Anthropology of Mathematics

This essay was long when originally published; now, on November 30, 2019, I have made it longer, in an attempt to clarify some points.

The essay begins with two brief quotations, from Collingwood and Pirsig respectively, about what it takes to know people. The Pirsig quote is from Lila, which is somewhat interesting as a novel, but naive about metaphysics; it might have benefited from an understanding of Collingwood’s Essay on Metaphysics. A recent article by Ray Monk in Prospect seems to justify my interest in Collingwood; eventually I have a look at the article. Ideas that come up along the way include the following.

  1. For C. S. Lewis, the reality of moral truth shows there is something beyond the scope of natural science.

  2. I say the same for mathematical truth.

  3. Truths we learn as children are open to question. In their educational childhoods, mathematicians have often learned wrongly the techniques of induction and recursion.

  4. The philosophical thesis of physicalism is of doubtful value.

  5. Mathematicians and philosophers who ape them use “iff” needlessly.

  6. One pair who do this seem also to misunderstand induction and recursion.

  7. Their work is nonetheless admirable, like the famous expression of universal equality by the slave-driving Thomas Jefferson.

  8. Mathematical truth is discovered and confirmed by thought.

  9. Truth is a product of every kind of science; it is not an object of natural science.

  10. The distinction between thinking and feeling is a theme of Collingwood.

  11. In particular, thought is self-critical: it judges whether itself is going well.

  12. Students of mathematics must learn their right to judge what is correct, along with their responsibility to reach agreement with others about what is correct. I say this.

  13. Students of English must learn not only to judge their own work, but even that they can judge it. Pirsig says this.

  14. For Monk, Collingwood’s demise has meant Ryle’s rise: unfortunately so since, for one thing, Ryle has no interest in the past.

  15. In a metaphor developed by Matthew Arnold, Collingwood and Pirsig are two of my touchstones.

  16. Thoreau is another. He affects indifference to the past, but his real views are more subtle.

  17. According to Monk, Collingwood could have been a professional violinist; Ryle had “no ear for tunes.”

  18. For Collingwood, Victoria’s memorial to Albert was hideous; for Pirsig, Victorian America was the same.

  19. Again according to Monk, some persons might mistake Collingwood for Wittgenstein.

  20. My method of gathering together ideas, as outlined above, resembles Pirsig’s method, described in Lila, of collecting ideas on index cards.

  21. Our problems are not vague, but precise.


When Donald Trump won the 2016 U.S. Presidential election, which opinion polls had said he would lose, I wrote a post here called “How To Learn about People.” I thought for example that just calling people up and asking whom they would vote for was not a great way to learn about them, even if all you wanted to know was whom they would vote for. Why should people tell you the truth?

Saturn eclipse mosaic from Cassini

With other questions about people, even just understanding what it means to be the truth is a challenge. If you wanted to understand people whose occupation (like mine) was mathematics, you would need to learn what it meant to prove a theorem, that is, prove it true. Mere observation would not be enough; and on this point I cite two authors whom I often take up in this blog.

  • In the words of R. G. Collingwood in Religion and Philosophy (1916, page 42), quoted in An Autobiography (1940, page 93) as well as in the earlier post here, “The mind, regarded in this external way, really ceases to be a mind at all.”

  • In the words of English teacher and anthropologist Verne Dusenberry, quoted by Robert Pirsig in Lila (1991, page 35), “The trouble with the objective approach is that you don’t learn much that way.”

I am going to say more about Collingwood and Pirsig:

  • Ray Monk, in a recent article, regrets that Collingwood (born in 1889) died early (in 1943);

  • I regret that Pirsig (born in 1928) did not make use of Collingwood’s existing work.

The main point, which both Collingwood and Pirsig recognize, is that there is more to the universe than meets the eye of empirical science.

I made the point while reading and writing here about the Iliad last September (2019). In Book XV and elsewhere, the gods are seen to bind themselves by oaths. That we humans can so bind ourselves means we recognize “The law of Human Nature, or of Right and Wrong,” which

must be something above and beyond the actual facts of human behaviour … besides the actual facts, you have something else—a real law which we didn’t invent and which we know we ought to obey.

Thus C. S. Lewis in The Case for Christianity (New York, 1950, page 18).

I am not here to make a case for Christianity as such. I question Lewis when, in addition to “the Materialist view and the Religious view,” he mentions (on his pages 22–3)

the In-between view called Life-Force philosophy, or Creative Evolution, or Emergent Evolution. The wittiest expositions of it come in the works of Bernard Shaw, but the most profound ones in those of Bergson … If … you want to do something rather shabby, the Life-Force, being only a blind force, with no morals and no mind, will never interfere with you like that troublesome God we learned about when we were children.

The bold emphasis is mine, here and throughout this essay, with one exception, which will be noted. I make bold the passage above, because I disagree with its implicit suggestion that what we learned as children must be right.

