Tag Archives: Wittgenstein

Articles on Collingwood

This article gathers, and in some cases quotes and examines, popular articles about R. G. Collingwood (1889–1943).

  • By articles, I mean not blog posts like mine and others’, but essays by professionals in publications that have editors.

  • By popular, I mean written not for other professionals, but for the laity.

Continue reading

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.

Continue reading



First posted May 17, 2018, this essay concerns Eugene Wigner’s 1960 article “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” I wrote a lot, which I now propose to summarize by section. (The meditations also continue in the next article.)

  • Some things are miraculous. Among Wigner’s examples are

    • that mathematics is possible at all, and

    • that “regularities” in the physical world can be discovered, as by Galileo and Newton.

    For Wigner, we should be grateful for the undeserved gift of a mathematial formulation of the laws of physics. This makes no sense theologically—and here I agree with the character Larry Darrell in Somerset Maugham’s novel The Razor’s Edge. Wigner’s idea that our mathematical reasoning power has been brought to perfection makes no sense to me either.

  • Everything is miraculous. Here I agree with Collingwood in Religion and Philosophy. A miracle cannot be the breaking of a natural law, since such a thing cannot be broken. A great artist like Beethoven follows no rules in the first place, or makes them up as he goes along; and he is like God in this way.

  • Natural law. That it cannot be broken is part of the very concept of natural law. Quantum phenomena and the theory of relativity have not in fact been brought under a single law; for Wigner, it may not be possible.

  • Mystery. Not only can we not define miracles, but (as we should have observed in the first place) we cannot even say when they happen. If like Wigner we call something miraculous, this means it cleanses our own doors of perception, in the sense of William Blake.

  • Definitions. In his treatment of miracle in Religion and Philosophy, Collingwood shows the futility of trying to define a term when you are not sure how to use it. He makes this futility explicit in The Principles of Art. If we are going to think about the use of mathematics in natural science, this means we ought to be mathematician, natural scientist, and philosopher; and not just “natural scientist,” but physicist and biologist, since if mathematics is effective in physics, it would seem to be ineffective in biology.

  • Being a philosopher. We are all philosophers, in the sense that Maugham describes in the story “Appearance and Reality,” if only we think. All thought is for the sake of action. This does not mean that thought occurs separately from an action and is to be judged by the action. We may value “pure” thought, such as doing mathematics or making music or living the contemplative life of a monk. This however moves me to a give a thought to the disaster of contemporary politics.

  • Philosophizing about science. For present purposes, compart­ment­al­ization of knowledge is a problem. So is the dominance of analytic philosophy, for suggesting (as one cited person seems to think) that big problems can be broken into little ones and solved independently. In mathematics, students should learn their right to question somebody else’s solutions to problems. In philosophy, the problems themselves will be our own. Philosophy as such cannot decide what the problems of physics or biology are, though it may help to understand the “absolute presuppositions” that underlie the problems. Philosophers quâ metaphysicians cannot determine once for all what the general structure of the universe is. This does not mean they should do “experimental philosophy,” taking opinion polls about supposedly philosophical questions. What matters is not what people say, but what they mean and are trying to mean. As Collingwood observes, metaphysics is an historical science.

For more on the last points, see a more recent article, “Re-enactment.” (This Preface added June 3, 2018.)

Continue reading

Boolean Arithmetic

Mathematics can be highly abstract, even when it remains applicable to daily life. I want to show this with the mathematics behind logic puzzles, such as how to derive a conclusion using all of the following premisses:

  1. Babies are illogical.
  2. Nobody is despised who can manage a crocodile.
  3. Illogical persons are despised.

The example, from Terence Tao’s blog, is attributed to Lewis Carroll. By the first and third premisses, babies are despised; by the second premiss then, babies cannot manage crocodiles.

George Boole, The Laws of Thought (1854), Open Court, 1940

Continue reading

Nature and Death

Thoughts on mortality and the evolution of the universe, occasioned by a funeral and by Collingwood’s Idea of Nature and Plato’s Phaedo

Cebeci, Ankara, 2016.05.17

When the husband of my second-grade teacher died, I wanted to pay my respects. My father took me to the funeral home, where I hid behind him as he greeted the family of the deceased. My teacher was not among them. When invited to view the body, I looked over and saw it, lying off to the side in an open casket. I had never seen the man when he was alive. I declined the opportunity to gaze at his lifeless form. Until I came to Turkey, this was my closest approach to the materiality of death—except for a visit to the medical school of the University of New Mexico in Albuquerque. There, as part of the laboratory program at St John’s College in Santa Fe, students viewed dissected human cadavers.

