My intermittent and slow progress through *The 25 Years of Philosophy* has brought me to the point (p 257) at which Förster raises the question in the title. Let others worry about the fact that any number of formulae can generate any finite sequence of numbers and rest satisified, for the moment, that such alternative formulae are often objectionably weird. My question is more prosaic. Förster claims that "there is no doubt that the path from the series to the formula lies in studying the *transitions*", and follows that up with a footnote saying that "an intellectual re-production of the transitions between 1, 1, 2, 3, 5, 8, 13, 21, is necessary is order to realize that, form the third element in the series onward, every number is the sum of the two preceding numbers; hence the next number must be 34, and we are dealing with the formula for the Fibonnacci series, *f*_{n} = *f*_{n-1} + *f*_{n-2}."

Ok. Let's grant that recognizing the recurrence relation is necessary to arrive at the formula for the series. But: why must the formula for the series explicitly be one that recapitulates the recurrence relation? Mightn't the *right* formula for the series be the closed form, *f*_{n} = (φ^{n} - ψ^{n})/√5? The two formulae agree everywhere, but so do the competing definitions of a circle on the one hand as "a figure in which all the lines drawn from the center to the circumference are equal to each other" and on the other as "a plane figure described by a line of which one end is fixed while the other is moveable", which is evidently something of moment, since only the second "is adequate and … expresses the efficient cause" (p 95). Certainly the formula given by the recurrence relation is the one we're apt to come up with first, is easier to calculate with, and corresponds to the way the Fibonacci sequence was actually made known in the first place. But the closed-form solution is actually pretty easy to understand (though it may have been hard to arrive at), and there are other cases in which we would likely consider the closed-form expression the right one and the recurrence relation strangely indirect.

Consider the following sequence, for example: 0, 1, 4, 9, 16, 25, 36 … Attendance to the transitions instantly reveals that the difference between successive numbers in the sequence are successive odd numbers, that is, that *f*_{0} = 0 and *f*_{n} = *f*_{n-1} + 2n-1. But I suspect that most people with the mathematical sophistication to recognize that would also be able to identify the following closed-form formula: *f*_{n} = n^{2}. Of course it is easy to get from the first of these to the second: the recurrence relation is simple enough that one can quickly see that *f*_{n} is the sum of the first n odd numbers. Since the sum of the n odd numbers is equal to the sum of the first n numbers plus the sum of the first n-1 numbers*, we have n(n+1)/2 + n(n-1)/2 = n^{2}.

In the case of the Fibonacci sequence we're unlikely to produce it by the closed-form expression, whereas we're unlikely to produce square numbers via the recurrence relation (a product of familiarity? I'd likely produce triangular numbers via the recurrence relation). But it seems to me that, quite aside from how one would write down the numbers, one grasps the sequence as a whole via the closed form better or more than via the recurrence relation. Why isn't *that* the formula for the sequence? After all, the recurrence relation defines each element of the series in terms of other elements.

* you can convince yourself of this fact, if you need to (I did), by gazing upon the following table:

1 2 3 4 5 ... 0 1 2 3 4 ... 1 3 5 7 9 ...

I suppose the more formulaic interpretation of this table is that the second column represents n, and the top n+1, so the third is 2n+1, i.e., the nth odd number.

"The two formulae agree everywhere, but so do the competing definitions of a circle on the one hand as "a figure in which all the lines drawn from the center to the circumference are equal to each other" and on the other as "a plane figure described by a line of which one end is fixed while the other is moveable", which is evidently something of moment, since only the second "is adequate and … expresses the efficient cause" (p 95)."

The passage on p.95 brought me up short when reading it, because I can't figure out what is wrong with the "inadequate" definition of a circle. I *think* he is using an example of Spinoza's there, but he doesn't cite him if he is. (If he's not using Spinoza's example, then I find it even more confusing: What the hell is "the efficient cause of a circle" supposed to mean, if Förster is speaking in his own voice in that passage?)

Posted by: Daniel Lindquist | September 29, 2012 at 07:44 PM

It reads very much as if it's a Spinozan example, so much so that I just assumed it was, but now I see you're right, it isn't explicitly attributed to him (and it comes up again on p 252 where again it isn't explicitly attributed to him). But at least the language of efficient causes is from Spinoza (note 3 on p 253), as, apparently, is the use of mathematical examples—it's just not clear that

thismathematical example is from Spinoza.I just assumed, at that passage on p 95, that it was indeed true that starting from the first definition of a circle you can't deduce all the properties of a circle, but it would have been nice if some specific property whose deduction the second definition does enable but the first doesn't had been given.

Posted by: ben w | September 29, 2012 at 09:19 PM