So... a few things have flashed by recently... let me take four of them in reverse chronological order, starting with the most inchoate, inarticulate moment of remembering/understanding that just happened as i stood up to take my coffee cup and pie plate to the "bus" station. i surveyed the crowd here in Zoka's and my attention was drawn to a woman having an animated conversation with her friend and then to a man in "business attire" standing by the spot for picking up drinks. Suddenly, i remembered that we don't have to be this way. We don't have to dress in these clothes or adopt these postures or inhabit these habits. We started, each of us, as children; we started, all of us, as funny, mostly hairless kinds of primates. So supple, so flexible, so curious and adaptable, we could go in a million different directions, we enjoyed nearly limitless possibility. And, even now, in this desperate position, on a planet warming up like the room in No Exit, we can make our way to a better place. Redemption is always possible. Until it isn't.
Last night... driving home from a work study group... i suddenly remembered a part of myself i had lost or been asleep to. i used to adopt this strangely relaxed, engaged, actively innocent and completely honest self-assessment, in realtime, in dialogue with someone who challenged me. It would often uncover terrifically unpleasant aspects of who i was and, if i'm honest, some cool things, too. i haven't been in this space in years. What's happened? Where have i been? What have i been doing? What caused me to lose touch with this way of being?
Listening to the media coverage of the Wolfowitz-World-Bank "story", i began to think about corruption in a different light. What if corruption is like oxidation? It happens; it's a process; and, a wide range of entities are all equally susceptible to it. There's nothing to get worked up about. If you need your piece of iron or steel to remain integral or reliable you must observe -- or submit it to -- processes that remove or prevent oxidation. Likewise, if you need individuals or organizations to enjoy a certain integrity or reliability, you must observe -- or submit them to -- processes that remove or prevent corruption. Even stainless steel knives need a certain amount of care. Likewise, even the most straight up people (or institutions) need a certain amount of care. In some sense, it's this regular attention to things that makes the kitchen or the church or the bank an attractive place to be.
The last one in this set is an observation about the notion of integral. When i learned calculus, i was taught the definite integral (of (very well-behaved) functions of a single variable) as a kind of sum. It added rectangles under a curve where the width of the rectangles was infinitessimally small. In symbols
∫
a b f(x) dx
is interpreted "intuitively" as an expression of the form
∑ f(x) * dx (sum up the height of the rectangle times the width of the rectangle)
i quote "intuitively" because it's not entirely intuitive what multiplication by an infinitessimally small quantity means (or what an infinitary addition is). One camp says that's just notation. What you're really calculating is a limit. That's all fine and good, if it weren't for the tremendous practical success of another camp that slings quantities of this form around just like they were numbers. It just took us a few hundred years (and one very smart mathematician) to figure out what this meant operationally. (And this little delta between when the notation developed and when we figured out what it meant is some evidence that "intuitive" should be marked in some way.)
i belabor the point because subsequent expansions, or elaborations of the notion of integral build on this intuition of integral as sum. Now, i want to point out that summing things means you've gotten pretty good at telling things apart -- because you don't want to double count. To borrow an example from Quine, if the man in the trenchcoat and fedora is Mr. Ordcutt, you don't want to count this individual as two men when counting the number of men you saw in a day. So, telling things apart, or dually, noticing when two things are really the same is an essential part of counting, which is an essential part of summing, which is the intuitive notion underlying the integral.
The next part of the observation is something that has built up over years of experience. i can sum it up in the slogan -- no set worth counting (measuring, really) has an elementary notion of equivalence. The examples are ubiquitous.
- Every effective representation of numbers -- be it in industrial application or a theoretical device -- has a very complex notion of equivalence
- Look at IEEE specs for real number computation
- Look at Conway's representations of numbers
- Look at Escardo's effective representation of reals
- Look at Integer arithmetic on a computer (including bounds and overflow)
- groups up to isomorphism
- topological spaces up to homeomorphism
- knots up to ambient isotopy
- processes up to weak bisimulation
- lambda terms up to contextual equivalence
As the Quine-propositional-attitudes example above is intended to illustrate, this is not just a mathematical oddity. Entities of concern to us in the everyday world (like other people) enjoy really complex properties related to identification and individuality. Giving this apparent fact of our experience its due in our symbolic representations of integration and summation is something that is not part of the ordinary mathematical account. Or is it?
What if the notation is really capturing a slightly different notion of measuring or counting? What if the expression
∫
a b f(x) dx
means something like: "measure of the
extension of the
rule described by f using a notion of
distinction up to dx"?
Now, let me try to unpack that phrase.
- f(x) -- The notion of function has wonderful history. It's trajectory through the canon of western literature starts with a very operational feel and flavor: people think about functions in terms of examples like polynomials or sines and cosines -- practical stuff for laying out your fields or figuring out where the bomb is going to fall ;-(. But, when set theory became the lingua franca of maths, the notion of function took on a very extensional, denotational flavor: it was a set of pairs enjoying certain properties. The view is something like the lookup table for all the values of the function laid out in its entirety -- which loses our sense and feel for the calculational aspect of the notion. As the the theory of computation began to develop, the operational and rule-like aspect came back into play. Then a remarkable thing happened, the theoretical computation community became cognizant of both aspects and the value of being able to relate the two views. One the one hand, there is a view of function as a rule, f, for calculating some information, f(x), given some information, x; and, on the other, there is a view of the function as it's extension -- some completely unfolded representation in which all the calculation has been done. Literature on the lambda-calculus gives a really sophisticated account of the relationships of these concepts: function-as-rule, extension-as-set-with-equivalence-relation, etc.
- dx -- category theory has been teaching us to be more explicit about the implicit maps lying around in our accounts of things, including the isomorphisms and up-to mappings, which give rise to notions of distinction as well as equality
Ok, now does all this history and philosophical background get us anywhere? Can we shut up and calculate, already? Or is this more sound and fury? Well, it turns out, if you're willing to pony up certain explicit codifications of what you mean by extension and the very act of summing itself, then you can give an effective interpretation of this view of the integral. That is, you can write down a computer program that will calculate your integral. Here it is:
[| ∫
C r d |](extension, empty, size, combine)
:=
foldl
(\acc class -> combine acc (size class))
(filter (\x -> (member [| C |] x)) (extension empty [| d |]))
(filter (\x -> (member [| C |] x)) (extension [| r |] [| d |]))
In English: the meaning of the expression "∫
C r d" is given in terms of maps, extension, empty, size and combine. It "sums", using the combine map, the "size" of the "equivalence classes" in the extension, starting with the empty extension as the initial seed. It filters out in terms of the (interpretation of) the boundary, C. Thus, we can make a distinction between a definite integral (as defined above), which is an operator from maps to "quantities", and an indefinite integral which is an abstraction in C and is an operator from maps to maps.
In some ways this is a bit of sleight of hand. i haven't really done anything except provide a highly parameterized notion of sum. The point, however, is to begin to get a proper accounting of all the pieces that go into a measure of a set, including an explict account of distinction and extension. Certainly, you can recover a version of the original notion of integral this way. i'm also guessing you can recover a version of motivic measure theory using this formulation.