What we learn as children may be right, but we should revisit it to make sure. Such is the case in mathematics, as for instance in the title subjects of “Induction and Recursion” in the De Morgan Journal (2012). In that article, I did not happen to take up Michael Spivak’s Calculus (Second Edition, 1980), though I have mentioned it a few times in this blog. I learned real mathematics from Spivak, but he confuses induction, strong induction, and being well-ordered, as properties of the natural numbers, in a way that John Baldwin calls “Pierce’s Paradox” in Model Theory and the Philosophy of Mathematical Practice: Formalization without Foundationalism (Cambridge University Press, 2018, page 38).

I say I learned real mathematics from Spivak, despite his confusion in a foundational matter, because I learned from him the possibility and desirability of straightening out such confusion and getting things right. C. S. Lewis has the example of the “law of Human Nature” as a reality that is not an object of study for natural science; my example is just mathematics. It may serve the natural sciences, but is not studied by them.

Such realities thus exist. As far as I can tell, a doctrine called physicalism denies this point, while trying to accommodate it. According to Daniel Stoljar in the Stanford Encyclopedia of Philosophy,

Physicalism is the thesis that everything is physical, or as contemporary philosophers sometimes put it, that everything supervenes on the physical. The thesis is usually intended as a metaphysical thesis, parallel to the thesis attributed to the ancient Greek philosopher Thales, that everything is water, or the idealism of the 18th Century philosopher Berkeley, that everything is mental. The general idea is that the nature of the actual world (i.e. the universe and everything in it) conforms to a certain condition, the condition of being physical. Of course, physicalists don’t deny that the world might contain many items that at first glance don’t seem physical—items of a biological, or psychological, or moral, or social nature. But they insist nevertheless that at the end of the day such items are either physical or supervene on the physical.

Let me first express my appreciation for the examples of older philosophers. Though I have not read Berkeley since college, I have been pleased to study and speak about Thales in the Roman theater in his hometown of Miletus.

In this blog I often quote certain older philosophers, both because they inspire me, and because they provide, for what I have to say, a touchstone, in roughly Matthew Arnold’s metaphorical sense:

… there can be no more useful help for discovering what poetry belongs to the class of the truly excellent, and can therefore do us most good, than to have always in one’s mind lines and expressions of the great masters, and to apply them as a touchstone to other poetry. Of course we are not to require this other poetry to resemble them; it may be very dissimilar. But if we have any tact we shall find them, when we have lodged them well in our minds, an infallible touchstone for detecting the presence or absence of high poetic quality, and also the degree oft this quality, in all other poetry which we may place beside them.

This is from Arnold’s essay “The Study of Poetry” in Volume 28, “Essays: English and American” (page 72) of the Harvard Classics. I have cut and pasted the text from Bartleby, but corrected against the scan at the Internet Archive to confirm two evident errors (missing “an”; “oft his” for “of this”), which I have tried to correct visibly above.

I do wish to disavow any suggestion that “poetic quality,” or beauty, is a property of certain objects that can be detected by physical means. I said this also in writing “On the Odyssey, Book I.” Quoting Collingwood as a touchstone, I said, “You can send a probe to another planet to detect oxygen, water, or amino acids; but it cannot detect beauty.”

Returning to the touchstone of Thales, we have nothing that he wrote. According to Herodotus, Thales predicted the eclipse that (we figure) must have happened in 585 b.c.e. Born two centuries after that, Aristotle attributes to Thales the thesis that the earth rests on water. Aristotle concludes that, for Thales, water is the first and only principle (ἀρχή) of things.

Aristotle may not have achieved of Thales the kind of understanding of a fellow human being that I set out to talk about here. Still, if only through his reputation, Thales seems to have initiated a research program, which continues today as the discipline called physics, or indeed as all of natural science, the word “natural” being in origin the Latin version of the Greek “physical.”

If the world rests on water, then, since water seems to be something we can understand, we have a chance to understand the world. This is what makes Thales’s thesis metaphysical to me; but then I take the view of Collingwood’s Essay on Metaphysics (1940), whereby metaphysics is the historical study of our absolute presuppositions. “Absolute” means these propositions are not answers to questions; they are a base or platform from which we can seek answers to questions, as we pursue our research, be this in physics, mathematics, history, or anything else.

Physicalism sounds like an attempt to shut down research—except perhaps for its qualifications involving the verb “supervene.”

I have to explain what I mean. Unfortunately words derived from the Latin verb supervenire, meaning “arrive on the scene,” are not really part of my vocabulary. The oldest example in the OED is from the 1594 Treatise of Conscience of Alexander Hume: “By reason of the cold supervenient winter, I was tyed to the bed.” In Stoljar’s example for explaining supervenience, the features of a printed photograph supervene on the dots of ink that make up the picture, because you can’t have a different picture without having different dots.

I understand the physical to be that which is studied by physics. By saying that everything supervenes on the physical, if you do not mean that everything is understood through physics, what can you mean? Stoljar gives an introductory answer (before going into “issues” that are “somewhat technical” and that newcomers are invited to skip over to reach the discussion of whether “physicalism is true”):

Physicalism is true at a possible world w iff any world which is a physical duplicate of w is a duplicate of w simpliciter.

