My whiteboards from Tuesday, November 3, 2015,
concerning Pappus of Alexandria,
in the course “Geometriler”
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.
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
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.)
I am writing from the Math Village, and here I happen to have read that Abraham Lincoln kept no known diary as such, but noted his thoughts on loose slips of paper. Admired because he “could simply sit down and write another of his eloquent public letters,”
Lincoln demurred. “I had it nearly all in there,” he said, pointing to an open desk drawer. “It was in disconnected thoughts, which I had jotted down from time to time on separate scraps of paper.” This was how he worked, the president explained. It was on such scraps of paper, accumulating over the years into a diaristic density, that Lincoln saved and assembled what he described to the visitor as his “best thoughts on the subject.”
Thus Ronald C. White, “Notes to Self,” Harper’s, February 2018. My own notes to self are normally in bound notebooks, and perhaps later in blog articles such as the present one, which is inspired by the 1960 article called “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” by Eugene Wigner.
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:
- Babies are illogical.
- Nobody is despised who can manage a crocodile.
- 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
In a 2012 post in this blog, I criticized a 2009 essay called “50 Years of Stupid Grammar Advice.” The putative advice was that of Strunk and White. However, what these two had written was not the elements of grammar, but The Elements of Style. They gave style advice by precept and example. The advice is good, if well understood. The critic should recognize that, as I wrote, “Rules of style are supposed to induce thinking, not obedience.”
All photos taken in Delphi, July 12–14, 2017
My two recent courses at the Nesin Mathematics Village had a common theme. I want to describe the theme here, as simply as I can—I mean, by using as little technical knowledge of mathematics as I can. But I shall talk also about related poetry and philosophy, of T. S. Eliot and R. G. Collingwood respectively.
With my presumptuous title, I imitate Arthur Danto’s What Art Is (2013), mentioned in my last post, “Some Say Poetry.” The book is fine, and I have learned from it; but Danto could have learned from Collingwood’s Principles of Art.
In a poetry review, a remark on being a student has drawn my attention:
In My Poets, a work of autobiographical criticism with occasional ventriloquial interludes, McLane recalls two “early impasses in reading,” freshman-year encounters with Charles Olson and Frank O’Hara. She writes about not “getting it” but wanting to get it, about a desire to get it that was left wanting by code-breaking and analysis and satisfied by hearing and feeling.
This is from the second half of a “New Books” column by Christine Smallwood, in the Reviews section of Harper’s, July 2017. After quoting Smallwood’s review, I want to say something about learning and creating, in poetry and also in mathematics.
Kuzguncuk, 2017.11.05
Executive summary (below) | Index to this series
After a break of half a year, I return to reading Collingwood’s New Leviathan. Being on holiday at an Aegean beach gives me the opportunity. While here, I may also return to Chapman’s Homer’s Iliad. Last winter I finished Part I of the New Leviathan, the part called “Man.” Here I continue with the first chapter of “Society.” I have reason to look at what Mary Midgley and Albert Einstein say about science. Collingwood’s investigation suggests a way of thinking about prejudice and discrimination.
Part II of the New Leviathan is “Society,” and the first two chapters of this, XIX and XX, concern the distinction between society proper and two more general notions. In Chapter XX, the more general notion will be community. In Chapter XIX, the more general notion has not got its own proper name, and so Collingwood denotes it by writing “society,” in quotation marks.
A “society” of chairs at the beach (Altınova 2017.08.31)
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
My question has two particular instances.
At a mathematical conference, can theorems “speak for themselves,” or should their presenters be at pains to help the listener appreciate the results?
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?
This is a foray into the mystery of how things happen, based the 164th of the 361 chapters of War and Peace. This chapter contains, in a one-sentence paragraph, a summary of Tolstoy’s theory of history:
Each man lives for himself, using his freedom to attain his personal aims, and feels with his whole being that he can now do or abstain from doing this or that action; but as soon as he has done it, that action performed at a certain moment in time becomes irrevocable and belongs to history, in which it has not a free but a predestined significance.
Polytropy 