I learned about Peter Turchin recently through his profile in the Atlantic by Graeme Wood. I had learned about the Atlantic article from historians on Twitter such as James Ryan, who does “Turkish history and other stuff,” according to his own Twitter profile, and who tweeted in response to Wood’s article,
This is really interesting research, but, uh, it is only history in the way that a particle physicist does history.
In response to that, a thread began:
Etymologically speaking, the asıl of a thing is its root. The Arabic root of the Turkish word means bitki kökü, “vegetable root,” according to Sevan Nişanyan’s Turkish etymological dictionary.
In the Iliad, why is Achilles so affronted by Agamemnon as to refuse to help the Greeks, even as their attack on Troy is becoming a defensive war, at the wall that they have erected about their own ships? If the answer is to be found through study, then Book IX of the Iliad is what to study.
Pacifism is properly pacificism, the making of peace: not a belief or an attitude, but a practice. Mathematics then is pacifist, because learning it means learning that you cannot fight your way to the truth. Might does not make right. If others are going to agree with you, they will have to do it freely. Moreover, you cannot rest until they do agree with you, if you’ve got a piece of mathematics that you think is right; for you could be wrong, if others don’t agree.
Being hosted by WordPress.com, and thus using the WordPress.org content management system, this blog has posts, pages, and media. You are now reading a post, as you can tell from its having an initial publication date in its address (as well as somewhere at the bottom). The contents of any post may still change. I mean to use this one as a directory for my pages and verbal media (namely pdf files).
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.
Large parts of this post are taken up with two subjects:
The notion (due to Collingwood) of criteriological sciences, logic being one of them.
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.
I continue with the recent posts about mathematics, which so far have been as follows.
“What Mathematics Is”: As distinct from the natural sciences, mathematics is the science whose findings are proved by deduction. I say this myself, and I find it at least implicit in an address by Euphemia Lofton Haynes.
“More of What It Is”: Some mathematicians do not distinguish mathematics from physics.
“Knottedness”: Topologically speaking, there is a sphere whose outside is not that of a sphere. The example is Alexander’s Horned Sphere, but it cannot actually be physically constructed.
“Why It Works”: Why there can be such a thing as the horned sphere.
When I first drafted the first post above, I said a lot more than I eventually posted. I saved it for later, and later is starting to come now.
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
If you roll out a lump of clay into a snake, then tie a string loosely around it, can you contort the ends of the snake, without actually pressing them together, so that you cannot get the string off?
You can stretch the clay into a Medusa’s head of snakes, and tangle them as you like, again without letting them touch. If you are allowed to rest the string on the surface of the clay, then you can get it off: you just slide it around and over what was an end of the original snake.