It is like a joke, the mathematical language here: the italicized variable, the abbreviation “iff.” The latter may be a convenient abbreviation for “if and only if”; but when your definiendum is X, and your definiens is Y, you can write your definition as “A thing is X if it is Y”; that a thing is X only if it is Y is implied by your making a definition in the first place. Thus for example Euclid says, in Heath’s translation,

When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right …

After “When,” Euclid need not add “and only when.” At first I thought he did need to, when I read him after learning calculus from Spivak; but according to the suggestion of the tutor (Samuel Kutler) of my freshman mathematics tutorial at St John’s College, such qualifications were implicit. Evidently I was convinced.

Not all mathematicians are convinced. At the beginning of their Model Theory (third edition, 1990, page XII), Chang and Keisler say,

The word ‘iff’ is used in all definitions that require it and is to mean “if and only if”

—and a few pages later (page 7),

A sentence 𝜑 is called valid, in symbols ⊧ 𝜑, iff 𝜑 holds in all models for 𝒮, that is, iff A ⊧ 𝜑 for all A.

The italicization of “valid,” as being the definiendum here, is a clue that the second eff of “iff” is not needed.

Looking up the example in my own copy of Chang and Keisler, I see I have noted an error of the kind I mentioned earlier. The authors say,

The … definition of the relation A ⊧ 𝜑 is an example of a recursive definition based on 1.2.2 [namely the recursive definition of 𝜑]. The proof that the definition is unambiguous for each sentence 𝜑 is, of course, a proof by induction based on 1.2.2.

The proof that the definition of A ⊧ 𝜑 is unambiguous is not merely a proof by induction based in the recursive definition of 𝜑. The recursive definition of sentences allows inductive proofs of properties of sentences: for example, the property that each sentence has the same number of right parentheses as left. However, unambiguity of the recursive definition of A ⊧ 𝜑 is not a property of 𝜑; it is a theorem that requires showing that each sentence is a sentence in only one way. All of this is treated in “Induction and Recursion,” where I make a case for why it is worthwhile to get these things straight. I think it is worthwhile, in part because I admire such work as Chang and Keisler’s and wish to build on it. My first teacher of mathematical logic had been Chang’s student; my doctoral advisor had changed fields from computer science to model theory after reading Chang and Keisler.

I do not reject the principle that all of us are created equal, even though the person who wrote it into the Declaration of Independence of the United States of America kept some of us as slaves.

Returning to Stoljar’s definition of physicalism, I say that the mathematical trappings can be jettisoned. One can just say that physicalism is true at a possible world if any physical duplicate of that world is a duplicate tout court. But then I object that physicalism was supposed to be a thesis about everything—a thesis that was true about everything; and “everything” would include any “possible world.”

Even if a kind of duplicate of our world made sense, I would have a couple of issues.

  1. We know what it means for a page of text to be a physical duplicate of another. We can verify the duplication by reading the pages or by aligning them and holding them up to the light. For one world to be a physical duplicate of another: the very meaning of this would seem to be a problem for physics. If the physicalist philosopher claims to have solved the problem already, he or she is telling scientists their business. This telling is the “scientific persecution” discussed by Collingwood in the first chapter of the New Leviathan (1942), the subject of five years of my blogging. I wrote a new article about Chapter I last July (2019), because I thought my original (January, 2014) was too long.

    Perhaps our philosopher is merely ignorant of what duplicating our physical world would involve. The first objection that Stoljar offers to “supervenience physicalism” is the possibility of

    one extra ammonium molecule located, say, on Saturn’s rings. It is natural to suppose that at [this imagined world], the distribution of mental properties is exactly as it is in the actual world — the presence of an extra molecule does not make that much of a difference.

    I don’t know how this is even an objection. It seems to ignore the logic of the given definition of physicalism, despite its mathematical appearance. By definition, physicalism says what happens—namely, nothing—when two worlds are physically the same. If they are physically different, they may or may not be different in other ways: physicalism does not seem to take a stand here. Stoljar’s ensuing discussion acknowledges this, but in what is to me a confused way. In any case, I’m not sure the number of ammonium molecules in a world is knowable, even in principle; in any case, the Uncertainty Principle would keep us from knowing precisely both where they were and how they were moving.

  2. If somehow we could have two physically identical worlds, where would the physicists be who were verifying this? Where would the truth of the verification be?

I mentioned beauty as being indetectable by a physical device. Perhaps there are truths that can be so detected. A probe sent to Saturn might be said to discover the truth (or some of the truth) about the planet: for example, that there is a moon, now called Daphnis, in the Keeler Gap of the A Ring. However, the first example I thought of here was that the rings of Saturn were not solid; then I found that this had been known from theory, long before any space shots. Apparently the existence of Daphnis could also be inferred from Earth.

In mathematics, the detection of truth cannot be made by anything but thought.