Continue reading

NL XIV: “Reason”

Index to this series

Summary added January 29, 2019, revised May 8, 2019. Practical reason is the support of one intention by another; theoretical, one proposition by another. Reasoning is thus always “motivated reasoning”: we engage in it to relieve the distress of uncertainty. Reason is primarily practical, only secondarily theoretical; and the reason for saying this is the persistence of anthropomorphism in theoretical reasoning: by the Law of Primitive Survivals in Chapter IX, we tend to think even of inanimate objects as forming intentions the way we do.

Reasons for adding this summary of Chapter XIV of Collingwood’s New Leviathan include

  • the tortuousness of the following post on the chapter,
  • the provocation of a Guardian column by Oliver Burkeman on motivated reasoning.

Says Burkeman, whose “problem” is apparently motivated reasoning itself,

One of the sneakier forms of the problem, highlighted in a recent essay by the American ethicist Jennifer Zamzow, is “solution aversion”: people judge the seriousness of a social problem, it’s been found, partly based on how appetising or displeasing they find the proposed solution. Obviously, that’s illogical …

On the contrary, how we reason cannot be “illogical,” any more than how we speak can be “ungrammatical.” Logic is an account, or an analysis, of how we do actually reason; grammar, of how we speak. Of course we may make errors, by our own standards.

Rogier van der Weyden (Netherlandish, 1399/1400-1464), Portrait of a Lady, c. 1460, oil on panel, Andrew W. Mellon Collection
Rogier van der Weyden (Netherlandish, 1399/1400–1464),
Portrait of a Lady, c. 1460, oil on panel
National Gallery of Art, Washington; Andrew W. Mellon Collection


There was a rumor that Collingwood had become a communist. According to David Boucher, editor of the revised (1992) edition of The New Leviathan, the rumor was one of the “many reasons why [that book] failed to attract the acclaim which had been afforded Collingwood’s other major works.” Continue reading

Thales of Miletus

This is about Thales of Miletus and what it means to study him. I am moved to ask what history is in the first place. It is a study of the freedom in which we face our conditions. Thales had his way of understanding the world, and we may benefit from trying to learn it.

“The Thaleses of the future are meeting in Didim, September 24, 2016”

“The Thaleses of the future are meeting in Didim,
September 24, 2016”

Continue reading

The Tradition of Western Philosophy

Note added October 16, 2018: Here I compare two projects of re-examining the philosophical tradition named in my title. The projects are those of

  • R. G. Collingwood in An Essay on Philosophical Method (Oxford, 1933);

  • Stringfellow Barr and Scott Buchanan at St John’s College in Annapolis, Maryland, beginning in 1937.

I review

  • how I ended up as a student at St John’s;

  • how Collingwood has been read (or not read) by myself and others, notably Simon Blackburn;

  • how Collingwood’s Essay is based on the hypothesis of the “overlap of classes.”

I say that Collingwood writes well. This is corroborated, in a sense, in the Introduction to the 2005 edition of the Essay by James Connelly and Giuseppina D’Oro. These editors say of two of Collingwood’s critics (namely M. C. D’Arcy and C. J. Ducasse),

both agreed that Collingwood’s language was imprecise, sometimes vague, and insufficiently analytical. This criticism was later echoed by A. J. Ayer in his Philosophy in the Twentieth Century where he remarked that ‘An Essay on Philosophical Method is a contribution to belles-lettres rather than philosophy. The style is uniformly elegant, the matter mostly obscure.’

At the end of the present post, I quote three elegant paragraphs from Collingwood, which begin:

Assumption for assumption, which are we to prefer? That in sixty generations of continuous thought philosophers have been exerting themselves wholly in vain, and have waited for the first word of good sense until we came on the scene? Or that this labour has been on the whole profitable, and its history the history of an effort neither contemptible nor unrewarded?

