Fibonacci fib?

Kevin gave us a nice report yesterday on the Fibonacci sequence, the  so-called golden ratio allegedly ubiquitous in nature. The phenomenon, originally noted in antiquity, seems to imply an ordered nature whose regularities defy coincidence. It stops short of demonstrating a divine sort of order, of course;  and even if there are pantheistic possibilities in the hypothesis, that's not really what most people mean by "God." Still, it's an impressive thing. But...

Not wishing to dash cold water, I nonetheless feel obliged to note that cold water has been dashed by others. To wit:

Fibonacci Foolishness. A search of the internet, or your local library, will convince you that the Fibonacci series has attracted the lunatic fringe who look for mysticism in numbers. You will find fantastic claims:

• The "golden rectangle" is the "most beautiful" rectangle, and was deliberately used by artists in arranging picture elements within their paintings. (You'd think that they'd always use golden rectangle frames, but they didn't.)
• The patterns based on the Fibonacci numbers, the golden ratio and the golden rectangle are those most pleasing to human perception.
• Mozart used f in composing music. (He liked number games, but there's no good evidence that he ever deliberately used f in a composition.)
• The Fibonacci sequence is seen in nature, in the arrangement of leaves on a stem of plants, in the pattern of sunflower seeds, spirals of snail's shells, in the number of petals of flowers, in the periods of planets of the solar system, and even in stock market cycles. So pervasive is the sequence in nature (according to these folks) that one begins to suspect that the series has the remarkable ability to be "fit" to most anything!
• Nature's processes are "governed" by the golden ratio. Some sources even say that nature's processes are "explained" by this ratio.

Of course much of this is patently nonsense. Mathematics doesn't "explain" anything in nature, but mathematical models are very powerful for describing patterns and laws found in nature. I think it's safe to say that the Fibonacci sequence, golden mean, and golden rectangle have never, not even once, directly led to the discovery of a fundamental law of nature. When we see a neat numeric or geometric pattern in nature, we realize we must dig deeper to find the underlying reason why these patterns arise...

Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and the golden ratio seems to be without foundation. 


And: one of Rebecca Goldstein's 36 arguments addresses the alleged power of mathematical mystery to elicit feelings of transcendental astonishment:

30. The Argument from Mathematical Reality

1. Mathematical truths are necessarily true. (There is no possible world in which, say, 2 plus 2 does not equal 4, or in which the square root of 2 can be expressed as the ratio of two whole numbers.)

2. The truths that describe our physical world, no matter how fundamental, are empirical, requiring observational evidence. (So, for example, we await some empirical means to test string theory, in order to find out whether we live in a world of eleven dimensions.)

3. Truths that require empirical evidence are not necessary truths. (We require empirical evidence because there are possible worlds in which these are not truths, and so we have to test that ours is not such a world.)

4. The truths of our physical world are not necessary truths (from 2 and 3).

5. The truths of our physical world cannot explain mathematical truths (from 1 and 4).

6. Mathematical truths exist on a different plane of existence from physical truths (from 5).

7. Only something which itself exists on a different plane of existence from the physical can explain mathematical truths (from 6). 8. Only God can explain mathematical truths (from 7).

9. God exists.

Mathematics is derived through pure reason — what the philosophers call a priori reason — which means that it cannot be refuted by any empirical observations. The fundamental question in philosophy of mathematics is: how can mathematics be true but not empirical? Is it because mathematics describes some trans-empirical reality — as mathematical realists believe — or is it because mathematics has no content at all and is a purely formal game consisting of stipulated rules and their consequences? The Argument from Mathematical Reality assumes, in its third premise, the position of mathematical realism, which isn't a fallacy in itself; many mathematicians believe it, some of them arguing that it follows from Gödel's incompleteness theorems (see the COMMENT in The Argument from Human Knowledge of Infinity, #30 above). This argument, however, goes further and tries to deduce God's existence from the trans-empirical existence of mathematical reality.

FLAW 1: The inference of 5, from 1 and 4, does not take into account the formalist response to the non-empirical nature of mathematics.

FLAW 2: Even if one, Platonistically, accepts the derivation of 5 and then 6, there is something fishy about proceeding onward to 7, with its presumption that something outside of mathematical reality must explain the existence of mathematical reality. Lurking within 7 is the hidden premise: mathematical truths must be explained by reference to non-mathematical truths. But why? If God can be self-explanatory, as this argument presumes, why then can't mathematical reality be self-explanatory — especially since the truths of mathematics are, as this argument asserts, necessarily true?

FLAW 3: Mathematical reality — if indeed it exists — is, admittedly, mysterious. But invoking God does not dispel this puzzlement; it is an instance of "The Fallacy of Using One Mystery to Pseudo-Explain Another." The mystery of God's existence is often used, by those who assert it, as an explanatory sink hole.