I write this now while many are suffering. Unfortunately that is always true.
What I am supposed to be focused on is virtue in the use of money. I shall get to this.
I write this now while many are suffering. Unfortunately that is always true.
What I am supposed to be focused on is virtue in the use of money. I shall get to this.
Here is an annotated transcription of a 1981 manuscript by Charles Greenleaf Bell (1916–2010) called “The Axiomatic Drama of Classical Physics.” A theme is what Heraclitus observed, as in fragment B49a of Diels, LXXXI of Bywater, and D65a of Laks and Most:
We step and we do not step into the same rivers, we are and we are not.
ποταμοῖς τοῖς αὐτοῖς ἐμβαίνομέν τε καὶ οὐκ ἐμβαίνομεν, εἶμέν τε καὶ οὐκ εἶμεν.
Bell reviews the mathematics, and the thought behind it, of
In a postlude called “The Uses of Paradox,” Bell notes:
Forty-five years ago I decided that when reason drives a sheer impasse into an activity which in fact goes on, we have to think of the polar cleavage as both real and unreal.
I like that reference to “an activity which in fact goes on.” In youth it may be hard to recognize that there are activities that do go on. We do things then, but that they will get anywhere may be no more than a dream. In any case, Bell himself goes on:
… that is a job as huge and demanding as Aristotle’s, and for me at 70, just begun.
“Look,” my friends say, “Bell’s been doing the same thing since he was 25. About that time he had a vision of Paradox as paradise, and he’s been stuck there ever since.”
To this post, I am adding this introduction in July 2021. I have returned to some of the ideas of the post, and I see that I left them in a jumble. They may still be that, but I am trying to straighten up a bit.
Beyond this introduction, the post has three parts. Part III takes up more than half of the whole post and consists of my notes on
In Anderson’s article I see – as I note below – ideas that are familiar, thanks to my previous reading of philosophers such as Robin George Collingwood, Mary Midgley, and Robert Pirsig. Henry David Thoreau may not exactly be one of those philosophers, but he is somehow why I came to write this post in the first place.
Here is a table of contents for the whole post:
This past spring (of 2020), when my university in Istanbul was closed (like all others in Turkey) against the spread of the novel coronavirus, I created for my students an exercise, to serve at least as a distraction for those who could find distraction in learning.
Note added, April 17, 2023: An account of the mathematics involved in the exercise would ultimately be published as: Pierce, D. (2021). “Conics in Place.” Annales Universitatis Paedagogicae Cracoviensis | Studia Ad Didacticam Mathematicae Pertinentia, 13, 127–150.
Causation seems commonly to be understood as a physical concept, like being a fossil. The paleontologist seeks the one right answer to the question of when a particular dinosaur bone became part of the fossil record; likewise readers of international news seem to think there is one right answer to the question of whether Donald Trump or Ali Khamenei caused the shooting down of Ukraine International Airlines Flight 752 on January 8, 2020.
There is not one right answer. If you are Trump, you caused 176 civilian deaths by attacking the Iranians and provoking their response. If you are Mitch McConnell, you caused the deaths by inhibiting the removal of Trump from office. If you are Khamenei, you did it by meeting Trump’s fire with fire.
Being a cause does not mean you deserve condemnation or praise: that is another matter.
After Descartes gave geometry the power of algebra in 1637, a purely geometrical theorem of Apollonius that is both useful and beautiful was forgotten. This is what I conclude from looking at texts from the seventeenth century on.
This is an appreciation of Gödel’s Incompleteness Theorem of 1931. I am provoked by a depreciation of the theorem.
I shall review the mathematics of the theorem, first in outline, later in more detail. The mathematics is difficult. I have trouble reproducing it at will and even just confirming what I have already written about it below (for I am adding these words a year after the original publication of this essay).
The difficulty of Gödel’s mathematics is part of the point of this essay. A person who thinks Gödel’s Theorem is unsurprising is probably a person who does not understand it.
In the “Gödel for Dummies” version of the Theorem, there are mathematical sentences that are both true and unprovable. This requires two points of clarification.
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.
Continue readingLast week I wrote about the Turkish Impressionist Feyhaman Duran, born in 1886. Now my subject is the Hungarian-French Op Artist born twenty years later as Győző Vásárhelyi. His “Rétrospective en Turquie” is at the Tophane-i Amire Culture and Art Center in an Ottoman cannon foundry.