## Wednesday, September 23, 2020

This is a response to a several things, but, ﬁrst of all, to a project of Andrea Sangiacomo’s, as reported in this article.[1]The article, which is clearly based on an interview with Sangiacomo, is available in both English and Dutch, but there is no indication I can see as to the relationship between the two versions nor as to the language(s) used in the original interview. I will mostly rely on the English version, for obvious reasons. As usual, I am ill-informed: I don’t know anything about Dr. Sangiacomo’s work besides what the article contains, and, given that such sources are not always very reliable, I can scarcely be said to know even that. I don’t know, for example, whether Sangiacomo has been accurately quoted, or whether his words, even when they are his, may have been taken out of context. So my remarks here should not be understood as criticism of Sangiacomo’s project, still less of him personally.

If not that, though, then what are they? Sangiacomo is reported to have said something to the point:

Many colleagues still consider philosophy to be purely a job for humans. And do not get me wrong, I generally agree with that. The most relevant philosophical insights are still being generated by people who reflect deeply.

The difference between human and non-human is not, as such, of much concern to me in this regard: if horses, oxen, or electronic computers can aid in the generation of relevant philosophical insights, then I am more than happy to share the glory of such with them. What does concern me is the difference between reflective and unreflective. To paraphrase Locke: I grant, AI’s will not come to deep reflection till they be more like humans; and I add, nor then, neither. Natural intelligence is fully capable of executing an algorithm.

I introduce this word, “algorithm” (which does occur, but not very prominently, in the article under consideration) with some hesitation, and not only because, as Popper somewhere notes, the word should really be “algorism.” What I want is a word for the thing non-human AI’s can deﬁnitely do, according to Sangiacomo, because to do that thing — to implement or execute what I am calling an algorithm — does not require deep reflection. “Algorithm” may or may not be the best choice for this, and I certainly don’t intend to import any of its technical content from theoretical or applied computer science. In what follows, it should taken to mean: a process that can be carried on without reflection. This I think an informative deﬁnition because, despite having read and been thoroughly confused by a whole canonfull of philosophers who all use the term “reflection” in various ways, I still ﬁnd the deﬁniens clearer than the deﬁniendum.

With only slight exaggeration, then, I would say: the present post expresses a reflection prompted by the output of a certain algorithm. One stage of the algorithm in question, namely the part that made Martin Lenz’s link to the article come up on my Facebook feed, was implemented by electronic computers. But, before that, I would guess it was a human who decided that the University of Gronigen needed a story about this under “About us $$>$$ Latest news $$>$$ News $$>$$ News articles,” and I know it was a human, by the name of Jorn Lelong, who actually wrote the story.[2]I ﬁnd with some Googling that Jorn Lelong is a freelance journalist based in Ghent, Belgium, who has authored various pieces (in English) about developments in technology research for various different outlets, and who also has a Twitter account (in Dutch). There is even a picture of him, so, barring deepfakery, he is deﬁnitely an actual human being — a kind of cool, cheerful looking young human being in a bright red hoody and dark sunglasses. There is no credit for a translator. Perhaps this means that Lelong wrote both the Dutch and the English versions. But I mean no disrespect in saying that I doubt the humans in question required any deep reflection to perform these tasks, and in fact I even imagine that the best current AI might be trained up to do them tolerably well. Moreover, the main factor in the decision to publish the story is presumably what gets mentioned at the very end, namely that Sangiacomo has received an ERC grant of 1.5 million for a follow-up research project. How much deep reflection is involved in awarding such grants? I hope Martin will not be offended if I add that, whatever reflection he may have engaged in upon reading the article, it was likely not an essential part of his decision to post the link. And, as for my own decision to follow that link: what chance has reflection in the face of clickbait? In conclusion, then, Sangiacomo (as reported) and I (as self-reported) agree thus far: we both hold that philosophy requires reflection, and yet we both concede that the output of an unreflective, algorithmic process can present a useful occasion for such reflection.

The question, however, is whether some such algorithms are superior to others. Sangiacomo, in conjunction with the Data Science team at Gronigen’s Centre for Information Technology, has come up with a new algorithm to do this, and plans to spend 1.5 million following up on this work, because he is unsatisﬁed with the results of a different algorithm, namely, the one whose output he characterizes as “a few great works by a few great authors.”[3]At least: these words are in quotation marks in the article, which I think is supposed to imply that they are due to Sangiacomo, rather than to Lelong. This algorithm, too, has various stages which have been implemented on different hardware. The ﬁrst and longest part was carried out by what you might call a neural network — anyway, it was a network of some kind, as Sangiacomo reportedly asserts. In a section of the article titled “The importance of a network,” Lelong writes:

Sangiacomo also found that social connections also have a signiﬁcant role to play in the popularity of certain philosophical theories. As such, in addition to the shifts in meanings, he also sought to trace the mutual relationships and networks of early modern philosophers.

Working over the course of some centuries, this network has brought a few great authors to prominence. The second and shorter part is carried out by the current institutions of philosophy, as part of the process of professionalization: students, professors, journal editors, etc., are trained up to recognize these few great authors and to produce and respond selectively to the work of philosophical scholars which is conﬁned, in a certain way, to a few great works by them.

That the ﬁrst, long part of this process was algorithmic in the sense in which I am using that term, which it to say, not properly reflective, is Sangiacomo’s point when he (reportedly) goes on to say:

A priori, there are no obvious reasons as to why some philosophical theories become much more popular than others. Throughout history you sometimes see that really crazy ideas become mainstream, while good ideas are side-lined. To explain that, you have to look at social connections, because people are the real driving force of history. In that respect, this project provides a lot of information.[4]Again, these words are in quotation marks in the article.

Now, the discovery that some ideas are “really crazy” (Dutch: erg gekke) even though they have “become mainstream” (Dutch: mainstream worden), is, in other words, enlightenment. As such, it requires, one would think, signiﬁcant philosophical insight, and must therefore depend on deep reflection. This calls up the image of an alternate version of history: a history in which the owl of Minerva gets up early in the morning and flaps around all day, pecking out the really crazy ideas and making sure that the good ones become mainstream. Sangiacomo, however, regardless of what he may or may not imagine, nevertheless does not propose either to apply reflection now, in order retrospectively to put such a history together, nor to try and institute the rule of reflection from now on. Instead, he offers a new algorithm.

But before considering Sangiacomo’s new algorithm: I have yet to describe the second, shorter stage of the old one. This stage, as it happens, is well describe by Martin Lenz himself in his own recent blog post:

People become enthusiastic if they recognise something. . . .  I think much the same is true of our talk of “great thinkers.” We applaud recognised patterns. But only applauding the right kinds of patterns and thinkers secures our belonging to the ingroup. . . .

This is why calling someone a “great thinker” is to a large extent self-congratulatory. It signals and reinforces canonical status. What’s important is that this works in three directions: it affirms that status of the ﬁgure, it affirms it for me, and it signals this affirmation to others. Thus, it signals where I (want to) belong and demonstrates which nuances of style and content are of the right sort.

