I’ve always found the definitions to be among the most interesting parts of a math book. If you can truly understand the definitions, and by understand I mean have some sense of “why” things are defined the way they are, I think you have a good sense of the conceptual foundations of whatever mathematical topic you are reading about. Sure, understanding the definitions will not ensure that you understand proofs of any difficult theorems, but it is definitely a necessary prerequisite to having a good understanding of the subject. A lot of careful thought goes into choosing good definitions for mathematical concepts, and I’d like to give an example of one which I think has a very nice definition: the integral.

The idea of “area under a curve” is a very basic one which even the ancient Greeks tackled from time to time. The concept makes so much sense that nobody felt the need to define it precisely until the 19th century. However, I feel like the intuitive properties which one would expect the integral to have make it a pretty simple concept to formalize. From the geometric formulas for area of squares, triangles, circles, and so on, I think there are three intuitively obvious properties which give rise to a good definition of the integral:

1. If A and B are two disjoint regions in the plane, area(A)+area(B) = area(A U B).

2. If the region A is contained in the region B, area(A) cannot be greater than area(B).

3. The area of a rectangle is base times height.

Although one cannot say that these are the “right” (in an objective sense) properties for the concept of area under a curve, all three of them feel intuitively obvious to me. At least with my intuitive prejudices, it would be misleading to apply the word “area” to something which did not satisfy these properties. Amazingly, these three properties are sufficient to give us a good definition. Formally, this is what we do:

A partition of an interval [a,b] is a finite sequence of values x_i such that

Each interval [x_i−1,x_i] is called a subinterval of the partition. A refinement of the partition

is a partition

such that for every i with

there is an integer r(i) such that

In other words, to make a refinement, cut the subintervals into smaller pieces and do not remove any existing cuts.

Let ƒ:[a,b]→R be a bounded function, and let

be a partition of [a,b]. Let

Lower (green) and upper (green plus lavender) Darboux sums for four subintervals

The upper Darboux sum of ƒ with respect to P is

The lower Darboux sum of ƒ with respect to P is

The upper Darboux integral of ƒ is

The lower Darboux integral of ƒ is

If U_ƒ = L_ƒ, then we say that ƒ is Darboux-integrable and set

the common value of the upper and lower Darboux integrals.

(Note: formal development pilfered from wikipedia.)

I hate language. Language is imprecise. Language is malleable. When I want to communicate precisely, I’d rather not have the ability to combine reasoning and verbal play. I might enjoy using verbal tricks for fun, but sometimes it is best to be forced to say exactly what you mean. Yet even in symbolic logic, we have a problem! Amid all of the “definition” “theorem” and “proof” of even the most rigorous treatise on logic or math, there is still the issue that the definition of a word or symbol requires the usage of other words and symbols. Of course, as in any remotely philosophical issue, it was Aristotle who first noticed this. However, nobody really has a solution for this problem! If you read Euclid’s elements (well, maybe not the entire series) and you interpret his so called “definitions” in a way that he did not intend, you are really reading something totally different from what he wrote!

Even worse, I’m not even attempting to define the words I use in this post! For some reason, I have a strange faith that some part of what I am writing is communicated through a bunch of pixels on your screen. As my intellectual hero of the moment Wittgenstein said: “Propositions cannot represent logical form: it is mirrored in them. What finds its reflection in language, language cannot represent. What expresses itself in language, we cannot express by means of language. Propositions show the logical form of reality. They display it.”

Thus, as much as I hope this blog to be clear and precise, I obviously cannot confront the flaws in my language in a blog. Most of the time, I will begrudgingly use ordinary language. From time to time, I will math it up or even use symbolic logic. I hope to talk about math (including philosophy of math), logical positivism, the efficacy of language, economics, empiricism, politics, and anything else where I feel like a rational point can be made. Maybe next time I’ll be more explicit about what my terms mean.