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Trick or Truth Essay Contest (2015)
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On Finding Meaning in the Language of Physics by Conrad Dale Johnson
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Author Conrad Dale Johnson wrote on Feb. 25, 2015 @ 16:38 GMT
Essay AbstractGalileo spoke of the universe as a book, written in the language of mathematics. Following his metaphor, we ask – why does the language of physics include so many different kinds of math, some simple, others extremely complex? Like any language, this one has both a formal structure and a web of semantic relationships, that give contexts of meaning to its terms and expressions. In physics the formal, mathematical structures have been studied in depth, and despite their difficulties, this aspect of the language is quite well understood. The semantic aspect of physics, though, remains unexplored. We tend just to take it for granted that the many terms that appear in the equations are physically meaningful – terms like space and time, mass and charge, etc. Yet none of these is observable or even definable by itself, apart from the contexts given by other terms in the language. Each physical variable and constant appears in many key equations, which together define its meaning in relation to other terms. This essay considers what it takes for a language to do this – to make all its expressions meaningful in terms of each other. This kind of semantic self-sufficiency is unique to physics, since in any other language, expressions have meaning primarily by referring to things beyond the language itself. But the language of the physical world is fundamental; there’s no deeper level of meaning to which it can refer. We consider the diverse mathematics involved in atomic structure to illustrate how physics is able to give meaning to the complex variety of facts and regularities on which everything else in the universe depends.
Author BioI’ve lived mainly in the US. My interest in the foundations of physics goes back to my graduate-school days at the University of California at Santa Cruz, where I earned my Ph.D. in the History of Consciousness, focusing on the evolution of Western philosophy and science. I’ve contributed essays on related topics to the FQXi contests since 2012.
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Ed Unverricht wrote on Feb. 25, 2015 @ 21:55 GMT
Hi Conrad Dale Johnson,
Enjoyed reading your essay. You caught me with this comment "What’s not at all understood is why the universe should be built on such strangely diverse mathematical structures." and "Physics has succeeded brilliantly in explaining a vast range of phenomena by uncovering their underlying mathematical structures."
And I agree with your comment "The question now becomes, how can we explain the peculiarly various architecture of the mathematical language itself?" where "the existence of atomic matter in our universe depends on all these very different mathematical structures."
Very nice ideas starting with "The formal structure of this language is represented in our equations; its vocabulary consists of all the parameters that appear in the equations." In the words of Schrödinger, "The world extended in space and time is but our representation."
Regards, Ed
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Author Conrad Dale Johnson replied on Feb. 26, 2015 @ 13:04 GMT
Ed – Thanks very much. I’m glad the basic question I raised in the essay seemed interesting to you. I only wish I were able to go further in suggesting how it might be answered!
Conrad
Edwin Eugene Klingman wrote on Feb. 26, 2015 @ 02:47 GMT
Dear Conrad Dale Johnson,
Your interesting essay focused on the physical world as unique in giving meaning to language and particularly mathematical language. Whereas languages are defined in basic terms, the physical world is a seamless web that presents all definitions in terms of itself, i.e., it's own self-consistent reality. The language may contain truth or falsity. The physical reality is true only. As you note, this truth is not the truth of a "proof", it is in the contextual or semantic meaning.
In one example you refer to
tracking a particle in a magnetic field. My essay focuses on finding the meaning of physical reality based on this very key experiment in physics and it's oversimplification in the standard (i.e., Bell) narrative. The interesting point is that his mathematical 'proof' is correct, but his physical theory that is being 'proved' is incorrect. I hope you will find the time to read my essay and comment upon it.
Whereas many interesting essays deal with the specifics of the mathematical language, I very much like your focus on the ultimate reality that underlies all descriptions, whatever the language, natural or mathematical. I believe that some of the more esoteric languages and narratives popular at the moment will fall by the wayside relatively soon, while there will be no change at all in the underlying physical reality, just better understanding of it.