The algorithm at this stage has a familiar purpose, one of the main things both artiﬁcial and natural intelligences are trained to do. It is an algorithm for recognizing patterns, and, more particularly, for recognizing places in a pattern: ﬁgures that stand out against a ground, enclosing what is inner and excluding what is outer. Some of the places, or topics, it is trained to recognize are great thinkers of the past, whereas others (metaphysics, philosophy of mind, etc.) are contemporary, but, one way or another, knowing how to recognize such topics; how to market oneself as belonging to one of them (or to the “intersection” of two of them); how, on the basis of this topic assignment, to enter into a mutually beneﬁcial group of researchers who own that topic, in which everyone is bound to cite each other’s works; how (if one is a journal editor) to determine the topic of a submission and ensure that it is refereed by members of the group who own that topic — all this, and more of the like, is what professionalization in academic philosophy currently entails.

Or — and this will make a difference to our understanding of what Sangiacomo proposes to do — is that description already somewhat outdated? In using the term “topic” I allude, on the one hand, to Kant and, via Kant, to Aristotle. More on that below. But I also, on the other hand, allude to Brian Weatherson’s fascinating History of Philosophy Journals: Volume 1: Evidence from Topic Modeling, 1876–2013. Unlike Kant and Aristotle, Weatherson cannot be held responsible for the word: “topic modeling” is just the usual terminology for the type of algorithmic technique he uses. But still, it is interesting to inquire into what “topic” ends up meaning for him, i.e. what “topics” the algorithm comes up with when applied to the contents of philosophy journals during the indicated period. It is particularly interesting in the present connection because, to quote the Wikipedia article just linked to:

HLTA [a method of topic modeling][5]HTLA stands for hierarchical latent tree analysis. Weatherson uses a different method, LDA (latent Dirichlet allocation). For some technical details and, should you be interested, links to more, see the Wikipedia article in question and Weatherson’s chapter on methodology. was applied to a collection of recent research papers published at major AI and Machine Learning venues. The resulting model is called The AI Tree. The resulting topics are used to index the papers at aipano.cse.ust.hk to help researchers track research trends and identify papers to read, and help conference organizers and journal editors identify reviewers for submissions.

So, in other words: the AI researchers themselves feel that their algorithm, when implemented by artiﬁcial intelligence, can fulﬁll at least part of the function I assigned to professionalized human philosophers above — or rather, it can fulﬁll at least part of the corresponding function in that ﬁeld. So an interesting question would be: will Weatherson’s related technique succeed in ﬁnding the corresponding kind of “topic” in philosophy, such that it, too, could be used for those purposes? The interesting answer, as I understand Weatherson’s results, is that his method can succeed in this respect, but only during a certain, relatively short period of the history of philosophy (and of the history of philosophy journals).

A perfect philosophical work, at whatever time and in whatever way written and/or published, ought to have a topic, a place, in a formal (logical) sense. That is: it ought to have some leading concept.[6]You might think the condition should instead be that the work have a leading judgment (a thesis). For Kant, however, such a unity of thesis depends on and presupposes a conceptual unity: “A judgment is the representation of the unity of the consciousness of different representations or the representation of their relation insofar as they make up a concept” (Jäsche Logic, §17, Ak. 9:101,5–7). For, as Kant says in the Amphiboly of the Pure Concepts of Reflection:

One can call any concept, any title under which many cognitions belong, a logical place [Ort]. Upon this is founded [gründet sich] the logical topics of Aristotle. (KrV A268/B324)[7]Kant, reasonably enough, treats τόποςtopos as the equivalent of Ort, and reserves Topik for a theory or system of such places (parallel to other terms such as Logik and Physik). But I will stick, somewhat reluctantly, to the more common practice of calling each individual logical place a topic. (Hume, following a different interpretation of the Topics, uses “topic” in yet another sense.)

The (logical) perfection of such a work would consist in the consequence of all its parts from, and their joint adequacy to, that single leading concept:

In every cognition of an object there is namely unity of the concept, which one can call qualitative unity, insofar as only the unity of the comprehension of what is manifold in the cognitions is thought thereunder, as, for example, the unity of theme in a play, an oration, a fable. Secondly, truth with regard to consequences. . . .  This one can call the qualitative plurality of the notes that belong to a concept as their common ground. . . . Finally, perfection, which consists in this, that this plurality, [taken] together, reduces, conversely, to the unity of the concept, and completely agrees to this [concept] and to no other, which one can call qualitative completeness (totality). (B114)

I hasten to add (in fact, this is the whole point of §12 of the B edition) that these three qualitative moments of quantity are necessary but not sufficient for reference, i.e. for “cognition of an object”; in other words, even a work that is logically perfect in this sense might lack a subject matter (what might be called an objective or material topic, although Kant never uses such expressions, to my knowledge). And then again, on the other hand, a work might be full of objective content and yet lack formal–logical perfection. This latter condition, roughly speaking, is what the methods of “topic modeling” assume: namely, that a given article will contain a mixture (in some determinate proportion) of the consequences (notes, characteristics) of various different logical topics. Words are used as proxies for such consequences, and a statistically recurring cluster of words is taken as the sign of an underlying conceptual unity from which all (and, ideally, only) the members of that cluster flow.

Even assuming, however, that all or most of the articles published in philosophy journals have at least such imperfect logical locality, it doesn’t follow that the topics located by topic modeling will be topics of the aforementioned kind, namely the topics that guide the current algorithm of professionalized philosophy. For one thing: if a topic is to be usable in that way, it must be, not only a logical place, but also, so to speak, a social or political one. It is not enough for it to comprehend, in a conceptual unity, what is manifold in certain cognitions; it must also comprehend, in a political unity, what is manifold in certain people. This is explicit enough in one part of the function of such topics: if the editor is to decide, based on the topic of a paper, who would be appropriate reviewers, or rather, to which reviewers that paper is appropriate, then there must be an identiﬁable group of people who have a right — in a broad sense, a property right — to the topic in question. This is the external deference to or recognition of the political structure which is owed by the journal editor; internally, the main political tie, is, as I have mentioned above, the implicit covenant to cite one another’s papers.[8]It came out recently, in certain infamous cases, that the editors of some journals had not paid this political deference to the owners of certain topics, e.g. trans studies, and, as a consequence, had accepted articles for publication that failed to cite the literature. No doubt these articles were also flawed in their content. I highly doubt they were based on deep reflection or contained many relevant philosophical insights. But people were quite right, in my view, to complain, not only about problems in content, but about the merely procedural errors in the acceptance process. This lack of deferral to experts was itself, indeed, an unjust act, given the way other topics are currently treated. Following the usual terminology, I will call such combined logico-political topics specialties. For topic modeling to pick up topics of the right kind, at a minimum, the constitution of professional philosophy — the constitution under which the body of writing to be analyzed was produced — must involve division into specialties.

