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FQXi FORUM

February 23, 2018

CATEGORY:
Trick or Truth Essay Contest (2015)
[back]

TOPIC: The Language of Nature by David Garfinkle [refresh]

TOPIC: The Language of Nature by David Garfinkle [refresh]

Galileo considered mathematics the language of nature. However, Wigner thought the effectiveness of mathematics in physics "miraculous" and noted that much of the mathematics needed for quantum mechanics had been previously developed by mathematicians for purposes having nothing to do with physics. I argue that Galileo's view is correct; but that the examples cited by Wigner in support of his view can be explained using two deep truths, one about mathematics and the other about physics. These truths are: (1) Since the advent of non-Euclidean geometry, new mathematics has been developed by abstracting and generalizing old mathematics. (2) New physical theories have old physical theories as limiting cases.

David Garfinkle is a Professor of Physics at Oakland University, in Rochester, Michigan. He has a BA in physics (Summa cum laude) from Princeton University and a PhD in physics from The University of Chicago. His field of research is Einstein's general theory of relativity, especially the study of spacetime singularities. He is the author (along with his brother Richard Garfinkle) of "Three Steps to the Universe" a book for general readers on black holes and dark matter.

Dear David,

I think your lucid essay gave a succinct and nearly complete answer to the theme question. The only thing I would add is that when presented with physical phenomena that lend themselves to the application of already discovered/invented mathematics, it still takes some imagination to 1)realize that there is "off the shelf" available mathematics that will be useful in describing these phenomena (it is easy to underestimate the flash of insight it takes after such usefulness has already been recognized), and 2) to apply the mathematics in such a way that otherwise unavailable new physical insights are gained.

The role of imagination is even more acute in those situations where there is no "off the shelf" mathematics available, and I am sure we agree that these situations occur in the physical sciences also quite often.

In my own essay, I focused the part devoted to answering the theme question on the latter kinds of situation, but if someone were to ask me about those involving the unreasonable usefulness of already existing mathematics, I would give essentially the same answer as you did.

Best wishes,

Armin

PS. I saw you at a recent Math Colloquium at UM about general relativity. As an expert in this field, what was your opinion about the new proposed definition of angular momentum?

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I think your lucid essay gave a succinct and nearly complete answer to the theme question. The only thing I would add is that when presented with physical phenomena that lend themselves to the application of already discovered/invented mathematics, it still takes some imagination to 1)realize that there is "off the shelf" available mathematics that will be useful in describing these phenomena (it is easy to underestimate the flash of insight it takes after such usefulness has already been recognized), and 2) to apply the mathematics in such a way that otherwise unavailable new physical insights are gained.

The role of imagination is even more acute in those situations where there is no "off the shelf" mathematics available, and I am sure we agree that these situations occur in the physical sciences also quite often.

In my own essay, I focused the part devoted to answering the theme question on the latter kinds of situation, but if someone were to ask me about those involving the unreasonable usefulness of already existing mathematics, I would give essentially the same answer as you did.

Best wishes,

Armin

PS. I saw you at a recent Math Colloquium at UM about general relativity. As an expert in this field, what was your opinion about the new proposed definition of angular momentum?

report post as inappropriate

Dear David Garfinkle,

Well written and interesting. The idea of Euclidean geometry as, in effect, a limiting case of more general geometries seems to me both relevant to the essay theme and insightful. The two paragraphs on page 4 beginning, "But why do we discover the limiting cases first?' are pearls.

Based on your essay, you might find mine interesting.

Best wishes.

Bob Shour

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Well written and interesting. The idea of Euclidean geometry as, in effect, a limiting case of more general geometries seems to me both relevant to the essay theme and insightful. The two paragraphs on page 4 beginning, "But why do we discover the limiting cases first?' are pearls.

Based on your essay, you might find mine interesting.

Best wishes.

Bob Shour

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As a layman I appreciate your concise and classical writing approach (readable). Another example of mathematics that was discovered before it was put to use is Ramanujan's 'mock modular functions' which is now used in black hole physics. More surprises await.

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Dear Prof. Garfinkle:

I agree with your clear argument that the common nature of discovery of new physics and math explains much of the mutual effectiveness of the two. I would take it one step further; some of this effectiveness is an illusion created by inappropriate application of abstract mathematical models in certain cases. The general acceptance of such a model can create an established scientific dogma, which may actually discourage the development of more appropriate models.

