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.
I continue with the mathematics posts, taking up, as I did in the last, material originally drafted for the first.
Designated for its own post, material can grow, as has the material of this post in the drafting. Large parts of it are taken up with
the notion (due to Collingwood) of criteriological sciences, logic being one of them;
Gödel’s logical theorems of completeness and incompleteness.
I have 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. This makes logic a criteriological science, since it seeks, examines, clarifies and limits the criteria whereby we can make deductions. As examples of this activity, Gödel’s theorems are 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.
I say that mathematics is the deductive science; and yet there would seem to be mathematicians who disagree. I take up two cases here.
Mathematics “has no generally accepted definition,” according to Wikipedia today. Two references are given for the assertion. I suggest that what has no generally accepted definition is the subject of mathematics: the object of study, what mathematics is about. Mathematics itself can be defined by its method. As Wikipedia currently says also,
it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions.
I would put it more simply. Mathematics is the science whose findings are proved by deduction.
This is a little about mathematics, and a little about writing for the web, but mostly about the nuts and bolts of putting mathematics on the web. I want to record how, mainly with the pandoc program, I have converted some mathematics from a LaTeX file into html. Like “Computer Recovery” then, this post is a laboratory notebook.
The mathematics is a proof of Dirichlet’s 1837 theorem on primes in arithmetic progressions. This is the theorem that, if to some number you keep adding a number that is prime to it, there will be no end to the primes that you encounter in this way.
The bulk of this post is a summary of the chapter on art in Collingwood’s Speculum Mentis: or The Map of Knowledge (1924). The motto of the book is the first clause of I Corinthians 13:12:
In the fall of 2017, I created what I propose to consider as being both art and mathematics. Call the art conceptual; the mathematics, expository; here it is, as a booklet of 88 pages, size A5, in pdf format.
More precisely, the work to be considered as both art and mathematics is the middle of the three chapters that make up the booklet. The first chapter is an essay on art, ultimately considering some examples that inspire my own. The last chapter establishes the principle whereby the lists of numbers in Chapter 2 are created.
