24 March 2018

Mathematical metaphors

Mathematics is, by its very nature, as disjoint from reality as is chess. But now and then some mathematical result can be interpreted in such a way that, in a receptive eye, it reflects some aspect of reality. Below, we consider some of these mathematical metaphors. 'Meta-phor' means 'to carry over' and this is exactly what happens.

Plutarch. De Iside et Osiride, section 56 (online here):
Now in that triangle the perpendicular consists of three parts, the base of four, and the subtense of five, its square being equal in value with the squares of the two that contain it. We are therefore to take the perpendicular to represent the male property, the base the female, and the subtense that which is produced by them both. We are likewise to look upon Osiris as the first cause, Isis as the faculty of reception, and Horus as the effect. For the number three is the first odd and perfect number, and the number four is a square, having for its side the even number two. The number five also in some respects resembles the father and in some again the mother, being made up of three and two.

 Guido Grandi. In 1703, Grandi published Quadratura circuli et hyperbolae (online here), concerning which we learn from Grandi's biography (here) the following:

One of Grandi's results in the Quadratura caused a lot of interest. He used the series expansion

1/1+x = 1 - x + x2 - x3 + x4 - ...
and putting x = 1, obtained 1 - 1 + 1 - 1 + 1 - 1 + ... = 1/2. However, he argued that (1 - 1) + (1 - 1) + (1 - 1) + ... = 0 + 0 + 0 + 0 +... . In the first draft of the work, Grandi claimed that since the sum of infinitely many 0s equaled 1/2 he had proved that God could create the word out of nothing. The censor allowed the mathematics to be published but required the removal of the comment that it showed God could create the word out of nothing. Grandi reluctantly agreed to remove the comment but many mathematicians across Europe discussed Grandi's result that 0 + 0 + 0 + 0 +... = 1/2.
The series, in geometrical form, is to be found on page 51 of Quadratura (Corollarium III to Propositio VII). Grandi complied with the censor, but defended himself vigorously in Riposta apologetica (1712, online here). The discussion concerning the challenged interpretation starts on page 153. The core of the mathematico-theological debate is the principle ex nihilo nihil: out of nothing you get nothing (p.157).

Back to the metaphor: Grandi viewed his 'equation' (which is mathematical nonsense according to today's standards)

0 + 0 + 0 + 0 +... = 1/2

as a symbol of 'Creation out of nothing'.

Kepler. In May 1608, Kepler wrote a letter to Joachim Tancke (Latin original here, English translation of the relevant passage here) in which he dwells on the "divine proportion". As always with Kepler, it's a mix of mysticism and science. Even so, the content is very interesting, because, for the first time in history, the golden and/or Fibonacci numbers are somehow linked to the physical world.

We start with five properties concerning the golden and Fibonacci numbers of which Kepler seems to have been (qualitatively) aware in writing the letter.

The first one is trivial, given that a/b=b/(a+b) is the geometrical definition of the golden section, and the subsequent equalities just repeat that definition. For (2): algebraically, φ is the irrational number (√5-1)/2. Properties (3) and (4) are in fact basic properties of convergents of continued fractions, applied to φ.  For a direct proof of (4) see e.g. ProofWiki, Cassini's Identity. Formula (5) is the case m-1=n of ProofWiki, Fibonacci Number in terms of smaller Fibonacci Numbers.

For Kepler, the divine proportion is the "model of (pro)creation", archetypus generationis, and he gives several reasons for this.

(a) what was formerly the larger part now becomes the smaller, what was formerly the whole now becomes the larger part, and the sum of these two now has the ratio of the whole. This goes on indefinitely; the divine proportion always remaining. I believe that this geometrical proportion served as idea to the Creator when He introduced the creation of likeness out of likeness, which also continues indefinitely. 

This is a theological interpretation of property (1) above, and far from convincing. The A4-format, with ratio √2:1, can be described in much the same way: folding it in half, the smaller dimension becomes the larger dimension of the next generation, the ratio remaining, and this too goes on indefinitely.

(b) I see the number five in almost all blossoms which lead the way for a fruit, that is, for creationand the number five, that is the pentagon, is constructed by means of the divine proportion.

