Dear Conrad,

It is indeed an important aspect of physics that you are pointing out, that is not usual to point out. However I would not exactly agree with the claim that "no other system has this sort of completeness, defining itself entirely in terms of itself. Mathematics in general certainly does not. Every branch of mathematics is built on certain primitive notions that are left...

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Dear Conrad,

It is indeed an important aspect of physics that you are pointing out, that is not usual to point out. However I would not exactly agree with the claim that "no other system has this sort of completeness, defining itself entirely in terms of itself. Mathematics in general certainly does not. Every branch of mathematics is built on certain primitive notions that are left undefined, such as “point” or “set”."

I do not see any difference here. Mathematics does have the ability of defining itself entirely in terms of itself. See my introduction to the foundations of mathematics where I sketch the idea of how it happens.

Points and sets are defined by their role, which is specified by an axiomatic theory, just like many physics concepts are defined by their role with respect to other concepts. More comments on this aspect are in my section 1.4.

We can have an intuition of sets just like physicists have an intuition of physical concepts, but this intuition does no more need to refer to physical experience than physicists have to imagine experiments to understand the theories of physics.

"It would defeat the purpose of pure mathematics, which is based on logical proof, to define all its elements in terms of each other; that would only make all its arguments circular."

Bad ways of defining things in terms of each other may indeed make arguments circular. But foundations of mathematics have been elucidated in a such way that all things are defined in terms of each other but yet it is a satisfying way. It does not fulfill all of Hilbert's dreams of self-justification, but it is much more valid, fruitful and self-sufficient than "circular arguments" in the usual sense of the expression.

"The language built into the physical world, on the other hand, is not about proving things; it’s about giving them contextual meaning"

Mathematics is also about giving things contextual meaning, and not only proving things.

"Even the language itself isn’t purely mathematical. (...) a quantity of mass is not at all the same as a quantity of space, or time, or even energy. What makes each of them different isn’t that it represents some absolute, “qualitative” reality that lies beyond the mathematical language. Rather, each term in the language has its special character because it plays a different role within the semantic web."

I do not see anything non-mathematical in these qualifications.

"a functional system, not a purely logical one". What is the difference ?

I do not see how your ideas can constitute an explanation for the absence of equation for the collapse of the wave functions. You seem to not have studied the measurement problem in much details. Supporters of the many-world interpretation "explain" this absence by taking the rest of equations seriously and considering that there is no collapse and that all possible measurement results coexist in parallel. I gave another explanation by a different interpretation in my own essay. I invite you to study works on decoherence.

"It’s a complicated combination of many particular types of math, very different in general relativity, quantum mechanics, quantum field theory and the Standard Model of particle physics. (...) why the universe should be built on such strangely diverse mathematical structures. They’re so far from making an elegantly unified formal system that the math of general relativity and of quantum theory seem hardly compatible"

The mathematical language is not so different between theories. It is almost the same concepts of tensorial fields, vector bundles and curvature, that are used in General Relativity and quantum field theory. General Relativity is known to be expressible in terms of the least action principle, which is also the condition allowing the treatment of other fields by quantum field theory. It is only when actually trying to process this quantization, that troubles appear in the details.

"The full mathematical basis for the stability of other atoms turned out to be very subtle, and wasn’t fully worked out until the 1970’s.(2) The Pauli exclusion principle plays a major part in the explanation"

The stability of the helium atom has essentially the same causes and structure as that of the hydrogen atom (with the "small difference" that orbitals cannot be written in the form of nice formulas but require dirty numerical analysis to be approached).

Then the Pauli exclusion principle plays a major part in the explanation, not of the stability of any individual atom (that is in principle as ensured as that of hydrogen), but:

- In the fact that the lowest energy state of atoms heavier than hydrogen and helium have electrons in higher orbitals than the lowest one (1s)

- in the fact that atoms "repel" each other when they are too close to each other, so that they can form molecules and other structures instead of just collapsing onto the space of a single atom.

"This complex interaction keeps the neutrons from breaking down, so long as they’re bound inside a nucleus."

Not exactly. Rather, the weak interaction provides the way for neutrons to break down if they are not bound inside a nucleus.

There are radioactive atoms, where neutrons break down (turn into protons) while they are bound inside a nucleus.

Usually what keeps systems stable is that they have a lower energy than other combinations into which the weak interaction would be able to exchange them ; reactions have no problem to release energy into the environment but cannot easily take energy from the environment at usual temperatures.

"In turn, having neutrons in the nucleus allows many protons to be bound together, despite the very strong electrostatic repulsion between their positive charges."

I did not study things in details but I guess this particular property could as well be reached without neutrons by adjusting the values of physical constants.

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Dear Conrad,

I finally had a chance to read all three of your FQXi essays and found that many of the themes you address are also those that I have spent time thinking and writing papers about. To just name two examples, on the issues of the context-dependence of "measurements" of physical quantities you may find this paper interesting, which deals exclusively with the measurement of time...

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Dear Conrad,

I finally had a chance to read all three of your FQXi essays and found that many of the themes you address are also those that I have spent time thinking and writing papers about. To just name two examples, on the issues of the context-dependence of "measurements" of physical quantities you may find this paper interesting, which deals exclusively with the measurement of time and space in special relativity:

http://fqxi.org/community/forum/topic/577

And with respect to the fact that all physical quantities derive their "meaning" from their relation to everything else in the semantic web of physics, you may find this paper interesting, in which the realization that the relationship of mass to the rest of the theory within the context of QM and GR is different leads to the proposition that mass in GR and mass in QM are two different quantities:

http://deepblue.lib.umich.edu/handle/2027.42/87999

But now on to some specific points in your paper:

You make a number of perspicacious observations, in particular, the fact that in physics all quantities are defined in terms of each other, that above and beyond the mathematical structure of our physical theories there is also a semantic structure layered on top, and the fact that there are very specific mathematical structures which exhibit a particular kind of interdependence is in need of explanation are important issues I have not seen raised elsewhere but can relate to very well.

For example, it bothers me when a high energy physicist says something like "we understand" such-and-such phenomenon (say, some symmetry like U(1) or SU(2)) because it betrays such an ignorance of the semantic layer (Feynman was a notable exception to this). On the other hand, I have been told by other physicists that my emphasis on what things "mean" in physics is more appropriate for philosophy than for physics. I personally strongly disagree with this point because I believe that only when we understand what things "mean" can we make theoretical progress in the absence of corresponding empirical progress. In other words, in the age in which the cost of building new experiments which probe nature at its deepest have become nearly prohibitive, the importance of understanding the "meaning" behind our fundamental theories was never greater than it is now.

I hope that the questions you ask will be considered by more physicists because the ratio of their visibility in discourse to their importance is extremely low.

Best wishes,

Armin

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