The Fregean Vision Of Language

Gottlob Frege

It's interesting, though not surprising, that Frege toiled in such obscurity. All of mathematical logic was suspect until it finally started bearing fruit by taming set theory and giving rise to computers. Mathematical logicians had a reputation of pedants even among mathematicians - and Frege was unusually careful and rigorous even by the standards of mathematical logician.

Bertrand Russell

Of course, Frege wasn't a complete unknown. Frege influenced Bertrand Russell and Peano to be more bold in their formalism. Obviously, Wittgenstein's early philosophy is entirely an attempt to draw out more philosophical consequences of Frege's methods and insights. His later philosophy is also deeply Fregean, though more critical than his fawning early work. I'll come back to this in a bit. Dedekind and Zermelo were aware of his work and held it in esteem. At the time, the analytic/continental philosophy distinction did not exist, so Frege actually had a good bit of influence on several "continental" philosophers. He was one of very few teachers Gershom Scholem respected. Scholem attempted to communicate Frege's ideas to Walter Benjamin, which seems to have been a bit optimistic on Scholem's part. Frege helped embolden Husserl to completely abandon psychologism in mathematics - which became a major plank in developing transcendental phenomenology. All this adds up to one thing: this is going to be one of those black and white pictures of dead men posts.

Gottlob Frege

Frege's analysis of mathematical language was a shining philosophical gem: it killed the mistaken Millian theory of psychological abstraction as a foundation for mathematics and seriously wounded Kant's related notion that arithmetic was synthetic (further work by Godel showed that it was not synthetic in another sense). His book Begriffsschrift may be the greatest technical piece of philosophical argument ever written. I'd like to spend a little time developing what his analysis would be in modern terms.

How do you define "a definition"? This is one of the most fundamental tasks one must take in logical analysis, but it can be surprisingly difficult to do. One answer would be naive atomism: each word (except the logical connectives) represents one idea and sentences are fusions of such ideas. This won't do. Some words are relations which gain meaning only when surrounded by other words. Some words are functions that gain meaning when given an argument. Examples include "My father's mother is gone.", "God's in His Heaven - all's right with the world!", "The cat is on the mat." and "The square of two is four.". The word 'square' in the sense used in the last example is obviously a function. The word "My" in the first is also a function, as is "father's". The word 'on' in the third example is a binary relation.

Frege's solution, which you might call "limited holism", was that each word gains meaning only in the context of the sentences in which it is used. The basic unit of meaning is the sentence in the following sense: only a sentence may be true or false. A "definition" is a rule that tells one how to go about using a word in a way that generates true sentences. When you observe a certain state of the world (in a very rough grained, perception/culture/etc mediated way), you may convey that state of the world to an English speaker by uttering "The cat is on the mat.". When young Pippa observes a certain state of the world (or, more accurately, passes without carefully observing it) she conveys this to the people of Asolo (including herself) by saying "God's in His Heaven - all's right with the world!". There is little syntactical difference between these sentences.

David Hilbert

 
Frege's vision was somewhat confused because he did not always carefully distinguish syntax and semantics. If symbols are "defined" in the sense above, then they have semantic content. The "definitions" of all the terms of a Fregean language give a model for a syntactical system. The well formed formulas of the syntactical system are given by the true sentences of any model of that system. This means that, for instance, if a formula can be shown to be well formed by purely syntactical means, then it must also be true in all models (I think Godel was the first to notice this). But a formal system may have multiple interpretations, more than one model. This was first formally recognized by Hilbert, who used it to demonstrate the relative consistency of different formal systems.

 Carl Gauss

However, even before Hilbert, this model theoretic vision was being used to do non-trivial mathematical and philosophical work. I'll try to explain how one can use model theory to prove that the parallel axiom is independent of the others. Start with all of Hilbert's Axioms considered purely as a formal system. Obviously, Euclidean geometry is one model of these axioms systems. Guided by that model, construct the following objects: great circles on a sphere and antipodal points on the same sphere. Call the great circles "line2s" and antipodal points "point2s". We know have new sentences that are concatenations of old primitives. If we take these sentences as our "primitives", then we find that line2s and point2s satisfy all the axioms ... except Euclid's axiom! When we get to that one, we find instead that given a line2 and a point2 not on that line2, any line2 passing through that point2 will intersect the given line2. Put aside the model for a moment. The syntax of line2s and point2s is just concatenations of earlier concepts. But we don't have to bring in the Euclidean models for these meaningless symbols. We can use elliptic geometry by itself as a model. They have two - really, infinitely many since there are so many Riemannian geometries - models. This shows that the two systems are relatively consistent - one is not contradictory unless they both are.

Ludwig Wittgenstein with his family (including his sister, a woman! Wooo)

I promised earlier to go over one of Wittgenstein's criticisms of the Fregean vision outlined above. Since we only observe people's behavior, we cannot in general know the rules that they are "really" using. Let's say that a highly educated person, like myself, is working on a programming project that puts them along side a brilliant self-taught programmer. At one point in the project, presumably as she's explaining something, she writes "1+1=2". We both agree that this sentence is true. At another point, I write on the same board "23,412,341,243+432,141,234=23,844,482,477" and she storms out of our work area and demands that I be fired. You see, she learned to add on a 32 bit machine. The correct version of the above sentence is clearly "23,412,341,243+432,141,234=2,369,645,997". Why should she have to work with an incompetent like me?

Physical behavior (including human behavior) is syntactical ("The world is the totality of facts, not of things."). We want to be able to attribute to this syntax certain semantics. For instance, I might want to interpret your sound making as a meaningful sentence. I have a model for your verbal utterances. But this model is not unique. The meaning of terms is a social process that can break down, as shown above.