We may distinguish here between finding the truth and confirming it. The former is harder, just as factorizing a number is harder than multiplying two numbers together. Finding prime numbers p and q from their product pq is more difficult than verifying that pq is their product, or so it seems: no strict proof is known, but this does not stop us from basing computer security on the principle.

Discovering the proof of a theorem may be hard, but once it is discovered, verifying that it is a proof is easy, or at least easier. However, the verification still takes thought. One hazard is overlooking mistakes, in the belief that verification should be easy.

In “On Being Given to Know,” I said a machine could not tell us what was true. I did not take up the objection that a computer, properly instructed, can at least verify a proof. In that case, we must still check

  1. that the human-readable proof has been properly rendered machine-readable, and

  2. that the computer, as a physical object, has nonetheless not failed to do what we intend.

Verification of the latter would be empirical, and thus not of the nature of a mathematical proof. It might be good enough for some practical purpose; but the mathematical ideal of proof would remain as something that, in principle, might not have been achieved.

Our brains are physical objects; but whether the truth of a theorem is in those brains is the point at issue. In any case, we were considering computer verification of a proof from given axioms. By Gödel, there are theorems about the natural numbers that do not follow from a previously identified list of axioms.

So again I say a machine cannot tell us what is true. Truth as such is not an object for physics or any other natural science; it is a product of science. I see this idea in the work of the philosopher who arises incessantly on this blog.

One reason why he arises now is Ray Monk’s recent article, “How the untimely death of R G Collingwood changed the course of philosophy forever” (Prospect, September 5, 2019)—changed for the worse, as the lede of Monk’s essay suggests:

The passing of this eclectic and questioning man in his prime allowed the narrower and more imperious Gilbert Ryle to dominate British philosophy. Had Collingwood lived, could the deep and damaging schism with continental thought have been avoided?

I have a related question: if Pirsig had read what Collingwood did live to write, would Lila have been a better book? Monk’s question and mine may have the same answer, be it yes or no. Collingwood already wrote a lot before he died, and people did read his work, and they continue to read it; however, sometimes professionals, even sympathetic ones, seem to have a vested interest in not getting Collingwood’s work, and I do not know how a longer life for the author could have changed this.

I propose that an academic schism may be the least of the problems that arise from not getting Collingwood’s work.

Collingwood died on January 9, 1943, and would have turned 54 on February 22. My source for the precise dates is Inglis, History Man: The Life of R. G. Collingwood (2009), pages 307 and 2 respectively. In his 1978 Introduction to Collingwood’s 1939 autobiography (page xii), Stephen Toulmin writes of what might have been:

Had he lived in a more cosmopolitan time—for instance, in the years following the Second World War, when visiting appointments would have taken him regularly to the United States—he might have felt less isolated, and have ended by writing less stridently.

Maybe so; but Americans could still read the published books, in lieu of listening to lectures by their author.

Toulmin has a footnote on Collingwood’s “intemperate attack on psychology” in An Essay on Metaphysics. This was published in 1940, though Toulmin gives the date of 1939, and according to him,

What passed itself off as the science of the human mind should be recognized, [Collingwood] declared, for what it alone had the intellectual power to be: namely, the history of ideas. (If we are to study how people think, we must look and see in what terms they think. Ergo: ‘cognitive psychology’ is the same thing as ‘conceptual history’.)

The problem may be mine, but Toulmin does not make much sense to me here. For Collingwood, as I understand him, a psychology called “cognitive” is a fraud; and history is already “conceptual,” in the sense that, in the words of An Autobiography (page 110), “all history is the history of thought.” I have to consider now whether Toulmin’s example should be added to the list given in “Re-enactment” of what I judge to be professional misunderstandings of Collingwood.

Collingwood’s “intemperate attack on psychology” was an attack on the confusion of thinking with feeling: a confusion that supported the rise of barbarism in the form of fascism and Nazism. Writing in the 1970s, Toulmin may have thought barbarism had been defeated in 1945; but if so, he was mistaken, as Collingwood himself had warned in the New Leviathan (1942):

27. 55. The real dialectic of harmonious co-operation between contradictory principles, theoretical and practical at once, which is the spectacle history presents to those who take part in it intelligently, is thus imagined as being replaced by a false dialectic of oscillating conflict between false abstractions.

Perhaps the second instance of “dialectic” here was meant to be eristic, unless this is just what the phrase “false dialectic” is supposed to mean. Collingwood had written, three paragraphs earlier, “Such a replacement of dialectical process by eristical process”—such a replacement whereby the French Revolution was seen as the final triumph of democracy over aristocracy—“is always illusory and always dangerous.”

As for Toulmin, the footnote in his introduction to Collingwood’s autobiography continues:

However, if his experience had been wider and his reading of psychology more charitable, he might have discovered that much the same arguments as his own had long since been current within the field of psychology itself: at least, ever since the turn of the century, when Wilhelm Wundt was writing about Volkspsychologie, or the relativity of “cognitive functions” to their cultural and historical contexts.

