Shminux, it may be that you will find that your concerns are substantially addressed by Joshua Landsberg's Clash of Cultures essay (2012), which is cited above.

"These conversations [are] very stressful to all involved ... there are language and even philosophical barriers to be overcome."

The entanglement(s) of hot-noisy-evolved biological cognition with abstract ideals of cognition that Eliezer Yudkowsky vividly describes in Harry Potter and the Methods of Rationality, and the quantum entanglement(s) of dynamical flow with the physical processes of cognition that Scott Aaronson vividly describes in Ghost in the Quantum Turing Machine, both find further mathematical/social/philosophical echoes in Joshua Landsberg's Tensors: Geometry and Applications (2012), specifically in Landsberg's thought-provoking introductory section Section 0.3: Clash of Cultures (this introduction is available as PDF on-line).

E.g., the above discussions above relating to "map versus object" distinctions can be summarized by:

Aaronson's Law of Ontic Mixing "We can't always draw as sharp a line as we'd like between map and territory".

as contrasted with the opposing assertion

Landsberg's No-Mixing Principle "Don’t use coordinates unless someone holds a pickle to your head"

As Landsberg remarks

"These conversations [are] very stressful to all involved ... there are language and even philosophical barriers to be overcome."

The Yudkowsky/Aaronson philosophical divide is vividly mirrored in the various divides that Landsberg describes between geometers and algebraists, and mathematicians and engineers.

Question Has it happened before, that philosophical conundrums have arisen in the course of STEM investigation, then been largely or even entirely resolved by further STEM progress?

Answer Yes of course (beginning for example with Isaac Newton's obvious-yet-wrong notion that "absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external").

Conclusion It may be that, in coming decades, the philosophical debate(s) between Yudkowsky and Aaronson will be largely or even entirely resolved by mathematical discourse following the roadmap laid down by Landsberg's outstanding text.

Quantum aficionados in the mold of Eliezer Yudkowsky will have fun looking up "Noether's Theorem" in the index to Michael Spivak's well-regarded Physics for Mathematicians: Mechanics I, because near to it we notice an irresistible index entry "Muggles, 576", which turns out to be a link to:

Theorem The flow of any Hamiltonian vector field consists of canonical transformations

Proof (Hogwarts version) ... <natural exposition>

Proof (Muggles version) ... <index-laden exposition>

Remark It is striking that Dirac's The Principles of Quantum Mechanics (1930), Feynman's Lectures on Physics (1965), Nielsen and Chuang's Quantum Computation and Quantum Information (2000)---and Scott Aaronson's essay The Ghost in the Turing Machine (2013) too---all frame their analysis exclusively in terms of (what Michael Spivak aptly calls) Muggle mathematic methods! :)

Observation Joshua Landsberg has written an essay Clash of Cultures (2012) that discusses the sustained tension between Michael Spivak's "Hogwarts math versus Muggle math". The tension has historical roots that extent at least as far back as Karl Gauss' celebrated apprehension regarding the "the clamor of the Boeotians" (aka Muggles).

Conclusion Michael Spivak's wry mathematical jokes and Eliezer Yudkowsky's wonderfully funny Harry Potter and the Methods of Rationality both help us to appreciate that outdated Muggle-mathematical idioms of standard textbooks and philosophical analysis are a substantial impediment to 21st Century learning and rational discourse of all varieties---including philosophical discourse.

Shminux, perhaps some Less Wrong readers will enjoy the larger reflection of our differing perspectives that is provided by Arthur Jaffe and Frank Quinn’s ‘Theoretical mathematics’: Toward a cultural synthesis of mathematics and theoretical physics (Bull. AMS 1993, arXiv:math/9307227, 188 citations); an article that was notable for its biting criticism of Bill Thurston's geometrization program.

Thurston's gentle, thoughtful, and scrupulously polite response On proof and progress in mathematics (Bull. AMS 1994, arXiv:math/9307227, 389 citations) has emerged as a classic of the mathematical literature, and is recommended to modern students by many mathematical luminaries (Terry Tao's weblog sidebar has a permanent link to it, for example).

Conclusion It is no bad thing for students to be familiar with this literature, which plainly shows us that it is neither necessary, nor feasible, nor desirable for everyone to think alike!

Thank you for your gracious remarks, Paper-Machine. Please let me add, that few (or possibly none) of the math/physics themes of the preceding posts are original to me (that's why I give so many references!)

Students of quantum history will find pulled-back/non-linear metric and symplectic quantum dynamical flows discussed as far back as Paul Dirac's seminal Note on exchange phenomena in the Thomas atom (1930); a free-as-in-freedom review of the nonlinear quantum dynamical frameworks that came from Dirac's work (nowadays called the "Dirac-Frenkel-McLachlan variational principle") is Christian Lubich's recent On Variational Approximations In Quantum Molecular Dynamics (Math. Comp., 2004).

Shminux, perhaps your appetite for nonlinear quantum dynamical theories would be whetted by reading the most-cited article in the history of physics, which is Walter Kohn and Lu Jeu Sham's Self-Consistent Equations Including Exchange and Correlation Effects (1965, cited by 29670); a lively followup article is Walter Kohn's Electronic Structure of Matter, which can be read as a good-humored celebration of the practical merits of varietal pullbacks ... or as Walter Kohn calls them, variational Ansatzes, having a varietal product form.

There is a considerable overlap between Scott Aaronson's "freebit" hypothesis and the view of quantum mechanics that Walter Kohn's expresses in his Electronic Structure of Matter lecture (views whose origin Kohn ascribes to Van Vleck):

Kohn's Provocative Statement In general the many-electron wavefunction for a system of electrons is not a legitimate scientific concept, when , where .

