Is God a Mathematician_ - Mario Livio [79]
This was quite interesting in itself, but Grassmann’s extension contained even more surprises. Note that if we were dealing with algebra instead of geometry, then an expression such as AB usually would denote the product A B. In that case, Grassmann’s suggestion of BA AB violates one of the sacrosanct laws of arithmetic—that two quantities multiplied together produce the same result irrespective of the order in which the quantities are taken. Grassmann faced up squarely to this disturbing possibility and invented a new consistent algebra (known as exterior algebra) that allowed for several processes of multiplication and at the same time could handle geometry in any number of dimensions.
Figure 46
By the 1860s n-dimensional geometry was spreading like mushrooms after a rainstorm. Not only had Riemann’s seminal lecture established spaces of any curvature and of arbitrary numbers of dimensions as a fundamental area of research, but other mathematicians, such as Arthur Cayley and James Sylvester in England, and Ludwig Schläfli in Switzerland, were adding their own original contributions to the field. Mathematicians started to feel that they were being freed from the restrictions that for centuries had tied mathematics only to the concepts of space and number. Those ties had historically been taken so seriously that even as late as the eighteenth century, the prolific Swiss mathematician Leonhard Euler (1707–83) expressed his view that “mathematics, in general, is the science of quantity; or, the science that investigates the means of measuring quantity.” It was only in the nineteenth century that the winds of change started to blow.
First, the introduction of abstract geometric spaces and of the notion of infinity (in both geometry and the theory of sets) had blurred the meaning of “quantity” and of “measurement” beyond recognition. Second, the rapidly multiplying studies of mathematical abstractions helped to distance mathematics even further from physical reality, while breathing life and “existence” into the abstractions themselves.
Georg Cantor (1845–1918), the creator of set theory, characterized the newly found spirit of freedom of mathematics by the following “declaration of independence”: “Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established.” To which algebraist Richard Dedekind (1831–1916) added six years later: “I consider the number concept entirely independent of the notions or intuitions of space and time…Numbers are free creations of the human mind.” That is, both Cantor and Dedekind viewed mathematics as an abstract, conceptual investigation, constrained only by the requirement of consistency, with no obligations whatsoever toward either calculation or the language of physical reality. As Cantor has summarized it: “The essence of mathematics lies entirely in its freedom.”
By the end of the nineteenth century most mathematicians accepted Cantor’s and Dedekind’s views on the freedom of mathematics. The objective of mathematics changed from being the search for truths about nature to the construction of abstract structures—systems of axioms—and the pursuit of all the logical consequences of those axioms.
One might have thought that this would put an end to all the agonizing over the question of whether mathematics was discovered or invented. If mathematics was nothing more than a game, albeit a complex one, played with arbitrarily invented rules, then clearly there was no point in believing in the reality of mathematical concepts, was there?
Surprisingly, the breaking away from physical reality infused some mathematicians with precisely the opposite sentiment. Rather than concluding that mathematics was a human invention, they returned to the original Platonic notion of mathematics as an independent world of truths,