In the absence of such a constitution, the method may well detect some topics, but these topics will not be specialties. It might still pick up the logical topics of individual works, even though, since these topics are not specialties, the information will not be usable by journal editors etc. in the contemplated ways. On the other hand, however, it might detect political topics directly, even though these are not specialties, i.e., not correlated with logical topics. A group of people who selectively read and mention each other, and selectively control one another’s ability to publish and otherwise flourish, will naturally develop some linguistic markers of their own. They would do so, I’m certain, even under an imagined constitution in which political topics were formed arbitrarily, say by lot, which means that topic modeling could easily discover even such purely political topics. Philosophy has never been so constituted, and I hope it never will be.[9]Some systems of academic patronage can contain elements of such a constitution, however. This may be what Sangiacomo has in mind when he (reportedly) talks about “the network of a well-known male author,” and also when he (reportedly) says that “people are the real driving force of history.” It’s not what you know but who you know, as they say. But philosophy has very often, in fact almost always, had another constitution that might produce, so to speak, an even stronger political signal. I refer to the system of philosophical schools, also known as sects: αἱρέσειςhaireseis. In this system, a few great authors — authors in the original sense, authorities — are treated, not as logical topics, conceptual themes that a philosophical work might be about, but rather as heresiarchs, mythical or historical originators and successive rulers of political topics that preserve themselves through time.

Maybe it seems that there is little to choose between our constitution of specialties and this older one of sects. Isn’t the latter just an alternate way of dividing up the same territory? But actually everything is different, because no sect ever owns its authorities. If anything, it is the other way around, although sovereignty should not be confused with property. In any case: this relationship, however it should be described, leaves the authorities still available to anyone else. The ancient Peripatetics and Stoics could not prevent, and had no right to prevent, the Neoplatonists from using Aristotle or Epictetus for their own purposes. The German Idealists might have many things against Schopenhauer, or the Marburgers against the Phenomenologists, and vice versa, but no one could coherently complain: Kant is ours. If Natorp gave a talk, and Husserl raised his hand during the Q&A, he would not have to begin his question with an apology: “This is not my ﬁeld, but …”. The authorities were no one’s ﬁeld. They remained a commons.[10]This is not to say that there can’t be problems about property rights to authorities. Problems will come up when we want to use an authority, e.g. Confucius, who belongs to an entirely different tradition, a tradition that has its own institutions of teaching and writing and need not necessarily welcome being appropriated by the Western academy (descended from Plato’s Academy). In general, a commons belongs to the village, not to the world.

And, sure enough, Weatherson does pick up at least two sects as “topics”: post-Kantian idealism (dominant in the earlier years of his journals) and ordinary language philosophy (peaking sometime in the 1950’s). Weatherson is well aware that these are not “topics” in the same sense as most of his others: he does call idealism a “school,” and he says of ordinary language philosophy that “what the model is ﬁnding is a style as much as a content,” although I think “school” (or “sect”) would be more appropriate in that case, as well. He explains here that, while the particular model that forms the basis of his book did not detect another sect, pragmatism, this sect did appear as a topic in many other runs. Since the reign of Analytic philosophy began with the capture of most of these journals by a single sect,[11]As discussed by Weatherson, ibid., referring in turn to an article by Joel Katzav and Krist Vaesen. most of the well-deﬁned sects in the postwar period do not appear in his model, or at best have a very small signal.[12]Clear examples of this would be Straussians and Cavellians, both of whom tend to publish elsewhere. Continental philosophy, while it could be regarded from the outside as one big sect, also continued to have a stronger sect-based constitution internally during much of this period, but, again, their publication mostly did not appear in these journals. The nature and evolution of Continental philosophy’s internal constitution are beyond my scope here, and in part beyond my knowledge (though I do have some ideas).

I have one last matter to consider from Weatherson, before getting back to Sangiacomo and, by way of him, back to Kant (and Hume). This is the nature of his Era 5 (1999–2013), which he describes somewhat anticlimactically as “dominated by a number of distinctive topics, such as reasons, vagueness, contextualism and Williamsonian epistemology and metaphysics.” I suppose Era 5 is all of that, but it is also the period which saw the appearance of the “bad topic” he discusses here and again here. This bad topic (“bad” in the sense that it is not the kind of result Weatherson wants) does not appear in the list he eventually settled on; it resulted from an attempt apply too many iterated “reﬁnements” to the same underlying model — I refer you to the above linked text for technical details. Here is Weatherson’s description of what happened:

One signature problem with the kind of text mining I’m doing is that it can’t tell the difference between a change of vocabulary that is the result of a change in subject matter, and a change of vocabulary that is the result of a change in verbal fashions. . . .

So after 100 iterations, we ended up with a model that wasn’t a philosophical topic at all, but was characterized by the buzzwords of recent philosophy.

To be speciﬁc, the words the over-reﬁned model gave most weight in identifying this topic were: accounts, role, commitment, commitments, account, proposal, constitutive, practices, challenge, typically, claims, worry, approach, relevant, project, focus, features, issue, appeal, provide. I blush to say that I have relied on some of these myself in the past and that, even after having seen the list, I sometimes ﬁnd it difficult to avoid them. Weatherson hesitates between two possible explanations of why a bad topic like this only appeared towards the end of his period:

1. There has been a linguistic revolution over the last generation, and philosophers now write in a very different style to how they wrote a generation ago.
2. This is an artifact of model building, and if you stopped the model at any time, and ran the same study I did, you’d get results like this. That is, doing what I did will get you weird results whenever there is linguistic drift, and there is always linguistic drift.

But, although I have not a shred of that meticulous empirical evidence which Weatherson requires and could supply, I don’t doubt for a moment that the answer is (1). This certainty of mine you should probably distrust: the old are always affronted by the language of the young, after all. I can only say how it seems to me. The rise of these buzzwords, namely, seems to me a sign that it is not only Weatherson’s algorithm that can’t tell the difference between subject matter and verbal fashion, but that, rather, the algorithms of professionalization themselves are beginning to have the same problem. We are beginning to train up producers and detectors of buzzwords.

Back to Sangiacomo, then. In place of the algorithmic — that is, as I am using the term: unreflective — process which has brought up a few works by a few great thinkers for our consideration, he offers a different algorithmic process which will bring up different pieces of text. Granted, there may be little reason to suppose, with Hegel, that the old algorithm, the algorithm of history, was suffused with the cunning of the spirit, such that it could be trusted always to output exactly what deep reflection requires. Whatever Hegel means by “the end of history,” it probably is not the kind of end we face now. Still, it would be reasonable to ask: what was wrong with the old algorithm, and how will the new one improve on it? Sangiacomo (reportedly) explains:

Our entire understanding of early modern philosophy is based on the works of ﬁve authors.[17]The ﬁve (or six) authors he counts, in the order in which Lelong lists them, are: Spinoza, Descartes, Hume, Locke, Newton “and, at least according to some, Kant.” A few other names, e.g. Leibniz, could probably have been added to this. So naturally, that is an incredibly distorted view. But what is the alternative?…

I realized that I would never be able to gain a complete picture of that period. Because if it turns out that there were actually as many as a thousand philosophers active during that time, what would be the advantage of adding ten to ﬁfteen names to the existing canon? We need new tools.

