Monday, October 24, 2005

Construction and Existence

Roshdi Rashed, in his "Analysis and Synthesis According to Ibn al-Haytham," writes:

We cannot better Zeuthen's word, when he writes: "the essential value of geometric construction resides in that in it must serve to demonstrate that that to which the determination of the construction leads, really exists."

Zeuthen's word in this case is the French translation of Geschichte der Mathematik im Altertum und Mittelalter, Vorlesungen von H.G. Zeuthen, Professor an der Universität Kopenhagen (The History of Mathematics in Ancient and Medieval Times, Lectures by Heronymus Georg Zeuthen, Professor at the University of Copenhagen), originally priced at 15 Marks, but available now for free from the University of Michigan Historical Mathematics Collection (see also the World Digital Mathematics Library). I don't know whether Zeuthen actually lectured in German. His first language was Danish, naturally. No translator is credited for the German publication of his lectures. (They have since been translated into many languages, though I cannot locate an English translation at this time.) Rashed's word is an English translation by Matheiu Marion.

So what does that sentence mean? Here is the German:

Die wesentliche Bedeutung der geometrischen Konstruktion liegt darin, dass sie zum Beweise dafür dienen soll, dass dasjenige, auf dessen Darstellung die Konstruktion ausgeht, wirklich existiert.

I would translate that as:

The essential significance of the geometric construction lies in that it should serve to prove that whatever the representation of the construction seeks to show actually exists.

I added "to show" to clarify what I think it means. Zeuthen may also be saying that "whatever the construction seeks to represent actually exists," which is not literally accurate, but certainly a meaning that can be grasped.

In his Aporias against Ptolemy, al-Haytham cautions that truths are immersed in uncertainties, and that we must take a critical attitude towards scientific authorities and towards our own minds, which are prone to error:

[T]he seeker after the truth is not one who studies the writings of the ancients and, following his natural disposition, puts his trust in them, but rather the one who suspects his faith in them and questions what he gathers from them, the one who submits to argument and demonstration, and not to the sayings of a human being whose nature is fraught with all kinds of imperfection and deficiency. Thus the duty of the man who investigates the writings of scientists, if learning the truth is his goal, is to make himself an enemy of all that he reads, and, applying his mind to the core and margins of its content, attack it from every side. He should also suspect himself as he performs his critical examination of it, so that he may avoid falling into either prejudice or leniency."

That translation was provided by Abdelhamid I. Sabra, who translated the first three volumes of al-Haytham's Optics into English, and is one of the two widely acknowledged authorities on al-Haytham (the other being Dr. Rashed). His name may also be familiar to some for his role in the ouster of Nadav Safran. Dr. Safran stepped down as director of Harvard's Center for Middle Eastern Studies after it was revealed that he failed to properly disclose a grant from the Central Intelligence Agency. Dr. Sabra was one of three professors in the program who had publicly called for Dr. Safran's resignation.

At present complete English translations of al-Haytham's opuscula as well as his magnum opus do not exist, and indeed many works may exist only in manuscript form, though, lacking a knowledge of Arabic, I can't say with certainty that they haven't all been published. Of the 96 works al Haytham is known to have authored, nearly half have perished.

At present? The European awareness of al-Haytham, since the first translations of his work appeared in Latin, has always been narrowly focused on optics. Alan Gilchrist, in a 1996 editorial for Perception, argues that "language differences, ahistoricism, aculturalism, and racism" have all played a role in the neglect of scientists like al-Haytham. "Science must be defended against these antiscientific trends," he continues. "Now more than ever we should protect and promote the international flavor of science. Domination of science by any one country is bad for science. Surely the time has come to celebrate diversity in the scientific as well as the cultural realm." To Gilchrist's list I would add the problem of patronage. We have to ask, Why are some scientists supported and not others? Why are some projects funded while others are not? Why are some works widely disseminated while others are ignored or suppressed? Why are some manuscripts copied while others are left to rot? Why are some works translated into many languages while others are not? In every instance we see that the objectivity of scientific knowledge is not a given; the scientist must contend with powerful rulers and institutions who seek to influence the course of investigations and mediate exchanges between the scientific community and the public. Thus I may take issue with Gilcrist. The "celebration of diversity" for its own sake will not further the scientific enterprise, and as a counterbalance to American hegemony it may be of dubious worth. It rather depends on who's footing the bill for the celebrations, to what purposes, and what impediments if any are placed before the scientist as a result of this patronage. This latter concern touches on broader issues of the institutional contexts in which knowledge is propagated, and how scientific knowledge is constituted. My concern is that some celebrations of diversity may, in actual practice, lead to a ghettoization of knowledge rather than a "democraticization" of knowledge, an openness of knowledge to any and all. But I say "some" advisedly, for I am not completely unaware of my own prejudices and leniencies. I regard the matter as open to scrutiny.

Al Haytham's Optics, it should be noted, was intended for a wide audience, whereas the Aporias was intended for advanced students. This goes directly to the problem of patronage. Critical, epistemological or philosophical works are less in demand that explanatory, encyclopedic works. As a consequence, authoritative truths are more likely to be confirmed than refuted. In the public consciousness, obviously, but also in the judgements of scientists, in the consesus of opinion that the working scientist relies upon to guide further inquiry. Perhaps most insidiously, a bias towards established, explanatory truths may engender repurcussions beyond the immediate sphere of ideas about science, serving to echo or sustain profoundly undemocratic sentiments within the larger society, and thus, in turn, inhibiting the future progress of science. It seems to me, then, that a first priority of the scientific community is to value the critical perspective, recognizing that even that was never given as such, but had to be imagined, proposed and debated. Nurtured.

The essential significance of the geometric construction lies in that it should serve to prove that whatever the representation of the construction seeks to show actually exists.

That's really quite a remarkable claim--assuming Zeuthen is correct in his appraisal. I will keep that in mind when I revisit Husserl's Origins of Geometry. (I may be taking a detour through Derrida's Genesis and Death so don't hold your breath.)

posted by Fido the Yak at 10:16 AM.


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