This was part of Wittgenstein's general criticism of (his interpretation of) Frege's idea of language. Fregean analysis works fine on many things - for instance those Hamiltonian systems I'm always talking about. I have a great example in terms of Sinai Billiards in particular that I don't now have the energy to go through. Even given this, Wittgenstein's example shows that it may not be very good at analyzing language - which is what it promised to be!

David Lewis

In order for a rule to be learnable, it must be (at most) recursive. In fact, it must be fairly efficiently learnable to gain any popularity. Bacteria and other primitive organisms signal each other, these signals may be very low on the Chomsky hierarchy. The theory of signalling in biology is well established. It's a particularly successful application of game theory, first applied by the philosopher David Lewis in his book Convention. This may be the single most successful theory started by a pure philosopher in the 20th century. The theory was actually perfectly rigorously described by Hume in the 18th century. The core insight is that conventions are not essentially linguistic, instead language is conventional. Successful communication is given by success in some other sense (biological fitness, utility, etc.). We only care about having different models insofar as they are inconsistent and even then only insofar as the inconsistency affects things. This is not what Frege promised. He promised too much. Wittgenstein was correct to point this out.

Donald Davidson

The signalling games considered by Lewis explicitly are very simple, though he gives a sketch of how to go about human language. If signaling rules are learnable, then they must be at most recursive in complexity. Living humans have an extremely advanced immune system that is a "model" in the Hilbert sense for a Fregean formal syntax. The immune system learns and communicates in a very complex way. Humans also have spoken language which is also at most recursive. Learning a language is (partially) gaining enough of a culture that one can apply enough of a model to the verbal syntax of other speakers that behavior can be coordinated. A command is true when it is obeyed, a question is true when its declarative translation is true, etc. The varieties of signalling syntax that can be understood so are called "languages". Other signals - such as the chin flick, the "get bent" gesture, the Bronx cheer, the middle finger, blushing, smiling etc. - cannot be interpreted so and are not languages (though they may be cultural - such as the middle finger - or biological - such as smiling determined signals). The signal game is larger than the language game.

Well, we've come a long way from the original, simple Fregean vision. I believe that vision is broadly correct, even with all the adjustments made above. Frege's seeming pedantry and perfectionism made him obscure in his life except to a few people who shared similar obsessions, but they gave birth to the modern, computational world.

Idealism And Modern Science: Space And Time

Kant, again

So, another post on High German Idealism. Before, I'd been pretty kind and polite, even being careful to point out good parts in Hegel. But today, I'd like to point out a major error in the philosophy of Kant, Schopenhauer and others, one that makes much of their exposition wrong as a matter of strict fact. This major error has to do with the division between the underlying "noumenal world"/World As Will and the phenomenal world of experience/World As Representation. Kant and Schopenhauer believe that spatial and temporal order of the world is part of human experience, but not the world in itself - this is simply completely wrong. That the world of experience is 3+1 dimensional is a deep fact about the underlying physical world in itself. Further, human perception does not, in fact, take place in a mostly "geometric" manner, by which Kant would mean specifically Euclidean geometry. This is not a minor flaw, but appreciating it requires far more technical apparatus than Kant and Schopenhauer had access to, even giving them the benefit of deep insight through dim appreciations. I'm going to go about this exposition quickly but carefully. First, I will again recapitulate the core Kantian argument so that it will be understood that Kant meant these terms literally. I'll give an example of a seemingly objective property of an object that is actually not mind-independent. After that, I will give a brief description of the correct modern understanding of these facts.

Kant began his career as a serious natural philosopher in an initially typical 18th century mold. He studied physics with great intensity and soon was setting very difficult physical problems for himself to solve. He had been a minor player in the debate over whether energy or momentum is conserved - unfortunately only publishing after it was understood that both were. This book, though confused by Cartesian concepts, also contains many important insights - such as the correct general explanation of inverse square laws. Kant appreciated that conservation of momentum implied that a collapsing cloud of particles would force the system to rotate and flatten out. Eventually, he argued, the cloud will condense into a star and planets. Kant used this to explain the two dimensionality of the solar system, and went much further than that. He argued that the solar system itself was part of a much larger scale condensation, what we now call the Milky Way Galaxy. At the time, this was a novel and innovative hypothesis. It turned out to be impressively correct. Such arguments shows that Kant was familiar with conservation laws and how they can be used to give powerful qualitative arguments. When Kant was 30, he won a prestigious prize for a demonstration that resistance to tides causes the rotation of planets to slow. This implicitly involves energy considerations and demonstrated that the solar system could not be infinitely old (an open question before this).

Kant's peaceful potential life as an eminent but minor Prussian physicist was ruined one day when he happened to read a book by David Hume, the greatest of all philosophers. Kant's research had convinced him completely of the correctness of Newtonian mechanics - classical mechanics to me and you. But Hume had a devastating and novel argument in favor of skepticism of what is called "induction" - essentially learning. Induction obviously cannot be justified by empirical knowledge, since learning from observation requires learning. No particular induction can be justified on general logical principles - since such an induction would be a general deduction. There are some truths - such as conservation of energy - that are either true or not about particular systems, we have to learn whether they are. So this kind of reasoning is not enough. Finally (and this was Hume's addition to the skeptical argument) induction cannot be justified inductively - that's a vicious circle!

Hume forced Kant to see how delicately his hypotheses leaned on conservation laws that he (and everyone at that time) barely understood and were certainly not necessarily true. In particular, Hume forced Kant to question whether we could learn the laws of physics. Kant spent years - decades - attempting to carefully develop both the system of the world and our knowledge of it into a coherent whole even given Hume's critique. In 1781, he hastily published a massive tome containing all of his work, after a health scare convinced him he would die unpublished if he did not.

