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.

Against The 70s

Someone not so recently asked a big question, "Why does the 1970's inflation matter so much to economic thought?". There are so many stories told about that period that it has passed into myth. It isn't really clear why, at least if you look at it through the lens of high powered macroeconomic theory. Supposedly it all has something to do with the Phillips curve. The Phillips curve is an expression of the observation that the rate of inflation is negatively correlated with the rate of unemployment. The Phillips curve hypothesis is that this correlation is stable (at least, holding institutions constant) and that the order of causation can run from inflation onto unemployment. The usual story is simplistic: "Back then we believed in the Phillips curve, but the 70's taught us not to do so.". That sound you hear is a vortex generated by philosophers of science rolling their eyes.

When you try to put meat on these bones, you find they break too easily. It isn't easy to find a high flying macro theorist who actually believed in a stable, exploitable Phillips curve. The classics certainly couldn't believe in such a money illusion; Keynes couldn't have argued for a stable inflation/unemployment trade-off (if it existed, why would we need G > 0 ?); Samuelson and Solow certainly didn't believe in it; nor did Fed Chairman at the time Arthur Burns. It's hard to find a single person that "believed" in the Phillips curve in the way it is said to have been.

Milton Friedman

These facts - and they are brute historical facts - have led some to believe in a conspiracy theory of the 70's. "Milton Friedman and the Chicago School convinced everyone that the 70's 'proved Keynes wrong because the Phillips Curve was wrong!1!' was all a lie and therefore modern macro is an illusion meant to disguise naked power grabs by The Enemy.". I put it in a way that you could see how fallacious such reasoning is, but I've seen it put almost as bluntly before.

It is strange, you have to admit, that such brilliant people would be snookered by it. Not just right wing radicals who want validation came out of this. Ed Phelps, Stanley Fischer, and Tom Sargent all saw ... something invalidated. But if it wasn't the Phillips curve, what was it?

Stevie Nicks

The first thing that you have to realize about the 1970s is that it was not, in fact, the 2010s. Nobody in 1976 - not Nixon, not Friedman, not Samuelson and not anyone in heterodox economics - was also in 2016. Everybody was groping blindly and if some had more insight than others, well we should be so lucky. The other thing to realize is that high falutin' macro theory is a tiny and rare thing. Nixon certainly never read a single work in the field and didn't have any friends that did. Ed Cole - one of the presidents of General Motors - could tell you a lot about the Chevy Corvair or Vega, but knew and needed to know nothing of the debate over large scale statistical models and certainly he had no opinions worth noting on anything as abstruse as the Cambridge Capital Controversy that was so distracting in the 60's. This is interesting given how macroeconomic his job was. Some basic facts from Wikipedia: GM at its height measured its profits in % of GDP. GM was the second largest employer in the world - behind only the entire Soviet state.

So, given that Cole was no expert in high macroeconomic theory for its own sake, what did he believe? Clearly, I can't read his mind. But I can give you a picture of what he likely believed, even if he would quibble with a nibble or two. If you are a fan of brief summaries, I'll give you one: "He thought he lived in the era of Mad Men.".

John Kenneth Galbraith

The person who most clearly put the vision to paper is J K Galbraith in his books The Affluent Society and The New Industrial State. Since this vision failed, it might seem that I came to pick on him, but I actually a lot of sympathy for him. He was trying his best and had a basically empirical outlook. He was basically right on backing imperfect competition. Even if he went too far and replaced it with something equally simplistic, Galbraith was right to question consumer sovereignty. He worried about the structure of the firm and capture of government legislation by business before it was cool. Galbraith was trying to think through ideas that don't formalize very easily. He was trying to get away from the myth of the rational consumer. Herbert Simon was working on similar ideas and did better work, but never anything macro related. Galbraith tried his hand and if he didn't succeed, well, did anyone?

Carlo Ponzi

Galbraith's method is developed in his second book, The Great Crash, 1929. This is a strange book for a modern to read. The first thing one notices is how little a role the year 1929 plays. The lengthy, hilarious section on the Florida real estate bubble is the best part of the book - but what all does it have to do with the Great Depression? This part of the book is an argument - I don't remember if it is explicit or implicit - that the market is not guided by rational consumers. The masses are irrationally attracted (that is to say, they will invest more than they would expect to receive if held down and forced to think it through) to the promise of easy money, even if it comes from Carlo Ponzi or Donald Trump. They are irrationally repelled by the slightest loss. They put good money after bad. They're moved by emotional displays from the wealthy. They do many things, but they do not maximize net present value like Irving Fisher taught us.

Sigmund Freud

Though the masses are not very good consumers, the advertisers and engineers, they're very good (supposedly). From this vision, the corporation emerges as the fundamental entity of economics. A corporation consists of four parts (my typology, not his): the capitalist, the engineer, the laborer and the advertiser. The advertiser has read his Freud and empirically studied the deep parts of human nature. As depicted in Freud On Madison Avenue, the advertiser designs the aesthetics of the car to be a giant phallus with a clitoral emblem on a vaginal grille. He can then determine exactly how much the irrational consumer will buy in aggregate. The engineer then designs the car as a functional item, which selects the costs of production. The laborer and capitalist then build the cars. The income is then divided among the four parts of the corporation by the labor contract (which is fixed by negotiation between the capitalist and labor, which in practice is represented by union officials). Note that it is not the profit that is divided, but the income. That doesn't matter in this system - income and profit are jointly decided by the engineering and advertising experts. The corporation never has to worry about society wanting less products in general, demand will be simply created by the government if it ever accidentally slackens.