The example that I present in my essay is Quantum Mechanics and the Hilbert Space Model. "Remove the Blinders: How Mathematics Distorted the Development of Quantum Theory" presents a simple realistic picture that makes directly testable experimental predictions, based on little more than Stern-Gerlach measurements. Remarkably, these simple experiments have never been done.

The accepted view of QM is that the physics (and mathematics) of the microworld are fundamentally different from those of the macroworld, which of course creates an inevitable boundary problem. I take the radical (and heretical) view that the fundamental organization is the same on both scales, so that the boundary problem immediately disappears. Quantum indeterminacy, superposition, and entanglement are artifacts of the inappropriate mathematical formalism. QM is not a universal theory of matter; it is rather a mechanism for distributed vector fields to self-organize into spin-quantized coherent domains similar to solitons. This requires nonlinear mathematics that is not present in the standard formalism.

Alan Kadin

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I agree with your clear argument that the common nature of discovery of new physics and math explains much of the mutual effectiveness of the two. I would take it one step further; some of this effectiveness is an illusion created by inappropriate application of abstract mathematical models in certain cases. The general acceptance of such a model can create an established scientific dogma, which may actually discourage the development of more appropriate models.

The example that I present in my essay is Quantum Mechanics and the Hilbert Space Model. "Remove the Blinders: How Mathematics Distorted the Development of Quantum Theory" presents a simple realistic picture that makes directly testable experimental predictions, based on little more than Stern-Gerlach measurements. Remarkably, these simple experiments have never been done.

The accepted view of QM is that the physics (and mathematics) of the microworld are fundamentally different from those of the macroworld, which of course creates an inevitable boundary problem. I take the radical (and heretical) view that the fundamental organization is the same on both scales, so that the boundary problem immediately disappears. Quantum indeterminacy, superposition, and entanglement are artifacts of the inappropriate mathematical formalism. QM is not a universal theory of matter; it is rather a mechanism for distributed vector fields to self-organize into spin-quantized coherent domains similar to solitons. This requires nonlinear mathematics that is not present in the standard formalism.

Alan Kadin

report post as inappropriate

Dear David,

We found your essay enjoyable and reasonably convincing: old physics and old maths relate to each other; new maths generalises from old maths, older physical theories are limiting cases of newer theories, and hence it is not surprising that the newer theory is built on some of the new maths. We were just wondering if there is any way to make this argument more quantitative / precise. In the sense: is there some tangible way to see that the abstraction / generalisation in mathematics precisely parallels the abstraction / generalisation in physical theories.

Maybe one could add here that in the development of a new physical theory, the transition from new data to a new mathematical formulation often involves an intermediate step - great conceptual leaps / unifications. Just to take some very simple examples: the unifying idea that the force that makes the apple fall to the earth is the same as the force that makes the moon go round the earth; that electricity and magnetism are two facets of the same force; the black-body radiation spectrum compels the proposal that radiation is emitted and absorbed in discrete units; the photoelectric effect suggests that the energy of radiation is quantised in units of its frequency, etc. It would seem that the introduction of a new concept is often an important intermediate step between the new data and the new mathematical description.

Thanks for putting up a very nicely written and lucid essay.

Best regards,

Anshu, Tejinder

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We found your essay enjoyable and reasonably convincing: old physics and old maths relate to each other; new maths generalises from old maths, older physical theories are limiting cases of newer theories, and hence it is not surprising that the newer theory is built on some of the new maths. We were just wondering if there is any way to make this argument more quantitative / precise. In the sense: is there some tangible way to see that the abstraction / generalisation in mathematics precisely parallels the abstraction / generalisation in physical theories.

Maybe one could add here that in the development of a new physical theory, the transition from new data to a new mathematical formulation often involves an intermediate step - great conceptual leaps / unifications. Just to take some very simple examples: the unifying idea that the force that makes the apple fall to the earth is the same as the force that makes the moon go round the earth; that electricity and magnetism are two facets of the same force; the black-body radiation spectrum compels the proposal that radiation is emitted and absorbed in discrete units; the photoelectric effect suggests that the energy of radiation is quantised in units of its frequency, etc. It would seem that the introduction of a new concept is often an important intermediate step between the new data and the new mathematical description.