Kant's system is not easily explained, partly because it involves such a complex mingling and careful separation of the world as it is and our perception of it. But it goes something like this. The world in itself consists of innumerable particles. The particles move around by exchanging energy and momentum with each other, according to specific - Newtonian - laws. But we cannot see the naked system of the world - instead we see a coarse grained, psychological, language drenched, enculturated, clothed system. In modern physics, we represent the whole system by a vector, and the motion of the particles are given by the so-called "Hamiltonian" of the system. This gives the fine-grained reality of the system, but the world of experience with fluids, pressures, etc. "exists" only as an approximation at a coarse grained higher level.

Homer

The whole world of experience may be coarse-grained, but that doesn't make it "subjective", in the usual sense. Purely physical systems interact on a coarse grained level. The usual thermodynamic functions are minimal statistics of systems, so that any reasonable description of the world must include some of them. They are "forced moves" in Dennett's words.

But very little of our representations of the world are forced! In fact, perception is highly dependent on language, culture and conditioning. The most famous example - first pointed out by John Locke - is color. To an English speaker, it seems to be an objective fact that the sky on a hot cloudless day is blue and the sea on that same day is also blue. But if I were to mention this to Homer, he would be shocked! How could the sun bright sky and the wine dark sea be "the same color"? The answer is that in my culture - the culture of English speaking people - we learned to divide up the spectrum of light in ways that some are called blue and others not.

 This is a somewhat dishonest way of living, color is not so simple, color perception even less than that.  To our culture - you and I are English speakers after all - black is white. It's plain to see that the dark blue sea is the same color as the bright blue sky. If we merely apply the same argument to grey, we see that black and white are obviously the same color.

How do we survive zebra crossings then? In spoken language, our culture simply partitions sufficiently dark greys into black and sufficiently light ones into white. It's hardly less arbitrary than most of life. Our non-linguistic experience of color that motivates most of our actions may be different. Generally, we try to keep life or death situations away from subtle color gradations.

Of course, much more than just color is part of the World As Representation that isn't grounded in the underlying physical world (Schopenhauer's World As Will) in a unique way. Psychologists study physical perception in the form of affordances. Beyond that there are complex social systems that include languages, governments, markets and all that goes with them (such as philosophy).

Emmy Noether

So, if so much of the world is ungrounded in the huge vector and the Hamiltonian rules that make up The Dang-An-Sich, what makes me so sure that space, time or spacetime is part of it? In order to understand this, you have to use tools far more modern than anything to which Kant had access. Kant understood that the rules of The Dang-An-Sich conserved certain global properties such as energy and momentum. This was not an easy thing to figure out, and he had to do it for himself. But in order to understand space and The Dang-An-Sich, one must understand the connection between conserved properties and symmetry. This could not have been done before group theory, it could not have been done before Lie groups and algebras, it could only have been done by someone who understood them both. The person who did so was Emmy Noether, and this alone would have made her one of the most important persons in mathematical physics. The fact that the theory of groups was entirely absent from physics before her makes her probably the most important person in the history of mathematical physics. Kant appreciated that conservation of momentum was a fact about physical systems, what he did not and could not have known that this equivalent to the existence of an symmetry operator on the laws of physics - on the Hamiltonian. This can be strengthened by Wigner's Theorem - not only must every conservation law give a differential symmetry, but the symmetry must take a very special form. To say that these theorems is the very foundation of modern physics would be to understate how central they are.

Let's look in particular at conservation of momentum, which suffices to give the philosophical flavor. It arises from the following symmetry: if every particle was moved in a way that keeps all the relative distances the same, then the relative motions of each particle wouldn't change. This symmetry operator defines the three dimensions of space. This is a fact about the Hamiltonian, a fact about The Dang-An-Sich and therefore has nothing to do with perception. It is not even a coarse grained fact, but applies on the microscopic level. Perception may take advantage of this organization, though it only does less than one might think. Actual perception is a lot more edge detection and topological relations, Euclidean geometric representation (with it's angles etc.) is learned.

The above argument has many slight alterations important to physics but not philosophy. The symmetry operators that defines actual physical space are called the Poincare Group and they give rise to geometry which is relativistic - not Euclidean. But these alterations, constrained as they are by Noether's and Wigner's Theorem, cannot alter the simple fact is that spacetime is part of the organization of the world in itself and the Kantian/Schopenhauerian thesis that it is not is simply incorrect.

Idealism And Modern Science: Intentionality

Immanuel Kant

Last time we talked about High German Idealism, I concentrated on giving an example of how it attempts to reconcile the physical portrait of the world with the world of experience and intuition. We constructed a loose picture of what I called The Dang-An-Sich, which was - roughly speaking - the entire universe. I used Kant's name, but it could also be called Schopenhauer's "World As Will" with no loss. I said that The Dang-An-Sich was "empty of content". There was no volumes and pressures, no fluids or gasses, no chairs or minds, etc. I showed where one could find proofs that, among other things, the basic thermodynamic functions such as volume and so forth could be shown to be "minimal statistics" of the behavior of The Dang-An-Sich. Therefore, they or functions of them will be in every living thing's description of the physical world. This is part of what is called in Schopenhauer's language "My Representation", which exists and is well formed even though The Dang-An-Sich cannot be directly probed. This gave us good examples of idealism and showed that their ideas were not empty of content.

Arthur Schopenhauer

Today I'm going to talk about some more philosophical concerns of Idealist philosophers. In particular, it can be shown exactly that Schopenhauer is correct when he says  the universe as a whole, The Dang-An-Sich, must be purposeless in some sense. That is, The Dang-An-Sich has a special property that means that it doesn't care at all what overall state it is in beyond an important technical detail. This demonstration implies that any system that does care about what state it is in, called by Husserl an "intentional system",cannot be the whole universe. Therefore, any subsystem of the Dang-An-Sich that has the property that it prefers some states to others must divide the universe into an inside and outside. This means that an idealist may not be "solipsistic", in a well defined sense.

