The Deeper Roles of Mathematics in Physical Laws by Kevin H Knuth

Cristinel Stoica

Dear Kevin,

I like how you identify ordering, the ubiquity of additivity, and symmetry, as fundamental concepts allowing us to quantify and to abstract mathematical structures from the physical world. I think your essay is compelling and well written, and sheds light on the unifying vision driving your research work from other of your articles.

Best wishes,

Cristi Stoica

Alexey/Lev Burov

Dear Kevin,

Your question

"To what degree are physical laws derivable and to what degree are they accidental, contingent or decreed by Mother Nature?"

inclines me to ask in return, what do you mean by "Mother Nature", who is able to decree the laws?

I hope you would agree that the entire set of laws cannot be "derivable", since when all of them are taken together, there are no laws anymore to derive anything from.

As far as the degree of chance, our essay refutes the option of them being fully accidental. That is because their selection from an ensemble of all mathematically possible sets of laws (=full-blown multiverse of Tegmark) by the weak anthropic principle is insufficient.

Regards,

Alexey Burov.

Noson Yanofsky

Thank you for an interesting essay.

Your fundamental question about additivity is nice.

Martin Gardner wrote ". . . if two dinosaurs met two other dinosaurs in a clearing there would have been four there even if no humans were around to observe them. The equation 2 2 = 4 is a timeless truth."

Your writing on symmetry is very relevant to my paper. Have a look.

All the best,

Noson Yanofsky

Steven Sax

Dear Professor Knuth,

Your essay is excellent and completely addresses this forum topic. I like how you differentiated between the order/symmetry aspects of mathematics and the quantification aspects of mathematics, and how you profoundly connected fundamental concepts of additivity, symmetry, and order. Your salient thesis of "much of the utility of mathematics arises from our choice of description of the physical world coupled with our desire to quantify it" is a hallmark of my essay, which also looks at the dynamic constraints physical explanation and mathematical representation have on one another. You clearly and rigorously explained how certain laws may be derived as constraint equations enforcing relevant symmetries, and I really appreciated the examples given. The three slit problem you explained succinctly and lucidly, and that's especially revealing how you can derive Feynman Rules via symmetries. I would like to check out your papers on deriving the math of flat spacetime and the quantum symmetrization postulate. Thanks too for the technical endnotes which were very resourceful. Furthermore, the dialogue you had with Tejinder Singh (as well as other comments here) which further developed the connection to physical understanding, was very enlightening. I instantly give this the highest rating.

Please take a moment to review and rate my essay as well.

Thanks, Steve Sax

James Hoover

Kevin,

As time grows short, I am revisiting essays I've read to assure I've rated them. I find that I rated yours on 3/7, rating it as one I could immediately relate to. I hope you get a chance to look at mine: http://fqxi.org/community/forum/topic/2345

Jim

Akinbo Ojo

Dear Kevin,

Congrats on a brilliant essay. Early in your essay, is the quote, "Familiarity breeds the illusion of understanding". We are more familiar with pencils, pennies, rocks, sticks, candy, monkeys, planets and stars, all of which are visible than we are with electrons, quarks, virtual particles, etc. I hope you agree on this?

Now your very interesting poser, ""Why is it that when I take two pencils and add one pencil, I always get three pencils? And when I take two pennies and add one penny, I always get three pennies, and so on with rocks and sticks and candy and monkeys and planets and stars. Is this true by definition as in 2+1 defines 3? Or is it an experimental fact so that at some point in the distant past this observation needed to be verified again and again?", and "Is it not remarkable that 6 sheep plus 7 sheep make 13 sheep; that 6 stones plus 7 stones make 13 stones?", applies to those things we can see. An unstated underlying assumption of 6 + 7 = 13 or 1 + 2 = 3 is that what exists cannot perish. This is the Parmenidean curse I discuss in my essay.

But suppose what exists can perish? Will 6 + 7 still equal 13?

It is certainly easier for a single fundamental particle to perish than for a sheep made up of billions of such particles to do so. Therefore, adding 6 fundamental particles to 7, may not sum up to 13. But because the perishing of a fundamental particle in a sheep does not change the concept of a sheep, although not a certainty, it is more likely that 6 sheep plus 7 sheep has a higher probability of being 13. Certainly, by far more likely than would be the case for a fundamental particle.

