Philosophy of Mathematics & Mathematical Practices in the Seventeenth Century Paperback – 29 Jul 1999
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Students of the history of mathematics and philosophers of mathematics will find this a valuable addition to the literature. (Choice)
Mancosu's book shows philosophical acumen as well as high technical competence―and it makes good reading even as it explores abstruse notions or involved technicalities. For historians of early modern mathematics, it is essential reading. (Isis)
Mancosu tells the story well and is good at bringing out significant points. (International Philosophical Quarterly)
This is a very carefully researched and documented analysis of the rich relationship between philosophy of mathematics and mathematical practice during the 17th century. (Mathematical Reviews)
Mancosu's scholarly book is very carefully researched, but it is also clearly written and fascinating to read. It is not to be missed by anyone with a serious interest in philosophy of mathematics. (Philosophia Mathematica)
The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmetic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting with the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relation between geometrical and epistemological theories of the infinite, and the Leibnizian calculus and the opposition to infinitesimalist procedures.In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics.
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This should perhaps be the end of the story, but Mancosu wants to milk it further. It seems to me that the Aristotelian perspective essentially played out its part with Barrow, as indeed Mancosu hints: "Although Barrow still appears willing to frame his argument in the context of the Aristotelian logical terminology, he begins with the basic presupposition that mathematics is the science par excellence. Nothing can be more remote from his perspective than the subtle scholastic distinctions that had characterized the Renaissance contribution to the Quaestio." (p. 23). Support for the view that this was the end of Aristotelian influence is offered by Mancosu himself (although to him it is evidence for the very opposite, as we shall see). At the same time as Barrow, Arnauld (1662) offered what appears from Mancosu's account to be an admirably clear and unprejudiced discussion of the issues involved. Consider for example his condemnation of proofs by contradiction: "Those kind of demonstrations which show that a thing is such, not by its principles, but by some absurdity which would follow if it were not so, are very common in Euclid. It is clear, however, that while they may convince the mind, they do not enlighten it, which ought to be the chief result of knowledge; for our mind is not satisfied unless it knows not only that a thing is, but why it is" (p. 101).
This sounds to me like a thinker who has freed himself of scholastic shackles, but Mancosu will have none of that and prefers instead to insist on projecting Aristotle onto Arnauld and others, so that he will be allowed to conclude that "a great part of the philosophy of mathematics in the seventeenth century was shaped by the Aristotelian notion of science" (p. 102). I would suggest changing "was shaped by" to something more like "has a fairly shallow connection to", as surely Mancosu is aware would be more accurate.
Knowing what pays the bills in today's academia, Mancosu continues: "But, as I said, I want to claim more. The Aristotelian theory of science was very influential far beyond the seventeenth century." (p. 102). This is supposedly proved by taking an abrupt 150-year leap to Bolzano. Bolzano is another outstanding example of a clear and independent thinker, but he was also a meticulous scholar, so once he happened to make a passing reference to Aristotle, which in Mancosu's tainted reading proves that "Bolzano clearly characterizes his project as reviving the Aristotelian distinction between proofs of the 'fact' and of the 'reasoned fact'" (p. 104). Mancosu also claims that his perspective explains "apparently queer pronouncements of Bolzano and Frege" (p. 105), when in fact there is nothing "queer" about them at all---these authors are discussing natural ideas with great clarity; it happens that similar ideas were perceived by Aristotle, but this adds nothing to our understanding of these obviously independent works. It is sad that mathematics should be infected by the humanities-post-doc disease of fancy "reinterpretations" that are plainly undermined already by the very evidence brought up in their support.
The last part of the book changes the topic to the philosophical issues raised by infinitesimal calculus. Here Mancosu tells the story straight with no tricks. Unfortunately, the story is not very interesting, because the philosophers involved are too incompetent. Perhaps the first to attack infinitesimals was Clüver, but "The debate with Clüver turned out to be rather unsatisfactory. Although Leibniz and Jakob Bernoulli had put great effort in trying to open a dialogue, it also turned out that Clüver simply did not have the technical skills and intellectual honesty required for such an exchange." (p. 158). The next critic, Nieuwentijt, was more competent, but not more so than that "Hermann, at that time a student of Jakob Bernoulli, ... rebuked Nieuwentijt point by point" (p. 164). The next critic was Rolle, a member of the Paris Academy of Sciences, but he too was rebuked by a nobody: "The reply to Rolle was written by a protégé of L'Hôpital, Joseph Saurin ..., who was not yet an academician" (p. 174), whereupon the academy issued what was essentially an "official condemnation of Rolle's memoir" despite this being "a flagrant contradiction of the spirit of the academy" (p. 175).
In particular Mancosu shows how the issue of causality, derived from Aristotle's philosophy of mathematics, was interpreted in the sixteenth century and seveteenth centuries, and how it generated problems of interpretation for those who wished to argue that mathematics was a genuine science, against the background of contemporary developments in mathematics which seemed to undermine this position. Mancosu includes, as an appendix, a translation of a work of the sixteenth century mathematician Biancani, in which there appears a fascinating list corrolating Euclid's demonstrations with the form of causation that each is supposed to embody. Why causality in mathematics was so important to sixteenth and seventeenth century thinkers is another question not addressed in the book, but clearly this seems to be an instance of a conceptual divide between the present (where such an issue is hardly even concievable) and early modern thinking in mathematics.
Mancosu also studies Bolzano's philosophy of mathematics and links his concerns with those of Aristotle, i.e., the need to look at mathematical proofs from within a causal framework. Since Bolzano is often linked with the beginning of something like a modern conception of mathematics, Mancosu's conclusions would seem to indicate that the Aristotelian concern with causality is also part of this modern conception. Yet in spite of this, the concerns of modern mathematics and a mathematician like, for example, Biancani, seem very different. Modern logicists are usually Platonists as well - Somehow, putting Biancani, Bolanzo, and Frege in the same boat seems ill-concieved.
Another issue of importance at the time was the status of proof by superposition, as used, for instance, in Euclid I, 4., where one figure is transposed over another and correlated with it. The validty of superposition as a form of proof was, along with more familiar worries about proofs by reductio ad absurdum (i.e. what would today be called non-constructive proofs) a focus of much concern.
There is much more in this book - a discussion of Kant's philosophy of mathematics and its differences from Bolzano's; a discussion of some pre-Berkelian critiques of the calculus; Descartes' algebraization of geometry; Cavalieri's attempts to carefully ground his theory of indivisibles in an axiomatic system. But the impression left by the whole book is that the concern with rigour in mathematics is not simply something that emerged with the Greeks and lay dormant until the well-known nineteenth century labours of Cauchy and others in analysis.