In An Autobiography, Collingwood does write about how Hobbes’s State is not Plato’s πόλις, because the cultural and historical contexts of the words are different. But such verbal matters have little to do with the argument in An Essay on Metaphysics (Chapters IX–XII) about the science of thought that psychology claims to be.

If beings do not think, they may nonetheless have their own purposes: they may be trying to do something. For the Greeks, things in nature were trying; for modern science, not. Scientists of any era are still thinkers though, and thinkers not only try to do things, but judge the success of the endeavor.

I tried to make the argument in “A New Kind of Science”; but here is how Collingwood states the case in An Essay on Metaphysics (pages 107–8):

A mind aiming at the discovery of a truth or the planning of a course of conduct will not only score a success or a failure, it will also think of itself as scoring a success or a failure; and since a thought may be true or false its thought on this subject will not necessarily coincide with the facts. Any piece of thinking, theoretical or practical, includes as an integral part of itself the thought of a standard or criterion by reference to which it is judged a successful or unsuccessful piece of thinking. Unlike any kind of bodily or physiological functioning, thought is a self-criticizing activity. The body passes no judgement upon itself. Judgement is passed upon it by its environment, which continues to support it and promote its well-being when it pursues its ends successfully and injures or destroys it when it pursues them otherwise. The mind judges itself, though not always justly. Not content with the simple pursuit of its ends, it also pursues the further end of discovering for itself whether it has pursued them successfully.

Thus for example in 1997 I set out to discover and prove a mathematical theorem. I judged myself successful, and for a while, so did my environment: my article was published in a peer-reviewed journal in 2003. Then a colleague found a counterexample and sent it to me. My earlier judgment had been unjust. I went back to the drawing board and found my error. A correct theorem and proof were published in 2014. I mentioned the incident in “Antitheses” (as well as in the Iliad post discussed above).

I don’t know how many persons are as conscious as mathematicians of the need to get right what they think, according to their own standards. These are ultimately the standards of everybody else who cares, as I suggested for example in “Şirince January 2018.”

Rod Dreher in The American Conservative was recently exercised by a document, ostensibly from the Seattle public schools, headed “K-12 Math Ethnic Studies Framework (20.08.2019).” One section reads as follows (with bold emphasis as in the source),

Where does [sic] Power and Oppression show up in our math experiences?

  • Who holds power in a mathematical classroom?
  • Is there a place for power and authority in the math classroom?
  • Who gets to say if an answer is right?
  • What is the process for verifying the truth?
  • Who is Smart? Who is not smart?
  • Can you recognize and name oppressive mathematical practices in your experience?
  • Why/how does data-driven processes [sic] prevent liberation?

These are all excellent questions, it seems to me, except perhaps for the last one, which I don’t understand. Students of mathematics ought in particular to think on the question of who decides what is right. I hope they will come to recognize that they themselves have the right to decide, along with the responsibility to resolve any disagreement with others’ judgments. Strictly, logic or the universe—or God, if you will—decides what is right; but we all in principle have access to the decision.

Power should be distinguished from both oppression and authority, and I don’t know whether the people in Seattle recognize this; the grammatical issue in their first quoted line suggests not. In the sense discussed by Collingwood in the New Leviathan, a society may use its power to force children to sit in a classroom, and the teacher may then use force on the students. This force may or may not be used oppressively. In time, at least, the classroom itself ought to be a society, in which the students grant a certain authority to the teacher. Ultimately they must all recognize the authority of the truth.

It may be a problem if Seattle teachers are leading their students in chants of, “We are all smart!” However, Dreher gives no justification for his assertion,

In the future, historians will look back upon the suicide of our civilization and will see this poison for what it is. In Seattle, the city’s public schools have decided that everything, even mathematics, has to be seen through the lens of oppression and racism … Eventually, bridges are going to start falling down. That too will be the fault of Whiteness.

That is the first mention of Whiteness in the article. Dreher’s self-consciousness as a White person may be an ongoing problem.

In saying “The mind judges itself,” because it is “Not content with the simple pursuit of its ends,” Collingwood is not speaking as a mathematician, whose judgments would have to meet universal standards. I’m afraid Collingwood is speaking mainly for himself.

How many of us avoid judging our work? In a mass mailing (possibly concerning career planning) to recent graduates of my college, before the days of email, a fellow alumnus wrote something like, “Apologies in advance for any errors, since I have no intention of proof-reading this letter.” Our college did not reveal grades unless students asked for them. Students did hear the oral judgments of their tutors twice a year; otherwise, it was up to students to judge themselves, but this didn’t mean they did it.