How he arrived at the ﬁgure of (as many as) a thousand “active” philosophers, I can only guess. But I can say a few things about this. First of all, if the knowledge we need about our history, in order to make reflection possible, consists in knowing the views of as many as a thousand different people, then there is no hope for us, because there is no way we can possibly know that. Secondly, and worse: as philosophy is now constituted, we, the thousands upon thousands of philosophers who are said to be active now, are very far from knowing the views even of those ﬁve authors. Each of these authors constitutes a specialty (or rather, as things now stand, a relative specialty, which is a generality to the various specialties lying under it). The owners of a specialty are alone authorized to say, to one another, in the journals, what each one of these authors said, and always on the condition that they cite one another. The rest of us are authorized, more or less, to supply caricatures of them for the purposes of jokes at department meetings, lectures to undergraduates, and chronically unfunny comic strips. Adding ten or ﬁfteen new history AOS’s would only make this situation worse. To be added to the canon, under our present constitution, is to be culturally appropriated: removed from the commons and brought under the corporate control of a specialty. After canonization, the ﬁgure in question is no longer known by us: from our point of view, they have been canceled. Canonization is appropriation is cancellation. Only those who are (still) too obscure to be claimed by any specialty can still be known by us at all.

What is the true fear, the true uneasiness, that ﬁnds expression as the fear that “our” “view” is “incredibly distorted”? There is an inconsistency between the way we understand contemporary philosophy (the classics of Weatherson’s Era 3, plus the vast output or Eras 4 and 5) and the way we — that is, the specialists among us — understand the history of philosophy. To know philosophy now, to be a professional, means (1) to know to which specialty one belongs, who else belongs to it, when to cite them, and so forth; and (2), increasingly, to be up on the latest verbal fashions, to know what buzzwords to emit. But no one knows anything parallel about philosophy before Era 3. It would be impossible to know (1), since specialties did not yet exist, but some historians, so-called contextualists, now supply a simulacrum by pretending that, say, Descartes, spent much of his time eagerly scanning the latest literature by every (active) member of his “ﬁeld” (which was …?), as if he lay under our obligation to read recent things so as to able to spew a list of name–date pairs into the appropriate footnote. But this leaves (2) unaccounted for, and (2) is what Sangiacomo’s algorithm promises to supply. Along with his collaborator, Christian Marocico, and others at the CIT, he “used ShiCo (Shifting Concepts through Time), an open source tool developed by the Netherlands eScience Center and the University of Utrecht, to analyse shifts in words in a historical context.” Here is how it worked out:

Of course, the tool needed a little tinkering. ShiCo was developed to record conceptual shifts in 20th-century newspaper reports. Data scientist Marocico adapted the model so that the algorithm was able to analyse 70,000 letters exchanged between philosophers and academics during the 17th and 18th centuries. . . .  They investigated which word associations were formed for certain scientiﬁc concepts, and how this changed over the years. Sometimes this produced remarkable results. “For example, we found that in the 17th century the English word ‘spirit’ was used in both a chemical and a religious context. In the 18th century, however, we see that the word has lost its chemical meaning. It illustrates how concepts change.”

Whether deep reflection, of the kind essential to philosophy, can ever be prompted by knowledge of verbal fashions — either contemporary verbal fashions or the verbal fashions of the 17th century — I honestly don’t know. The spirit, as that term is used in a Hegelian context, rather than a chemical or a religious one — the spirit, whose perfections non comprehendere, sed quocunque modo attingere cogitatione possumus, is always more cunning that we expect; it always overflows our concepts, shift them as we might. It is said to have revealed itself in cracked turtle shells, in images on toast, in words written on subway walls and tenement halls, in riddle games in the dark at the roots of the Hithaeglir, in comic strips, in superhero movies, in Facebook memes, in a brutal command to wipe out Amalek — perhaps, who can say, it reveals itself in philosophical buzzwords, as well. But, if we are to act on God’s apparent, presumptive, antecedent will, rather than his secret, decisive, consequent one, I would have to say that this method does not look promising.

What about the old algorithm, then — let’s say, the old algorithm as it operated (or: in one of the many ways it operated) before the rise of our current specialties? What reason is there, Hegel’s reasons aside, to expect anything useful from that? “Why,” to quote the title of Lenz’s blog post cited above, “would we want to call people ‘great thinkers’?” To tell the truth, although I have often used that phrase in the past, I am growing tired of it. To present day ears, at least, it sounds like an award we are giving to some people, as if we were doing something for them, rather than: that they did something for us, namely, made us possible. I am pleased to ﬁnd that Justin E.H. Smith makes some remarks along the same lines in a newsletter I received as I was writing this, which I will quote, since he writes so much better than I do. Speaking of demands that we diversity our canon, he says:

An implicit premise … is that it is a good thing for a thinker to end up on a course syllabus, a sort of posthumous prize handed out to the dead by the living. Thus it is presumed that to study the old canon, as we inherited it, the canon of dead white men, is to honour these dead white men.

But scholarship is not a fan club, and to read Descartes or Kant is an undertaking that is entirely neutral with respect to whether they are praiseworthy as thinkers (a fortiori whether they are praiseworthy as people). When we study the history of philosophy we want, ﬁrst, to know what happened, and second, to know how what happened shaped the world we inherited. This is not a celebration, but a solemn duty.[19]I received this essay as an edition of Smith’s newsletter, to which I urge you to subscribe, but for now you can see it online without a subscription here.

I disagree with some of this,[20]I agree that at least some of the complaints about the old canon arise from a thought like this: that assigning someone’s book is like building a monument to them (which, however, does rather undermine the meme about, “Won’t they be surprised to hear that we can learn history from books?”). But I doubt that all or most of the impetus for a new, more diverse canon can be traced to same source. What students want to see on the syllabus is not someone to whom an award is due, but someone who “looks like them,” who reminds them of themselves. And, in some sense, as I will go on to say, that is exactly what they should want. I can’t think that what is really desirable in this respect is to ﬁnd someone in the reading list the same color or gender as yourself, but I can see that feeling alienated by the color and gender you exclusively do ﬁnd might be an obstacle. We all face such obstacles, but some more than others. We should try to remove them if we can. and with some of the other things Smith says in his essay, but I agree with the main point: canonization is not a reward. So, although I continue to think that most of the canonized, if not all, were indeed praiseworthy, and in particular that they were praiseworthy as (ﬁnitely!) wise and good — wiser and better than me, at any rate — I would rather avoid calling them great anything. Let me call them my authorities.