My best regards,
Edwin Eugene Klingman
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Author Conrad Dale Johnson replied on Mar. 3, 2015 @ 18:17 GMT
Dear Edwin Eugene,
Thanks for your note. The way you're talking about language -- as a means of describing an underlying reality -- is very reasonable. But the point of my essay is that the physical reality itself functions as a unique kind of language, in making each of its constituent elements meaningful in terms of other elements.
It's not just self-consistency that makes this possible. In fact, I'm not sure it's necessary for all the different mathematical structures we find in physics to be strictly consistent. For example, it might be that the large-scale gravitational structure of spacetime and the sub-microscopic quantum structure of particle interaction can't be described within the same mathematical framework. What's important is that each of them contributes something different to the semantic web that physically defines and measures everything.
Thanks again -- Conrad
David Lyle Peterson wrote on Feb. 27, 2015 @ 02:48 GMT
Dear Conrad,
I liked some of your concepts: math is built into the physical world and is more than just a convenient tool, articulating our universe, speaking in the language of physics (``it has nothing beyond itself to depend on or refer to’’), interrelated definitions, measurement provides context, complex web of interaction, and emphasis on stable structured matter. As you say, the mathematical language of the physical world is pretty well understood. But for the quantum world, I’m sure you know that the interpretation of the math is becoming increasingly contentious – and that is also part of understanding. And unlike math, Nature is the owner of definitions for physics; so physics has to continually improve its definitions to match (quantum state, photon, electron, vacuum,…) —and you mentioned measurement. The math is precise, but terms are often unclear. E.g., Sometimes, measurement means a projection (magnetic orientation, polarization) and sometimes collapse to a point with no surviving state.
On the whole, nice essay!
Regards, Dave.
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Author Conrad Dale Johnson replied on Mar. 3, 2015 @ 19:00 GMT
David --
I appreciate your interest, and I'm really glad you were able to make some sense out of all those ideas. You're certainly right that even when the math is well understood, the interpretation is still highly problematic, especially regarding measurement. A large part of the problem is that measurement isn't a single well-defined process; it involves very different kinds of procedure depending on what you’re measuring. This is something I discussed in
my 2012 FQXi essay and again
in 2013.
Thanks very much -- Conrad
susanne kayser-schillegger wrote on Mar. 2, 2015 @ 02:17 GMT
Dear Conrad,
rather impressed by your insightful words I want to cite one sentence:
" So the world is in a very deep sense mathematical, yet not in the sense imagined by the long tradition of philosophical speculation that extends from Pythagoras and Plato to Max Tegmark. What’s fundamental in this universe is not its mathematical pattern per se, but what all these highly diverse kinds of 5
patterning are able to accomplish."
I wish you luck in continuing your quest for a deeper understanding of math and physics being complementary.
Best
Lutz
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Anonymous wrote on Mar. 4, 2015 @ 00:06 GMT
Interesting essay. The language may lead to misunderstanding.
Take for example scientists saying that the universe is this or is that...
To say "IS" is to commit oneself to the description of what this thing is by itself .i.e metaphysics / ontology; not the experience we have of it.
Be more appropriate when talking physics/physicality to say that things "appear" to be.... Then, one does not pretend knowing something, when in fact he doesn't.
Good luck,
Marcel,
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Christopher Duston wrote on Mar. 7, 2015 @ 19:41 GMT
Hello Conrad,
It is interesting how our essays superficially seem to tackle the same set of questions, but arrive at drastically different places! I particularly liked how you incorporated the idea that "[...] nothing is meaningful in and of itself; meanings always depend on the possibility of other meanings." This is certainly the strength of your essay - it's the linguistic equivalent of "background independent".
Having said that, I have two major reservations:
1) Since we don't have a complete understanding of the universe, it seems we can't be sure our choice of parameters which we assign meanings (mass, charge, etc) are correct, or even remotely reasonable. The mass of an electron shows up in many places, but at the end of the day this is simply a parameter we picked, and without a complete theory showing how this parameter is really connected to the unified whole, I would be worried about deluding ourselves. In some sense, by using the background independence you've removed the "invariance of meaning". If the very definition of the quality you are talking about is not fixed, how can this idea be logically consistent? Isn't this something like saying "meaning only has meaning if it has meaning", and leave it up to the observer to supply their own definition of meaning?