All the best in the competition. If you have the time, you may take a look at my essay where I elaborate more on this.

Regards,

Akinbo

Kevin Knuth

Dear Akinbo

Thank you for your kind words.

Your comments focus on the idea that some things can perish and others cannot, and that this is central to the fact that 6 7 = 13.

I think that this is absolutely correct. In my essay, I call this concept closure and introduce the very same problem by combing a lighted match, a paper napkin and a glass of water. The order in which you combine these objects matter because if you combine the lighted match with the paper napkin, both will perish; whereas if you combine the lighted match with the glass of water they do not.

However, if you consider the constituent atoms, you have closure since atoms joined with atoms leads to collections of atoms.

So closure is the additional important concept here, along with ordering, commutativity and associativity.

All the best

Kevin

pjackson

Dear Kevin,

I think you've really nailed the topic, uplifting and inspiring me after reading so much poor and flawed analysis and philosophizing. Your essay presently lies just above mine and is well worth it's place (as I think does mine of course!)

I don't even have any questions to ask you, but hope we might discuss the 'application' of your work in terms of the important consistent insights I hope mine offers if you manage to read it.

Congratulations for a brilliant job.

Peter

Kevin Knuth

Dear Pankaj

Thank you for your comments.

I am sorry if you misunderstood my intention.

I did not try to argue which was more fundamental: physics or mathematics. Physics is a description of the natural world. Mathematics is a language that describes things in two very different ways: relationships such as symmetries and quantification with numbers.

I worked to show, explicitly, that the concept of consistent quantification constrained by symmetries leads to specific mathematical laws. (in the example given, it was additivity)

You seem to be claiming that quantum mechanics may not be governed by symmetries and thus not quantifiable via mathematics. Don't give up too fast! We have shown that the Feynman rules for manipulating quantum amplitudes is indeed derivable from fundamental symmetries:

Origin of complex quantum amplitudes and Feynman's rules

Philip Goyal, Kevin H. Knuth, and John Skilling

Phys. Rev. A 81, 022109 - Published 11 February 2010

http://journals.aps.org/pra/abstract/10.1103/PhysRevA.81.022109

and

Quantum Theory and Probability Theory: Their Relationship and Origin in Symmetry

Philip Goyal, Kevin H. Knuth

Symmetry 2011, 3(2), 171-206; doi:10.3390/sym3020171

http://www.mdpi.com/2073-8994/3/2/171

These are proofs---not arguments.

Indeed QM *can* be derived and from consistent quantification constrained by symmetries!

Kevin Knuth

Dear Jose

Thank you for your comments.

You state that

"Your reductionist approach, I think, coincides with that of mine. I feel that the only law in mathematics is the law of addition; it is not an axiom, but a rule, the eternal rule that cannot be violated even by an omnipotent creator."

I agree and disagree.

I do not think that addition is the only law. Additivity (or any invertible transform of additivity) arises from ordering, closure, commutativity and associativity. I agree that even an omnipotent being could break it, unless of course, he or she or it broke one of those conditions. Then it would be broken. But one could not keep those conditions and break additivity.

I believe that there are fundamental symmetries and relationships and that these place strong constraints on any attempt to consistently quantify phenomena. These constraints give rise to constraint equations (enforcing these basic properties) which we interpret as physical laws.

Kevin Knuth

Dear Jayakar

Thank you for your comments.

I am not sure what you mean when you say that

"Symmetry and Asymmetry are the basic kind of abstracts for both Physical and Mathematical entities"

Physical things sometimes have symmetries. This simply means that one can make a correspondence between some aspect of one physical system and some aspect of a kind of transform of that physical system.

You also note that

"quantization limits the value with 1 and returns 0 on absence of a quanta"

This is an example of a quantification scheme mapping a 0 to one state and a 1 to another.

Basically, what I argue (and explicitly demonstrate) in my essay is that the act of quantification (assigning numbers to states or things or systems or aspects of systems) is often constrained by symmetries and that these constraints result in constraint equations, which can be interpreted as physical laws.

