In the Footsteps of Pierre Duhem: How a Modern Theory of Value Relates to XIX Century Physics

Abstract

Having in the last 40 years focused on a function generated from a theory of morphogenesis called Elementary Catastrophe Theory (ECT) [19], the Swallowtail, to show its fitting the most respected landmark criteria of Utility or Value in Economics I felt the need to explain the resolution of the controversy surrounding the shape of this function via a polynomial solution against the trend of mainstream literature [10]. A little noticed theorem dating back to 1963 by Rene Thom, the author of ECT [22], set the ground for the development of ECT while linking it to the thermodynamic Potentials (Entropy and Internal Energy) as the XIX century ended up characterizing them with an exclusive feature: Differentiability [2]. The importance of the synthesis attempted by P. Duhem comes from not assuming any atomism without precluding it while seeking a unified field theory between gravitational, electrical, magnetic and chemical theories based on Newton law as reinforced with Coulomb law for electrical charges [26]. One of my basic finding was the negativity of the Schwarz’ derivative the highest order (the third) differential invariant coming from the heart of Pratt’s paper on risk in Economics [31] as a characteristic displacing concavity. My result so far has been that only for a polynomial representation we have a theory of self-observation fitting uniquely with the procedure of Stimulus-Response known since 150 years under the title of Psycho-Physics and opening up vistas to the world of Linguistics [32].

Since it’s a survey and review paper I am following Sir Peter Medawar suggestion to present the interactions of people and ideas in producing results, a kind of extended mind storming. The synthesis achieved brings a new paradigm completely substituting to the expected utility hypothesis already outliving its usefulness for several generations.

Appendix

Annex I

The double slit experiment. In the small, the big surprise is that the very same function than for our visual system, the Swallowtail, is predicted to occur at the level of the synapse by the same theory (E.C.T.). But if one considers that resonance is really a two way street one could admit that it works in both directions up and down. We quote R. Thom directly: [19].

“I would not have carried these very hypothetical considerations so far if they did not give a good representation of the behavior of nervous activity in the nerve centers, where an excitation (called a stimulus in physiology) remains relatively canalized until it results in a well-defined motor reflex; here the role of diffusion seems to be strictly controlled, if not absent. It is known (the Tonusthal theorem of Uexkull) that, when the associated first reflex of a stimulus is inhibited by artificially preventing the movement, there is a second reflex which, if inhibited, leads to a third reflex, and so forth. This seems to suggest that diffusion of the excitation is, in fact, present, but that, as soon as the excitation find an exit in an effective reflex, all the excitation will be absorbed in the execution of this reflex. This gives a curious analogy with the mysterious phenomenon of the reduction of a wave packet in wave mechanics.” pp. 149–150 [5].

Briefly, considering the general 5^th degree polynomial, we see on its graph in the positive quadrant that a necessary condition for self-duality is that the local Max and Min, respectively Min and Max to each other, are confounded at the same point. It is an analog to von Neumann’s Mini Max concept. With symmetry toward the origin this condition becomes a sufficient one (Fig. 1).

Fig. 1.

The general quintic polynomial

$${\text{V}} = \frac{1}{5}\mathop {\text{x}}\nolimits^{5} \text{ - }\frac{2}{3}\mathop {\text{x}}\nolimits_{0}^{2} \mathop {\text{x}}\nolimits^{3} + \mathop {\text{x}}\nolimits_{0}^{4} x$$

Notice the start with the highest exponent stressing the fact of the Polynomial representation.

This was written over 50 years ago. By now many Quantum Scientists like Henry Stapp [5] firmly believe the following empirical facts related in Jose R. Dos Santos latest book” La clé de Salomon” pp. 452:

“In general, these quantum jumps are only possible in spaces whose width is equivalent to seven atoms. But in rare cases they could happen in widths reaching 180 atoms at a maximum. Well it so happen, by sheer coincidence or not, that the width of the synaptic slit is precisely 180 atoms. Since electrons are in constant movement, they could try billions of times to cross the synaptic tissue during the one thousandth of a second that the electrically polarized synapse takes to get activated. This will bring their rate of success in the quantum tunnel for that width to 50%. By studying with scrutiny the structure of a synapse one realized, again with a strange coincidence, that its architecture is perfectly suited to exploit an effect of quantum tunnel when an impulse arrive at a synapse, the slit becomes electrically polarized and this powerful electric field allows the quantum tunnel phenomenon. That is why one can assume that the wave function collapses in the synapses when a thought occurs, and that from this phenomenon consciousness emerges.”

