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So I'm currently writing an article regarding a priori axiomatic systems and the nature of inference (I'm not a physicist but a philosopher doing philosophy of science), one of my main texts is "Dynamics of reason" by Michael Friedman which, to summarize, explains that empirical relationship themselves are determined by the theoretical framework, specially the mathematical principles within which the theory is established (following the more known Thomas Kuhn), on the other hand, I'm also using Von Neumann's quantum logic as an example of how a logical system can be tailored to fit certain epistemological needs of an object, but for everything to fit neatly with Friedmann's idea, I of course need to speak of the specific conditions or axioms that possibilitate quantum mechanic as a scientific endeavor (the text itself uses Newtonian and relativistic physics as the main examples, and only makes a passing mention of its applicability to quantum physics).

So essentially I'm looking for a current and updated text about the conceptual foundations on the field, not necessarily absolutely complete and specific, not very extensive if possible, introductory maybe, but that establishes the main principles correctly (I have in fact a text from Von Neumann called "The mathematical foundations of quantum mechanics" which seems conceptually perfect, though I'm afraid of it being too outdated); regarding the mathematical complexity, as logic is one of my main interests I usually deal with abstract algebra, model, order and set theory (of course, not claiming any deep expertise, but I feel comfortable with its proofs and procedures), I know calculus and differential equations though to be honest, is something I haven't dealt with for years, however, given the topic, detailed proofs are necessary (but I'm of course, not expecting to solve anything or make the proofs myself).

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    $\begingroup$ The foundations of QM are ~100 years old. There are many good recommendations for texts on this, but I will give my two cents for a book I don't see recommended enough: "Lectures On Quantum Theory: Mathematical And Structural Foundations" - Chris J Isham. It is newer than many fundamental texts. I believe the lectures are able to be found online. There last chapters are very informative as they address conceptual issues of QM and logic. I think these lectures were to be used as a first formal QM course. Also see: physics.stackexchange.com/questions/12175/book-recommendations $\endgroup$
    – Fire
    Dec 1, 2020 at 21:23
  • $\begingroup$ @Fire thanks! I have been checking the contents of the book, and seems like an excellent fit, as it does seems to put interest in the logical estructure and formalism of both classical and quantum mechanics, and even has a sub-chapter about the epistemological discussion of realism vs instrumentalism. I'll probably check R. Shankar's introduction too as per the book recommendations page. $\endgroup$
    – Vagoltof
    Dec 1, 2020 at 22:52

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I would like to suggest Are Quanta Real? by J.M. Jauch: A relatively short fictitious dialogue between three main characters, written in the style of Galileo's "Dialogue Concerning the Two Chief World Systems". Virtually no mathematics, but doesn't compromise on philosophical rigor; and offers a fantastic exhibition of some of the main active interpretations of quantum mechanics.

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    $\begingroup$ Thanks, it does really seems very helpful, I read the introduction (and have of course, read the Galilean original) and while it seems to not be to centered in the specific logical and mathematical framework that I need to address, it certainly will give me direct insight both of the actual possibilities of epistemological interpretation of the mathematical and empirical results (as I'm most likely not fit to do) and also work as a intermediary between the philosophical and the physical concepts. $\endgroup$
    – Vagoltof
    Dec 3, 2020 at 18:22
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Foundations of quantum mechanics has been a little regarded field by physicists, and it is plagued by the plethora of false "interpretations", many of which do not even respect the mathematical structure of quantum mechanics. Those which do usually add little, and generally regurgitate in different words, and with less mathematical precision, things already said by von Neumann without understanding what von Neumann actually said.

Partly this is because von Neumann's book, while accurate enough, is not exactly an easy read. Partly it is also because physicists are mostly more concerned with whether their calculations give correct empirical predictions than with underlying mathematical structure. The book I used to learn quantum mechanics (after stalling on less mathematical treatments) was Jauch's Foundations of Quantum Mechanics. It is closely based on von Neumann and gives a sensible account of quantum logic (unlike some treatments I have seen), but it has been out of print for some time and may be difficult to get hold of.

Von Neumann's book is still vital. It is outdated in certain respects, it is non-relativistic, the projection postulate has been superseded, there are tighter versions of the no-hidden-variables theorem (including in Jauch) and perhaps most importantly it does not give a derivation of the general form of the Schrodinger equation from axiomatic principles (nor does Jauch). Although it is known that the general form of the Schrodinger equation can be proven from the probability interpretation, it is not easy to track down a rigorous derivation.

I wrote my own books with an intention of rectifying these omissions. The Large and the Small is conceptual. The Mathematics of Gravity and Quanta is mathematically rigorous, and should contain everything you need. I think you will find that the account fits the scheme you have in mind reasonably well. Axioms are established from general principles concerning the nature of measurement as a physical process, and empirical predictions do indeed follow from the mathematical structure so defined.

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  • $\begingroup$ Thanks for you insights about the idiosyncrasies of the field, it's definitely something that I'll keep in mind, also regarding about what I need to be careful with in Von Neumann's text which I'll address for organic purposes. I read what I could in the preview of your book and it seems excellent, it provides that bottom-up foundation that any philosopher of science hopes for research, and does not shy away from the central philosophical discussions necessary both for the mathematical and empirical endeavor, they are definitely a must have for my future inquiries. $\endgroup$
    – Vagoltof
    Dec 3, 2020 at 19:01
  • $\begingroup$ Thanks for your response. You may also be interested in academia.edu/40217917/Mathematical_Implications_of_Relationism, which contains in condensed form the core arguments, with material from published papers in appendices. $\endgroup$ Dec 3, 2020 at 19:20

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