W R Hamilton

The fundamental thing about the universe as a whole, The Dang-An-Sich, the thing that distinguishes it from any other object is this: it does not interact with anything outside of itself. I will talk about a universe that consists of many, many classical particles. Each particle has a position and momentum at a particular time, so that the entire system can be seen as a vector in a very high dimensional space. This space is called "phase space" and its points are the states of the system. Any particular fact about the system at a given time is a function of the position and momentum of (at most) every particle. There are few essential changes to this picture if we move to quantum mechanics, except the dimension of the space is infinite and the algebra of dealing with the functions is different.

The laws of physics do not depend upon time, which can be derived from the first fact. Any system where the laws of physics depend upon time can be expanded as a subsystem of one where the laws of physics do not, but the universe is not a subsystem of a larger system. Therefore, laws of physics of the universe are time independent. If the laws of physics of a system are time independent, the system described conserves energy. Therefore, the entire evolution of the system is given by the level curve of a so-called "Hamiltonian" function. These functions were named after their discoverer - the above pictured William Rowan Hamilton, based on his work with optics (and Lagrange's equally foundational work). I will throughout call an energy conserving system a Hamiltonian system.

But what is a Hamiltonian? Recall that we've just proved an essential physical fact about a system - it has constant energy. The system can change phase only by moving energy around - between its particles, for instance. The Hamiltonian function captures all of the flow of energy within a system. From a given state, the amount of energy it takes to get to a neighboring state by changing the position or momentum of one or another particle (including that - unique! - neighboring state which requires no energy change) gives the change in the Hamiltonian. As before, if energy is conserved, then the system moves on the level curves of the Hamiltonian.

The most simple Hamiltonian is that of a harmonic oscillator. The idea is of a particle bobbing up and down, as on a spring. As the velocity goes up, the particle gets a little farther (closer) from (to) equilibrium. This causes some of the energy to move from (to) the spring and restore . As a result, the level curves are simply ellipses. We can similarly find the results for pendulums and many other system. Most Hamiltonian systems cannot be solved exactly, but wander around state space almost randomly. Much like a fractal, such curves (nearly) fill the volume of state space.

There are many important facts about Hamiltonians. For instance, their level curves (constant energy trajectories) of a Hamiltonian never intersect, so that no two identical systems will be in the same state unless they also have the same energy. Classically, they can get as close as they like, however quantum mechanics forces a discrete separation. Energy is therefore a macroscopic "state function". There is no cheating here, since non-dependence of the laws of physics everywhere is not a local property, we shouldn't be surprised that one derives global properties from it.

Possibly the most important fact about Hamiltonian systems is what is called Liouville's Theorem (notice, again, there is a proof in the quantum mechanical case as well). This means that a cloud starting points of always has the same "volume" as each point moves on its own curve. Looking at the above example. If one draws a circle of starting points on the above graph and lets follows the lines, the ellipses will stretch and bend but never grow or shrink. This means that, in particular, it is never the case that the circle grows or shrinks. This is perfectly general.

Liouville's Theorem implies that there are no stable equilibria for a Hamiltonian system. In the oscillator example, the system stays still if the spring is left at rest, but every perturbation no matter how small means the system moves forever. Since the universe is a Hamiltonian system, it has no stable equilibrium states. This means that the evolution of the universe cannot be "toward" some final state. The Dang-An-sich, the universe in itself, has no preferences among states. It just wanders around state space. It is not only empty not only of content, but it also has no goals.

Edmund Husserl

Edmund Husserl is often called the "father of phenomenology", supposed to be an exact philosophical science of all perception. Husserl was originally a mathematician trained by no less than Leopold Kronecker and Karl Weierstrass. Like many of Weierstrass's students, he was acutely sensitive to foundational issues in mathematics. This lead him into philosophy, where he was inspired by the philosopher and co-founder of psychology Franz Brentano (you might have heard of another one Brentano's students - Sigmund Freud). Brentano was a Catholic priest and took from the Scholastic's interpretation of Aristotle and Aquinas the idea that conscious is always directed at something. One can be conscious of one's surroundings or of one's goals or (most importantly for the Scholastics) of God, but not conscious in general. As G K Chesterton said in Orthodoxy "The worship of will is the negation of will ... because the essence of will is that it is particular.".

Husserl claimed to invent a psychological/philosophical/transcendental method of achieving absolute certainty by "bracketing" each little bit of sense-data and examining it, disregarding questions of its existence. Every time we bracket a blob of sense-data, either 1) we discover it's content is identical with something we already are certain exists or 2) our world grows by one object (More on this in a bit). Why? We may be absolutely certain that we exist and the existence of an object toward which consciousness is directed toward. If it can be known that it is not an object that we were previously aware of, then it is a new object. Therefore, we can supposedly - very slowly! - build a build a world of absolute certainty.

There are flaws with this idea. A system which is directed may not be conscious. Alfred North Whitehead said that it was a profound mistake to think about what we are doing. Not only may the majority of the activities of a system that is conscious be only scarcely directed by consciousness, some of the activities we value most may be barely conscious. This was pointed out by Heidegger to Husserl, who ignored it. The "bracketing" process is vague on how we can learn enough about a piece of sense-data to absolutely know it consists of an object about which we do know absolutely know, kicking that whole important process over to science per se. It isn't clear whether bracketing is psychological or transcendental. Husserl himself changed his mind about this - initially he thought it was psychological, later transcendental. Husserl was a Christian (a Lutheran), but it isn't clear how to treat things we have no sense-data of - like the divine.