I hope that this helps explain my stance.

Kevin Knuth

Dear Allen

Thank you for your comments!

You write:

"What happens if there is a union ( or battle ) of Knuth and t'Hooft..."

I am not sure what a union would look like, but if it were a battle, he would probably simply beat me silly. ;)

I am sorry that your question is not making much sense to me. However, I get the feeling that you are asking something rather deep that has not quite settled in your mind.

Perhaps you can try to rephrase it?

I would be very interested in understanding what you are considering.

My guess is that you are worried about my reliance on an observer that actively quantifies aspects of the universe, and reconciling that with a view of the universe as some kind of cellular automaton that simply marches forward according to some set of "bit flipping" rules.

Is something like this the questions?

Kevin Knuth

Dear Rob

Thank you for your comments and questions.

I am not completely sure that I understand what you find confusing. Somehow, we must be talking past one another.

Let me give this a try...

I do not really talk about regrouping, which I feel has connotations of grouping objects to form different objects.

When I discussed combining objects, I spent a little time discussing closure, which is the idea that on combining sets of objects, the objects do not change or combine or annihilate or anything like that.

I do talk about sets, but one only needs to consider that we are considering sets of things that we want to describe quantitatively. That is, we will map sets of things to real numbers.

The point is that if you have sets of things that you want to quantify, and these things obey closure on combination, they can be ordered in some way, and they obey both commutativity and associativity on combination, then the only consistent means that one can conceive of to quantify such sets of objects is via some invertible transform of additivity. This is why adding the quantities of things when you combine them works (as long as you have closure, they can be ordered and as long as the combining rule obeys commutativity and associativity).

Does this help?

Kevin

pjackson

Kevin,

Thanks for your kind comments on my essay, and links. I've now managed the arXiv paper but the 2nd link is dead, could you please re-post, direct or here with the 'link help page' above.

I found the arXiv paper excellent and, as you say, giving a core relationship the sound mathematical formalism required. I checked on our 'It from bit' conversation and we similarly agreed. I've pushed on with many other aspects since then. I don't know if you did go back to my (2010/11) '2020 Vision' essay which identifies some. (I've also identified ex NASA Edward Dowdyes 'Extinction Shift' hypothesis from 1996 as an early version of the same basic dynamic.)

I hope you've found some time to explore my papers too. This arXiv paper also gives a broadish overview, http://arxiv.org/abs/1307.7163 and you should also find the (HJ published) consequential cyclic cosmology paper interesting. A_CYCLIC_MODEL_OF_GALAXY_EVOLUTION_WITH_BARS I do hope we can collaborate on development and presentation as you're far better placed to make headway against the entrenched mainstream direction than John and I are. (Ted Dowdye seems a poor communicator and a loner, the curse of advancing understanding!)

Peter

Kevin Knuth

Hi Peter

Thanks for the links to your papers.

My paper on emergent spacetime is here at JMP, 2014, 55(11), p.112501.

http://scitation.aip.org/content/aip/journal/jmp/55/11/10.1063/1.4899081

But you can also get it on the arxiv at:

http://arxiv.org/abs/1209.0881

My Contemporary Physics paper, which addresses both spacetime and fermion physics is on the arxiv as well:

http://arxiv.org/abs/1310.1667

Cheers

Kevin

Philip Gibbs

Kevin,

I have reread your fascinating essay. The question of whether 1 2 = 3 is experimental is both profound and provocative. One answer is that for concrete representations of the problem it is empirical, but there is a mathematical abstraction that serves as a model which can be derived from axioms. With the abstract model in hand we just need to check which physical situations comply with the model. This of course just raises deeper questions about how we know the axioms and whether they were invented to fit the physics or they exist in their own right.

I agree very much about the importance of symmetry. After all without temporal and spatial symmetry we could not form any fixed physical laws. I like the way you see symmetries in algebraic principles. I think that this is where they must emerge from

Daniel Moskovich

Dear Kevin,

I very much enjoyed your essay. Your argument strikes me as Kantian- you are arguing for innate categories of perception (mathematics) projected onto reality (physics). Have you thought about things explicitly in these terms? How would you compare your argument to the philosophy of Kant?

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