Dear Victor,

Thank you for your interest in my work.

I believe that, as time goes on, logical independence will become important in applied mathematics.  The problem is knowing how mathematics connects with logic.  And what must be understood is that the mathematics itself generates logic, separate from any logic that might stem from principles of economics or physics that we place on top of the maths.  This means that the maths cannot be regarded simply as a tool for the calculation of quantities, but as a process that is part of the system being modelled. The mathematics is not simply a passive conveyor of information; it is an active conveyor

My approach has been to treat the mathematics of quantum theory as absolutely pure mathematics, without attributing any meaning whatever.  I replicate formulae in quantum theory by deriving directly from axioms of elementary algebra. In this way I can determine whatever mathematical information is within them.

For some time I have been interested in the concept of money, from a mathematical viewpoint.  If I was not working on quantum theory, I would be studying money.  To me, existence of money would seem to dependent a self-referential definition.  And that self-reference must have impact on the money value over time.  What I am suggesting is that rapid fluctuations in money values might not only come from principles of economics, or from   complexity, but the maths itself.

I believe the thing to do with Gödel is to embrace him.  There is plenty of evidence that the effects of Gödel apply to physical machines and to Nature. What you need for your investigations is not his Incompleteness Theorems.  You need the Soundness Theorem and Completeness Theorem.

Your possible/necessary logic is officially known as a modal logic.   For an overall picture of logics including 3-valued logics, which might affect your work, I recommend Susan Haack, Philosophy of Logics.  

Most of all, I recommend the book I cite in my paper, by Stabler.   I also cite Chaitin. From his work and Turing, you can see the physical work cannot be exempt from Gödel.  Also for 3-valued logic, I recommend Reichenbach and also Putnam. I cite all these.

Keep in touch,  Steve. 

My third axiom is a generalization of Ludwig von Mises' Regression Theorem, which is a self-referential definition of money.  Unfortunately, because of a dispute with his older brother Richard, who was renowned in aerodynamics and famously argued against Kolmogorov's axiomatic theory of probability, Ludwig's work had a decidedly anti-mathematician tone.  Also, some of his results, like originary interest, were wrong.  Thus, he was largely dismissed and I am alone today in attaching any value to his Regression Theorem.

I mention my debt to Mises here:  http://axiomaticeconomics.com/axioms.php

"There is an easy plausibility argument for axiom #3: Interest is calculated as a percentage of the amount owed. Fixing this percentage (using the same percentage throughout the calculation) is a special case of the percentage varying slightly every day. Thus, the fact that people have calculated interest in this way throughout history, and done so unquestioningly, implies that they should accept my axiom as self-evident. There are other conceivable ways of calculating interest (which I consider in the appendix of my book) but, as far as I know, nobody has ever thought to even ask this question. Thus, axiom #3 actually meets the dictionary’s more restrictive second definition of axiom, “a universally accepted principle or rule.”

"Ludwig von Mises was not a mathematician – he actually despised us – but, in his own vague way, he made the same claim about money. Money is valued today because it had value yesterday and it had value yesterday because it had value the day before. In this way, Mises’ Regression Theorem traces the value of money back to when it had value primarily for its use. But that is as far as he got; not being a mathematician, he had no way of knowing that the distribution of people’s valuations of their first unit of the monetary unit is log-normally distributed.

"I went beyond Mises by recognizing that his Regression Theorem is not a theorem but an axiom and that it applies to all phenomena, not just money."

I was not sure what you meant when you asked if economics is chaotic.  But if you meant, is the change in the value people place on money from day to day chaotic, I would answer no.  If it were chaotic, the distribution of people's valuations of money would not be a well-behaved distribution with moments of all orders.

But the discussion in the 2013 addendum to my Simplified Exposition shows that price fluctuations obey an inverse power law, which has a fat tail that could be seen as evidence of chaotic behavior.

Note that these are different things.  The former is a distribution of people; where any individual stands in the distribution is dependent only on personal factors.  The latter is a distribution of prices; where each price stands in its distribution is dependent on what the entire distribution of people looked like at the moment when the price was set.

Dear Victor,  Regarding necessary versus possible but not necessary information, you might like to have a look at this short article:

https://www.researchgate.net/publication/284030021_The_logical_difference_in_quantum_mathematics_separating_pure_states_from_mixed_states

In that paper, there is very clear logical distiction, asserted by the mathematics.