2) When you talk about atoms being fundamental to the functionality of the physical world, it seems like you are really referring to how we experience the physical world - to the act of measurement. We have no idea if space and time make sense in the absence of stable atoms - all we know is that they do make sense in the presence of them. This further suggests to me that correct interpretation of "meaning" in this context requires input from an observer.
In any case, I think the self-consistency shown in this essay is exemplary, and it's basic thesis is excellent. I wish you the best of luck in the contest.
Chris
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Author Conrad Dale Johnson replied on Mar. 9, 2015 @ 17:56 GMT
Chris – thanks very much for your comments. I did read your essay and note that it was worth reading twice – I’ll do that soon.
You’re right of course that all of physics is provisional, even the choice of parameters. And since our current physics was developed to stay as close as possible to the 19th-century picture of fields and particles, it is quite likely that we’re...
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Chris – thanks very much for your comments. I did read your essay and note that it was worth reading twice – I’ll do that soon.
You’re right of course that all of physics is provisional, even the choice of parameters. And since our current physics was developed to stay as close as possible to the 19th-century picture of fields and particles, it is quite likely that we’re “deluding ourselves” in some ways. Yet the current language of physics is empirically so successful that its vocabulary is surely at least “remotely reasonable”.
I see why you make the connection with “background independence”, as opposed to relying on some absolute framework to give meaning to physical parameters like mass or charge. But for me the deeper issue is the interdependence of background-contexts. For example, we need spacetime to measure distances and velocities and masses, but spacetime itself is dynamically dependent on masses and their motion. Physicists who want a background-independent theory see this interdependence, but still look for a mathematical formulation that abstracts from it. Spacetime as well as mass and charge, etc. are supposed to emerge from purely mathematical relationships within a “unified whole”.
Even if this ultimately succeeds, though, they still have to be able to derive all the variety of interdependent observables from their equations. So the question will remain, as to how such a system is able to make all its parameters meaningful in terms of each other.
You’re right also that being physically meaningful has to do with being measurable and observable – and this is the functionality that atoms support. And I suppose in some other universe, space and time might be measurable in some way that doesn’t depend on stable entities that move around and relate to each other as atoms do. The discussion here is very much relative to our particular world. But I think that any world in which any kind of measurement is possible would be subject to the same basic constraint, i.e. that observables have to be meaningfully definable in terms of other observables.
Best regards,
Conrad
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Luca Valeri wrote on Mar. 11, 2015 @ 12:30 GMT
Hi Conrad,
I'm glad you wrote an essay in this contest. I read your essay before I wrote mine. So it might not be a coincidence, that there might be some parallels between our essays, although I tried to actively avoid the word 'meaning'.
But it might also be unavoidable to have some parallels between the essays, since yours point at the heart of the problem: how terms (physical or mathematical) accuire meaning.
That meaning is accuired in physics by the relation of the terms themselves would correspond to Heissenbergs 'Closed Theories'. Heisenberg states that the history of physics is a succession of closed theories, where the older theories are limiting cases of the newer ones. But also, that terms of the newer ones can only be understood by the terms of the older ones. Von Weizsäcker uses the term 'sematical consistency'.
A kind of circularity seems unavoidable. The means of verification or falsification of the theory (measurements) are described by the very same theory, that should be falsified or confirmed. This disturbed Lorenzen who tried to build a proto physics, where the basic terms like space, mass etc. are defined by operational procedures (for example the rubbing of a stone against other stones to get a flat space), that are independent of the physical theory.
I discuss this and other issues in my essay (hopefully soon to be shown), that I had to write really fast and I had no time to go into the details, but that I hope we find the time to discuss in the forums.
Best regards
Luca
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Author Conrad Dale Johnson replied on Mar. 11, 2015 @ 14:33 GMT
Hi Luca,
I’m very glad to hear from you, and to find that an essay of yours is in the queue.