Collingwood and perhaps many others and I prefer the second assumption; and in this we may seem to follow Daniel McCarthy in “Modernism & Conservatism” (The American Conservative, September 25, 2012), an essay recently promoted on Twitter (which is why I return now to this post). The freedom embraced by modernism may drive one to conservatism, as apparently it did T. S. Eliot. McCarthy quotes Donald Livingston:

The true philosopher recognizes that philosophical reflection consistently purged of the authority of the pre-reflective leads to total skepticism. In this moment of despair, hubristic reason … becomes impotent and utterly silent. It is only then that the philosopher can recognize, for the first time, the authority of that radiant world of pre-reflective common life in which he has his being and which had always been a guide prior to the philosophic act.

McCarthy comments on this,

Once reason has disestablished everything, including its own authority, what remains? The ground beneath your feet, the social order of which you are a part—things predicated not on any theory but on their immediacy. This is the profound conservatism to be realized from modernism.

One may find this conservatism in some students and faculty at St John’s College; I think it is not inevitable, and Collingwood hasn’t got it, for all his admiration for Eliot.

A recent theme of this blog has been juxtapositions, especially of paintings, as in the articles “Pairing of paintings” and “More pairings” (both from July, 2013).

In this article I juxtapose two texts, from the 1930s. Both of them decry current intellectual troubles. Both find a solution in a return to the intellectual tradition.

  1. One of the texts is American: the “Bulletin of St. John’s College in Annapolis 1937–38”. This is currently was available in a pdf image of a 2004 reprint. (The link died, but the Internet Archive saved the content, most recently from June 22, 2013.)

  2. The other text is British, from 1933: R. G. Collingwood’s Essay on Philosophical Method, especially its final section, which I quote at the end of this article.

St John’s College is my alma mater, and I use that term seriously, for its meaning of “foster mother.” R. G. Collingwood is my favorite philosopher. The College lately has difficulty attracting enough students. Collingwood does not attract many readers. I think both of these situations are unfortunate. I cannot propose that bringing together Collingwood and the College will help either one to become more popular. But it may help interested persons to understand what St John’s College is all about. Not that the College is about one thing, and Collingwood is about the same thing. By one account that I have heard, the College was changed in the 1950s, and not for the better, under the leadership of the author of Greek Mathematical Thought and the Origin of Algebra.

The 1937 “Bulletin” of the College announced the New Program. Because of this Program, I chose to attend the College in 1983. Looking back from thirty years later, I would articulate my reason for attending the College as follows. I was living in a tradition, whether I liked it or not, and I wanted to know what it meant. I wanted to know what it really was. I thought the tradition needed questioning: here I was fired up by Robert Pirsig’s philosophical travel book, Zen and the Art of Motorcycle Main­tenance. (I recently wrote about this book and others in another blog article, “Books hung out with.”)

Before the Pirsig influence though, in my tenth-grade geometry class, I was dissatisfied with our textbook. I wished we would just read Euclid. This is what I ended up doing at St. John’s, although by that time I knew a lot more mathematics, perhaps too much. My article “Learning Mathematics” concerns my last two years of high school. Before that, in tenth-grade geometry, I don’t think I had a clear reason for disliking our textbook; I was mainly offended by the condescending tone, the sense that the text was written for children. I did find Euclid in the library, and I read some of him. Having spent a lot more time with him now, I have clearer reasons for disliking the tenth-grade geometry text; but that is a matter for another article, not yet written. (The text was Weeks and Adkins, A Course in Geometry: Plane and Solid, Ginn and Company, Lexington MA, 1970; I refer to it in “On Commensurability and Symmetry,Journal of Humanistic Mathematics, Volume 7, Issue 2, July 2017, pages 90–148.)

I never heard of Collingwood at St. John’s. This is not because he was too young. My senior language tutorial at the College spent some time with Collingwood’s slightly younger contemporary, Wittgenstein. We read from his Philosophical Investigations. Even in the freshman language tutorial, after reading my first essay, my tutor recommended the Philosophical Investigations, because of a similarity of style as well as general theme. Both Wittgenstein and I gave our readers little clue about where we were going.

Where I am going now is, as I said, the last section of Collingwood’s Essay on Philosophical Method. But I shall go there a bit slowly. You can skip ahead if you want, to the long block of quoted text at the end of this article.