Article The logical difference in quantum mathematics separating pur...

Dr. Faulkner–

Thank you for informing me about modal logic, which I was unaware of.

A brief biography:  After graduating from high school in 1984, I studied philosophy on my own, reading Immanuel Kant, Ernst Cassirer and Willard Quine.  The indices of Quine’s books indicate that he never wrote about modal logic.  But he was writing in the 1950s and, from what I now read on the internet, modal logic did not become popular among logicians until the 1970s.  Perhaps if I had studied philosophy in college I might have heard of it.

Kant uses the term modality in his table of the pure concepts of the understanding, though Cassirer (p. 171) criticizes Kant’s organizational scheme:

“Delight in comprehensive architectonic structure, in the parallelism of the art form of systemization, in the monolithic schematism of concepts seems to play a greater part in the detailed working out of the doctrine of the categories than is proper.”

I agreed with Cassirer in this and rejected Kant’s architectonic structure.  But my own three-term system of logic was mostly in agreement with Kant and opposed to Quine, whom I took to be canonical.  In high school I had been a fan of Quine, but over the next four years I would reject almost all of what he had written about logic.  I defined three senses of the term truth and I felt that the fundamental flaw running through Quine’s work was that he conflated them.

During this period of my life I considered myself a philosopher and was only incidentally interested in economics or quantum mechanics.  Four years out of high school I had an epiphany:  supply and demand is wrong; it should be stock and price.  The basic form of the demand distribution came to me in a flash of insight, but I quickly realized that my knowledge of mathematics was not sufficient to define it rigorously.  So I enrolled in college and majored in math.  I felt I knew enough about philosophy to write the first part of my book about foundations and so in college I focused only on the math I would need to write the second part of my book about theory; I never formally studied philosophy.

I did take a course on quantum mechanics in college and was at the head of my class, though this was largely because I knew more math than the physics students and because I had the wit to focus only on the equations and to just accept that there is no visual representation of an atom.  The failing of quantum mechanics students is their hope that someone will clarify things for them by drawing a picture of an electron in its orbit.  Their saving is the realization that such pictures create confusion, not clarity.

But that was a long time ago and I do not now know enough about quantum mechanics to comment on your work except to say that I believe that we are in agreement regarding the importance of not conflating the different senses of truth.

REFERENCES

Cassirer, Ernst.  1981.  Kant’s Life and Thought.  New Haven and London:  Yale University Press

Victor, I am both a mathematician and a philosopher, and I work a lot with Logic, which is my preferred part. 

I have just studied Godel's proof of God and you can see the result of that in the book On Religion, which is found on my profile here.

I think that the aim of Logic is not confusing people, but making things easier and nicer instead. 

Priest, like you, loved formalizing anything. 

The problem with that is that you end up working in the abstraction, but you then want to come back to reality in the end and, whilst the first move could be acceptable, if things were ever done properly, the second move is usually impossible to to accept. 

Basically, I have written a lot about this by now: We can fit a reduced universe into a large one, but there are lots of things that escape logic in human language/discourse and the reverse move is not usually OK.

The sentences you wrote are easily dealt with without us changing them into symbols and, therefore, the need to do that does not exist. We are not simplifying if doing that either, we are actually complicating and probably missing the point, so that that is also not advisable. 

What Prof. Faulkner says is true and it actually occurred to me as I read your writing, but in a much more informal way. 

If you read more of my work, you will understand why this concern you seem to be having is actually equivocated for Mathematics. Creating new systems of logic, as Priest does, is fun, but may also be completely useless activity. 

Apart from learning the concepts involved in the modelling of the systems and applying those, there is nothing to it. 

If you want them to be useful, you must make sure you find a situation for which we feel the necessity of coding things. In second place, as you have probably noticed, you should also check if the system already exists. 

Remember that we must improve things with them, not confuse people even more or lead them to mistake in reasoning.

Priest has a book with almost all non-classical systems and you could actually use his book, since it works like an encyclopedia, to find out if a system already exists or not. 

I think he finished this book in 2000, after he went through it with me. 

I am flattered. But, reluctantly, I must admit that I am neither a professor nor a PhD.

Steve.