But one of the important assumptions, that the above concept of intentionality (interpreted in a highly minimalistic way) always implies that there is at least two "objects" is rigorously true. It follows from Liouville's theorem above. A system that prefers a given, for example, temperature, it must have an outside. This is not a trivial factoid - it is seen in real physics of Hamiltonian "thermostats". These can be checked theoretically and numerically. One can also consider "barostats", etc. that prefer states with particular values of other thermodynamic potentials.

Since human beings are - among other things - thermostats and barostats, they may not be closed systems. Therefore, one may not be The Dang-An-Sich by oneself. This shows that there should be no idealist solipsists.

I have stated all of this without reference to the higher level phenomena of actual experience. I left out the "minimal statistics" state functions (other than energy) such as pressure, volume, etc. These state functions can be described as functions on every possible state. We can then define a "macrostate" as the set of states such that all the state functions are the same for each state (or "microstate") in that set. Here the story actually gets a bit more complex. It turns out there are some macrostates that have a lot more microstates in them than others. Since "most" Hamiltonians wander around phase space almost at random, we can see that a Hamiltonian system will (probably) spend (almost) all of its time at the unique macrostate with maximum entropy. This can be made much more precise, of course.

It is not clear to me yet how this relates to the simple story of Schopenhauer and his followers (such as Heidegger and Sartre). It is philosophically important that The Dang-An-Sich has no direction, but it is not so clear that the non-intentionality of My Representation follows from any principle. I would like to take up this some time later, but no promises.

Karl Marx At The Sunset Of Classical Economics

Adam Smith

For such a famously humble man, Adam Smith left a complicated legacy. A friend and student of David Hume and Francis Hutcheson, Adam Smith took it upon himself to inquire upon the Nature and Causes of the Wealth of Nations since the wealth of a nation had so much obvious bearing upon the quality of life of its inhabitants.

David Ricardo

Within Wealth Of Nations, there are mixed up two very different forms of analysis which are not easily reconciled. On the one hand, there was analysis of the "state of nature" and "stationary state", both concepts learned from Hume. In these states, there was a labor theory of value and life was static. On the other hand, there was the dynamic life of growth that he saw in England and America, where he applied a more lax "Supply and Demand" analysis. In so doing, Smith was the father of both classical economics as developed by Ricardo and neoclassical economics as developed by Alfred Marshall.

Thomas Malthus 

The stationary state is one in which land and capital have been distributed and innovation is low enough to be negligible. It is a very unpleasant place: "Though the wealth of a country should be very great, yet if it has been long stationary, we must not expect to find the wages of labour very high in it.". Without movement or innovation, capitalism would then take on many of the aspects of feudalism - Adam Smith examined the late Chinese Empire as a model stationary state.

Without innovation, wages would become a decreasing function of population. This state is often now called "Malthusian", even though it was first developed by Smith and its reality and immanence was not questioned by any classical economist (including Marx). Ricardo (and Malthus) differed from Marx in their predictions about the stability of the stationary state. All agreed that it was, in Marx's famous words, the center of gravity about which capitalist prices moved. Ricardo and Malthus believed that it would be stable, and that life would be horrible forever.

Karl Marx

I promised in a previous post to explain Marx. This can now be done. The price of a good or service (I will say "task") in the stationary state is called by Marx its "value" and by Adam Smith its "natural price". Again, in the stationary state technique, resources and factor cost is fixed for each firm. It is easily seen that in this case, aggregate output is solely a function of the quantity of employees hired. In addition, with fixed technology, a given level of employment for a given task fixes the return for that task. Since the price of a task is  simply the ratio of the total income each unit rewards to the count of units, price in the stationary state is solely a function of labor allocation. This is the "labor theory of value" stated in a fashion both simple and acceptable to anyone.

One can go further than this. In the stationary state, each child born is born to a given task. That, under capitalism, laborers beget laborers, and Irishmen Irishmen was unquestioned by Marx (he despised it, which is different than questioning its truth). This means that price was, in fact, a function of total population. One could not ask for a stronger labor theory of value than the value of every task being solely a function of total population!

Marx went so far as propose a theory that part of total population was left idle in the stationary state, what he called "reserve army of labor". This is no small achievement, it was probably the first quantitative theory of unemployment!

Leon Walras

What about the neoclassical economists? They fit into this scheme just fine. Let me explain. I think it is likely that the Smith-Ricardo-Malthus description of the stationary state that culminated in Marx is correct. Karl Marx also insisted constantly throughout his life (in all three volumes of Capital, for instance), that supply and demand determines the fluctuation of prices. (Wage Labour and Capital suggest that he had in mind something like a cobweb model in mind, but that is a digression)

What does this mean? From a mathematical point of view, Marx's thoughts are a special case. It holds when land, technique, resources and factor cost are all fixed and laborers will be distributed in a unique way between different industries. Marx would, if he were not in a nasty mood, agree whole heartedly! Marx's point is that the Ricardo-Malthus analysis (he would be upset to see Malthus there) is the economically central case. Not the more mathematically general or logically most important. Marx believed that this was a globally asymptotically stable equilibrium, the only results of the economic laws of gravity was to approach it. This is a perfectly logically consistent way of viewing the world. Marx would be pleased with his "critic" who would point out that we can also think of other societies that have different equilbria - that's why Marx was a communist after all!

Robert Solow

Recall that I said Adam Smith had two legacies. The classical economists expanded on his Humean analysis of two possible static equilibrium. But there was another, less developed but more original side to Adam Smith. This was the Adam Smith of supply and demand; the Adam Smith that noticed that England, the United States and probably all of Europe is nowhere close to a stationary state. The Adam Smith who was perfectly capable of noticing that demand curves slope down.

