Cistercian Numerals

In the 13th century, Cistercian monks worked out a system of numerals in which a single glyph can represent any integer from 1 to 9,999:

https://commons.wikimedia.org/wiki/File:Cistercian_digits_(vertical).svg
Image: Wikimedia Commons

Once you’ve mastered the digits in the top row, you can represent tens by flipping them (second row), hundreds by inverting them (third row), and thousands by doing both (fourth row). And now you can combine these symbols to produce any number under 10,000:

https://commons.wikimedia.org/wiki/File:Cistercian_numerals.svg
Image: Wikimedia Commons

The monks eventually dropped the system in favor of Arabic numerals, which reached northwestern Europe at about the same time, but it was being used informally elsewhere as recently as the early 20th century.

Maverick

The ancient Chinese philosopher Gongsun Long appeared to claim that a white horse is not a horse:

Is ‘a white horse is not horse’ assertible?

Advocate: It is.

Objector: How?

Advocate: ‘Horse’ is that by means of which one names the shape. ‘White’ is that by means of which one names the color. What names the color is not what names the shape. Hence, one may say ‘white horse is not horse.’

Objector: If there are white horses, one cannot say that there are no horses. If one cannot say that there are no horses, doesn’t that mean that there are horses? For there to be white horses is for there to be horses. How could it be that the white ones are not horses?

Advocate: If one wants horses, that extends to yellow or black horses. But if one wants white horses, that does not extend to yellow or black horses. Suppose that white horses were horses. Then what one wants [in the two cases] would be the same. If what one wants were the same, then ‘white’ would not differ from ‘horse.’ If what one wants does not differ, then how is it that yellow or black horses are acceptable in one case and unacceptable in the other case? It is clear that acceptable and unacceptable are mutually contrary. Hence, yellow and black horses are the same, one can respond that there are horses, but one cannot respond that there are white horses. Thus, it is evident that white horses are not horses.

Interpretations vary; one explanation is that the conundrum blurs the distinction between identity and class, exploiting an ambiguity in the Chinese language — certainly the expressions “white horse” and “horse” do not have identical meanings, but one can refer to a subset of the other.

Whether the philosopher was serious isn’t clear. His other paradoxes include “When no thing is not the pointed-out, to point out is not to point out” and “There is no 1 in 2.”

More trouble with horse color.

03/08/2024 UPDATE: A Swedish Facebook meme of 2012: Horses are a fruit that does not exist. (Thanks, Mikael.)

Chronological Order

By Lee Sallows: If the letters BJFGSDNRMLATPHOCIYVEU are assigned to the integers -10 to 10, then:

J+A+N+U+A+R+Y     = -9+0-4+10+0-3+7     =  1
F+E+B+R+U+A+R+Y   = -8+9-10-3+10+0-3+7  =  2
M+A+R+C+H         = -2+0-3+5+3          =  3
A+P+R+I+L         = 0+2-3+6-1           =  4
M+A+Y             = -2+0+7              =  5
J+U+N+E           = -9+10-4+9           =  6
J+U+L+Y           = -9+10-1+7           =  7
A+U+G+U+S+T       = 0+10-7+10-6+1       =  8
S+E+P+T+E+M+B+E+R = -6+9+2+1+9-2-10+9-3 =  9
O+C+T+O+B+ER      = 4+5+1+4-10+9-3      = 10
N+O+V+E+M+B+E+R   = -4+4+8+9-2-10+9-3   = 11
D+E+C+E+M+B+E+R   = -5+9+5+9-2-10+9-3   = 12

Similarly, if -7 to 7 are assigned SROEMUNFIDYHTAW, then SUNDAY to SATURDAY take on ordinal values. See Alignment.

(David Morice, “Kickshaws,” Word Ways 24:2 [May 1991], 105-116.)

Kürschák’s Tile

https://commons.wikimedia.org/wiki/File:K%C3%BCrsch%C3%A1k%27s_tile.svg

Hungarian mathematician József Kürschák offered this “proof without words” that a regular dodecagon inscribed in a unit circle has area 3. If the circle is inscribed in a square, the resulting figure can be tiled by triangles of two families — 16 equilateral triangles whose sides are equal to those of the dodecagon and 32 isosceles triangles with angles 15°-15°-150° and longest side 1. The area of the large square is 4, and the triangles that make up the dodecagon can be rearranged to fill 3 of its quadrants (see the video below). So the area of the dodecagon is 3/4 of 4, or 3.

(To see that the square and the dodecagon can be tiled as claimed, see Alexander Bogomolny’s discussion here.)

(Gerald L. Alexanderson and Kenneth Seydel, “Kürschak’s Tile,” Mathematical Gazette 62:421 [October 1978], 192-196.)

Revolution

[Wittgenstein] once greeted me with the question: ‘Why do people say that it was natural to think that the sun went round the earth rather than that the earth turned on its axis?’ I replied: ‘I suppose, because it looked as if the sun went round the earth.’ ‘Well,’ he asked, ‘what would it have looked like if it had looked as if the earth turned on its axis?’

— G.E.M. Anscombe, An Introduction to Wittgenstein’s Tractatus, 1959

A Fuss Budget

https://commons.wikimedia.org/wiki/File:Honor%C3%A9_Daumier_-_The_Loge_(In_the_Theatre_Boxes)_-_Walters_371988.jpg

Francis Galton quantified boredom. At a tedious meeting in 1885, he observed that the distance between neighboring heads in the listless crowd began to vary:

When the audience is intent each person forgets his muscular weariness and skin discomfort, and he holds himself rigidly in the best position for seeing and hearing. As this is practically identical for persons who sit side by side, their bodies are parallel, and again, as they sit at much the same distances apart, their heads are correspondingly equidistant. But when the audience is bored the several individuals cease to forget themselves and they begin to pay much attention to the discomforts attendant on sitting long in the same position. They sway from side to side, each in his own way, and the intervals between their faces, which lie at the free end of the radius formed by their bodies, with their seat as the centre of rotation varies greatly.

He wasn’t able to estimate this numerically, but he did find another measure: He counted about 50 fidgets per minute in each section of 50 people. “The audience was mostly elderly; the young would have been more mobile.” He urged “observant philosophers” at dull meetings to estimate “the frequency, amplitude, and duration of the fidgets of their fellow-sufferers” in hopes that “they may acquire the new art of giving numerical expression to the amount of boredom expressed by the audience generally during the reading of any particular memoir.”

(Francis Galton, “The Measure of Fidget,” Nature 32:817 [June 25, 1885], 174–175.)