There is much to criticize in the above system. The biggest problem is a very strong difference that is assumed to exist between the irrational common consumer and the tiny echelon of experts that control them. There's no way to get around this, there should be no apologetic for it and there is no question into which class Galbraith put himself. It is assumed, not proven. When pushed on this point, Galbraith would fall back on his endless supply of jokes about irrational consumers. In his economics and his novel, this was all but explicit. The only question was whether people Galbraith would be allowed by his fellow elites to make the masses a comfortable world. There is no freedom to give them.

The role of the state is very strong in Galbraith's mind. I've already mentioned maintaining demand at a full employment level. Another is managing labor unions. The labor unions face a macroeconomic prisoner's dilemma. Imagine two labor unions that have the choice of either asking for higher wages or keeping them stagnant. If they both keep stagnant, then the price level stays the same and everyone is well off. If just one asks for an increase, then its members are much better off. But if both ask for an increase, relative wages stay the same but the money price of everything goes up. Therefore, people are worse off. (This actually happened in England in the 70's) This is one of the state's tasks as countervailing power against the large corporations (and their unions). The simplest and most dangerous way of doing this is wage controls - Galbraith never bothered to ask for another one. Price controls in general follow from the same argument on advertisers instead of laborers.

 Panasonic Space Age Television

Outside of the secluded world of high economic theory, much of what I just exposited was uncontroversial. Arthur Burns, the head of the Federal Reserve at the time, believed much of it. Galbraith's books were best sellers. Further, it was in the general culture. To be an adman in the 60's was to be the king of the world.

What happened in the 70's was the fall of the whole idea. For the first time, the mandarins in charge were forced to admit that demand management was non-trivial. The incoherent system of monetary policy, price supports, unpredictable government regulation and massive war time spending interacted with oil-induced supply shocks and changing culture demand shocks to pull aggregate demand in every direction. The net result was gas queues, stagflation and malaise. The role of the failure of price supports, while not shocking to any theorists, bolstered Friedman's claims that markets were a necessary part of demand management (he didn't put it like this). Friedman prestige didn't come from just getting it right, it came from how he got it right. No high powered macroeconomist denied the possibility of stagflation - but Friedman happened to have the perfect combination of being against price supports, for rule based monetary policy, for smooth government regulation (okay, I'll be fair. He was against government regulation in general) and having done deep work about how the economy can smooth over shocks. It was exactly what people needed in 1971 (well, government regulation in general is arguable, but certainly not basically arbitrary price controls).

The markets upset the Galbraithian vision in a deep way. Recall that GM's deeply learned advertisers decide on how much sales they make this year. The human mind cannot resist the sexual allure of their automobiles ... supposedly. But the oil crisis meant that people wanted smaller, more efficient cars, not rolling slabs of steel. The Japanese entered the automobile market with cars that people wanted and GMs sales declined. Wasn't GM supposed to control sales?

Another example. When GE designed a television, they decided the sales. They knew - knew - what people wanted in a TV: they wanted a wood exterior and a durable stainless steel frame. They wanted furniture (I know they thought this, because I've talked with the people who built them at the time). When Japanese companies started exporting cheaper, lighter all plastic televisions, GE was sure no American would want one. As it turned out, those "irrational" consumers resisted the psychological allure of the expensive American furniture and just bought cheap, functional boxes. This is not how the system was supposed to work! Didn't these consumers know that they were irrational?

The important thing about the 70s is that they seemed to show that irrationality could only be pushed so far. Consumer preferences can't be written off as a minor addendum. The point of Friedman about inflation expectations in the Phillips curve is minor. People had been writing about inflation expectations since monetary analysis began. What was influential was his whole approach. It was part of a general tendency to move toward rationality - game theory began to go deeper into traditional economic realms (such as industrial organization). And damn it, if you presume consumers act basically according to their preferences, then price controls and shocks cannot coexist. Friedman and others hammered on these points as far as they could. And then they were pushed even further...

You see, through the lens of the 70s experience outlined above, purely rational expectations economics starts to look good. However, once the "t"s were crossed and the "i"s dotted, the New Classical school that supposedly took them as their basis didn't live up to its promises. I mean this explicitly - they promised to pass statistical tests that they did not. The New Classicals stopped using these statistical methods because "they were rejecting to many 'good' models". Perhaps this movement toward "rationality" was itself irrational. In many ways, by the time the 80s ended, the rationality revolution of macroeconomics was spent. In other fields (the aforementioned industrial organization, for instance), the move to rationality bore better fruit.

Even with all these qualifiers, it was still the 70's that forced people in power to consider the consumer as an autonomous human being. And that is why it looms so large in our thought.

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.