The early neoclassical models of Cournot, Alfred Marshal and Leon Walras were one period models. As a result, their relationship with the theories of Ricardo, Malthus and Marx were terribly unclear. Austrian economists had a particular interest in extending these models to many period, however this line of thought had its greatest culmination in an American economist - Irving Fisher. However, this only increased the murkiness of the relationship of neoclassical models to the Marxian stationary state. The reason was that the Fisher/Austrian models had "periods of time" (and therefore an interest rate), but no growth per se.

The first theory of stable, exogenous growth was proposed by Robert Solow and Trevor Swan in 1956. (I will ignore endogenous growth, which is irrelevant to Marx) Possibly the best introduction to the model and neoclassical thinking about it is here, but Solow's original paper is excellent. Finally we had a model that could accept changes in the supply of labor and the intensity of capital! The analysis of this model may seem to lead to a very Marx-esque solution, asymptotically only technical change produces growth. Since - as noted above - in the long run of Marx there is no innovation, this is okay. But as a matter of fact, this model, and virtually all modern growth models, propose a path of steady growth with no end in sight! There is no cap on the population, no steady state wage stagnation and capital can be accumulated unlimitedly! There are Solow models with fixed inputs, such as land, but the possibility of unlimited growth is baked into the analysis.

How is this possible? Where Marx set economic growth to zero in the steady state, Solow looks to a more economic and less mathematical definition. An economy's growth is called "balanced" if its capital stock divided by its total output is constant. That means each machine is asked to stamp goods at only a certain, reasonable rate. Holding a ratio still means that the numerator and denominator go to infinity - as long as they do so at the same rate. What makes Solow's analysis brilliant is that he is able to find mathematical meaning to that fact and get a quantitative model out of it! The result is clear: economic center of gravity is at infinity, and value in Marx's sense is simply unimportant.

Marx (and his hated enemy, Malthus) would think this analysis is foolish. The crash is going to come - either in population (Malthus) or capital (Marx). This brings us to the sunset of classical economics. As the years and generations - much more than a century - has gone by, this stylized fact and great prediction has failed and failed again. I don't say that the day is over for it, but its light is going out...

Unexamined Life: What Have I Learned From Philosophy?

I have read a lot of philosophy, perhaps more than the subject deserves and perhaps less. Philosophy is odd, everything is obvious to a philosopher but nothing between philosophers. Philosophy has been prepped for the trash pit many times, by religious fundamentalists, by over-reaching scientists, by mad governments, by great philosophers and - most fatally - by many bored readers.

Albert Einstein

The Pythia said "No one is wiser than Socrates.". In his day, he could walk up to what we would call a scientist and defeat them - clearly, nobody knew more than him. Much of the disrepute of modern philosophy is rooted in this no longer being true. No matter who you think the greatest living philosopher is (and I doubt that is a definite description), you probably wouldn't go on to say that they are the smartest person in the world. You'd probably admit Terence Tao is at least a little better at math.

But I didn't put Einstein up there because I think he was smarter than Husserl. The disrepute of philosophy comes from another, related source: philosophers aren't our deepest thinkers any more. Einstein has had more influence on us how we think about time than Carnap or Heidegger, more on how we think about space than Bergson or Whitehead. Nietzsche's philosophers of the future don't call themselves philosophers.

I do think philosophy has an important place. What makes humans unique is that they can understand what they do (this, of course, is a philosophical opinion!). In many instances, philosophy is just thinking about what we do. When you read a paper like (Alchian, 1950) or (Krugman, 1996), you are seeing philosophical argument. In this case, there is complete agreement, but I would say - another economist, another philosopher might disagree.

Daniel Dennett

So, what have I learned from philosophy? What philosophers - and "philosophers" - have influenced me? The answer is simple: Daniel C Dennett III. I'm going to leave him to another post as being too important. I will admit I have been more influenced by the technical/logical philosophers of the so-called "analytical" school. It isn't that I think they are smarter, just as a mathematician their work is often more directly relevant to my daily life. They are almost all black & white, dead men. I should include female philosophers such as Susan Haack & Deborah Mayo. In fact, the only reason I didn't include Mayo because I left her book in America. I'm leaving continental philosophers for another post.

Jaako Hintikka

Despite my fascination with intuitionist/constructive approaches, their philosophical views I find less interesting. The math is neat, but the philosophy is weak. The philosopher that has influenced my view of mathematics the most is Jaako Hinitikka. Unfortunately, he was not as cool as the above picture makes it seem. Hintikka developed what are called "Game Semantics" for quantifiers. This is the best explanation for why classical analysis has the structure it does. The reason is that classical analysis is based on non-refutable arguments. It's best explained with an example.

Let's say that I claim a given function, perhaps the angle of a shower knob and the equilibrium temperature of the water coming out of the shower head, perhaps the solution to a DE, is continuous. What does this mean? One might say that it takes on the value of the limit on that point. But this is not the point of view of classical analysis. In classical analysis, the important thing about my claim is that you can't disprove it. Let's say we know the function takes on a value at a certain point - we know by, say, measuring the heat of the water when the knob is at a certain angle. When I say it is continuous, that means you can't truthfully say it doesn't get close to that value when the angle is close. If it could, you could say it's getting near some over value. We can call the difference between the measured value and the "error". But any given amount of error is too much, I can always just sneak by it. You can't prove that there is a jump, therefore the function is continuous.

Hintikka gave formal rules for interpreting any sentence from classical mathematics like this. That is by itself a huge deal.