I hadn’t thought about the circularity of meaning between observation and theory, though that’s certainly another important aspect of the theme. The theories only have meaning by explaining observations, and the observations would be meaningless data (or would never even be made) apart from the theories.
It’s understandable that the concept of meaning is avoided in physics, since it has such a strong connotation of subjectivity and anthropocentrism. That’s unfortunate, since it makes it hard to think about how the physical world itself makes all its components meaningfully definable and observable.
The heart of the problem is a certain understanding of what it means to be fundamental, that goes back to the beginning of our intellectual tradition. We tend to assume that whatever’s truly basic in the world must be something that doesn’t need any basis itself, that just
is in some absolute sense. Mach imagined that science could be built on pure, raw sense-data; you mention the idea that it could all be reduced to simple operational procedures. The FQXi contests are full of attempts to derive the physical world from mathematical axioms. One way or another, it seems there must be some sort of absolute starting-point.
I find the alternative difficult to articulate – that everything needs a basis, a context in which it can make a definable difference to other things. So, everything also has to contribute to contexts in which other things make a meaningful difference. That’s a very abstract way of putting it, though.
I look forward to reading your essay, no matter how rapidly written, and will surely find time to discuss it.
Thanks – Conrad
Sophia Magnusdottir wrote on Mar. 16, 2015 @ 14:35 GMT
Hi Conrad,
I think your essay is the first I've read in this contest that gave me something new to think about, thanks for that :) I never really thought about the self-referentialness of physics in the sense that you discuss it.
I am somewhat puzzled that you're not elaborating on the question of dimension (not spatial dimension, but dimension of parameters). These are, in some sense, that what distinguishes math from physics, that what makes it "real", if you wish. This has always been, still is, a great mystery to me. In my essay, of course, I have argued that for the pragmatist it doesn't really matter.
You should think over your opinion about the anthropic principle though. It isn't true that it's useless, this is just incorrect. That our theories must be so as to allow life to exist does put a constraint on the theories, and this can actually be used to obtain limits on certain parameters.
Let me add that your essay is very well and fluidly written :)
-- Sophia
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Author Conrad Dale Johnson replied on Mar. 17, 2015 @ 14:41 GMT
Sophia,
Thanks very much. You’re right, if we want to understand how the physical world defines itself, the big outstanding question is why there all these parameters with these particular dimensions. If I could have done it convincingly, I would have tried to sort this out in the essay. But this is one of those questions that’s so obvious and yet so difficult that even the most...
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Sophia,
Thanks very much. You’re right, if we want to understand how the physical world defines itself, the big outstanding question is why there all these parameters with these particular dimensions. If I could have done it convincingly, I would have tried to sort this out in the essay. But this is one of those questions that’s so obvious and yet so difficult that even the most abstract theoreticians are pragmatic enough to avoid it.
I entirely agree with you that this is the key difference between math and physics – that in physics we find all these curious differences between mass and charge, space and time, baryon and lepton numbers, etc. But the old way of thinking about them as “qualitative” differences between different substances is hardly useful. And they are all quantities, all related to each other mathematically, so I can see why someone might want to leap to Max Tegmark’s conclusion. So, even though the difference between math and the real world is perfectly evident to all non-physicists, I don’t think we can be clear about it unless we ask a different kind of question. Instead of “what’s the world made of?” or “what is reality?” I think we need to ask, “what’s the world doing?” and “what does it have to do, for anything to be meaningfully real?”
As to the anthropic principle, what I really object to is its vagueness. With just the same logic I could say, our theories must be such as to allow me, Conrad, to exist. That’s very true, but it brings in no new constraint on fundamental physics. Neither do we need any special constraints on our theories to allow for human beings on Earth... so instead, we talk about what’s needed for the existence of life. But again, since life might be able to exist in other forms, it’s still very vague what physics this actually requires. So I argue in my essay, let’s ask instead what kind of physics is able to produce atoms. Then at least we’re in the right ballpark. And this also connects to the question about how the universe is able to define all its own facts and parameters.
