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.

Kantian Origins Of Peircean Frequentism

Immanuel Kant

Charles Sanders Peirce was an early "evolutionary" philosopher. He believed that while our knowledge was now imperfect, correct science would - as a whole - learn to reduce those imperfections. He was a serious student of German Idealism (famously, he studied philosophy by reading one page of Critique of Pure Reason a day). He also helped found statistics, experimental psychology, modern logic and much else. Today, I want to look into how his interest in philosophy and statistics cross-bred.

CS Peirce

How much of an evolutionary philosopher was Peirce? He went so far as to define "truth" as the outcome of an ideal scientific process. For instance, imagine we didn't know Peirce's first name. We could look it up in a book, you say. That's the best scientific practice, therefore that's the truth. Let's be more extreme. Say that, for some reason, direct records of his first name had been lost. At first we would only know that his name is in a certain set. By our knowledge of human language we know that his name isn't "Hmxfrzt". By historical considerations we can eliminate "Cao Pei" and "Christina". Through long search and careful philological textual criticism, eventually we figure out it was probably "Charles". Therefore, it is true that Peirce's first name was "Charles".

This is eccentric because we normally think of "Charles" as being Peirce's first name because of actions done in the past (namely, his being named by his father), not because it is an outcome of actions of philologists of the future. This definition will even have an important effect in his statistical prescriptions.

Peirce's definition didn't come out of thin air. To recapitulate: On the one hand, he was an experimental scientist inspired by his work in physics, psychology, etc. On the other hand, he was a serious Kant-inspired philosopher. In particular, Kant's image of the sensible world of experience and the unknowable world of things-in-themselves was an inspiration to Peirce as a statistician. The world we can see, hear, smell, taste & feel is called the "phenomenal world" (as in, it's where phenomena occur), the deeper underlying world is called the "noumenal world" (we'll get to why in the next paragraph).

How do we gain knowledge of the noumenal world? Remember that this is the old days, before some young Germans questioned Newton & Euclid. So most people believed we did have knowledge of the underlying world-in-itself. Kant did not. Kant believed that the phenomenal world was basically psychological and sociological. Human beings evolved to perceive the abstract world-in-itself in Newtonian/Euclidean ways, he thought. We - our society - adopted conventions constrained by those evolved capacities. This mode of thought was further developed by Schopenhauer and I've covered it on this blog before.

Peirce (and, earlier, Hegel) disagreed with Kant. They hoped that perception of the things-in-themselves would turn out to be solid and objective rather than subjective and biological. Hegel defined truth as the outcome of a long social process - one which, unfortunately, only existed in his mind. Peirce defined, as we saw above, as the outcome of a convergent scientific process - processes that he then went out and tried to do.

In Peirce's theory, the real world-in-itself is a set of interacting (possibly/often non-measurable) facts and relations between these facts. These facts can be constants, such as the 19 parameters of the Standard Model, or they can be variables, such as the total population of a country or temperature. These facts can be basic, like energy, or "emergent", like temperature. That underlying world could only be approximated sadly phenomenal studies. Therefore, even crafty experiments surrounded the true values (of, say, the fine structure constant) with error bars. Pierce called these error bars the "probable error", today we call the equivalent notion "confidence interval". Peirce's work is, in many ways, the beginning of statistics.

Peirce first developed his statistical ideas when studying the experimental errors of using pendulums to study the acceleration due to gravity, but it is equally valid to consider coin-flips. The facts of a given sequence of coin-flips are statistically related to the underlying reality of governing the coin. In the case of coin-flips, we can appeal to Bernoulli's theorem to prove that the scientific best practice leads to The Truth, the coin-in-itself.

This is a mathematical version of the general example I gave above, when we learned Peirce's first name. You then might again notice that Peirce's definition of truth is eccentric. Mathematically, one must posit a true value and prove convergence toward it. I think Peirce would reply that this is a mathematical convenience and the truth was the reverse, a coin is known as fair from the throwing. Peirce developed this definition in scientifically relevant ways. For instance, he would say that Bayesian methods are not scientifically relevant unless paired with a robust convergence proof. One can construct instances in which a Bayesian procedure does not converge. From Peirce's point of view, this would mean that for such agents, the truth is meaningless.

So we see how philosophy affected statistics. Peirce's forward looking definition of truth ruled out Bayesianism, his love of Kant made Frequentism attractive. Notice that these are logically quite separate!

All this would have been by itself interesting, but Peirce actually went further. He gave a specific quantitative guide to such reason in his "Note on the Theory of Economy of Research". The essence of Peirce's reasoning is here. Peirce's discovery is even more remarkable because not only did he notice the parallel with the ratio of marginal utility - he also did so in 1879, making him the among the first important American Marginal theorists of any kind!

Given the importance of Marginalism in his thought, one should not be surprised when he says: "The truth is a kind of efficiency.". Surely someone who could say that can be called a pragmatist.

