This book has nothing in particular to say. It fills its pages with unimaginative, thoroughly neutral, semi-encyclopaedic surveys of one branch of mathematics after another, one philosophical debate after another, and so on, while offering next to nothing by way of synthesis or interpretation.



I shall criticise Gray for being rather more uncritical than befits a historian in his acceptance of party-line modernism. The tenet of party-line modernism that I shall focus on is the myth that history shows that intuition must be abandoned since it leads to "false" results. It is an important task for historians to reject such propaganda abuses of history as the fabrications that they are; but unfortunately Gray is somewhat rubbing the back of the establishment in this case.



A typical statement of the myth in question is the following passage, where Gray is supposedly quasi-paraphrasing Perron:



"Spatial intuition is a very frequent source of error, especially when it is used to supplant proofs, as, for example, in proofs of the intermediate value theorem. 'Intuition is a crude instrument that lets us make out true relationships only imprecisely' (p. 204), and this is particularly so of our understanding of curves, which may fail in all sorts of ways to have the intuitive properties one suspects." (p. 275)



The propaganda myth is that intuition leads one to suspect that curves should have certain properties while they really don't. Rather, the problem is that the intuitive notion of "curve" does not correspond precisely to the formal mathematical notion. So the "error" referred to above is not at all an error of intuition; it is the error of stupidly taking intuition to apply to formal objects.



All of this is spelled out explicitly by Perron himself on the very page that Gray is referring to above (204). But you will have to go read the original article to find that out, for Gray omits it, thus skewing Perron's point to agree with party-line modernism. (In other contexts, however, Gray does quote people making the exact same point (almost verbatim) as Perron; namely Pierpont on p. 229 and Felix Klein on p. 197.)



Gray's discussion of the Dirichlet principle is similarly skewed. Weierstass's "decisive" criticism of this principle constituted, according to Gray, "evidence, it would seem, that a mixture of physical intuition and mathematical naivety was capable of leading mathematicians astray" (p. 75). But again intuition is being blamed for something that was not its fault. In fact, the Dirichlet principle is perfectly true, and it was only by extrapolating a particular formal generalisation of it (which no one claimed was intuitively obvious) that Weierstass was able to construct a so-called "counterexample."



But perhaps the clearest example of Gray's party-line tendencies is his enthusiastic approval of the ludicrous propaganda history of arch-establishmentarian axe-grinder Ernest Nagel:



"Nagel argued [that] the use of duality in projective geometry plays havoc with intuition and, he argued, opens the door to purely logical reasoning. ... I think [this claim] is on the mark" (p. 19).



It is baffling how Gray can accept such nonsense. The leading systematiser of duality in projective geometry was Jakob Steiner, the most intuitively inclined mathematician of all time. And Gray himself quotes Enriques as saying that "projective geometry refers to intuitive concepts, psychologically well defined" (pp. 122, 363) and Klein agreeing that it is "always intuitive" (p. 123).



Another illustration of Gray's underhand attack on intuition concerns Euclidean geometry. Pasch wrote accurately that "Elementary geometry cannot only be reproached for its difficulties, but also for its incompleteness and obscurities ..." From here Gray concludes: "[Pasch's] criticism of elementary, intuitive geometry from the standpoint of late nineteenth century criteria of rigor was typical." (p. 118). Note Gray's sneaky insertion of the word "intuitive." Pasch did not use this word, and he had good reason not to. Sure enough, Euclid's Elements contains numerous flaws from a formal point of view; for example, the triangle congruence "theorems" should really be axioms and so on. But it makes no sense to blame intuition for this. It is plainly a flaw of the Elements qua formal system.



Now perhaps some party-liners might object that it was intuition that tricked Euclid into making this mistake. To this I have two replies. First, I would say that we have intuitions about geometry, not about axiomatic structures. Secondly, I would ask if, in the opinion of this person, there could ever be any mistake in mathematics that he would attribute to formalism rather than intuition. Because if this is not a mistake of formalism then I do not know what such a mistake would look like, whence it would appear that one runs the risk of simply defining "intuition" as "that thing that causes all errors whatever in mathematical reasoning."