Thanks again for your encouraging comments – Conrad
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Sylvain Poirier wrote on Mar. 24, 2015 @ 14:42 GMT
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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Author Conrad Dale Johnson replied on Mar. 25, 2015 @ 14:47 GMT
Dear Sylvain –
Thanks for reading and commenting so extensively.
Your notion of a cyclically self-referential foundation of mathematics is interesting and apparently very unusual. The brief notes on this topic in my essay reflect a mainstream view that seems sensible to me, but I’m willing to believe there are other possibilities. Even so, I think there’s a basic difference...
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Dear Sylvain –
Thanks for reading and commenting so extensively.
Your notion of a cyclically self-referential foundation of mathematics is interesting and apparently very unusual. The brief notes on this topic in my essay reflect a mainstream view that seems sensible to me, but I’m willing to believe there are other possibilities. Even so, I think there’s a basic difference between defining something mathematically and measuring something physically. It’s hard to imagine any real-world situation in which this difference wouldn’t be obvious. As to the difference between the logical structure of mathematics and the “functionality” of the physical world – I tried to describe this at more length in my
2013 FQXi essay.
My paragraphs on quantum measurement are too brief to reflect any of the complexity of the subject. Of course decoherence is a highly developed theory at this point, and though it doesn’t by itself account for unique outcomes, there are a number of attempts to model the “collapse” mathematically. So my statement that “no equation exists” for this isn’t quite true; I should say no such approach is generally accepted. But my point here is that the difference between an unobserved state and an observed one has to do with the adequacy of the physical context to define a value for a given parameter... not with any specific process of “collapse” that would need its own mathematical model. Both my 2013 and
2012 essay develops this thought further.
You’re right that the essay ignores many very important commonalities in the mathematics of the various physical theories. Sophia Magnusdottir also mentioned this in the thread to
her essay. But my point is that the deep differences between these mathematical structures also seems to be necessary to support a universe like ours. So the quest for unification in physics needs to be supplemented by an analysis of what’s required to makes all these different kinds of information meaningful in terms of each other.
As to atomic structure, I agree with your notes. My rough summary was intended only to show how much of the complexity of physics is involved in the existence of these “basic” building-blocks.
Thanks - Conrad
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Richard Lewis wrote on Apr. 2, 2015 @ 10:08 GMT
Dear Conrad,
I found your essay to be very clear and I agree completely with your emphasis on the importance of language.
Physics depends greatly on the correct use of language to describe reality. Mathematics is to some extent less dependent on the English language as it has a language of mathematical symbols with its own rules.
I find that there is a need for greater attention to resolving the nature of reality at the descriptive level rather than, for example, the quantum theory acceptance of a lack of agreed physical interpretation.
As an example in your essay you mention the idea of 'which slit the photon passed through' in the interference experiment. It seems totally clear to me that the photon is a real physical wave that passes through both slits.
It is only when we have resolved at the descriptive level, the nature of light, of mass and charge and found a framework for unification that we can begin to apply mathematics effectively and build back in the ideas of quantum theory and the standard model within the new conceptual framework.
I have tried to make a start with this and my essay 'Solving the mystery' covers the main ideas.
It would be great if we could devise a 'language of physics' which was English and yet as precise as mathematics to construct a physical description of reality in a structured way. This would somehow allow us to agree on a valid description which then feeds into the mathematical modelling.
Regards
Richard
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Joe Fisher wrote on Apr. 3, 2015 @ 16:31 GMT
Dear Dr. Johnson,
I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.
All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.
Joe Fisher
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Alexey/Lev Burov wrote on Apr. 3, 2015 @ 20:40 GMT
Dear Conrad,
I see your accent on meanings as interesting. One way to put meanings in a metaphysical center is by means of a transcendental mind as a source of meanings. Another way is to assume something like full-blown multiverse of Tegmark, where all logically possible options are realized. I tried to see what is your view on that but did not succeed. In
our essay the Tegmark's multiverse is refuted, but you might think differently.
Regards,
Alexey Burov.
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Armin Nikkhah Shirazi wrote on Apr. 20, 2015 @ 19:15 GMT
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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