Though Peirce had a chance to become one of the great economists of his time, he didn't take it up. In addition to the above, he was also the first to state the axiom of transitivity of preferences (he had to be - he also invented relational algebra). Interestingly for the proto-frequentist, he was also the first to measure systemically subjective probabilities and among the first to rigorously define probability in terms of economic decisions. Unfortunately, he rarely took the time to find deeper implications of his economic thoughts (the above being the only exception to this rule). Certainly, his rival Simon Newcomb (interestingly, the rivalry, while well-attested, was unknown to Peirce...) would not have appreciated it.

Karl Marx

All that brings me to the next 19th century philosopher/economist to explain: Karl Marx.

Classical Thermodynamics From "Intuitive" Symmetries? Part 1

In my last post, I promised to talk about Paul Samuelson's paper "Conserved Energy Without Work Or Heat". Now I will do so. Even earlier I had earlier promised a post about Disney Princesses. While I have a variety of observations, I haven't yet put them together into a theme.

First, a word about our author.

P Samuelson

Paul Samuelson is one of the father's of modern economics. Even more than with more famous economists like Keynes or Friedman, one can divide economics into a pre- and post-Samuelson quite easily. Paul Samuelson's most important work was his dissertation, Foundations of Economic Analysis. In that book, he provided perhaps the first completely mathematically clear explanation of what it is economists where doing. When an economist says that, for instance, a given tax is good or bad, what he or she means is that changing something given, the rest of the economy will eventually adjust and where it will settle will be better or worse than where it is now. A good example (with applications to finance) can be found here - and in innumerable other places! Samuelson made many other advances in almost every area of economics. Much can be said about his "scientific personality". He considered himself a child genius well into his 80's. He was immensely concerned with every aspect of scientific work, empirical, theoretical, philosophical, historical and pedagogical. Unlike many economists of his stature and influence, Samuelson almost never outright dismissed an economist or an economic theory, always describing them as containing nuggets of truth - of course, truths that he himself has formally obtained. His scientific ideal was that of classical thermodynamics, where the foundations were clear, the applications enormous and the empirical validity impeccable. Some have objected to his appreciation for classical thermodynamics, but most of these complaints are ill founded. To the extent he was inspired by classical thermodynamics, classical thermodynamics is inspiring. (Aside: it is not at all correct to believe that just because pre-Samuelsonian economists mostly used geometry and post-Samuelsonian economists mostly used algebraic notation that one was less mathematical. J.C. Maxwell's Theory of Heat was written almost entirely with geometry, and this example can be multiplied.)

Now on to the paper itself. This paper has multiple goals. One is a slick derivation of deep aspects of physics (esp. thermodynamics) from a few qualitative empirical regularities. One is to present the argument that one could have - in an alternate universe - completely missed the deep connections of thermodynamics to Newton's Laws. I am going to play loose with language, sometimes I will say "amount of heat" and other imperfectly defined things in order to bring these ideas closer to everyday experience. If this confuses anyone, then I will do another version which I am more careful (or they can read Samuelson's original).

(Aside 2: This idea, in a loose philosophical way, might be connected to "microfoundations" debate economists sometimes talk about. If microfoundations can be seen to be like statistical mechanics, then macroeconomics is like thermodynamics. This paper would then be an example where a simple empirical regularity is all that is needed to establish deep economic laws, rather than investigation of the deepest parts of the consumer's psyche. This argument is unimpressive, and Samuelson would never have dreamed of making it, but you can think about it if you like.)

Now, on to the deep parts of the paper. The main empirical principle of this paper is that if you put two hot things in contact, then they will equilibrate. A cold drink will turn warm in the hot summer afternoon. This principle is purely qualitative, but quantitative measures will fall out of it. Notation will simplify things. Our Principle is that "The Temperature of System 1 and the Temperature of System Two go to the Equilibrium Temperature in System 1 and the Equilibrium Temperature in System 2.". This is a mite cumbersome. Instead we say \( (t_1 ; t_2) \rightarrow (t_eq ; t_eq) \). We assume that \(t_eq\) is a function of the initial conditions.

Brief considerations as to measurability of heat are given, but these are minor enough that the assumption that the only important conclusion is that the final temperature is to be a function of the initial temperatures. This paper is written as an alternate history past Carnot, so we aren't interested in chemical, gravitational, electromagnetic, etc forces (more on aside 2: one of the many arguments against the above aside is that one couldn't do this without "microfounding" heat. Is this true? Discuss.). We consider the effects of heat by itself.