Scott's essay would (as it seems to me) be stronger if it referenced the views of Kohn (and Van Vleck too) ... especially given Walter Kohn's unique status as the most-cited quantum scientist in all of history!

Walter Kohn's vivid account of how his "magically" powerful quantum simulation formalism grew from strictly "muggle" roots---namely, the study of disordered intermetallic alloys---is plenty of fun too, and eerily foreshadows some of the hilarious scientific themes of Eliezer Yudkowsky's Harry Potter and the Methods of Rationality.

In view of these nonpareil theoretical, experimental, mathematical (and nowadays) engineering successes, sustained over many decades, it is implausible (as it seems to me) that the final word has been said in praise of nonlinear quantum dynamical flows! Happy reading Shminux (and everyone else too)!

Shminux, there are plenty of writers---mostly far more skilled than me!---who have attempted to connect our physical understanding of dynamics to our mathematical understanding of dynamical flows. So please don't let my turgid expository style needlessly deter you from reading this literature!

In this regard, Michael Spivak's works are widely acclaimed; in particular his early gem Calculus on Manifolds: a Modern Approach to Classical Theorems of Advanced Calculus (1965) and his recent tome Physics for Mathematicians: Mechanics I (2010) (and in a comment on Shtetl Optimized I have suggested some short articles by David Ruelle and Vladimir Arnold that address these same themes).

Lamentably, there are (at present) no texts that deploy this modern mathematical language in service of explaining the physical ideas of (say) Nielsen and Chuang's Quantum Computation and Quantum Information (2000). Such a text would (as it seems to me) very considerably help to upgrade the overall quality of discussion of quantum questions.

On the other hand, surely it is no bad thing for students to read these various works---each of them terrifically enjoyable in their own way---while wondering: How do these ideas fit together?

Gjm asks "Along what vector field V are you taking the Lie derivative?

The natural answer is, along a Hamiltonian vector field. Now you have all the pieces needed to ask (and even answer!) a broad class of questions like the following:

• Alice possesses a computer of exponentially large memory and clock speed, upon which she unravels the Hilbert-space trajectories that are associated to the overall structure , where is a Hilbert-space (considered as a manifold), is its metric, is its symplectic form, is the complex structure induced by , and are the (stochastic,smooth) Lindblad and Hamiltonian potentials that are associated to a physical system that Alice is simulating. Alice thereby computes a (stochastic) classical data-record as the output of her unraveling.

• Bob pulls-back onto his lower-dimension varietal manifold (per Joseph Landsberg's recipes), upon which he unravels the pulled-back trajectories, thus obtaining (like Alice) a classical data-record as the output of his unraveling (but using far-fewer computational resources).

Then It is natural to consider questions like the following:

Question For hot noisy quantum dynamical systems (like brains), what is the lowest-dimension varietal state-space for which Bob's simulation data-record cannot be verifiably distinguished from Alice's simulation data-record? In particular, do polynomially-many varietal dimensions suffice for Bob's record to be indistinguishable from Alice's?

Is this a mathematically well-posed question? Definitely! Is it a scientifically open question? Yes! Does it have engineering consequences (and even medical consequences) that are practical and immediate? Absolutely!

What philosophical implications would a "yes" answer have for Scott's freebit thesis? Philosophical questions are of course tougher to answer than mathematical, scientific, or engineering questions, but one reasonable answer might be "The geometric foaminess of varietal state-spaces induces Knightian undertainty in quantum trajectory unravelings that is computationally indistinguishable from the Knightian uncertainty that, in Hilbert-space dynamical systems, can be ascribed to primordial freebits."

Are these questions interesting? Here it is neither feasible, nor necessary, nor desirable that everyone think alike!

JLM, the mathematically natural answer to your questions is:

• the quantum dynamical framework of (say) Abhay Ashtekar and Troy Schilling's Geometrical Formulation of Quantum Mechanics arXiv:gr-qc/9706069v1, and

• the quantum measurement framework of (say) Carlton Caves' on-line notes Completely positive maps, positive maps, and the Lindblad form, both pullback naturally onto

• the varietal frameworks of (say) Joseph Landsberg's Tensors: Geometry and Applications

Textbooks like Andrei Moroianu's Lectures on Kahler Geometry and Mikio Nakahara's Geometry, Topology and Physics are helpful in joining these pieces together, but definitely there is at present no single textbook (or article either) that grinds through all the details. It would have to be a long one.

For young researchers especially, the present literature gap is perhaps a good thing!

The dynamicist Vladimir Arnold had a wonderful saying:

"Every mathematician knows that it is impossible to understand any elementary course in thermodynamics."

This saying is doubly true of quantum mechanics. For example, the undergraduate quantum physics notion of "multiply a quantum vector by " is not so easy to convey without mentioning the number "." Here's how the trick is accomplished. We regard Hilbert space as a real manifold that is equipped with a symplectic form and a metric . Given an (arbitrary) vector field on , we can construct an endomorphism by first "flatting" with and then "sharping" with , that is . The physicist's equation thus is naturally instantiated as the endomorphic condition .

The Point To a geometer, the Lie derivative of has no very natural definition, but the Lie derivative of the endomorphism is both mathematically well-defined and (on non-flat quantum state-spaces) need not be zero. The resulting principle that " is not necessarily constant" thus is entirely natural to geometers, yet well-nigh inconceivable to physics students!

Thank you gjm. To the best of my understanding, (1) all markup glitches are fixed; (2) all links are live; and (3) an added paragraph (fourth-from-last) now explicitly links dynamic-J methods to Scott's notion of "freebits".