How can reading an authority prompt deep reflection? What is “reflection”? It is reflective, reflex. I recognize a pattern, all right, something familiar, but the pattern, the παράδειγμαparadeigma, is familiar, not because I have previously picked it up with my buzzword detector, but because I myself am its image, ut esset tanquam nota artiﬁcis operi suo impressa; nec etiam opus est ut nota illa ﬁt aliqua res ab opere ipso diversa. This kind of pattern recognition is called recollection, ἀνάμνησιςanamnesis, and it cannot be performed by algorithm: not because machines, in the end, really only know how to count; not because πᾶν τὸ πρὸς ἑαυτὸ ἐπιστρεπτικὸνpan to pros heauto epistreptikon ἀσώματόν ἐστινasomaton estin; not (if I may allude to a certain harasser) because recollection depends on the actual physical–chemical properties of actual human brains — but simply because recollection is reflection, and, as you will recall, I have deﬁned an algorithm as a process that does not require reflection. To see an electronic computer perform ἀνάμνησιςanamnesis would surprise me no more than to see a human being do so, which is to say: it would surprise me a great deal, and always again surprise me even if I were to see it always and everywhere I looked (although, sadly, there is little prospect of that). But I would also know, of the computer or the human, that they could have performed this recollection only thanks to a reminder. The sign, the trace that can remind me in this way leads always beyond my shifting concepts, לעילא לעילא מכל ברכתא ושירתאle-ʿeila le-ʿeila mi-kol birchata ve-shirata, and is hence everywhere beyond needing any praise from me. But I need everything from it.

I close by coming back to Kant, and from Kant to Hume. What did Kant think was wrong in Hume? Not, alas, that he was a racist: Hume’s racism, if anything, thanks to its tentative, empirical nature, would scarcely have seemed racist enough for Kant. But he did think Hume had committed an error or two. The worst one, as it happens, was his failure to reflect, or rather, his failure to reflect in the right way. Reflection (I will assert here without evidence) is the (logical) topic Kant takes up in the Analytic of Principles, in its third Hauptstück, Of the Ground of the Distinction of all Objects in general into Phenomena and Noumena.[21]The distinction itself between phenomena and noumena is also a local distinction: it distinguishes what is within the bounds of experience (phenomena) from what is without (noumena). These two places constitute what Kant calls transcendental topic — or, in the terminology I have adopted here: they are the two transcendental topics (only, of course, that the second one is empty). Accordingly, he begins there by raising a question about everything that has preceded. Granted that everything you say is true, Kant, why tell us? (For, it must always be recalled: there is a negative duty not to lie, not, absurdly, a positive duty to say everything that is true.)

If, then, through this critical investigation, we learn no more than what, even without so subtle an inquiry [Nachforschung], we would anyway have executed [von selbst wol würden ausgeübt haben] in the merely empirical use of the intellect, then it seems that the advantage which one draws from it is not worth the effort and the preparation. (A237/B296)

Leaving aside the conclusion, where does the premise of this objection come from? Who says that we would have done the same thing, anyway, without this subtle inquiry? Who indeed. Who says, for example, that we would have employed the law of causality, that every event must have a cause, even if we not only could not demonstrate the truth of that judgment, but could not even show title to the concepts it contains? The answer is: Hume. Hume says that. So what does the whole apparatus of the Transcendental Analytic contribute that was missing in Hume? If we must, anyway, use the concept of a cause (when, for example, we speculate, rightly or wrongly, about the causes of certain historical–geographical trends), what good to us is a demonstration of the Second Analogy? Kant answers:

the intellect when occupied with its merely empirical use, which does not reflect [nicht nachsinnt] upon the sources of its own cognition,[22]Nachsinnung is not the official equivalent Kant later gives for reflexio (rather: Uberlegung — see A260/B316). But I feel the translation is justiﬁed, given that he is talking here about what he goes on to call “transcendental reflection”: investigating the relation of one’s cognitions to the faculties that are their “sources.” can indeed progress very well, but there is one thing it cannot achieve, namely, to determine of itself the limits of its use, and to know what may lie within or without its whole sphere.

What Kant thinks Hume failed to do — or perhaps did incorrectly, so as to obtain a negative (skeptical) result — was transcendental reflection; and he adds that Hume was therefore led to say (as, indeed, Hume does say quite clearly) that reason cannot set its own limits (i.e., transcendentally locate itself). Which, from Kant’s point of view, is an inﬁnitely bad error, both theoretical and practical: reason’s theoretical self-limitation is the destruction of knowledge that makes room for faith, and reason’s practical self-limitation is self-legislation, autonomy. For Kant, Hume is guilty, not of making some particular moral error, but rather of asserting that morality, as such, is impossible.

So why would Kant want to call Hume a “great thinker” (or words to that effect)? Or, as I would rather say: why would Kant take Hume as an authority?

Because Hume is perhaps the most ingenious [the richest in spirit, geistreichste — not, one assumes, in the chemical sense] among all skeptics, and incontrovertibly the most eminent with respect to the influence which the skeptical procedure can have upon the awakening of a thoroughgoing rational self-examination [einer gründlichen Vernunftprüfung], it may well be worth the trouble to represent, insofar as it is appropriate to my aim, the course of his conclusions and the strayings [Verirrungen], which nevertheless began on the track of the truth,[23]It is difficult to translate this passage correctly. The idea is that, when we follow the course or path (Gang) of Hume’s reasoning, we will ﬁnd that, at certain points, he takes off after the tracks or traces (Spur) of the truth, but nevertheless goes astray (verirren). of so insightful and estimable a man. (A764/B792)

Hume supplied the place — the authoritative topic, so to speak — of Kant’s awakening.

I freely confess: the recollection [Erinnerung] of David Hume was just that which, many years ago, ﬁrst interrupted my dogmatic slumber and gave to my investigations in the ﬁeld of speculative philosophy an entirely different direction. (Prolegomena, Ak. 4:260,6–9)

Hume himself — wrong, inﬁnitely wrong as he may have been — was for Kant the trace of truth, the trail, the spoor, the path that leads right inward. In Kant’s terminology: one might say that Hume, in his life and thought, constituted a symbolic, indirect exhibition of the transcendental idea of freedom. The recollection of Hume was, then, for Kant, the recollection of his own true self, of his transcendental ego. To name a tower after such a one, even a monument as high as the moon; to put him on a syllabus, or even, like some notice about the plagiarism policy, on every syllabus; to hire a fantastic young scholar who specializes in his work, or even to hire a whole army of such young scholars, each more fantastic than the last — all this is nothing, not the smallest part of what we owe him, or, rather, what we would owe him, if he needed anything from the likes of us.[24]Here I am speaking about what we might owe Hume in common, not about what is owed to him by those specialists who have appropriated him and used him as a means, extracting the value of his labor to further their own careers. But these latter will have to make their own reckoning.

