Customer Review

March 23, 2016
The main thesis of this book may be summed up as follows:

"The secondary literature, particularly the older one, mostly suggests that mathematical rigour and unassailable foundations were not of paramount importance during this period [of the early development of infinitesimal methods]---that the fruitfulness of the new methods and notions balanced, as it were, the logical deficiencies in their foundations. We would like to suggest here that just the opposite may be the truth. That it was precisely because mathematicians cared a lot about questions of rigor and foundations that they set out to modify Cavalieri's theory." (24)

"In the second half of the seventeenth century we find many leading mathematicians, such as Pascal, Wallis, Sluse, and Barrow, willing to grant full legitimacy to the method of indivisibles. In the same period a growing number of authors asserted the essential equivalence between proofs involving indivisibles and proofs patterned according to the classical exhaustion method. In Pascal's well-known words, 'everything which is demonstrated by the true rule of indivisibles can also be demonstrated with rigour and in the manner of the ancients; ... one of these methods does not differ from the other except for the manner of speaking'. This meant that the method of indivisibles had gained a new status: it was now able to provide legitimate demonstrations, which was something that few mathematicians would have dared to say in the 1630s and 1640s. ... Beneath this significant modification of the logical status of the method of indivisibles lies the substitution of infinitesimals for classical indivisibles." (20)

Thus, instead of asserting the perfect rigour of his methods, Cavalieri instead offers the considerable weaker defence of comparing it to similarly liberal methods in algebra, such as the use complex numbers:

"I have used an artifice similar to that often used by Algebraists for solving problems. For, however ineffable, absurd and unknown the roots of numbers be, they none the less add, subtract, multiply and divide them; and they are convinced of having sufficiently met their obligation when they have succeeded in finding out from the given problem the result which was required." (16)

As well as an argument by induction:

"I do not think it is useless to notice ... that most of the things that were disclosed by Euclid, Archimedes, and others, have also been demonstrated by me, and that my conclusions agree perfectly with their conclusions. This must be an evident indication that I have gathered true things in the basic principles [of my method]---even though I know that from false principles truths can be sophistically deduced. But that that had happened to me in so many conclusions [otherwise] proved geometrically, I would deem to be absurd." (17)

But the problem with this argument is evident. To establish the reliability of Cavalieri's method, it is not enough to verify that it leads to many known truths, but also that it does not lead to any falsehoods. Indeed, it is precisely where we do not have the independent verification of ancient mathematics that we want to use the new method.

And indeed the method unfortunately leads to some paradoxes, as Cavalieri himself was aware (25). For instance, consider a triangle. Take one of its sides as a base and drop the perpendicular through the third vertex. The triangle is thus split into two halves, which are generally unequal. But with Cavalieri's method they can be "proved" equal all the same. For run a ruler parallel to the base through the triangle. At any given stage it intersects the sides of the triangle in two points, each of which determines a perpendicular to the base of equal height. Altogether, each of the two sub-triangles are "made up" of these lines. And since these constituent lines are equal at every stage, the areas of the two sub-triangles must be equal as well, which is absurd.

Clearly the problem here is that the lines have "different thickness" if we think of them as infinitesimals, so we see indeed that the move from indivisibles to infinitesimals was called for foundationally, just as the above thesis statement says.
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