Related art —
See also the previous post.
For the Church of Synchronology —
* For related remarks, see posts of May 26-28, 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."
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….
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
The books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.
“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
‘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
See also Finite Geometry and Physical Space.
Related material from MacTutor—
The paper by J. W. Shirley, Binary numeration before Leibniz, Amer. J. Physics 19 (8) (1951), 452-454, contains an interesting look at some mathematics which appears in the hand written papers of Thomas Harriot [1560-1621]. Using the photographs of the two original Harriot manuscript pages reproduced in Shirley’s paper, we explain how Harriot was doing arithmetic with binary numbers.
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
of Number Concepts in Early Modern England (Springer, 2002),
(Continued from February 19)
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.
"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
Continuous transformations to
Euclidean geometry to
Euclidean quantities to
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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