Annex II

The missing link to fulfill R. Thom last wish. “Going back to the last attempt by Pr Maurice Allais to axiomatize the utility function, I noticed 2 new axioms that Pr Allais added, after reviewing his 1952 experiments and analyzing the diverse reactions for over a generation, more precisely in “The expected utility hypotheses and the Allais paradox”, 1979:

Axiom (VI): Axiom of invariance and homogeneity of the index of psychological value and

Axiom (VII): Axiom of cardinal isovariation.” [14]

After analyzing two cases, the log linear and the non-log linear approximation one, he finds an excellent fit with a behavior verifying his axioms, up to approximation to the errors due to psychological introspection. In the meantime I sent my first paper, “the essential tension” seeking his feedback, thinking I was dealing with a prominent psychologist who set up the “Allais Paradox” at the Paris colloquium of 1952. He sent me, a few weeks later, the first English version, I was to learn later, of any of his works, the 1979 book with O. Hagen summarizing many centuries of dealings with the subject since before the Physiocrats. I learned later that Pr Allais, considering his disappointment from not having won the Nobel freshly set up in Economics in 1969, was considering emigrating to the USA! Indeed his own daughter told me later that he and his wife spent few months considering seriously the pros and cons of such a drastic move at the age of 60! Putting this in perspective one could understand the determination of such a character. Since he was finally awarded the Prize in 1988 for work done in 1943, he acknowledged his own handicaps. Indeed he was publishing only in the French language during and after WWII. Secondly all his memoirs were no less than 100 pages long. The conclusion was that all the intelligent people knowing French could plagiarize him at length during decades without being caught and without giving him any credit! Add to that the fact that he decided to redo his second preparatory year for entrance at the prestigious Ecole Polytechnique because although admitted from the first attempt he was not the “major”, or the first of his class, which he ended up with again at the exit exam. When one considers the risk of not being accepted that the competitors face with the odds of 10 to one, one measure the caliber of such straight bullets!

So when he sent me the book he also wrote a short note: “To William M. Saade with my cordial homage” I was flabbergasted because I had not forgotten yet my French language. I asked myself “what did I do to deserve such an honor?” I shared the news with the guru of Decision Analysis at my department, E.E.S. The next thing I remember was Pr Arrow, from the Economics department, approaching me with the statement: “If I don’t put my nose in it nobody will”!

Looking back at that time, I understand why ECT seemed like a glove fitting perfectly the hand:

“First: the new conceptualization of ECT brings an essential feature under the name of critical point or Catastrophe point, fitting the bill for a reference point, but the novelty being that it shows on the numerator side, instead of the denominator one, retained since the time of Weber-Fechner.

Second: the basic problematic, also from the time of Weber-Fechner, as to the search for an origin and scale of the logarithmic shape inherited from the time of Bernouilli, precisely represented by his two new axioms, was addressed with success by ECT, by the hypothesis of “diffeomorphism”. Now, the final punch line to relate it to our problem resides in the methodology brought about by ECT. Indeed, one recalls from classical Analysis, two basic results:

• 1. Any function could be approximated to any degree (of approximation) by a polynomial.

• 2. The Taylor series expansion cannot be sure to converge. And even when it does, it’s not sure the convergence will be to the original function which led to the local Taylor expansion [26].

ECT formalism had, a decade earlier, solved this problem in an original fashion.

Inspired by his thorough correspondence with Waddington and Zeeman about physiology, (chreods…) R. Thom embarked himself and others on a research work leading to the emergence of potential functions in finite number, the famous seven catastrophes, corresponding to a combination of the dimensions of two spaces, the control and state spaces. It was the first time the control space was introduced from the outset in conjunction with the state space and I identified it with the physical space corresponding to the brain wiring. Hence I focused on the dimension 3 in order to have a chance to have a measurement by the human brain corresponding to an observable result.

The basic procedure of ECT states that, starting from the “germ”, highest term of the Taylor expansion at which one wishes to stop the approximation, a diffeomorphic change of coordinates leads to a UNIQUE representation with “determinacy”, i.e. exact polynomial. And this is possible only by the introduction of the catastrophe point which insures both the “structural stability” and the “analytical continuation” i.e. the passage from the local to the global. It was the striking answer to the problem dormant since the XIX century as if Weber-Fechner had commandeered R. Thom to do that job a century earlier!

How could this connect with the logarithmic shape? Taking the Taylor expansion of the neural quantal function:

$${\text{Log}}(1 + {\text{x}}) = {\text{x}} - {{x^{2} } \mathord{\left/ {\vphantom {{x^{2} } 2}} \right. \kern-0pt} 2} + {{x^{3} } \mathord{\left/ {\vphantom {{x^{3} } 2}} \right. \kern-0pt} 3} - {{x^{4} } \mathord{\left/ {\vphantom {{x^{4} } 2}} \right. \kern-0pt} 4} + {{x^{5} } \mathord{\left/ {\vphantom {{x^{5} } 2}} \right. \kern-0pt} 5} - \ldots$$

(1)

Retaining the “germ” ${{x^{5} } \mathord{\left/ {\vphantom {{x^{5} } 5}} \right. \kern-0pt} 5}$, corresponding to the dimension 3 by definition, since the germ has to have two degrees higher than the dimension of the control space, ECT leads to a function, called “self-intersection curve”, defined by the cancellation of its first and second derivatives at the critical(catastrophic)point. This was the perfect topological translation of a non-decreasing function with its derivative reaching its strict minimum (zero) at the reference point, parameterizing the control space, by definition. Symmetry towards the origin completes the representation from – to + infinity. The most striking aspect which hit me was the remark of Wigner about “The incredible precision of mathematics in describing the real world”. But still what we have here is the incredible continuity in human reasoning across the centuries finally leading to a conclusion of the primordial place of the human in his world [24].”