Category Archives: Logic

Mathematics and Logic

Large parts of this post are taken up with two subjects:

  1. The notion (due to Collingwood) of criteriological sciences, logic being one of them.

  2. Gödel’s theorems of completeness and incompleteness, as examples of results in the science of logic.

Like the most recent in the current spate of mathematics posts, the present one has arisen from material originally drafted for the first post in this series.

In that post, I defined mathematics as the science whose findings are proved by deduction. This definition does not say what mathematics is about. We can say however what logic is about: it is about mathematics quâ deduction, and more generally about reasoning as such. This makes logic a criteriological science, because logic seeks, examines, clarifies and limits the criteria whereby we can make deductions. As examples of this activity, Gödel’s theorems are, in a crude sense to be refined below, that

  • everything true in all possible mathematical worlds can be deduced;

  • some things true in the world of numbers can never be deduced;

  • the latter theorem is one of those things.

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Why It Works

The last post, “Knottedness,” constructed Alexander’s Horned Sphere and proved, or sketched the proof, that

  • the horned sphere itself is topologically a sphere, and in particular is simply connected, meaning

    • it’s path-connected: there’s a path from every point to every other point;

    • loops contract to points—are null-homotopic;

  • the space outside of the horned sphere is not simply connected.

This is paradoxical. You would think that if any loop sitting on the horned sphere can be drawn to a point, and any loop outside the horned sphere can be made to sit on the sphere and then drawn to a point, then we ought to be able to get the loop really close to the horned sphere, and let it contract it to a point, just the way it could, if it were actually on the horned sphere.

You would think that, but you would be wrong. 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

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When is a help a hindrance? The Muses have provoked this question. They did this through their agents, the cicadas, who sang around the European Cultural Center of Delphi, during the 11th Panhellenic Logic Symposium, July 12–5, 2017.

     Cicada, European Cultural Center of Delphi, 2017.07.15     

Cicada, European Cultural Center of Delphi, 2017.07.15

My question has two particular instances.

  1. At a mathematical conference, can theorems “speak for themselves,” or should their presenters be at pains to help the listener appreciate the results?

  2. When the conference is in Greece, even at one of the country’s greatest archeological sites, does this enhance the reading of ancient Greek texts, or is it only a distraction?

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NL VII: “Appetite”

Index to this series

How can we compare two states of mind? This is the question of Chapter VII of The New Leviathan. The answer is contained in the chapter’s title. “Appetite” is a name, both for the chapter and for the fundamental instance of comparing a here-and-now feeling with a “there-and-then” feeling. We compare these two feelings because we are unsatisfied with the former, but prefer the latter.

It would seem then that appetite is at the root of memory. Thus we are among the ideas of the opening verses of The Waste Land of T. S. Eliot, who attended Collingwood’s lectures on Aristotle’s De Anima at Oxford (and was just a year older):

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NL V: “The Ambiguity of Feeling”

Index to this series

Feeling differs from thought. Thought is founded in feeling; thought is erected on feeling; thought needs feeling. Thought needs feelings that are strong enough to support it. But thought itself is not strong (or weak); it has (or can have) other properties, like precision and definiteness. Thought can be remembered and shared in a way that feeling cannot.

The New Leviathan is a work of thought. One might say that a work of thought cannot properly explain feeling. Collingwood himself says this, more or less, in Chapter V, even in its very title: “The Ambiguity of Feeling.” Continue reading

NL II: “The Relation Between Body and Mind”

Index to this series

I continue making notes on The New Leviathan of R. G. Collingwood (1889–1943). Now my main concern is with the second chapter, “The Relation Between Body and Mind”; but I shall range widely, as I did for the first chapter.


Some writers begin with an outline, which they proceed to fill out with words. At least, they do this if they do what they are taught in school, according to Robert Pirsig:

He showed how the aspect of Quality called unity, the hanging-togetherness of a story, could be improved with a technique called an outline. The authority of an argument could be jacked up with a technique called footnotes, which gives authoritative reference. Outlines and footnotes are standard things taught in all freshman composition classes, but now as devices for improving Quality they had a purpose.

That is from Zen and the Art of Motorcycle Maintenance, chapter 17.

Does anybody strictly follow the textbook method of writing? Continue reading

NL I: “Body and Mind”

Index to this series. See also a later, shorter article on this chapter

The Chapter in Isolation

“Body and Mind” is the opening chapter of Collingwood’s New Leviathan. The chapter is a fine work of rhetoric that could stand on its own, though it invites further reading. In these respects it resembles the first of the ten traditional books of Plato’s Republic, or even the first of the thirteen books of Euclid’s Elements. The analogy with Euclid becomes a bit tighter when we consider that each chapter of The New Leviathan is divided into short paragraphs, which are numbered sequentially for ease of reference.

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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.


Preface (January 17–18, 2019). This essay is built around two extended quotations from Collingwood.

  1. One is from the posthumous Idea of History (1946) with the core idea, “people do not know what they are doing until they have done it.”
  2. The other is from An Essay on Philosophical Method (1933), about how logic is neither a purely descriptive nor a purely normative science.

The quotations pertain to the title subject of psychology for the following reasons.

  1. Psychological experiments show that we may not know what we are doing until we have done it.
  2. Psychology is a descriptive science.

Psychological experiments can tell us about what we do, only when we presuppose the general applicability of their findings. This is true for any descriptive science. Philosophy demands more. A philosophical science like logic is categorical, in the sense of the second listed quotation, because it is what Collingwood will later call criteriological. I go on to discuss criteriological sciences as such in “A New Kind of Science,” but not here.

Here I suggest examples of not knowing where one’s life is going. A simpler example would be making art. By the account of The Principles of Art (1938), this is something we do all the time, as for example when we utter a new sentence. We do not know what the sentence is going to be, until it is said. On the other hand, we do somehow guide its utterance. See the quotation about painting at the end of “Freedom.

Collingwood discusses categorical thinking for the sake of explaining the Ontological Proof, which I go on to analyze myself in later articles. Meanwhile, the present essay ends with a look at Graham Priest’s dismissive treatment of the Proof.

The original purpose of this article is to record a passage in The Idea of History of R.G. Collingwood (1889–1943). I bought and read this book in 2001. I was looking back at it recently, because I was reading Herodotus, and I wanted to see again what Collingwood had to say about him and other ancient historians.

The passage that I want to talk about reminded me of some psychological experiments whose conclusions can be overblown. Writing before those experiments, Collingwood shows that the similar conclusions can be drawn, in more useful form, without the pretence of a scientific experiment.

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