Wittgenstein

The first philosopher to genuinely fascinate me was Wittgenstein. I spent a month reading and re-reading his Tractatus Logico-Philosophicus, trying to find the meat hidden on its austere bones. On the technical side, Wittgenstein co-invented the truth table within it. On the more philosophical side, it pointed to a metaphysical semantics of the new logics and set theory. On a deeper level, it pointed to a world beneath language and made sense of the idea that there was more on Heaven and Earth than in our philosophies. In his later work, Wittgenstein would attack the metaphysical parts of the Tractatus, on the grounds that even if the metaphysics was true they had nothing to do with why we believed that they were the case. This attack, laid out in Philosophical Investigations, naturalized language in a way nobody had seen since Hume. Only after this book could we go back and see how wise Hume was. I don't think that the attack affects the value of the Tractatus metaphysics as a semantics of set theory, but certainly no one will take them without a grain of salt anymore...

Thomas Schelling

Thomas Schelling's Micromotives And Macrobehaviors is one of the greatest books on social philosophy I ever read. Along with the Tractatus, it took me apart and put me back together a smarter and wiser person. It is hard for me to summarize, but I don't feel bad - it is hard for him to summarize too. Often we associate game theory with rigorous mathematical analysis - on the grounds that if Von Neumann was doing, so must everyone else. Schelling was given a Nobel Memorial Prize for contributions to game theory, but he never used math in any deep way in his work (contrast with the other winner, Robert Aumann). The important thing for Schelling was that game theory forced the user to consider the effects of his actions on others, and theirs on himself. What mattered to Schelling, in other words, is the notion of an "equilibrium". Game theory then is as much Hume & Kant as Luce & Raiffia. There is no formal "game theory" in Micromotives and Macrobehaviors, but there are hundreds of examples of equilibrium arising from social interactions giving results paradoxical and straightforward.

It was perhaps Hegel who was the first to recognize the importance of self-negating equilibria. Every society comes with it a set of norms & expectations. But every society so far has had some norms that force in conflict with the expectations. Eventually society adjusts its norms to remove this contradiction. Each society is out of equilibrium and therefore history matters to it. Unfortunately, in Hegel, these notions are tied to a history that is, in most matters of fact, false. I will give the Marxist version: In perfect competition, the wages to labor will be subsistence wages (this assumption was common to all classical economists) and technical change will tend to deepen capital (this is a norm). Capitalists expect capital deepening to profit them (this is an expectation), but will find in the long run they can't all deepen against each other (this is called fallacy of composition). Instead, capitalists will be paradoxically trapped underneath a mountain of less profitable goods. This is a Hegelian contradiction, a self-negating equilibrium. Marx's proof used the labor theory of value instead of the fallacy of composition, but it amounts to a different gloss on the same thing. Marx might have been smart enough to make that argument by himself, but I needed Schelling.

David K Lewis

I feel kind of odd putting David K Lewis on this list. He's too important to me to ignore, but my disagreements with him are part of what made me keep reading philosophy. He was the first academic philosopher that I liked, not just the philosophical aspects of a scientist or economist. Lewis made permanent contributions to mathematics, but I have to say - I basically am completely uninfluenced by them. I've never once started a proof with megethology in mind. Lewis is most famous for his adapting Wittgenstein's "The world is everything that is the case" classical logic semantics for modal logic - the so-called "possible world" semantics. They nearly single handedly brought metaphysics back into academia. Lewis (and, to a greater extent, his ally Daniel Dennett and, to the optimum extent, linguist/mathmatician Noam Chomsky) helped bring analytic philosophy out of behaviorism. But Lewis's solution was Bayesianism, which is not to my own taste (though it is important and I'm glad someone was working on it).

Of his ideas, his concept of a coordination game has been the most important and influential. He claimed to have been inspired by Thomas Schelling above. I first read about it in his book Convention, which is also where it was invented. I've gone over this before. His definition of value as what we "desire to desire" is something I've been thinking about recently. I like the way that it reduces the theory to preference theory (for the classic reference on preference theory, see Debreu's Theory of Value - Debreu's "Value" is value in another sense) without impoverishing the value part of the apparatus. The words of Kant are very comprehensible: what we should desire to desire is those desires that are coherent for the population. The words of Bentham too: what we should desire to desire is the satisfaction of the most individual desires. Hegel (and Marx) pointed out: our desires generally conflict, society is out of equilibrium. You could, if you want, do all social philosophy this way.

In fact, one reason I like Lewis is his congenial approach to formalization. Formalizing philosophical concept of value by putting it in terms of (in some sense "reducing" it to) the preference theory of economics allows us to sharpen and clarify the philosophical differences of old - but shouldn't try to artificially "solve" them. This seems to me to be a right way to go about things.

Robert Nozick

Okay, so if I felt odd about David K Lewis, I have to say this about Nozick: I've never read his big book on political philosophy. I've read an article attacking Ayn Rand and another one attacking "Austrian" Economics, but hey, easy targets. I'll come out and say it: Nozick tried to make a philosophical explanation of what we call "libertarianism" or "classical liberalism". I won't address whether his argument fails since, again, I haven't read it. I kind of doubt that reason/philosophy alone can make a political argument look good - logic shorn of evidence tells you nothing about reality.

What I find most interesting about Nozick is his last book, Invariances. In this book, Nozick develops a novel explanation of what is "objective". The usual philosophical gloss is that something is "objective" iff we could conceive of a completely physical description. This does not match what we usually mean by the word. When someone is pointing a gun at me, I would say they are objectively being a menace. I do not mean that there is some physical description of him being a menace. What I mean - according to Nozick - is that their being a menace is invariant over the variations relevant to the conversation. This ties down the notion of objectivity to Wittgenstein/Lewis idea of language games.

