Is God a Mathematician_ - Mario Livio [83]
One of De Morgan’s most important contributions to logic is known as quantification of the predicate. This is a somewhat bombastic name for what one might view as a surprising oversight on the part of the logicians of the classical period. Aristotelians correctly realized that from premises such as “some Z’s are X’s” and “some Z’s are Y’s” no conclusion of necessity can be reached about the relation between the X’s and the Y’s. For instance, the phrases “some people eat bread” and “some people eat apples” permit no decisive conclusions about the relation between the apple eaters and the bread eaters. Until the nineteenth century, logicians also assumed that for any relation between the X’s and the Y’s to follow of necessity, the middle term (“Z” above) must be “universal” in one of the premises. That is, the phrase must include “all Z’s.” De Morgan showed this assumption to be wrong. In his book Formal Logic (published in 1847), he pointed out that from premises such as “most Z’s are X’s” and “most Z’s are Y’s” it necessarily follows that “some X’s are Y’s.” For instance, the phrases “most people eat bread” and “most people eat apples” inevitably imply that “some people eat both bread and apples.” De Morgan went even further and put his new syllogism in precise quantitative form. Imagine that the total number of Z’s is z, the number of Z’s that are also X’s is x, and the number of Z’s that are also Y’s is y. In the above example, there could be 100 people in total (z 100), of which 57 eat bread (x57) and 69 eat apples (y 69). Then, De Morgan noticed, there must be at least (x y z) X’s that are also Y’s. At least 26 people (obtained from 57 69 100 26) eat both bread and apples.
Unfortunately, this clever method of quantifying the predicate dragged De Morgan into an unpleasant public dispute. The Scottish philosopher William Hamilton (1788–1856)—not to be confused with the Irish mathematician William Rowan Hamilton—accused De Morgan of plagiarism, because Hamilton had published somewhat related (but much less accurate) ideas a few years before De Morgan. Hamilton’s attack was not at all surprising, given his general attitude toward mathematics and mathematicians. He once said: “An excessive study of mathematics absolutely incapacitates the mind for those intellectual energies which philosophy and life require.” The flurry of acrimonious letters that followed Hamilton’s accusation produced one positive, if totally unintended, result: It guided algebraist George Boole to logic. Boole later recounted in The Mathematical Analysis of Logic:
In the spring of the present year my attention was directed to the question then moved between Sir W. Hamilton and Professor De Morgan; and I was induced by the interest which it inspired, to resume the almost-forgotten thread of former inquiries. It appeared to me that, although Logic might be viewed with reference to the idea of quantity, it had also another and a deeper system of relations. If it was lawful to regard it from without, as connecting itself through the medium of Number with the intuitions of Space and Time, it was lawful also to regard it from within, as based upon facts of another order which