Pleasing a teacher is a better reason to study than avoiding the sting of a ruler on the fingers or a riding crop on the buttocks. I suggested this in “All You Need Is Love.” Ultimately one needs to learn to please oneself. Robert Pirsig takes this problem up in Zen and the Art of Motorcycle Maintenance (1974), where he refers to a 1961 letter to a fellow teacher. I recently discovered the text at Henry S Gurr’s website. The colleague addressed is Edith Buchanan in the Department of English Language and Literature, University of New Mexico. Pirsig writes her how hard it is for students to get beyond trying to do what the teacher wants. The emphasis is mine:

I gather from newspapers that there is a great amount of complaint at present about Freshman English instruction and that many departments are looking for new answers.

The answer presented here is in the disguise of an old answer, so that at first it doesn’t appear very new. The problem being fought is the old problem that is renewed each time a student brings in a rewritten paper saying, “Is this what you want?” The question seems ordinary enough to the student but every time one tries to answer it honestly it becomes a frustrating and subtly maddening question. An instructor often gets the feeling that he [sic] could spend the rest of his life telling the student what he wanted and never get anywhere precisely because the student is trying to produce what the instructor wants rather than what is good.

One also notices that on many of these occasions the particular student is as frustrated and angered as the instructor. The student keeps trying to figure out how to please the instructor and to his [sic] way of thinking, the instructor doesn’t seem to know himself. The student turns in a rambling paper. He is told he needs better organization and should make an outline. He goes to work, makes an outline and writes a new story that follows the outline but is told the story is too dull. He goes to work, tries to brighten it with choice bits of liveliness and brings it in. He is then told the story sounds too artificial. He begins to look at the instructor with a deep feeling of estrangement. He decides in his own mind that from the evidence available it is clear that he is talking to an incompetent instructor. He goes his separate way with little accomplished and the cause of English composition has fallen another tiny step backward.

I suspect that the particular problem involved in this situation is a deep one, a fundamental problem that pervades all teaching of English composition and perhaps all teaching. Because instructors are compelled to say what they want they do say what they want, and when they do, they force the students to conform to artificial molds that destroy ideas that students have on their own. Students who go along with their instructors are then condemned for their inability to be creative and take a stand of their own or produce a piece [of] writing that reflects a student’s own personal standards of what is good.

Pirsig’s proposed solution, briefly (all told it has seven steps), is to have students judge one another’s work in the manner described in Zen and the Art.

Two decades earlier, after a stroke in January, 1941, Collingwood decided to resign as Waynflete Professor of Metaphysical Philosophy. He got his future wife pregnant in February, began divorce proceedings with his current wife, and “throughout all these weeks he was reworking and polishing the New Leviathan in order to send Kenneth Sisam the finished manuscript in August” (Inglis, page 304).

Ryle became Collingwood’s successor in the professorship, and according to Monk, philosophy departments in Britain would ask Ryle whom to hire.

Throughout those departments, British philosophers propagated Ryle’s sense that he and his colleagues were doing philosophy in a way that broke sharply both with philosophers of the past, and with those from other countries. Their way was the better way, and philosophy from earlier times and other places wasn’t really worth bothering with.

How did they know their way was better, if indeed it was? It could be better, or at least there may be no need to listen to the past: this is said by another of my touchstones, young Henry David Thoreau, in the “Economy” chapter of Walden:

Age is no better, hardly so well, qualified for an instructor as youth, for it has not profited so much as it has lost. One may almost doubt if the wisest man has learned any thing of absolute value by living. Practically, the old have no very important advice to give the young … I have lived some thirty years on this planet, and I have yet to hear the first syllable of valuable or even earnest advice from my seniors …

These assertions are prefaced with irony:

No way of thinking or doing, however ancient, can be trusted without proof. What everybody echoes or in silence passes by as true to-day may turn out to be falsehood to-morrow, mere smoke of opinion, which some had trusted for a cloud that would sprinkle fertilizing rain on their fields. What old people say you cannot do you try and find that you can. Old deeds for old people, and new deeds for new. Old people did not know enough once, perchance, to fetch fresh fuel to keep the fire a-going; new people put a little dry wood under a pot, and are whirled round the globe with the speed of birds, in a way to kill old people, as the phrase is.

Thoreau still has the spirit of an old person. Later in “Economy,” he explains how he does not actually relish being whirled round at the speed of birds:

As with our colleges, so with a hundred “modern improvements”; there is an illusion about them; there is not always a positive advance. The devil goes on exacting compound interest to the last for his early share and numerous succeeding investments in them. Our inventions are wont to be pretty toys, which distract our attention from serious things. They are but improved means to an unimproved end, an end which it was already but too easy to arrive at; as railroads lead to Boston or New York …

… I have learned that the swiftest traveller is he [sic] that goes afoot. I say to my friend, Suppose we try who will get there first. The distance is thirty miles; the fare ninety cents. That is almost a day’s wages. I remember when wages were sixty cents a day for laborers on this very road. Well, I start now on foot, and get there before night; I have travelled at that rate by the week together. You will in the mean while have earned your fare, and arrive there some time to-morrow, or possibly this evening, if you are lucky enough to get a job in season. Instead of going to Fitchburg, you will be working here the greater part of the day. And so, if the railroad reached round the world, I think that I should keep ahead of you …

When I go to quote Thoreau, or Collingwood, or Pirsig, I may not know how to stop. So it is with Monk’s article, where one can read of what helps make Collingwood fascinating to some of us, while others may be resentful.

