The Reactionary Epicurean, as far as I can tell the only other mathematically inclined Yalie on the blogosphere, has posted a very nice rebuttal; of my first post, “Undefined Terms.” (While you’re on his blog, check out the video about Mobius Transformations for a nice bit of beautiful math.) While I intended the post primarily as a semi-ironic statement criticizing the possibility of my blog being as logical as I would want it to be, I still would like to counter a few of his claims.
The Reactionary Epicurean makes the assertion: “I’d submit the possibility that the form of communication has an effect upon the act of perception itself, leading us to the conclusion that Nietzsche, Hofstadter, and Marshall McLuhan were right all along: “The medium is the message.”" This is one of the primary reasons I distrust poetry, for example. Indeed, our perception is altered by their utterly non-deductive nature. As a form of art unto itself, I might enjoy poetry, but I consider it dangerous to look for any “truth” therein. In a world where people were cognizant of language’s power to manipulate and consciously avoided using it, I posit that we would be rid of many stupidly destructive ideas. (Religion is the best example, in my opinion.)
I also would like to dispute his simplistic gathering of Nietzsche and Douglas Hofstadter in the same camp. In the introduction to Godel, Escher, Bach, Hofstadter gives a very good one sentence summary of his views on language- “Meaningless symbols take on meaning in spite of themselves.” I certainly agree that, as Hofstadter also says, “In form, there is content.” However, I do not think this gives imprecise, flowery language a green light. To me at least, it provides evidence of the terrifying powers of a society which communicates illogically.

I used to think that a question that was empirically meaningless was simply not worth asking.  One cannot articulate a well-defined answer without descending into a slew of vague buzzwords, so I did not think there was anything to this.  Such questions include the “meaning of life”  “nature of love” “what it is to be a good person” and essentially any question which we respond to primarily in an emotional way.  However, I think there is a place for such questions even in a logically precise philosophy.  The catch is that one does not answer these questions with a statement or proposition- one must answer with an action, habit, or way of life.

At the risk of appearing obsessed, I’d like to return to Wittgenstein’s “Tractatus Logico-Philosophicus.”  One essential point in this book is that what language communicates cannot be referred to with language.  A series of propositional statements each have their own logical sense, but they communicate in another way as well.  The propositions also illustrate something, and this something cannot be referred to directly in language.  I think this distinction is at the heart of the merit of an illogical, unanswerable question.  The questions do not entail a precisely formulated statement which refers to something, but they can have the ability to illustrate something indirectly.  Asking a “philosophical question” is a suggestive tool to make us reflective about the world, and I think we must respond to them accordingly- we must respond through the world.

Thus, if someone asks me “what is the meaning of life,” or something else equally cheesy, I think the best way to respond is to simply live.   Given that “meaning” is itself a notoriously ill-defined term (again, that which language illustrates it cannot describe),  I think answering such a question directly reduces to a useless play with words.  That which we consider “meaning” is not something which we can logically deduce, it is in some sense an emergent property of living well.

From this point of view, I think the philosophy of existentialism is rather clearly misguided.  We cannot sit around and consciously choose what we consider meaningful.  To do so reduces to little more than trying to supply a definition to the word meaning, and such introspective philosophical techniques often reduce to little more than playing with language.   That which we can speak of we must speak of precisely, and that which we cannot must be approached by other methods.  To live in a fulfilling way, we must simply live in a fulfilling way, there is nothing more to it.  Our conscious mind does not have complete power to decide what our subjective, unconscious mind values (I’ve tried), even though we may have some ability to consciously manipulate how we think.  To imagine that our perception of meaning is totally our choice is to mistakenly overestimate the power of our ego.  We are a product of our unspeakably ill defined thoughts just as much as our unspeakably ill defined thoughts are a product of us.

At the end of my last post, I pointed out that model building is inherently not metaphysical.  Because the only strict criteria for the success of a model is that it fits empirical data, it says nothing about how the world “got that way.”  From the stark, positivistic definition of the word “meaning,” I would like to argue that anything more is meaningless.

For the rest of this post, I will define a statement to be meaningful if it asserts something empirical about the world.  The statement “The earth is round.” is meaningful because one can test this assertion by viewing the earth from space (assuming your definition of earth and round are the conventional ones).  The statement “the earth is beautiful” is not directly meaningful, since there is no obvious definition of the word “beautiful” which allows one to conduct an experiment to test this assertion.  Any type of question with the word “why” in it cannot have a complete, meaningful answer.  For example, conservation of energy is “why” a pendulum never swings higher than the point from which it was dropped, but one is then left with why energy is conserved.   Empirical tests can only see if something does or does not happen- data does not provide a narrative.  To answer a why question scientifically, one eventually must state the fundamental, unproven assumptions of a model.

It is precisely for this reason that I think we can do nothing better than build a model of reality.  Any meaningful theory will tell us how to interpret data, so we cannot even in principle be certain that we are not making simplifying assumptions in how we measure something.  Even outside the realm of science, we rely on informal models.  Marxism provides a (falsified to the extent it is falsifiable?) model of how history develops, as do many other political ideologies.  None of the narrative portions of such political ideologies are meaningful in my strict definition of the word.  Even in every day life, we informally model the behavior of others and act according to our predictions.  Such models clearly have unfalsifiable portions (what precisely does it mean to have a “bad day” for example), which must on some level be meaningless.  Thus, in all endeavors, I believe that model building is the best we can do.  We can use our models as a way of thinking about the world, but it is meaningless to say that our models have anything to with “how things really work.”

Given that I am strongly considering a career in academic economics, I am very much a believer in mathematical modeling. Economic interactions are really complicated, and I personally don’t have the intellectual power to even attempt to understand how the economy “really is.” Thus, if I want to develop an understanding of something, I must make simplifying assumptions. Simplifying to the point of triviality (kind of like building a model :-0 ), this is my outline of how we build a model of something in the world.

1. We assume there is a “real world out there,” and proceed to make observations.

2. We perhaps notice some regularities in how stuff happens.

3. We choose some fundamental assumptions (useful lies, one could say) which allow us to build a theory.

4. We derive the consequences of our assumptions to make predictions about how the world works.

5. We do, or tell some empiricist to do because we are lazy theorists, an experiment or look at a data set and see if our ideas cut the mustard.

From this process, we get a mathematical model (yay). A falsifiable model will spit some numbers out which are supossed to be measurable in the real world, which we can then test. The model will then enter one of two categories- possibly right (since it has not yet been falsified), or wrong. As time goes on, this process of falsification allows us to keep the level of bullshit in our models pretty low. A falsifiable model has to be right in some very general sense of the word in order not to be thrown out. However, this does not mean that a not yet falsified model describes the world as it “actually is.” For example, no sane economist would ever actually tell you that there is some time independent thing called a utility function or objective measure of happiness which your behavior necessarily maximizes. One could only say that assuming that people maximize utility allows one to make useful inferences about the world.

To me, this is the largest philosophical stumbling block in the process of mathematical modeling. I could very well model, say, continental drift by the actions of the tooth fairy and get a falsifiable model which makes good predictions. From the austere perspective of mathematical modeling, such a model is exactly as good as any other which makes the same predictions. From a positive, empirical point of view, a model reduces to simply making assumptions to make predictions. Any assumption we make must in some sense be false, as any simplification of reality is, but this does not matter. There is no metaphysics in mathematical model building! In part two, I will hope to extend this claim and thereby touch on some fundamental issues in my logical positivism.

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.