George Peacock: Treatise on Algebra (1830)

 

(1791–1858)

On a surface level, Peacock’s Treatise on Algebra seems a rather humdrum textbook on algebra, with no particularly original innovations to be found in it. Yet, what is most important in the treatise are not any purely mathematical results, but the more philosophical considerations on the nature of algebra and its relation to other parts of mathematics and especially arithmetics.

What Peacock is particularly interested in is the question whether algebra should be somehow based on arithmetics, which is based on concrete groups of objects, like seven apples. Peacock’s answer is a resounding no: arithmetics is far more constrained, allowing e.g. no negative numbers, since you cannot produce a group of, say, minus seven apples, although negative numbers are a common occurrence in algebra.

Not restricted by characteristics of arithmetical numbers, what is to determine what rules are to be assumed in algebra? Peacock makes the bold suggestion that these rules are just assumed: algebra concerns only symbols, and we could in principle choose any rules to govern our calculations in it.

Still, Peacock does not yet reach the modern notion of algebra, where we can have many different systems of calculation with different rules. Instead, he thinks that algebra should be especially the most general system of calculation, where all sorts of calculation are possible. Thus, he accepts not just calculations leading to negative numbers, but also roots of negative numbers, which we nowadays call imaginary numbers. Peacock goes even so far as to suggest that such seemingly nonsensical symbols like 0/0 could be used in a meaningful manner (although he mentions this possibility only in passing, he is referring to certain readings of the infinitesimal calculus).

Peacock still wants that algebra would be of practical use. Here, he suggests that the rules of algebra should at least correspond to the rules of arithmetics at least in those cases where the calculations and their results make arithmetical sense. Then again, he continues, results of algebra could also be used in other mathematical sciences, where some of them might be of relevance. Thus, negative numbers do make sense, for instance, when speaking of debt or of movement to an opposite direction. Even imaginary numbers can have concrete meaning, as describing movement perpendicular to a line.

Algebra becomes then, for Peacock, like a general tool for mathematical problem solving. When a concrete problem is given, it is turned into algebraic symbols: what quantities are known and what are unknown that are supposed to be determined in terms of the known quantities? After doing the algebraic manipulation of the symbols, one must still interpret the results. Often the context of the problem restricts this interpretation, and although algebra would lead to a number of possible results, only some of them might make sense in the context. Sometimes no result suggested by algebra would make sense, and then the problem itself will be impossible.