… Unlike Ryle and his disciples in the analytic school, Collingwood took a deep interest in both the history of his subject and the work of philosophers from the European continent, being especially infuenced by two Italians, Benedetto Croce and Giambattista Vico.

His intellectual range was astonishing. In philosophy itself, Collingwood made important contributions to aesthetics, the philosophy of history, metaphysics, the philosophy of language, and the understanding of philosophical method. He had important things to say about how each of these contributes to our understanding of ourselves. There was some commonality in his philosophical interests, and in the spirit in which he pursued them, with the incomparably more famous Wittgenstein (one of my biographical subjects). Outside philosophy, he did important work in archaeology and history, and was recognised as one of the country’s leading authorities on Roman Britain, writing the volume devoted to it in the Oxford History of England. In addition, he was an extremely accomplished musician, a talented painter and a gifted linguist, able to read scholarship in French, Spanish, German, Italian, Latin and Greek. He also wrote one of the most fascinating (if decidedly odd) autobiographies ever published …

Much of Monk’s information comes from that autobiography, though not all; I don’t know Monk’s source for saying that Collingwood “was such an accomplished violinist that he thought seriously for a while of becoming a professional musician.” For contrast, Monk quotes words of Bernard Williams about Ryle, who “affected an amiable Philistinism, which to some degree was also genuine: ‘No ear for tunes,’ he was disposed to say, if music was mentioned.”

Monk recalls Collingwood’s account in An Autobiography (pages 29–30) of walking past the Albert Memorial when working in London during the Great War of 1914–18. The thing was hideous. When I saw it for myself, ninety years later, I agreed. Probably Pirsig would agree; in Lila, writing of himself in the third person as Phaedrus, who is sailing down the Hudson River, he expresses antipathy for Victorian values, at least as they made their way to America (pages 97, 108):

A depression always came over him when he came East like this, but the oldness and abandonment weren’t the only reasons for it. He was a Midwesterner and he shared the prejudices of many Midwesterners against this region of the country. He didn’t like the way everything gets more stratified here. The rich start looking richer and the poor start looking poorer …

He remembered his graduate school adviser, white-haired Professor Alice Tyler, at the beginning of her first lecture on the Victorians saying, “This is the period of American history I just hate to teach.” When asked why, she said, “It’s so depressing.”

Victorians in America, she explained, were nouveau riche who had no guidelines for what to do with all their sudden wealth and growth. What was depressing about them was their ugly gracelessness: the gracelessness of someone who has outgrown his own codes of self-regulation.

Back in England, maybe the Albert Memorial fulfilled the purpose of the designer, George Gilbert Scott, whom Monk describes as “the noted Victorian architect and—as it happens—the great-uncle of Gilbert Ryle.”

Collingwood does not give us his ultimate conclusions—“disappointingly,” says Monk:

we are left wondering whether he ever found a way of thinking about the Albert Memorial that did not entail regarding it as a failure.

One should get over the disappointment. An Essay on Metaphysics is about how metaphysics ought to be done, and the essay ends with three examples. The Principles of Art (1938) is an application of An Essay on Philosophical Method (1933); it discusses what art criticism should be, and you can see such criticism in what Collingwood says about T. S. Eliot. Ultimately you have to be your own critic, and that is what Collingwood leaves you to be, as regarding the Albert Memorial. He is trying only to give the outlines of his thought before he dies. He tells us,

I will not try to describe everything I went through in what, for many months, continued to be my daily communings with the Albert Memorial. Of the various thoughts that came to me in those communings I will only state one: a further development of a thought already familiar to me.

The rise of fascism gives Collingwood’s work some urgency. Toulmin may have had some excuse for not recognizing this; I don’t know about Monk.

Monk does review the “further development” that Collingwood refers to. It is the logic of question and answer, whereby every question is based on a presupposition, and every presupposition is either an answer to a previous question or an absolute presupposition. My own account is in “On Causation”; by Monk’s account,

To understand a work of art, a person, a historical epoch or a religion is, so to speak, to “get inside its mind,” to see the world through the eyes of people using a different set of presuppositions to our own. If we try to understand others using only our own presuppositions, we will always fail.

If Collingwood can get inside Scott’s mind; and we, Collingwood’s; then we ought to be able to get inside Scott’s mind directly.

Monk observes Collingwood’s similarity to Wittgenstein, who was but three months younger:

Consider this statement from Collingwood’s Principles of Art: “One does not first acquire a language and then use it. To possess it and to use it are the same. We only come to possess it by repeatedly and progressively attempting to use it.” Ask any philosopher who wrote that and they would almost certainly guess Wittgenstein. And Collingwood’s notion of an absolute presupposition bears an obvious and striking similarity to the things Wittgenstein says in On Certainty, things like: “the questions that we raise and our doubts depend on the fact that some propositions are exempt from doubt, are as it were like hinges on which those turn.”

