Related art —

See also the previous post.

For the Church of Synchronology —

See also *this* journal on July 17, 2014, and March 28, 2017.

*** For related remarks, see posts of May 26-28, 2012.**

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*Josefine Lyche bowling (Facebook, June 12, 2012)*

A professor of philosophy in 1984 on Socrates's geometric proof in Plato's *Meno* dialogue—

"These recondite issues matter because theories about mathematics have had a big place in Western philosophy. All kinds of outlandish doctrines have tried to explain the nature of mathematical knowledge. Socrates set the ball rolling…."

— Ian Hacking in *The New York Review of Books* , Feb. 16, 1984

The same professor introducing a new edition of Kuhn's *Structure of Scientific Revolutions*—

**"Paradigms Regained"** (*Los Angeles Review of Books* , April 18, 2012)—

"That is the structure of scientific revolutions: normal science with a paradigm and a dedication to solving puzzles; followed by serious anomalies, which lead to a crisis; and finally resolution of the crisis by a new paradigm. Another famous word does not occur in the section titles: *incommensurability*. This is the idea that, in the course of a revolution and paradigm shift, the new ideas and assertions cannot be strictly compared to the old ones."

The *Meno *proof involves inscribing* diagonals *in squares. It is therefore related, albeit indirectly, to the classic Greek discovery that the diagonals of a square are *incommensurable *with its sides. Hence the following discussion of incommensurability seems relevant.

See also von Fritz and incommensurability in *The New York Times* (March 8, 2011).

For mathematical remarks related to the 10-dot triangular array of von Fritz, diagonals, and bowling, see *this* journal on Nov. 8, 2011— **"Stoned."**

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Jamie James in *The Music of the Spheres
*(Springer paperback, 1995), page 28—

Pythagoras constructed a table of opposites

from which he was able to derive every concept

needed for a philosophy of the phenomenal world.

As reconstructed by Aristotle in his *Metaphysics*,

the table contains ten dualities….

Limited |
Unlimited |

Of these dualities, the first is the most important;

all the others may be seen as different aspects

of this fundamental dichotomy.

For further information, search on *peiron + apeiron* or

consult, say, *Ancient Greek Philosophy* , by Vijay Tankha.

The limited-unlimited contrast is not unrelated to the

contrasts between

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The books pictured above are *From Discrete to Continuous* ,

by Katherine Neal, and *Geometrical Landscapes* , by Amir Alexander.

**Commentary—**

“Harriot has given no indication of how to resolve

such problems, but he has pasted in in English,

at the bottom of his page, these three enigmatic

lines:

‘Much ado about nothing.

Great warres and no blowes.

Who is the foole now?’

Harriot’s sardonic vein of humour, and the subtlety of

his logical reasoning still have to receive their full due.”

— “Minimum and Maximum, Finite and Infinite:

Bruno and the Northumberland Circle,” by Hilary Gatti,

*Journal of the Warburg and Courtauld Institutes* ,

Vol. 48 (1985), pp. 144-163

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**See also Finite Geometry and Physical Space.**

**Related material from MacTutor—**

Harriot and binary numbersThe paper by J. W. Shirley, Binary numeration before Leibniz, Leibniz [1646-1716] is credited with the invention [1679-1703] of binary arithmetic, that is arithmetic using base 2. Laplace wrote:-
However, Leibniz was certainly not the first person to think of doing arithmetic using numbers to base 2. Many years earlier Harriot had experimented with the idea of different number bases…. |

For a discussion of Harriot on the discrete-vs.-continuous question,

see Katherine Neal, *From Discrete to Continuous: The Broadening
*

pages 69-71.

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**(Continued from February 19)**

The cover of the April 1, 1970 second edition of *The Structure of Scientific Revolutions *, by Thomas S. Kuhn—

This journal on January 19, 2011—

If Galois geometry is thought of as a paradigm shift from Euclidean geometry,

both images above— the Kuhn cover and the nine-point affine plane—

may be viewed, taken together, as illustrating the shift. The nine subcubes

of the *Euclidean* 3×3 cube on the Kuhn cover do *not* form an affine plane

in the coordinate system of the *Galois* cube in the second image, but they

at least *suggest* such a plane. Similarly, transformations of a

non-mathematical object, the 1974 *Rubik * cube, are not *Galois* transformations,

but they at least *suggest* such transformations.

See also today's online *Harvard Crimson* illustration of problems of translation—

not unrelated to the problems of *commensurability* discussed by Kuhn.

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"These passages suggest that the Form is a character or set of characters

common to a number of things, i.e. the feature in reality which corresponds

to a general word. But Plato also uses language which suggests not only

that the forms exist separately (*χωριστά* ) from all the particulars, but also

that each form is a peculiarly accurate or good particular of its own kind,

i.e. the standard particular of the kind in question or the model (*παράδειγμα* )

[i.e. *paradigm* ] to which other particulars approximate….

… Both in the *Republic* and in the *Sophist* there is a strong suggestion

that correct thinking is following out the connexions between Forms.

The model is mathematical thinking, e.g. the proof given in the *Meno*

that the square on the diagonal is double the original square in area."

— William and Martha Kneale, *The Development of Logic*,

Oxford University Press paperback, 1985

Plato's paradigm in the *Meno* —

Changed paradigm in the diamond theorem (2×2 case) —

Aspects of the paradigm change* —

Monochrome figures to

colored figures

Areas to

transformations

Continuous transformations to

non-continuous transformations

Euclidean geometry to

finite geometry

Euclidean quantities to

finite fields

Some pedagogues may find handling all of these

conceptual changes simultaneously somewhat difficult.

* "Paradigm *shift* " is a phrase that, as John Baez has rightly pointed out,

should be used with caution. The related phrase here was suggested by Plato's

term *παράδειγμα** * above, along with the commentators' specific reference to

the *Meno* figure that serves as a model. (For "model" in a different sense,

see Burkard Polster.) But note that Baez's own beloved *category theory*

has been called a paradigm shift.

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