour blanket or generic depreciation of female faculty rather than giving thoughtful and reliable answers to each item separately. At least this is indicated by the only data we have showing a breakdown of the items T1-T5 one by one (Table B3). Unfortunately, we have such data only for graduate student instructors. There is reason to think that this is the instructor group that most confirm the authors’ thesis of gender bias. For the bias against female faculty “is larger for mathematical courses and particularly pronounced for junior women” (abstract). This could be due to stereotype bias. Alternatively, it could be due to gender bias in favour of women in graduate student recruitment. The fact that evaluations are lowest among junior female instructors and in mathematical fields would then be a reflection of the fact that these fields have lately been very aggressive in recruiting women at all costs.

Another argument for the authors’ interpretation is the fact that the gender bias is “driven by male students’ evaluations” (abstract). If female faculty were genuinely worse, wouldn’t female students too recognise this? Maybe. But an alternative explanation could be that female faculty are especially supportive of female students, so that the differing evaluations by student gender reflect a genuine difference in the quality of instruction received. The authors themselves note that this is by no means an outlandish hypothesis: “Female students receive 6% of a standard deviation higher grades in non-math courses if they were taught by a female instructor compared to when they were taught by a male instructor. … This might be evidence for gender-biased teaching styles.” (30) Note also that it is easy to imagine how T3 in particular could reflect such bias.

One reason to think that the students are not entirely off the mark in their evaluations is how their judgement develops over time. “The bias for male students is smallest when they enter university in the first year of their bachelors and approximately twice as large for the consecutive years. For female students, we find that only students in master programs give lower evaluations when their instructor is female, but not otherwise.” (30) You would think that students would get better rather than worse at judging teaching quality in the course of their education.

Here’s another point:

“Strikingly, despite the fact that learning materials are identical for all students within a course and are independent of the gender of the section instructor, male students evaluate these worse when their instructor is female.” (3)

Two possible explanations suggest themselves:

(a) The students are blinded by bias and cannot evaluate the course materials objectively. They let their predjudice against the female instructor cloud their judgement even on this question which had nothing to do with her.

(b) Female instructors were less good and hence unable to highlight and bring out positives and insights in the course materials, thereby making the course material seem less good. Hence lower evaluations of instructors and course materials go hand in hand.

Of course the authors suggest (a). But the supposed logic behind this is somewhat dubious. If male students hate women, shouldn’t their evaluation of the textbook be based on the gender of the textbook author? If they are driven by and seek to express their dislike of the female instructor, and the textbook was written by a male author, shouldn’t they rate the textbook higher rather than lower, so as to convey that it was the particular instructor rather than the course materials that were at fault? In fact, if the students had done precisely this, then that too could have been used as evidence of their blatant gender bias. Thus two completely different outcomes could both be spun as clear evidence of gender bias. This suggests that we should be careful before jumping to the conclusion that the data confirms our favoured hypothesis.

]]>> Students rated the male identity significantly higher than the female identity, … demonstrating gender bias. (291)

How can this be? Simple: the authors were so determined to prove gender bias that they decided to cheat and move the goalposts, as they admit in a footnote:

> While we acknowledge that a significance level of .05 is conventional in social science and higher education research, … we have used a significance level of .10 for some tests where: 1) the results support the hypothesis and we are consequently more willing to reject the null hypothesis of no difference; 2) our hypothesis is strongly supported theoretically and by empirical results in other studies that use lower significance levels; 3) our small n may be obscuring large differences; and 4) the gravity of an increased risk of Type I error is diminished in light of the benefit of decreasing the risk of a Type II error. (288)

In other words: We decided to call things significant even when they’re not if: 1) it agree with what we already decided in advance that the results of the study should be; 2) everyone already knows we’re right anyway so we don’t need any of that pesky “scientific evidence” stuff (even though finding such evidence is ostensibly the whole purpose of our paper); 3) our study is so ridiculously small that the results could mean anything; and 4) we may be dead wrong but on the other hand maybe we’re not.

Only with this sham, unprecedented definition of significance did the authors manage to find a so-called “significant” pro-male bias.

In any honest universe, the title of the paper would be “Failure to Expose Gender Bias in Student Ratings of Teaching” and the abstract would say “Students did not rate the male identity significantly higher than the female identity, … demonstrating that no gender bias can be inferred.”

The study does, however, demonstrate one clear and undeniable bias, namely that of the authors in favour of their preconceived hypothesis.

Actually, the authors were not even content with this amount of cheating. They lie even more when they say:

> These findings support the argument that male instructors are often afforded an automatic credibility in terms of their … expertise. (300)

In reality, they specifically asked the students to rate how “knowledgable” their instructor was, and the results (299) showed no significant gender effect even with the authors’ sham definition of significance. Since this didn’t stop them from concluding the exact opposite, one wonders why they bothered gathering any data at all.

]]>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 (update: here it is).

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 and 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 uses the same terminology and thinks there is “no doubt” that it 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, 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 it 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.

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