Nozick is interested in the implications for ethics - Could there be "objective ethics"? This is less interesting to me, but I'll go through it nonetheless. Nozick tries to build up from a libertarian state to a democratic state using the idea that some values (desired desires) can be better served by democracy/market mixture than a "pure" market. Rawls, another ethical/political philosopher, proposes his "veil of ignorance" argument as basically Nozickian objective ethics. I think that shows that there are, in fact, too many Nozickian objective ethical systems. These examples can be multiplied until and beyond one reaches the count of ethical philosophers. Nothing says that one definition of relevant variations is the right one.

My plan now is to do three more posts like this. One on Daniel Dennett, the philosopher who influenced me the most. Then a third post on classical and continental philosophers, who are important and deserve mentioning even if they haven't struck my fancy.

First color picture of the post

Also, you may or may not see a review today. I've been having internet troubles and am having a hard time watching videos.

Hume and Edgeworth

Or: The Consistency of The English Philosophy!

Hume has been called "one of the most important philosophers in the English language", his skepticism and empiricism inspiring - in positive and negative ways - whole swaths of philosophy starting in the 19th century and continuing to this day. Hume is best known for his attack on the connection between empirics and metaphysics. He argued that we have no way of demonstrating empirically that an event necessarily causes another event. This is of great importance, and both the conclusion and the argument are pregnant with possibilities. He would later clarify that we do have reason to act on the belief that an event causes another event, their (near) constant conjugation instills in a belief in (nearly) necessary cause. This positive solution - often ignored by philosophers - flowered into the associationist school of psychology, game theory, the Bayesian school of statistics.

Francis Edgeworth was an English economist, the author of Mathematical Psychics. In this book, he attempted to take the Jevonian gloss on Utilitarianism a firmer mathematical foundation. Modern economics, with its utility functions etc., is a descendent of this line of thought (which sprung in many places at several times) but is not intrinsically tied to it. Honestly, I haven't read Edgeworth's work in detail, but I have read a little bit of Mathematical Psychics and some of his stats papers for something I wrote on the history of statistics. In Mathematical Psychics, Edgeworth develops an ingenious device for thinking through bilateral trade now called the Edgeworth Box.

I will use this graphical method to illustrate a famous passage of Hume on co-operation. A modern thinker might think that it is obvious that a non-linguistic, behavioral definition of convention and co-operation was possible, and indeed since Lewis it has been standard to found the concept of linguistic meaning on pre-linguistic ideas of co-operation. In Hume's time, it was not so obvious. Hume had to explain "... [C]onvention is not of the nature of a promise: For even promises themselves, as we shall see afterwards, arise from human conventions.". Therefore, Hume gave this as illustration: "Two men, who pull the oars of a boat, do it by an agreement or convention, tho’ they have never given promises to each other.". This is a very important point! Not only do Lewis and Hume, and biologists following him, tell us that this is how meaning got into languages ("In like manner are languages gradually establish’d by human conventions..."), it was immediately used by Hume to explain how property got into society:

"Nor is the rule concerning the stability of possession the less deriv’d from human conventions, that it arises gradually, and acquires force by a slow progression, and by our repeated experience of the inconveniences of transgressing it."

These passages of Hume are pregnant with theory, and the modernity of the theory is sometimes surprising. Hume's theoretical stances - which are those of evolutionary game theory if I may be anachronistic - run deep. Property is, he says, some sort of evolutionary strategy, one with advantages and disadvantages. Hume, obviously, believes the advantages outweigh the disadvantages. This is a story about property that can extend beyond humanity (Hume was wrong to deny this), and it has been used - by Maynard Smith and others - to examine the phenomena of nesting in animals.

Let's analyze one of these pieces, with the more modern equipment of an Edgeworth Box. Two men, Mr Blue and Mr Green, pull the oars of a boat. They must paddle the same speed in order to avoid moving in a circle. Even if these men do not speak the same language, they can and will co-ordinate. We will ask more than Hume does explicitly here (he makes more assumptions implicitly elsewhere), we will ask that the men understand that you cannot go faster than the slower paddler (we don't assume that they know the others strength). The possible speeds they can go are a set of real numbers, setting up an axis:

That gives the following box as the range of possibilities:

Our assumption about their preferences gives them Leonteif indifference curves. For Mr Blue, his indifference curves are:

And for Mr Green:

Putting these together, we get:

Let's say they just start rowing at some speed. That means they get something like the following:

Anywhere inside the square of which that dot is the corner is better for both! A simple way to think about it is that the faster rower knows to slow down, but not slower than the slower rower, and the slower rower knows to speed up, but not faster than the faster rower. Eventually, the rowers will come to a corner on both curves. Here, neither rower can improve by himself. This is a stable situation! Here is one possible solution:

Edgeworth pointed out that this is not the only solution. In fact, there is a continuum of solutions called the "contract curve" or "core":

This is the Edgeworth analysis of Hume. Hume's correctness is not in doubt in this manner of thinking. Hopefully this shows both the depth of Hume's thinking and how it relates to modern ideas. I wouldn't mind if it helped one understand the modern ideas a little better too. There are more extensions that can be made (what happens as the quantity of rowers climbs? What if they can only imperfectly measure the others speed? What kind of equilibrium did we obtain?). Notice that Hume didn't make any explicit assumptions about the nature of their preferences, yet the Edgeworth explanation explicitly assumes convex preferences. Can non-convexity be made sense of here? What other interpretations of Hume are possible - do any of them attack the substance of this translation?

Incidentally, I had a devil of a problem making the images for this post. Matlab decided some of the lines I drew just weren't good enough for her. Awful thing it is, when I turned off the axis, the invisible axis was over the lines I actually drew. Goddamn thing. Some of these images are corrected, some not. The ones which were not may change if I come back later.