But do we need him! What have I just been saying, all this time, but that our dogmatic slumber now appears terminal? “Any prospect of awakening or coming to life to a dead man makes indifferent all times and places. The place where that may occur is always the same, and indescribably pleasant to all our senses” (Walden, 5.9). He is, incontrovertibly, the heresiarch most eminent with regard to what we require. We cannot receive him now, cannot receive anyone: Hume and all the others were already canceled the moment they were transformed from authorities into specialties. To quote someone else who had a morally wrong view here and there: only a god can save us. But, if and when the day should come, it has already been prophesied: et servus meus David princeps in medio eorum.

## Monday, September 7, 2020

This is the second in a series of posts about Locke’s theory of space.

At the end of the previous post, I had posed the question, how the combination of two simple ideas — namely, the ideas of space and of limit or ﬁnitude — can yield, rather than a single mixed mode, an inﬁnite number of distinct simple modes. I began my answer by saying that the idea of ﬁnitude, in this case, given what Locke takes to be its empirical origin, is the idea of what is left behind when a part of body moves away from another part. Or, as I also put it: the “obvious portions of extension” can affect us with the idea of a ﬁnite empty space because of the divisibility of body. I pointed out, however, that this explanation addresses directly only the question of quantitative diversity of our ideas of spaces: these can be of any size, relative to one another, because body is inﬁnitely divisible. What is less clear is how to explain their speciﬁc qualitative diversity: why, although they can be of inﬁnitely many different shapes, there are tight constraints on what these shapes may be (e.g. the so-called triangle postulate, which requires that the interior angles of a triangle be equal to two right angles).

Figure 1: New superﬁcies where before there was a continuity.

To begin dealing with this latter problem, let’s take a closer look at the process of division, which, recall, Locke deﬁnes as “by removing the Parts one from another, to make two Superﬁcies, where before there was a Continuity” (2.13.13) (ﬁg. 1). As the new superﬁcies TK moves away from the new superﬁcies HX, T moves along a line BA which passes through H, while K moves along a second line DZ which passes through X. I have drawn both the lines BA and DZ as straight[1]Locke, along with most geometers throughout the Western history of the subject, uses “line” to mean something which may (or perhaps must) be of ﬁnite length and which may be either straight or curved. In other words, this is roughly what we now call a “curve,” although it probably is not helpful to think of it as a set of points parameterized by an interval of the real numbers. For Locke, then, “straight line” is not a pleonasm. because that evidently is how Locke thinks of them, but also because it helps bring out the similarity between this ﬁgure and two other ﬁgures which I will now proceed to discuss.

Figure 2: The sucker in a pump. Detail from a diagram of Boyle’s third air-pump. Credit: Wellcome Collection. Attribution 4.0 International (CC BY 4.0). Accessed on Wikimedia Commons.

The ﬁrst (ﬁg. 2) shows something Locke mentioned in a passage I quoted in my previous post: the sucker of a pump. Speciﬁcally, it shows a sucker in the third of the famous air pumps constructed by Robert Boyle. Locke was part of Boyle’s circle in his Oxford years, so it makes sense that, when he thinks about an empty space, he thinks about a body being divided in this particular way: one piece sliding away from the other so as to leave behind a space which, if no other body moves into it, will remain empty. That the case of a perfect vacuum pump is at least conceivable shows, according to Locke, that our general idea of the process of division involves no supposition about the existence of any other bodies, although, obviously, if there are any other bodies around, their motions must be such as not to resist it. Somehow, from the mere idea of the original body, together with the ideas of the two pieces formed from it, we can understand that this process is possible.

Figure 3: MS Paris Bibliothèque Nationale arabe 2457, detail of fol. 157v. Source gallica.bnf.fr / Bibliothèque nationale de France.

The second (ﬁg. 3) is perhaps a bit more unexpected. This diagram comes from one of the two proofs of Euclid’s parallel postulate due to Thābit ibn Qurra.[2]For a printed version of this text, with French translation, see R. Rashed and C. Houzel, “Thābit ibn Qurra et la théorie des parallèles,” Arabic Sciences and Philosophy 15 (2005):9–55 (available online here for those who have access); I will quote from their printed text, since I can’t easily read the manuscript. For an English translation see A.I. Sabra, “Thābit Ibn Qurra on Euclid’s Parallels Postulate,” Journal of the Warburg and Courtauld Institutes 31 (1968):12–32. I learned of this work from De Risi, to be cited below. The diagram pertains to Thābit’s proof of his ﬁrst lemma, namely:

Any two straight lines which are in one plane, and there go out through what is between them two [further] straight lines that meet [the ﬁrst two], and [these second two lines] are equal and contain with one of the ﬁrst two lines equal angles on one side: any two perpendiculars which fall on that line from two points of the other line [of the ﬁrst two] are equal. (Rashed and Houzel, p. 29, ll. 11–14)

Figure 4: My version of Thābit’s ﬁgure.

That is (ﬁg. 4): if, for example, between the straight lines AB and GD, one can draw two equal straight lines HW and AG such that HWD = AGD, then all lines like TK — straight lines drawn between a point T on AB and a point K on GD, and perpendicular to GD[3]In Euclidean geometry, it will, of course, also follow that TK is perpendicular to AB, since the hypothesis is sufficient to show that AB and GD are parallel. But Thābit does not assume that, since the proof of it would depend on the parallel postulate that he is trying to establish. — are equal to one another. For the purposes of the proof, Thābit introduces (though he makes this explicit only later on) auxiliary points Z and X, deﬁned as the points of intersection with DG of perpendiculars dropped from A and H, respectively.[4]That we can do this is guaranteed by Elements 1.12, which is independent of the parallel postulate. He then begins:

We imagine [نتوهمnatawahham] that a solid contains the line AG and the segment GZ of the [straight] line GD, such that they are within it, and that that solid moves, in its entirety, from the side of G to the side of D, by one straight simple motion, along the direction [استقامةistiqāma] of the line GD, and that, moreover, there is the like of the lines AG, GZ drawn in it, remaining in it, with [shapes] like their shapes. (Rashed and Houzel, loc. cit., ll. 19–22)

He then proceeds to show that, as the solid carries the triangle AGZ (or its “like,” مثالmithāl — Sabra translates “analog,” which is loose but not unreasonable) along, the line GZ will move along GD until eventually G coincides with W and Z coincides with X, and that, at that time, A, having moved along the straight line AB, will coincide with H. On this basis he is able to show that, as A passes through any random point T on AB, Z will pass through the intersection K with GD of the perpendicular dropped from T. Hence any such perpendicular TK is equal to AZ, and so they are all equal to one another, QED.

