In Chapter I of The New Leviathan, we stipulated that natural science, the “science of body,” must be free to pursue its own aims. But we ourselves are doing science of mind, and:
1. 85. The sciences of mind, unless they preach error or confuse the issue by dishonest or involuntary obscurity, can tell us nothing but what each can verify for himself by reflecting on his own mind.
All of us can be scientists of mind, if only we are capable of reflection: Continue reading
Here is an image that I made when preparing the article Self-similarity nine months ago. The image appeared as a draft in the list of all of my articles on this blog. Here it is officially:
I continue making notes on The New Leviathan of R. G. Collingwood (1889–1943). Now my main concern is with the second chapter, “The Relation Between Body and Mind”; but I shall range widely, as I did for the first chapter.
Some writers begin with an outline, which they proceed to fill out with words. At least, they do this if they do what they are taught in school, according to Robert Pirsig:
He showed how the aspect of Quality called unity, the hanging-togetherness of a story, could be improved with a technique called an outline. The authority of an argument could be jacked up with a technique called footnotes, which gives authoritative reference. Outlines and footnotes are standard things taught in all freshman composition classes, but now as devices for improving Quality they had a purpose.
That is from Zen and the Art of Motorcycle Maintenance, chapter 17.
Does anybody strictly follow the textbook method of writing? Continue reading
Preface (January 17–18, 2019). This essay is built around two extended quotations from Collingwood:
The quotations pertain to the title subject of psychology for the following reasons.
Psychological experiments can tell us about what we do, only when we presuppose the general applicability of their findings. This is true for any descriptive science. Philosophy demands more. A philosophical science like logic is categorical, in the sense of the second listed quotation, because it is what Collingwood will later call criteriological. I go on to discuss criteriological sciences as such in “A New Kind of Science,” but not here.
The Lady and the Unicorn
(image from Wikimedia)
This is mostly reminiscences about high school. I also give some opinions about how mathematics ought to be learned. The post originally formed one piece with my last article, “Limits.”
I learned calculus, and the epsilon-delta definition of limit, in Washington D.C., in my last two years at St Albans School, in a course taught by a peculiar fellow named Donald J. Brown. The first of these two years was officially called Precalculus Honors, but some time in that year, we started in on calculus proper.
This is about limits in mathematics: both the technical notion that arises in calculus, and the barriers to comprehension that one might reach in one’s own studies. I am going to say a few technical things about the technical notion, but there is no reason why this should be a barrier to your reading: you can just skip the paragraphs that have special symbols in them.
Looking up something else in the online magazine called Slate, I noted a reprint of an article called “What It Feels Like to Be Bad at Math” from a blog called Math With Bad Drawings by Ben Orlin. Now teaching high-school mathematics, Mr Orlin recalls his difficulties in an undergraduate topology course. His memories help him understand the difficulties of his own students. When students do not study, why is this? It is because studying makes them conscious of how much they do not understand. They feel stupid, and they do not like this feeling. Continue reading
From the poster depicting a few von Neumann natural numbers, I created this animation. The moving image no longer depicts natural numbers in the sense of the poster, since there is no infinite descending chain of natural numbers. There is an infinite ascending chain of them; but the poster does not actually depict such a chain as nested circles. So running the animation in reverse would not give a correct suggestion of the original poster, even if it were of infinite size. Continue reading
See the next article, “Self-similarity,” for an animation of the image here.
I have long been fascinated by von Neumann’s definition of the natural numbers (and more generally the ordinals). In developing axioms for set theory, Zermelo used the sets 0, {0}, {{0}}, {{{0}}}, {{{{0}}}}, and so on as the natural numbers. Here 0 is the empty set. Zermelo’s method works, but is not so elegant as von Neumann’s later proposal to consider each natural number as the set of all natural numbers that are less than it is, so that (again) 0 is the empty set, but also n + 1 = {0, 1, …, n}.