Thanks for putting up a very nicely written and lucid essay.

Best regards,

Anshu, Tejinder

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Dear Professor Garfinkle,

Enjoyed your essay, learned some things about Galileo’s view right off the start, "*the image of nature and its laws as a book and asserted that that book was written in the language of mathematics*".

Your comment "*.. abstraction and generalization generated a great number of new mathematical objects and led to another pursuit of modern mathematicians: classification. For each new type of object (group, vector space, manifold, Lie Algebra, etc.) one would aim to produce a complete classification of all possible objects of that type.*" and argument that follows was very clear and thought provoking.

I hope you get a chance to have a look at my essay here. I try to build on these classifications and provide visual models of objects that match the proven mathematical models of the particles of the standard model and would be very interested in your comments.

Best of luck in the contest, you deserve a high rating and thank you for the essay.

Regards,

Ed Unverricht

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Enjoyed your essay, learned some things about Galileo’s view right off the start, "

Your comment "

I hope you get a chance to have a look at my essay here. I try to build on these classifications and provide visual models of objects that match the proven mathematical models of the particles of the standard model and would be very interested in your comments.

Best of luck in the contest, you deserve a high rating and thank you for the essay.

Regards,

Ed Unverricht

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Dear David,

In your concise, well argued essay, I think you perfectly answer Wigner's question when you say:

"Thus the new mathematics was related to the old mathematics, which was in turn related to the old physics. But why was the new mathematics just what was needed for the new physics? Here the answer has to do with the fact that old physical theories are limiting cases of new physical theories."

There's not much more to say... as long as we interpret this year's question as the relationship between known (or potentially known) mathematics and the observable (or potentially observable) universe. But there's another deeper question (perhaps too deep for science, and destined to remain in the realm of philosophy): what is the relationship between "all of mathematics" (in the limit that would be accessible to an infinitely intelligent mathematician) and the totality of all that physically exists?

Thank you for your essay, and good luck in the contest!

Marc

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In your concise, well argued essay, I think you perfectly answer Wigner's question when you say:

"Thus the new mathematics was related to the old mathematics, which was in turn related to the old physics. But why was the new mathematics just what was needed for the new physics? Here the answer has to do with the fact that old physical theories are limiting cases of new physical theories."

There's not much more to say... as long as we interpret this year's question as the relationship between known (or potentially known) mathematics and the observable (or potentially observable) universe. But there's another deeper question (perhaps too deep for science, and destined to remain in the realm of philosophy): what is the relationship between "all of mathematics" (in the limit that would be accessible to an infinitely intelligent mathematician) and the totality of all that physically exists?

Thank you for your essay, and good luck in the contest!

Marc

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Dear David,

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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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

report post as inappropriate

Dear David,

I enjoyed reading your essay. Indeed, non-Euclidean geometries originated from a question of pure mathematical, or rather logical interest, and constituted the (temporary) departure of mathematics from physics. Axiomatic systems emancipated mathematics, making it its own raison d'être, and leading to the maths that, unexpectedly, will be needed someday by quantum theory and general relativity. Your arguments demystify this apparent miracle. If I understand well, it goes like this: physics and mathematics were once siblings, then they developed independently, and then, their progenies turned out to be cousins. Add to this that mathematics evolved by generalization, and physics was bound by Ockham's razor to find first limiting cases, this effectiveness is no longer unreasonable. I liked very much your writing and your arguments.

Best wishes,

Cristi Stoica

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I enjoyed reading your essay. Indeed, non-Euclidean geometries originated from a question of pure mathematical, or rather logical interest, and constituted the (temporary) departure of mathematics from physics. Axiomatic systems emancipated mathematics, making it its own raison d'être, and leading to the maths that, unexpectedly, will be needed someday by quantum theory and general relativity. Your arguments demystify this apparent miracle. If I understand well, it goes like this: physics and mathematics were once siblings, then they developed independently, and then, their progenies turned out to be cousins. Add to this that mathematics evolved by generalization, and physics was bound by Ockham's razor to find first limiting cases, this effectiveness is no longer unreasonable. I liked very much your writing and your arguments.

Best wishes,

Cristi Stoica

report post as inappropriate

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