If you look in Vincenzo De Risi’s informative introduction to his edition of Sachheri’s Euclides vindicatus,[5]Gerolamo Saccheri, Euclid Vindicated from Every Blemish, ed. V. De Risi, tr. G.B. Halsted and L. Allegri (Birkhäuser, 2014); ISBN 978-3319059655. you will ﬁnd Thābit described as “arguably, the creator” of a proof of the parallel postulate “from motion,” which De Risi describes as follows:

Beginning in the Middle Ages, there was yet another hurdle: the incorrect but very tempting argument involving the employment of motion in geometry. Let the end point of a line segment slide continuously along a straight line, and let the segment remain perpendicular to the line while moving: then the other end point will trace out a line, undoubtedly equidistant from the given straight line; and it was evident to all that the line generated in this way could not be other than a straight line itself. . . .  For many centuries the genetic character of this procedure deceived countless mathematicians, who could not explain how such a simple and uniform movement might generate a curve — in fact, it would be hard for anyone lacking an account of curvature, that is, anyone before Gauss, to provide such an explanation. (p. 9; the description of Thābit as “arguably, the creator” is in n. 9 there)

But, although this bit of ultra-whig history may well apply to the later versions of the proof, for example the versions of Clavius and Borelli that are criticized by Saccheri, it does not apply to Thābit’s (arguably) original version. For, ﬁrst of all, Thābit assumes, not that a line (segment) is moving, but that a solid is moving, and carrying a triangle along with it. True, he does indicate, in his introductory remarks, that the use of a solid is strictly unnecessary, and to envision, therefore, a proof more along the lines of Clavius’ or Borelli’s. But he refrains from taking that course because to imagine this kind of motion of a line along another line would be, he says, “an unfamiliar thing,” and “it is not certain but that the line, in its transport [نقلةnuqla],[6]Forms of this root, نقلn-q-l, are, at least today, the (Modern Standard Written) Arabic equivalents for English “transport” or “transportation” (although نقلةnuqla apparently now means “migration”). Such present day equivalences often have their roots in medieval translation traditions, and I would like to think it no coincidence that Thābit uses the term here in what is (arguably) history’s ﬁrst discussion of parallel transport. The latter expression (for which, I see towards the bottom of this page, the current Arabic is indeed نقل متوازيnaql mutawāzīy), originates, I believe, with Tullio Levi-Civita, “Nozione di parallelismo in una varietà qualunque e conseguente speciﬁcazione geometrica della curvatura riemanniana,” Rendiconti del Circolo Matematico di Palermo 42 (1917):173–215, reprinted in his Opere matematiche. Memorie e note (Bologna: Nicola Zanichelli, 1954–75) 4:1–39, where the phrase “il trasporto di direzioni parallele lungo una qualsiasi curva” occurs on p. 3. Moreover, I note that Levi-Civita had an early interest in classical languages, and that his high school mathematics teacher, the number theorist Paolo Gazzaniga, recalled that his precocious student produced “an attempted demonstration of the parallel postulate that ran, elegant and impeccable, up to an inadvertent ﬁnal admission, in which the petitio principii was nestled” (U. Amaldi, “Tullio Levi-Civita,” in Levi-Civita, Opere, 1:ix-xxx, p. x). (I owe both of these references to Iurato and Ruta, to be cited below. My knowledge of Italian consists entirely of half-remembered high school Spanish, assisted by Google Translate.) Nevertheless, I don’t know how to supply any of the missing steps between Thābit and Levi-Civita, and it’s quite possible that نقلةnuqla here should be rendered “translation.” Thābit, presumably, is not thinking of the distinction between segments in the underlying space and vectors in the tangent bundle that would decide which he means, and meanwhile نقلةnuqla can be used for φοράphora, e.g. at Top. 4.2.122b27 (Arabic here), where Boethius (PL 64:945A) has latio. might change its shape or position,” and so he instead begins with “a well-known thing which I have premised concerning the solid,” namely with the following:

For any solid that we imagine to be moved in its entirety to one side by one simple direct motion: any point of it will be moved directly, and will delineate in its passage a straight line along which it passes, and as for the straight lines that are in [the solid], those of them that are in the direction of its motion will also pass along a straight line, whereas those of them that are in other than the direction of its motion will not do so. (Rashed and Houzel, loc. cit., ll. 5–9)

Following what seems to be the tradition in this ﬁeld (historiography of the parallel postulate), we might call this “Thābit’s axiom.” Although the motivation he offers for it may be a little difficult to follow, the axiom itself is clear enough, and he applies it in his proof to show exactly what, according to De Risi, generations of mathematicians couldn’t bring themselves to doubt: namely, that the point A traces a straight line as it moves.

Figure 5: The line (segment) αβ starts out coincident with AB and moves left along EF until α coincides with C. From Wallis, De algebra, p. 676.

But could Locke possibly have known anything about this text, which is extant in only two manuscripts, and which did not appear in print until the 20th century? Somewhat surprisingly, the answer is that he could have. Not that he likely read an obscure Arabic manuscript himself: although he was required to attend lectures on the Arabic language as part of his senior studentship at Christ Church, I have no reason to think[7]Based on my usual half-ass research. that he ever became very competent at it. The man who gave those lectures, however, Edward Pococke (or Pocock), was Locke’s favorite professor at Oxford. And we know that Pococke, around this time, was collaborating with another teacher of Locke’s, John Wallis (also a close associate of Boyle’s and his fellow adversary of Hobbes), in the study of Arabic proofs of the parallel postulate. At any rate, when Wallis later published his own proof,[8]In De postulato quinto et quinta deﬁnitione libri 6 Euclidis disceptatio geometrica, an appendix to Wallis’s De algebra tractatus (Oxford, 1693). The main book is a Latin version of Wallis’s 1685 Treatise of Algebra, but this appendix is based on a lecture that he delivered on July 11, 1663, i.e. while Locke was still at Oxford. he published it along with Pococke’s Latin translation of another medieval Arabic proof, this one due to the great Persian polymath Naṣīr al-Dīn al-Ṭūsī.[9]The published proof is now thought, on what evidence I don’t know, to be actually the work of a disciple. But Pococke also translated and shared with Wallis a version still thought to have been written by al-Ṭūsī himself: see De Risi, Introduction to Euclid Vindicated, p. 10 n. 12 and p. 16 n. 23. There is, moreover, some indication that Pococke might have shared knowledge of a different manuscript work of Thābit’s with Wallis. This is highly conjectural, but: (1) Pococke spent three years in Constantinople in the late 1630’s, collecting books and studying rare manuscripts; (2) MS 4832 of the Library of Aya Sofya Museum in Istanbul contains a letter in which Thābit states, without proof, a certain generalization of the Pythagorean theorem; (3) the ﬁnal three chapters of Wallis’s Treatise of Angular Sections, which were composed in 1665, are based on what is in effect the same result.[10]See C.J. Scriba, “John Wallis’ Treatise of Angular Sections and Thâbit ibn Qurra’s Generalization of the Pythagorean Theorem,” ISIS 57 (1966), pp. 56–66. Scriba himself does not actually assert, or even conjecture, that Wallis knew of Thābit’s work, though, thanks to a misreading of his paper in G.G. Joseph, The Crest of the Peacock: Non-European Roots of Mathematics (Penguin, 1990, ISBN 978-0691135267), p. 337, he has ended up as the indirect source of this assertion in the Wikipedia article on Wallis (and, it case it isn't obvious: it is by following that same trail in the other direction that I arrived at Scriba's article). Note that the same manuscript, Aya Sofya 4832, also happens to contain one of the extant copies of Thābit’s other proof of the parallel postulate. Finally, and most suggestively, Wallis’s own proof of the parallel postulate, although based on a different principle (now known, of course, as “Wallis’s axiom”), does nevertheless involves a segment transported along a straight line from one position to another, and so still contains a version of our same familiar picture (ﬁg. 5). So there is at least some possibility that Pococke and Wallis knew of Thābit’s proof, and, if they did, Locke might well have heard about it from one or both of them.

