De Morgan, Augustus - Trigonometry and Double Algebra - On symbolic algebra
Type of Spiritual Experience
Although De Morgan’s views on spiritual experience are of great interest, another area which is of immense importance spiritually is his work on symbols. The spiritual world in its communication with us, uses symbols almost exclusively. If one receives words, then one is getting thoughts from another human being on the mental radio that is telepathy. But anything of importance or of real note is symbolic. As such, much of De Morgan’s theories laid the ground work for a true understanding of spiritual concepts, which by their very nature have no equivalent in the physical.
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. 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:
A description of the experience
Trigonometry and Double Algebra - “On symbolic algebra"
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 A, B, C, 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 A = B. It indicates that the two symbols have the same resulting meaning, by whatever steps attained. That A and B, if quantities, are the same amount of quantity; that if operations, they are of the same effect, etc."
[MATHEMATICAL MONOGRAPHS BY ALEXANDER MACFARLANE
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 modifed. 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 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.
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.