Collingwood’s words are inspiring in their defense of the sixty generations of continuous philosophical thought that we are the beneficiaries of. At least, Collingwood’s words are inspiring, if you are already open to something like the New Program of St John’s College.

Collingwood writes well. At least I think so. It is a reason why I call him my favorite philosopher. Perhaps it is a non-philosophical reason; I am not sure. But I have twelve books on my shelf published under Collingwood’s name (some posthumously). I have read them all, in some cases several times. I did get bogged down in The Philosophy of Enchantment, which consists of manuscripts not properly edited for publication by Collingwood himself. For some reason I have not finished The Idea of Nature. This book, however, in its 1960 paperback edition, like the 1958 paperback edition of The Principles of Art, has never gone out of print. Other Collingwood books have been brought back into print with long editorial introductions, and appendices from the Collingwood archive.

There is even a recent biography of Collingwood: History Man by Fred Inglis. This is favorably reviewed by Simon Blackburn, who also favorably reviews Collingwood himself. By the way, I think Blackburn is mistaken to say that, for Collingwood, our “absolute presuppositions” are knowable by future generations, but not by ourselves. These absolute presuppositions are the proper subject of metaphysics, and as I read Collingwood’s Essay on Metaphysics, we can know our presuppositions; it is just difficult. (See “What It Takes,” May, 2018.)

“If Collingwood is as acute and interesting as I have suggested,” writes Blackburn, “how does it happen that he is largely a minority interest?” Blackburn thinks Collingwood boasts about his abilities, and this puts readers off. I don’t see it, myself. Maybe you have to be part of the British scholarly elite to see it. Collingwood wrote an autobiography, and perhaps such an endeavor needs an author who thinks highly of himself. In the autobiography, Collingwood tells how he has had to part intellectual company with all of his Oxford colleagues. It takes nerve to go out on your own; therefore Collingwood is implicitly telling us he has this nerve.

It also took nerve on the part of Stringfellow Barr and Scott Buchanan to create the New Program of St. John’s College. One of Collingwood’s complaints about his philosophical colleagues is that they do not properly read their predecessors. At best they select isolated passages in order to refute them. I don’t think this is exactly Barr and Buchanan’s issue with American education of their time. But their recommendation is the same as Collingwood’s: to go back to the sources, neither disputatiously nor worshipfully, but critically in the best sense.

In the Introduction to his Essay on Philosophical Method, Collingwood sets his work in a line that includes Socrates, Plato, Descartes, and Kant. I attempt a brief summary of Collingwood’s introductory summary of the contributions of these four.

  1. Socrates recognizes that philosophical knowledge is already in us; the proper method for bringing it out is not observing, but questioning. In this way, philosophical knowledge resembles mathematical knowledge, as the character of Socrates shows in Plato’s Meno. Here, by being questioned, a person raised as a slave in Meno’s house is led to discover that, to double a given square, he needs to construct a square on the given square’s diagonal.

  2. Mathematics and philosophy are nonetheless different. Plato understands this. Collingwood observes,

    Mathematics and dialectic are so far alike that each begins with an hypothesis: “Let so-and-so be assumed.” But in mathematics the hypothesis forms a barrier to all further thought in that direction: the rules of mathematical method do not allow us to ask “Is this assumption true? Let us see what would follow if it were not.” Hence mathematics, although intellectual, is not intellectual à outrance; it is a way of thinking, but it is also a way of refusing to think.

    The meaning of “hypothesis” here is not clear, be it according to Collingwood or Plato. I think Plato has not seen the possibility of a systematic development of mathematics such as is found in Euclid. Collingwood has seen it, but the understanding of it has changed over the generations. I do like Collingwood’s saying that mathematics is a way of refusing to think. Learning mathematics does mean learning not to think about some things. A student of mine once could not learn linear algebra properly, because he thought that no more than three spatial dimensions were possible.

    In any case, philosophy allows and indeed requires the questioning of hypotheses. This is Plato’s contribution to method: “the conception of philosophy as the one sphere in which thought moves with perfect freedom.” But this still does not distinguish philosophy from mathematics; it seems only to broaden the scope of the same kind of thinking.