I don’t think it necessary to insist that he did, however. The main point is what Thābit’s procedure — what the correctness of his procedure — shows about the relationship between the motion of bodies and the possible simple modes of space. For there is no error in Thābit’s proof, or, at least, no mathematical error, but there is a reason De Risi takes it for granted that there must be one: namely, that, as we now know, non-Euclidean geometries are possible. Since a proof of the parallel postulate amounts to showing that, at least in one important sense, a non-Euclidean geometry is not possible, it is natural to suspect that any such proof must contain an error.

What is it, however, that we know, under the title “that non-Euclidean geometry is possible”? This is actually one of those simple questions that gets harder and harder to answer the more carefully you think about it. One way to get at the problem is to imagine that two people are disputing: A maintains that Euclid is right about geometry, while B says that Euclid is right except insofar as his system depends on the parallel postulate, which, according to B, is false. From the point of view of contemporary (“pure”) mathematics (a point of view originally stemming, in large part, from the acceptance of non-Euclidean geometry), it may seem difficult to understand what A and B could be arguing about. Euclid is “right” about his geometry — that is, his theorems are consequences of his axioms[11]Plus some additional axioms which he doesn’t state. — but he is not, obviously, “right” about some other geometry — that is, not all his theorems are consequences of some other set of axioms. Isn’t that the end of matter? But there is still one way a contemporary mathematician can understand the dispute: namely, if we assume that A thinks B’s position is inconsistent, and vice versa. The consequences of an internally inconsistent set of axioms are not worth studying. The hero of our story, considered in this light, will be Eugenio Beltrami, who showed, in 1868, that B’s system (what we now call “hyperbolic geometry”) is consistent if and only if A’s is. It follows that neither A nor B will be able to win the debate on these terms.

If we see the history of thought on this question, up until the time of Gauss, Lobachevsky, and Bolyai, as a misguided attempt to stave off one half of Beltrami’s breakthrough, or, in other words, as one long attempt on A’s part to ﬁnd a contradiction in B, then it will be evident that every proof produced during that history must involve an error of one kind or another. Either A must commit some actual sophism, or A must have induced B to accept some extra assumption that goes beyond Euclid’s — which, given our statement of the dispute, would be cheating. This whole story, however, fails to address a different question about A and B — a question raised explicitly by Beltrami’s model-theoretic method, and, moreover, a type of question familiar to Locke, namely: whether A and B are even talking about the same thing. In other words, although Beltrami showed that hyperbolic “geometry” involves no logical absurdity, and is in that sense possible, he did not show that it is, even possibly, geometry. In light of this question, the point of proving the parallel postulate would not be to show that B is inconsistent, but rather to show that, when B says that two “straight lines” which intersect the same “straight line” so as to form less than two right angles on the same side may nevertheless never meet on that side, B cannot be talking about straight lines. Then this will involve exactly what looked like cheating before: namely, to make explicit whatever it is about straight lines — necessarily, something Euclid fails to say — that makes it clear that B must be changing the subject. The newly explicit constraint might take the form of a new deﬁnition of “straight line,” which will show that B’s view involves a contradiction in terms: Leibniz tries such a procedure in at least one place.[12]See J. Heis, “Leibniz versus Kant on Euclid’s Axiom of Parallels,” pp. 11–12; De Risi, Leibniz on the Parallel Postulate and the Foundations of Geometry: The Unpublished Manuscripts (Birkhäuser, 2016, ISBN 978-3319198620), pp. 96–7. Both Heis and De Risi say that Leibniz’s proof fails, but in the text they are discussing, the In Euclidis πρῶταprōta (which is unﬁnished and contains many erasures and restarts) it is not even clear how many attempted proofs there are and where they begin or end. Assuming De Risi’s §7 (p. 170) contains the end of a proof, the question of whether it is successful may depend on exactly what Leibniz means by “uniform” there and in §3 (p. 168). Or it might instead take the form of a new geometrical axiom or postulate which Euclid ought to have stated and which B ought to accept, the addition of which will make B’s position untenable: this, for example, is Wallis’s procedure. Or, ﬁnally, it might state a relationship between lines and some extra-geometrical concept. For example — and this brings us back to Thābit, and to Locke — a relationship between straight lines and the motions of bodies.

Now, to be clear, I should add immediately that Thābit himself would probably not accept the qualiﬁcation “extra-geometrical.” The ﬁrst part of his introduction is devoted to arguing, based in part on the authority of Euclid, that motion is a legitimate geometrical concept, and مجسمmujassam, although derived from the same root as جسمjism, “body,” is the standard word for “solid” in a geometrical context, the equivalent of στέρεονstereon = solidum.[13]E.g. Elements book 11, deﬁnition 1 begins in Arabic الشكل المجسم هو al-šakl al-mujassam huwa: see, e.g., MS Upsala Universitetsbibliotek O. Vet. 20, fol. 157r. (This part of the Arabic Elements remains unedited.) Hence he likely believes that he is using “solid” in, as Locke would put it, “[the] acceptation … which Mathematicians use it in” (Essay 2.4.1). But, whatever Thābit may think, Locke deﬁnitely agrees with Aristotle and Kant, among many others, that the movable in space is body — corporeal substance — and hence, by deﬁnition, beyond the reach of pure mathematics, whereas “The Parts of pure Space, are immovable” (2.13.14). For Locke, then, Thābit’s axiom, if it is true, is not a purely geometrical truth, but rather expresses a relation between geometry and physics.[14]Levi-Civita may also have been thinking about the relationship between geometry and physics when he developed the theory of parallel transport: see G. Iurato and G. Ruta, “The role of virtual work in Levi-Civita’s parallel transport,” Archive for History of Exact Sciences 70 (2016):553–66.

To return, then, to my main topic. I have already shown — or, well, anyway, asserted — that, according to Locke, the diversity of simple modes of space depends entirely on the connection between ﬁnite space and body. Suppose, then, that Thābit’s axiom expresses something evidently true about bodies, namely that the movement of their parts must look like ﬁg. 4, or, in other words, that the lines BA and DZ, in ﬁg. 1, must be straight. Then whatever follows from Thābit’s axiom, including at least the so-called triangle postulate, will act as a constraint on the qualitative diversity of those simple modes.

In a future post I will try to explain why Locke considers Thābit’s axiom to be evidently true, and, indeed, intuitive (as opposed to demonstrative). And I will also try to show that he indeed contemplates the use of exactly that principle in constraining the possible shapes of spaces.