Philosophy of Number, part 7: First and Second Intentions

Klein’s most concise definition of a symbol is that it is a “second intention” concept interpreted as a “first intention” concept. To clarify, he explains that there are, according to the medieval academics, two basic conceptual levels upon which we work. The first is the level of “first intentions.” Those are ideas that refer to things. When we use a word for an object, like “tree” or “dog” or “spaceship,” that’s a first intention, because it refers to a thing in the world. In the realm of numbers, a first intention idea would be a counting number, like one, two, three, five thousand, or whatever other number could be used to count things. But we can also further abstract from the first to the second intention by taking an attribute of an object and referring to it, resulting in an idea about an idea about an object. We can talk about “speed,” “color,” “numerability,” and so on, without bringing in any particular objects that are fast, red, or in groups of 6, for instance. In mathematics, then, a particular number is a first intention idea, while a symbol like a or x is a second intention idea, because it represents only the possible determinacy of a number, without actually being a number itself. (Klein, 1985, p 58-63)

The oddity is that in algebraic equations we operate on these indeterminate ideas of “any possible number” as though they were themselves numbers. So in an expression like axn=y, a is symbolic of any number, x or y of any variable, and n of any power, yet they can be operated upon as though they were wholly determinate numbers. It is likewise possible to find a numerical value for other second intention ideas like velocity, distance, time, energy, mass, and so on, and operate upon that value. Even beyond that, we could generalize the possibility of finding a numerical value for these ideas in the same way we generalize the possibility of a number simply, so that we’re dealing with the relationships between v, d, t, e, m and so on to produce formulas that do not describe any particular objects with any particular motion, but instead symbolize the relationship between any possible objects and motions. The result would be what Klein means by a symbolic science.

Since the seventeenth century we have tended to assume that symbolic mathematics is the natural and necessary way in which to look at all things numerable. There must be a general object to match the generality of scientific method. The symbol is precisely the “general object” of arithmetical operations. Descartes also developed a “general object” for geometry in his coordinate system. However, Klein argues, not only is the treatment of symbolic “general objects” not obvious, but the Greeks, despite centuries of mathematical advancement, never did take that step, but always dealt with determinate numbers of something (though in theoretical arithmetic and logistic they were numbers of purely intellectual monads) and determinate geometric objects (Klein, 1985, p 17). The manner of being with which these objects and especially the monads exist prompted a long and complicated philosophical debate which I’m not going to dwell on here.

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