This botanical observation (even extended to Fibonacci numbers in general) is by no means a general rule. Moreover, the presence of 5 in nature need not be the result of the golden ratio mysteriously at work, no more so than that a four-petaled flower is the trace of √2 or a circular flower head reveals the presence of π.

(c) there exists between the movement of the earth and that of Venus, which stands at the head of generative capability, the ratio of 8 to 13 which comes very close to the divine proportion. 

The astronomical fact is (approximately) true: Venus orbits the sun in 224.7 days, the earth in 365.24, and the quotient is 0.61521... while 8/13=0.61538... (though φ=0.61803...). Astronomically, this is a "coincidental near ratio of mean motion". (Here you can read more about it, and about the many simple ratios of integers that result from perfect orbital resonance.) Kepler, as much an astrolo-gist as an astrono-mer, identifies the second planet from the sun with the eponymous goddess of love and fertility, and this makes further comments superfluous.

(d) the earth-sphere is midway between the spheres Mars and Venus. One obtains the proportion between them from the dodecahedron and the icosahedron, which in geometry are both derivatives of the divine proportion; it is on our earth, however, that the act of procreation takes place.

Here, the astronomical references to the dodecahedron and the icosahedron are irrelevant, because Kepler's model of the solar system (based on the medieval idea "because we know six planets, there are six planets") was completely wrong. Even the astrology involved seems very twisted, as it deduces some property of "the planet of procreation" from the two neighbouring planets.

 (e) the image of man and woman results from the divine proportion.  

Kepler rediscovers, in his own way, the sequence of Fibonacci numbers


with the peculiarity that the two numbers 1 must be considered as different, viz. as the minor and the major part of the number 2 decomposed. Strange, but let's read on. He emphasizes the properties (2) and (3) stated above, which have in fact nothing very special and could be adapted (with a different sequence of integers) for any irrational number. Leaving aside the first 1, the Fibonacci numbers are said to be alternatively male and female. The argument is as follows. If you consider any Fibonacci number, its square will be 1 more or 1 less than the product of its two neighbours (the preceding and the following one); see property (4) for the correct formula. Now a "square +1" is a square with a surplus (like a body with a penis) while a "square -1" is a square with a cavity (like a body with a vulva). When the two copulate, a new Fibonacci number results; this is our formula (5) above. Don't forget to add 1 to the one square (resulting in a male) while subtracting it from the other square (resulting in a female).

To make things perfectly clear, Kepler adds a diagram showing how the Fibonacci numbers 3 and 5 copulate and produce the Fibonacci number 34 as their offspring.

As for the sexes involved: 3 is "male" (Fibonacci neighbours 2 and 5, product 10, a square +1), while 5 is "female" (neighbours 3 and 8, product 24, a square -1). Kepler writes of these illustrations: observe that you clearly see a penis here and a vulva there. When they copulate (illustration at the bottom) the surplus 1 of the one square fills the defect 1 of the other square, here resulting in the Fibonacci number 34. (Note that, while the parents in the formula are approximate squares, the offspring is just the plain number; moreover, being odd-indexed, it will always be the same Keplerian gender.) We almost have two divine proportions Kepler writes. In fact, we almost have two golden rectangles: in the first case a 2x5 rectangle of 10 squares, in the second case a 3x8 rectangle of 24 squares.

In this mix of math and myth, (e) is clearly a metaphor because human bodies don't look like squares at all. On the other hand, (b) and (c) are scientific observations, one qualitative and one quantitative. In connecting golden and Fibonacci numbers and rectangles to physical reality, Kepler is 250 years ahead of Zeising. But, unlike Zeising, Kepler leaves (human) art alone, and is well aware of the distance separating maths and reality: 
I also play with symbols; I have started a small work: "Geometrical Cabbala"; it deals with the ideas of the things of nature in geometry. Only I play in such a way as to never forget that I am playing. For nothing is proven with symbols; in the philosophy of nature no secrets are unveiled by geometrical symbols.
 Herz-Fischler (A Mathematical History of the Golden Number, Dover 1998, p. 171) writes 
Even in Kepler's case the statements glorifying the DEMR [division in extreme and mean ratio] are always related to the mathematical properties.
Are they? We fail to see any mathematical reference in Kepler's statement, under (c), about the golden ratio orbits.