Tag Archives: Stephen Toulmin

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 (as in a particular definition of physicalism) use “iff” needlessly.

  6. A pair of mathematicians who use “iff” needlessly 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.”

Continue reading

Re-enactment

Executive summary (added October 6, 2018). Historian Niall Ferguson praises Collingwood as a philosopher of history, while showing no sign of understanding Collingwood’s actual philosophy. This provokes me. My comments are in the following sections.

Presupposition

By Collingwood’s account, there is a science of our absolute presuppositions, be these in natural science or in politics. The science of absolute presuppositions is metaphysics, and it is an historical science, because absolute presuppositions do change with time.

Thinking

The historian’s job is to know the thoughts of the past. Leo Strauss disagrees with Collingwood over how one goes about this; but he would seem to agree with Collingwood that what is to be known is thought, as distinct from feeling.

Failures

I gather here some examples (in addition to Niall Ferguson’s) of what I think are failures to understand Collingwood (this gathering is on-going).

History

Ferguson reviews a book in which twenty historians try to recover the feeling of certain historical events. By saying that for Collingwood, “the historian was limited to reconstructing recorded thoughts,” Ferguson errs in two ways.

  1. Those thoughts need not be “recorded,” but anything can be used as evidence for a thought, if one knows how to use it.

  2. Feelings from the past can come down to us, only if they have been converted to thoughts.

See also “The Ambiguity of Feeling.”

Experiment

To know whether “an individual act altered the course of history,” Ferguson does recognize that we need to know more than past feelings. For him, “We need to imagine what would’ve happened if the act in question had not happened.” However, we cannot say where any particular thought is going to go, until we see where it does go, by thinking it. In this sense, every thought alters the course of history. Neither then can we say where a thought would have gone. In this way, history is different from natural science.


Presupposition

A theme of my last two articles here (namely “What It Takes” and, before that, “Effectiveness”) is the value of metaphysics, as being concerned with such problems as the following:

  • Physics has not been able to reconcile its theories of the infinitesimally small and the infinitely large.

  • America has not been able, in the words of Martin Luther King, to live out the true meaning of its creed, that all of us are created equal.

In a technical sense, these problems may not belong to natural science or political science as such. Considered as diseases, whether of the body politic or of the “body scientific,” the problems may not be curable, either by the body’s own immune system, or by remedies from outside. What is needed may be something resembling psychoanalysis, so to speak, or what Collingwood actually calls metaphysical analysis. This is an examination of absolute presuppositions, or the fundamental assumptions that have heretofore been left unquestioned. The analyst—the metaphysician—may suspect what those assumptions are; but the patient must confirm the suspicion, or else discover the assumptions independently. In any case, the patient will not be cured without agreeing that there is a disease.

Cures do happen, because absolute presuppositions change. Continue reading