Monk concludes by urging the canonization of Collingwood:

It took a long time for English-speaking philosophers to realise how wide the gulf was between the spirit that guided Wittgenstein’s work, and that which informed the analytic movement in philosophy during Ryle’s reign … thinkers ignored by the Rylean tradition, such as Kierkegaard, Schopenhauer and [Nietzsche], have now resumed their rightful place in the canon. This can only engender a richer philosophy. Collingwood would be cheered by that thought. He, too, should now be afforded his own rightful place in that canon.

Had Collingwood been canonized, would Pirsig have read him? I assume he didn’t actually read him; otherwise he would have remarked on the commonalities that I observe. A difference is that, for Pirsig in Lila (page 71),

Metaphysics is what Aristotle called the First Philosophy. It’s a collection of the most general statements of a hierarchical structure of thought. On one of his slips he had copied a definition of it as “that part of philosophy which deals with the nature and structure of reality.” It asks such questions as, “Are the objects we perceive real or illusory? Does the external world exist apart from our consciousness of it? Is reality ultimately reducible to a single underlying substance? If so, is it essentially spiritual or material? Is the universe intelligible and orderly or incomprehensible and chaotic?”

Pirsig keeps his notes on slips of paper; I keep some of mine here in my blog posts. I have learned from Collingwood to understand metaphysics as an account of convictions that may seem to be answers to such questions as Pirsig lists; but the point of the convictions is to guide us in our thinking. They are the absolute presuppositions mentioned above, and thus they are not actually answers to questions. If we can come to see them as answers, then they have ceased to be absolute. In a crisis, this may be desirable.

Clarifying these matters beyond what Collingwood does would be a worthy task. I imagine Pirsig might have helped. In Zen and the Art he identifies Quality as being, like the Tao, the origin of everything. To replace the conventional “subject-object metaphysics,” in Lila he develops a “metaphysics of quality.“ Some people take it as a guide to living. Pirsig takes it as an analysis of existence (pages 124–5):

Actually the issue before him was not whether there should be a metaphysics of Quality or not. There already is a metaphysics of quality. A subject-object metaphysics is in fact a metaphysics in which the first division of Quality—the first slice of undivided experience—is into subjects and objects. Once you have made that slice, all of human experience is supposed to fit into one of these two boxes. The trouble is, it doesn’t. What he had seen is that there is a metaphysical box that sits above these two boxes, Quality itself. And once he’d seen this he also saw a huge number of ways in which Quality can be divided. Subjects and objects are just one of the ways.

The question was, which way was best?

It’s too bad Pirsig didn’t read An Essay on Philosophical Method concerning the overlap of philosophical classes. He continues:

Different metaphysical ways of dividing up reality have, over the centuries, tended to fan out into a structure that resembles a book on chess openings …

Phaedrus had spent an enormous amount of time following what turned out to be lousy openings. A particularly large amount of this time had been spent trying to lay down a first line of division between the classic and romantic aspects of the universe he’d emphasized in his first book. In that book his purpose had been to show how Quality could unite the two. But the fact that Quality was the best way of uniting the two was no guarantee that the reverse was true—that the classic-romantic split was the best way of dividing Quality. It wasn’t. For example, American Indian mysticism is the same platypus in a world divided primarily into classic and romantic patterns as under a subject-object division …

We can, if we wish, divide up what we know. We don’t know everything; and as Pirsig has already observed, we are not going to know everything if we maintain a certain attitude. Quoted at the beginning of this article, Pirsig’s recollections of the words of his departmental colleague Dusenberry continue (pages 35–6, ellipses in source):

There’s this pseudo-science myth that when you’re “objective” you just disappear from the face of the earth and see everything undistorted, as it really is, like God from heaven. But that’s rubbish. When a person’s objective his attitude is remote. He gets a sort of stony, distant look on his face.

The Indians see that. They see it better than we do. And when they see it they don’t like it. They don’t know where in hell these “objective” anthros are at and it makes them suspicious, so they clam up and don’t say anything …

Or they’ll just tell them nonsense … which of course a lot of the anthros believe at first because they got it “objectively” … and the Indians sometimes laugh at them behind their backs.

I’ll finish with an anthropological remark of Collingwood in An Autobiography (pages 32–3):

People will speak of a savage as “confronted by the eternal problem of obtaining food”. But what really confronts him is the problem, quite transitory like all things human, of spearing this fish, or digging up this root, or finding blackberries in this wood.

People might as well say that mathematicians are confronted with the eternal problem of proving theorems, when what confronts us is how to axiomatize (if it exists) the model companion of the theory of fields with m commuting derivations, or how to derive, in an affine plane, Desargues’s Theorem from Pappus’s Theorem in their “parallel” versions.

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  1. By On the Idea of History « Polytropy on October 20, 2019 at 6:48 am

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    […] “Anthropology of Mathematics.” […]

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