  3. Whatever method Descartes uses in his own thinking, the method he tells us about is again excessively mathematical.

  4. Kant sees this, but his answer is to distinguish methodology from philosophy itself. First work out your method, and then do your philosophy with it. Such a separation is untenable.

What then is Collingwood’s contribution? Formally, his Essay on Philosophical Method is built up on the hypothesis of an overlap of classes. There is no such overlap in mathematics. A straight line is not curved. You may say that it is curved, because a straight line is a circle of infinite radius, and all circles are curved; but in this case, the difference between straight and curved has become the difference between finite and infinite.

Neither do the classes of empirical science overlap. An animal is not a plant. Collingwood acknowledges that there are borderline cases. I am not sure, but I think the biologist’s response to such cases is to improve the classification system so that such cases can be accommodated. The ideal remains the same: to divide the world of living things into classes, so that every living thing belongs unambiguously to one and only one of those classes.

Philosophy goes astray when it tries to classify the world in this way. Collingwood himself went astray in his first book, Religion and Philosophy. Recognizing there that religion, theology, and philosophy had something in common, he concluded that they were the same thing. But they are not the same, as he understands later; they are overlapping classes.

All philosophical classes overlap with others. The notion sometimes sounds absurd. Maybe it is a rhetorical trick; but it has good results. The class of what you agree with must overlap with the class of what you disagree with. If you are a philosopher, you cannot simply explain what is wrong with somebody else’s work; you have to do the same work better. This is similar to the point of a Friday-night lecture given by the Dean of St John’s College, Santa Fe campus, in the fall of 1985.

What I remember most clearly from Robert Neidorf’s lecture is that if on page Y of a book you find a sentence contradicting a sentence on page X, it doesn’t mean the book is wrong. A new student did not like this. He objected to Mr Neidorf’s criticisms of formal logic. I think the student may have been an Objectivist. He missed the irony in Mr Neidorf’s having been the author of a textbook called Deductive Forms: An Elementary Logic.

Collingwood has a lot more to say in his essay, all resting on the hypothesis of the overlap of classes. He checks his conclusions against the thoughts of the great philosophers, who are read at St John’s. (One exception is Spinoza, who was not read in my day.) This checking is the kind of hypothesis-questioning enjoined by Plato. But it seems circular. How can philosophy advance, if it needs to be confirmed by what has already been done?

Collingwood suggests that there is even a double circularity. If I understand him, he means roughly that using the tradition to confirm the philosophy erected on it is one circle; but appealing to the tradition at all is another circle, since it requires the assumption that there is a tradition. This objection is in the next-to-last section of Collingwood’s book. The last section is Collingwood’s “oblique” response; I end with this:

Assumption for assumption, which are we to prefer? That in sixty generations of continuous thought philosophers have been exerting themselves wholly in vain, and have waited for the first word of good sense until we came on the scene? Or that this labour has been on the whole profitable, and its history the history of an effort neither contemptible nor unrewarded? There is no one who does not prefer the second; and those who seem to have abandoned it in favour of the first have done so not from conceit but from disappointment. They have tried to see the history of thought as a history of achievement and progress; they have failed; and they have deserted their original assumption for another which no one, unless smarting under that experience, could contemplate without ridicule and disgust.

Yet it is surely in such a crisis as this that we should be most careful in choosing our path. The natural scientist, beginning with the assumption that nature is rational, has not allowed himself to be turned from that assumption by any of the difficulties into which it has led him; and it is because he has regarded that assumption as not only legitimate but obligatory that he has won the respect of the whole world. If the scientist is obliged to assume that nature is rational, and that any failure to make sense of it is a failure to understand it, the corresponding assumption is obligatory for the historian, and this not least when he is the historian of thought.

So far from apologizing, therefore, for assuming that there is such a thing as the tradition of philosophy, to be discovered by historical study, and that this tradition has been going on sound lines, to be appreciated by philosophical criticism, I would maintain that this is the only assumption which can be legitimately made. Let it, for the moment, be called a mere assumption; at least I think it may be claimed that on this assumption the history of philosophy, properly studied and analysed, confirms the hope which I expressed in the first chapter: that by reconsidering the problem of method and adopting some such principles as are outlined in this essay, philosophy may find an issue from its present state of perplexity, and set its feet once more on the path of progress.