A thought experiment. Consider a bowl of hot soup kept in contact with a container of cool apple sauce. The bowl and the container are strong, they do not melt or flex because of the heat. They are kept in a insulating lunch bag, so that they don't lose heat to the environment. A sort of drawing of this situation:

What will happen? By our principle founded on common observation, the substances will come to be the same temperature. Of course, there is more to consider than just the temperature. I drew the above as if the soup and apple sauce were in equal volumes, but if I had enormously more soup or enormously more apple sauce, then the one with enormously more volume would barely notice the change due to the other. There might be other dependencies, but Samuelson follows Carnot wisdom that the main action can be captured considering only the interaction of volume and heat - and sometimes heat alone! Symbolically: $$(t_1,v_1 ; t_2,v_2) \rightarrow (t_eq,v_1; t_eq,v_2)$$ $$t_eq = f(t_1,v_1 ; t_2,v_2)$$ The first part is read "The Temperature and Volume of System 1 and the Temperature and Volume of System Two go to the Equilibrium Temperature and Original Volume of  System 1 and the Equilibrium Temperature and Original Volume in System 2.". Obviously, this doesn't depend on how the apple sauce and the soup are oriented, since the bag is being thrown around all day anyway. This implies that \(f(t_1,v_1 ; t_2,v_2)= f(t_2,v_2; t_1,v_1 )\). Today, I will concentrate solely on what can be concluded from experiments of this type alone, but in Part 2 I will introduce a more complex experiment involving pistons.

These experiments can be compounded arbitrarily. For instance, we can put four substances together:

And the above reasoning still applies. The arrangement can be manipulated so that red and yellow equilibriate while orange and burgandy equilibriate, then those two are placed next to each other and the whole system is allowed to equilibriate. Alternately, the arrangement can be manipulated so that red and orange equilibriate while yellow and burgandy equilibriate, then those two are placed next to each other and the whole system is allowed to equilibriate. Either way, the system comes to the same temperature. It is easy to arrange the volumes (actually, specific volumes) to be the same. In this case we have the simple symbolic expression for the above: \(f(f(t_1,t_2);f(t_3,t_4))=f(f(t_1,t_3);f(t_2,t_4))\). It is read "The equilibrium temperature of the equilibrium temperature of the first two substances touching the second two substances is equal to the equilibrium temperature of the equilibrium temperature of the first and third substances touching the second and fourth two substance.". You can start to see why we invented this notation!

Now we add a couple new assumptions. These assumptions are unobtrusive and intuitive, but they might be wrong and must be stated. We have already extensively discussed the existence and symmetry of the function \(f \), which takes the system set up and gives the equilibrium temperature. We also assume that the function \(f \) has the property that if two substances have the same temperature, then the equilibrium temperature is that temperature. (more on aside 2: if we were interested in "microfoundations" right now, then this would be a statement about the nature of an equilibrium - namely that it is an equilibrium!) Otherwise, purely mental divisions could make physical changes. Finally, we assume that if you perform the same experiment, but you make one of the substances hotter before hand, then you always get a hotter equilibrium. It is these properties of heat, perhaps, that lead to the idea that heat was an independent substance!

These assumptions give a remarkable conclusion, the existence of a sort of energy function! Perhaps a better name would be a caloric function (this does not mean, of course, that there is any such physical substance!). A brief verbal argument can be made. I said before that the above that the latter assumptions make heat seem like a substance, since if you add more of it, more comes. The amount of that substance is the caloric. The equilibrium temperature is derived by averaging the amount of caloric in both substances (notice averaging, not summing. This becomes clearer when the mathematical argument is fully expounded). What is remarkable is the role of symmetry in the proof. If the function were not symmetric, we'd have no reason to think that the equilibrium temperature could be found by averaging equilibrium temperatures of partitions. It was this fact that is the foundation of this theory! By choosing references, one can convert this equation into a standard internal energy function. We have begun the battle of recovering classical thermodynamics from simple symmetry arguments.

Phew! I have covered much ground and only barely scratched the surface of this paper! I need to go through and show how this caloric function is found, give more examples, and there's a an entire second experiment! These posts will come every Sunday. If there's particular aspects of this argument that interest you, drop me a comment. Before I go, however, there are a couple things I would like to highlight. First of all, notice that we derived an energy function, but I didn't check anything about its form. For instance, I didn't make any assurances that it was always positive. It is this reason I prefer to call it a caloric function, even if I risk misinterpretation that this somehow vindicates the physical concept of caloric. More importantly, I want to highlight that this argument makes no reference to Newton's or Schrodinger's laws. Physicists have a deep rooted appreciation for the laws of thermodynamics, and it is arguments like these that gives substance to those feelings. No matter what the universe is like, the laws of thermodynamics will apply as long as those fundamental symmetries are observed. This does not establish that the laws of thermodynamics are universal, but a non-physicist might wonder why they are believed to be and arguments like this will help the intuition in that regard.

Finally, a social observation. The laws of thermodynamics are wildly misused in bad science, bad philosophy and even bad politics. I have seen irresponsible writers on the internet imply and argue that they make action on global warming impossible, often with naive importations into economics or politics. What I hope is that when reading this, you absorb some of the actual intuition of this science, rather than the slogans such people use. When someone claims an application of thermodynamics, check first to see if even first principles - such as these - apply. If it is not obvious how, then they are not obviously right. See you next week!