Augustus de Morgan - Mathematical Work

Mathematical Work

De Morgan was a brilliant and witty writer, whether as a controversialist or as a correspondent. In his time there flourished two Sir William Hamiltons who have often been confounded. One was Sir William Hamilton, 9th Baronet (that is, his title was inherited), a Scotsman, professor of logic and metaphysics at the University of Edinburgh; the other was a knight (that is, won the title), an Irishman, professor at astronomy in the University of Dublin. The baronet contributed to logic, especially the doctrine of the quantification of the predicate; the knight, whose full name was William Rowan Hamilton, contributed to mathematics, especially geometric algebra, and first described the Quaternions. De Morgan was interested in the work of both, and corresponded with both; but the correspondence with the Scotsman ended in a public controversy, whereas that with the Irishman was marked by friendship and terminated only by death. In one of his letters to Rowan, De Morgan says,

Be it known unto you that I have discovered that you and the other Sir W. H. are reciprocal polars with respect to me (intellectually and morally, for the Scottish baronet is a polar bear, and you, I was going to say, are a polar gentleman). When I send a bit of investigation to Edinburgh, the W. H. of that ilk says I took it from him. When I send you one, you take it from me, generalize it at a glance, bestow it thus generalized upon society at large, and make me the second discoverer of a known theorem.

The correspondence of De Morgan with Hamilton the mathematician extended over twenty-four years; it contains discussions not only of mathematical matters, but also of subjects of general interest. It is marked by geniality on the part of Hamilton and by wit on the part of De Morgan. The following is a specimen: Hamilton wrote,

My copy of Berkeley's work is not mine; like Berkeley, you know, I am an Irishman.

De Morgan replied,

Your phrase 'my copy is not mine' is not a bull. It is perfectly good English to use the same word in two different senses in one sentence, particularly when there is usage. Incongruity of language is no bull, for it expresses meaning. But incongruity of ideas (as in the case of the Irishman who was pulling up the rope, and finding it did not finish, cried out that somebody had cut off the other end of it) is the genuine bull.

De Morgan was full of personal peculiarities. On the occasion of the installation of his friend, Lord Brougham, as Rector of the University of Edinburgh, the Senate offered to confer on him the honorary degree of LL. D.; he declined the honour as a misnomer. He once printed his name: Augustus De Morgan, H - O - M - O - P - A - U - C - A - R - U - M - L - I - T - E - R - A - R - U - M (Latin for "man of few letters").

He disliked the provinces outside London, and while his family enjoyed the seaside, and men of science were having a good time at a meeting of the British Association in the country he remained in the hot and dusty libraries of the metropolis. He said that he felt like Socrates, who declared that the farther he was from Athens the farther was he from happiness. He never sought to become a Fellow of the Royal Society, and he never attended a meeting of the Society; he said that he had no ideas or sympathies in common with the physical philosopher. His attitude was possibly due to his physical infirmity, which prevented him from being either an observer or an experimenter. He never voted at an election, and he never visited the House of Commons, or the Tower of London, or Westminster Abbey.

Were the writings of De Morgan published in the form of collected works, they would form a small library, for example his writings for the Useful Knowledge Society. Mainly through the efforts of Peacock and Whewell, a Philosophical Society had been inaugurated at Cambridge; and to its Transactions De Morgan contributed four memoirs on the foundations of algebra, and an equal number on formal logic. The best presentation of his view of algebra is found in a volume, entitled Trigonometry and Double Algebra, published in 1849; and his earlier view of formal logic is found in a volume published in 1847. His most distinctive work is styled a Budget of Paradoxes; it originally appeared as letters in the columns of the Athenæum journal; it was revised and extended by De Morgan in the last years of his life, and was published posthumously by his widow.

George Peacock's theory of algebra was much improved by D. F. Gregory, a younger member of the Cambridge School, who laid stress not on the permanence of equivalent forms, but on the permanence of certain formal laws. This new theory of algebra as the science of symbols and of their laws of combination was carried to its logical issue by De Morgan; and his doctrine on the subject is still followed by English algebraists in general. Thus George Chrystal founds his Textbook of Algebra on De Morgan's theory; although an attentive reader may remark that he practically abandons it when he takes up the subject of infinite series. De Morgan's theory is stated in his volume on Trigonometry and Double Algebra. In the chapter (of the book) headed "On symbolic algebra" he writes:

In abandoning the meaning of symbols, we also abandon those of the words which describe them. Thus addition is to be, for the present, a sound void of sense. It is a mode of combination represented by ; when receives its meaning, so also will the word addition. It is most important that the student should bear in mind that, with one exception, no word nor sign of arithmetic or algebra has one atom of meaning throughout this chapter, the object of which is symbols, and their laws of combination, giving a symbolic algebra which may hereafter become the grammar of a hundred distinct significant algebras. If any one were to assert that and might mean reward and punishment, and, etc., might stand for virtues and vices, the reader might believe him, or contradict him, as he pleases, but not out of this chapter. The one exception above noted, which has some share of meaning, is the sign placed between two symbols as in . It indicates that the two symbols have the same resulting meaning, by whatever steps attained. That and, if quantities, are the same amount of quantity; that if operations, they are of the same effect, etc.

Here, it may be asked, why does the symbol prove refractory to the symbolic theory? De Morgan admits that there is one exception; but an exception proves the rule, not in the usual but illogical sense of establishing it, but in the old and logical sense of testing its validity. If an exception can be established, the rule must fall, or at least must be modified. Here I am talking not of grammatical rules, but of the rules of science or nature.

De Morgan proceeds to give an inventory of the fundamental symbols of algebra, and also an inventory of the laws of algebra. The symbols are 0, 1, +, −, ×, ÷, and letters; these only, all others are derived. His inventory of the fundamental laws is expressed under fourteen heads, but some of them are merely definitions. The laws proper may be reduced to the following, which, as he admits, are not all independent of one another:

1. Law of signs. + + = +, + − = −, − + = −, − − = +, × × = ×, × ÷ = ÷, ÷ × = ÷, ÷ ÷ = ×.
2. Commutative law. a+b = b+a, ab=ba.
3. Distributive law. a(b+c) = ab+ac.
4. Index laws. ab×ac=ab+c, (ab)c=abc, (ab)d= ad×bd.
5. a−a=0, a÷a=1.

The last two may be called the rules of reduction. De Morgan professes to give a complete inventory of the laws which the symbols of algebra must obey, for he says, "Any system of symbols which obeys these laws and no others, except they be formed by combination of these laws, and which uses the preceding symbols and no others, except they be new symbols invented in abbreviation of combinations of these symbols, is symbolic algebra." From his point of view, none of the above principles are rules; they are formal laws, that is, arbitrarily chosen relations to which the algebraic symbols must be subject. He does not mention the law, which had already been pointed out by Gregory, namely, and to which was afterwards given the name of the law of association. If the commutative law fails, the associative may hold good; but not vice versa. It is an unfortunate thing for the symbolist or formalist that in universal arithmetic is not equal to ; for then the commutative law would have full scope. Why does he not give it full scope? Because the foundations of algebra are, after all, real not formal, material not symbolic. To the formalists the index operations are exceedingly refractory, in consequence of which some take no account of them, but relegate them to applied mathematics. To give an inventory of the laws which the symbols of algebra must obey is an impossible task, and reminds one not a little of the task of those philosophers who attempt to give an inventory of the a priori knowledge of the mind.