## Tag Archives: mathematics

### The Peace of Liberal Education

The wall of Dolmabahçe Sarayı, January 11, 2015

The occasion of this article is my discovery of a published Turkish translation of Collingwood’s Speculum Mentis or The Map of Knowledge (Oxford, 1924). Published as Speculum Mentis ya da Bilginin Haritası (Ankara: Doğu Batı, 2014), the translation is by Kubilay Aysevenler and Zerrin Eren. Near the end of the book, Collingwood writes the following paragraph about education, or what I would call more precisely liberal education. The main purpose of this article then is to offer the paragraph to any reader who happens to stop by.

### Bosphorus Sky

This is about the morning of Thursday, December 18, 2014, a morning I spent by the Bosphorus, thinking mostly about poetry, and photographing the sky.

### Şirince 2014

This is about our second visit to the Nesin Mathematical Village in Şirince this year. The first visit was to attend the Summer School Around Valuation Theory, May 22–26. Now we have come back to teach, as usual, in the Turkish Mathematical Society Undergraduate and Graduate Summer School. This time we are teaching not just one week, but two: July 14–27. My own course, as several times in the past, is on nonstandard analysis. Each course meets every day but Thursday, two hours a day.

The Math Village only increases in beauty every year, as I mean to suggest by posting a few photographs below.

I shall also state an opinion. The summer school here in the Village used to receive some funding from TÜBİTAK (the Scientific and Technological Research Council of Turkey); but apparently this funding is no longer forthcoming. Nonetheless, the Nesin Mathematical Village is the kind of venture that governments at their best will support. Certain Libertarians, desiring minimal government, still want government to maintain property rights, so that citizens can make money. Other people look to government to create jobs more directly. But money by itself is worthless, and some jobs are more worth doing than others. I do not say that the Nesin Mathematical Village should be supported for the technological gains that mathematics can make possible. I say that participating in the activities of the Village is itself a gain. What are you going to do when your basic animal needs are satisfied? You could do a lot worse than spend time on mathematics.

### NL V: “The Ambiguity of Feeling”

Index to this series

Feeling differs from thought. Thought is founded in feeling; thought is erected on feeling; thought needs feeling. Thought needs feelings that are strong enough to support it. But thought itself is not strong (or weak); it has (or can have) other properties, like precision and definiteness. Thought can be remembered and shared in a way that feeling cannot.

The New Leviathan is a work of thought. It might be said that a work of thought cannot properly explain feeling. Collingwood more or less says this in Chapter V, even in its very title: “The Ambiguity of Feeling.” Continue reading

### NL III: “Body As Mind”

Index to this series

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

### NL II: “The Relation Between Body and Mind”

Index to this series

I continue making notes on The New Leviathan (1942) of R. G. Collingwood (1889–1943). Now my main concern is with the second chapter, “The Relation Between Body and Mind”; but again I shall range widely.

### Preliminaries

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:

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.

Thus Robert Pirsig, Zen and the Art of Motorcycle Maintenance, ch. 17. Does anybody strictly follow the textbook method? Continue reading

### Learning mathematics

This is mostly reminiscences about high school. I also give some opinions about how mathematics ought to be learned. This article originally formed one piece with my last article, “Limits”.

I learned calculus, and the epsilon-delta definition of limit, in Washington D.C., in the last two years of high 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. Continue reading

### Limits

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

### Self-similarity

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

### The von Neumann natural numbers: a fractal-like image

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 $n+1=\{0,1,\dots,n\}$. Continue reading