## Tag Archives: Euclid

### On Gödel’s Incompleteness Theorem

This is an appreciation of Gödel’s Incompleteness Theorem of 1931. I am provoked by a depreciation of the theorem.

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. Continue reading

### Victor Vasarely

Tophane-i Amire, 2017.03.25

Last 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.

### The geometry of numbers in Euclid

This is about how the Elements of Euclid shed light, even on the most basic mathematical activity, which is counting. I have tried to assume no more in the reader than elementary-school knowledge of how whole numbers are added and multiplied.

How come 7 ⋅ 13 = 13 ⋅ 7? We can understand the product 7 ⋅ 13 as the number of objects that can be arranged into seven rows of thirteen each.

Seven times thirteen

If we turn the rows into columns, then we end up with thirteen rows of seven each; now the number of objects is 13 ⋅ 7. Continue reading

### Equality Is Not Identity

I want to record here an account by Collingwood of Aristotle’s theory of knowledge. The passages quoted below are relevant, both to something I have learned from reading Euclid with students, and to the considerations of consciousness that led to my recent article “Body and Mind.” Continue reading

### 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.

### 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 I: “Body and Mind”

Index to this series

“Body and Mind” is the opening chapter of Collingwood’s New Leviathan (1942). The chapter is a fine work of rhetoric that could stand on its own, though it invites further reading. In these respects it resembles the first of the ten traditional books of Plato’s Republic, or even the first of the thirteen books of Euclid’s Elements. The analogy with Euclid becomes a bit tighter when we consider that each chapter of The New Leviathan is divided into short paragraphs, which are numbered sequentially for ease of reference.

### The Tradition of Western Philosophy

Note added October 16, 2018: Here I compare two projects of re-examining the philosophical tradition of the title. The projects are those of

• R. G. Collingwood in An Essay on Philosophical Method (Oxford, 1933);
• Stringfellow Barr and Scott Buchanan at St John’s College in Annapolis, Maryland, beginning in 1937.

I review

• how I ended up as a student at St John’s;
• how Collingwood has been read (or not read) by myself and others, notably Simon Blackburn;
• how Collingwood’s Essay is based on the hypothesis of the “overlap of classes.”

I say that Collingwood writes well. This is corroborated, in a sense, in the Introduction to the 2005 edition of the Essay by James Connelly and Giuseppina D’Oro. These editors say of Collingwood’s critics M. C. D’Arcy and C. J. Ducasse,

both agreed that Collingwood’s language was imprecise, sometimes vague, and insufficiently analytical. This criticism was later echoed by A. J. Ayer in his Philosophy in the Twentieth Century where he remarked that ‘An Essay on Philosophical Method is a contribution to belles-lettres rather than philosophy. The style is uniformly elegant, the matter mostly obscure.’

At the end I quote three elegant paragraphs from Collingwood, which begin:

Assumption for assumption, which are we to prefer? That in sixty generations of continuous thought philosophers have been exerting themselves wholly in vain, and have waited for the first word of good sense until we came on the scene? Or that this labour has been on the whole profitable, and its history the history of an effort neither contemptible nor unrewarded?

We prefer the second assumption; and in this we may seem to follow Daniel McCarthy in “Modernism & Conservatism” (The American Conservative, September 25, 2012), an essay recently promoted on Twitter (which is why I return now to this post). The freedom embraced by modernism may drive one to conservatism, as it did T. S. Eliot. McCarthy quotes Donald Livingston:

The true philosopher recognizes that philosophical reflection consistently purged of the authority of the pre-reflective leads to total skepticism. In this moment of despair, hubristic reason … becomes impotent and utterly silent. It is only then that the philosopher can recognize, for the first time, the authority of that radiant world of pre-reflective common life in which he has his being and which had always been a guide prior to the philosophic act.

Once reason has disestablished everything, including its own authority, what remains? The ground beneath your feet, the social order of which you are a part—things predicated not on any theory but on their immediacy. This is the profound conservatism to be realized from modernism.

Perhaps one may find this conservatism in some students and faculty at St John’s College; it is not inevitable, and Collingwood hasn’t got it, for all his admiration for Eliot.

A recent theme of this blog has been juxtapositions, especially of paintings, as in the articles “Pairing of paintings” and “More pairings” (both from July, 2013).