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

Kevin Knuth

Essay Abstract

Many have wondered how mathematics, which appears to be the result of both human creativity and human discovery, can possibly exhibit the degree of success and seemingly-universal applicability to quantifying the physical world as exemplified by the laws of physics. In this essay, I claim that much of the utility of mathematics arises from our choice of description of the physical world coupled with our desire to quantify it. This will be demonstrated in a practical sense by considering one of the most fundamental concepts of mathematics: additivity. This example will be used to show how many physical laws can be derived as constraint equations enforcing relevant symmetries in a sense that is far more fundamental than commonly appreciated.

Author Bio

Kevin Knuth is an Associate Professor in the Departments of Physics and Informatics at the University at Albany. He is Editor-in-Chief of the journal Entropy, and is the co-founder and President of the robotics company Autonomous Exploration Inc. He has 20 years of experience in applying Bayesian and maximum entropy methods to the design of machine learning algorithms for data analysis applied to the physical sciences. His current research interests include the foundations of physics, autonomous robotics, and the search for and characterization of extrasolar planets.

Download Essay PDF File

James Hoover

Kevin,

The two roles, symmetries and calculation, and your class examples help provide more clarity to your points. I wonder about your opinion of equations derived through educated guesses and/or trial-and-error. Such derivative functions must always be suspect in modeling and subjected to testing and peer review. Look at BICEP2. Your conclusive remarks seem to make this point when subscribing to a deeper understanding of math roles in physics regarding order, symmetries, and effectiveness.

Enjoyed your essay, feeling that your cautions are well-founded and your examples and descriptions were a great aid to clarity.

Jim

Kevin Knuth

Dear Jim

Thank you for your kind and thoughtful comments. I am very glad that you found my examples and descriptions to be a "great aid to clarity." I felt that it was risky to include such detail in an essay. However, without concrete examples, I feared that the ideas presented in the essay would be perceived as mere speculation rather than something that was born of insight and could be backed up with meaningful and relevant examples.

As for equations arrived at through educated guesses or trial-and-error, this is difficult. In any initial exploration, this is really all that one has. Much of physics was developed this way. But once one better understands the interconnections, one can begin to put the pieces of the puzzle together with greater insight. To some degree, this is where we are now in the history of physics. But it is unclear what aspects of physics are derivable from deeper principles and which are contingent (accidental or perhaps even dictated by decree). The distribution of the cosmic microwave background (which you note) is very likely contingent, just as is the current relative positions of South America and Africa. While some things are just not derivable, some are.

Thank you again.

Kevin

Ian Durham

Hi Kevin,

As I said in my e-mail to you, this is one of the best things you've written. I think it is a fantastic essay and captures some really deep insights.

Cheers,

Ian

Kevin Knuth

Dear Ian

Thank you for your high praise. I really appreciate it.

As I noted in my email reply to you, these are questions that I have had for a long time (many of us have) and regarding the specific problem of additivity, I was pleased to have answered it myself (back in 2003) when studying various derivations of probability theory related to Cox and Jaynes. It became clear to me that much of the universality of these additive laws came from the quantification of symmetries, such as associativity. This went on to seriously influence my close colleague John Skilling, which led to our subsequent research with Philip Goyal on deriving the Feynman rules of QM, which Philip (employing symmetry-constrained quantification) continues to take further. It has also allowed me to derive the calculus of questions, which is related to information theory (not mentioned in the essay, and only now having been polished is being written up into its final form). And of course, this has led to the emergent space-time work that I have done with Newshaw Bahreyni published in the Journal of Mathematical Physics just last year (and to be presented next week with my new student James Walsh at the Beyond Spacetime Workshop in San Diego). For me these insights into the role that mathematics plays in physical laws have been central to my research in foundations over the last 10 years.

Despite this, I have never really found an opportunity to write up what I had learned in general about the relations between mathematics and physics. I had intended to try to write something on the concept of consistent quantification and its use in deriving physical laws for a journal like American Journal of Physics. Though, as I learned more and came upon more powerful examples, it became difficult to imagine how to put it all into one paper.

This contest was perfect because it dealt with exactly that problem and I could write it in essay form, which I think is more powerful in this case than a scientific paper. The essay contest is wonderful as it provides people with the opportunity to convey what they think even to the point of speculation. This is lost in modern scientific writing.

Tejinder Singh

Dear Kevin,

You have a fascinating and brilliant essay deriving the `ubiquity of addtivity' from the underlying symmetries of commutativity and associativity, and from ordering. It was a pleasure reading the essay and we will need to read it again for better grasp.

We are not clear though, about what stance you finally adopt with regard to the relation between symmetries and physical laws of motion. We are accustomed to relating symmetries to conservation laws, but perhaps not in full generality to the equations of motion and the force laws themselves [say Newton's second law, and his inverse square law of gravitation]. Would you say that eventually we must understand force laws also from symmetries: where does one draw the line - how much from symmetry, and how much from experiment? For instance, it is not obvious how symmetry can dictate that equations of motion be first order or second order in time.

Also, are you suggesting that in general one should seek to derive mathematical axioms [say those of Euclidean geometry] from symmetries?

Questions apart, it was fun reading your very well-written essay.

Regards,

Anshu, Tejinder

Kevin Knuth

Dear Tejinder

Thank you for your very generous comments and questions. I appreciate the opportunity to answer them.

There is much more that I would have liked to have said about the role that these subtle symmetries play in physical law. Your focus on the physical laws of motion is a very interesting example, and you are right that I did not speak specifically to that. I decided to keep to my focus on the fact that at least SOME (perhaps surprising) laws were derivable from fundamental symmetries in this way, and to give clear and specific examples so that I could put meat on the bones of the argument, which otherwise might have appeared to be something originating from speculation rather than well founded theory. It is too easy to let an essay drift into speculation, or to have it interpreted as such. For this reason, when we know things for certain, it is important to make that crystal clear. That is where my effort was spent. For that reason, I believe that it was more important to focus on the very basic and critical ideas that I present there rather than to address how far they could possibly go.

Now part of the difficulty of physics is that it is not clear from the outset, which physical laws are derivable (must be) or contingent (accidental or dictated by decree or design). Without a derivation in hand, any statement I could make would be pure speculation. As mentioned above, I worked to avoid this. As an example, for the ancient Babylonians (as well as the Egyptians), the number pi (ratio of the circumference of a circle to the diameter) was an experimentally derived number. As such, a Babylonian could have argued that pi was contingent, and that there could be other universes with different values of pi. However, the Greeks were able to derive pi from more fundamental geometric concepts, and to them pi was derivable. From this perspective, arguments about the contingency of pi are specious. We are currently in a similar situation with the number known as the fine structure constant (alpha = 1/137), which dictates the strength of the electric force. Is it contingent or derivable?

So what about the physical laws of motion and forces? Can these be derived from fundamental symmetries, such as associativity? That is a question that is of great interest to me at the moment. I alluded to some of my recent work in this direction in the essay (just before the Conclusions) where I cited (Knuth, 2014 (arXiv:1308.3337); Knuth & Bahreyni, 2014 (arXiv:1209.0881); Knuth, 2015 (arXiv:1310.1667)). In these works, we have shown that simple symmetries of causally ordered sets of events constrain any attempts at quantification resulting in the mathematics of space-time. My graduate student James Walsh, has recently derived Newton's second law in this context(arXiv:1411.2163). The difficulty, is that we have been able to do this by adopting a particular model of things influencing other things. So that explicit discussion of this in the essay, would have required invoking a specific model of events and their relationship to one another. This would have been far outside the scope of the essay distracting the reader from the very simple (and critical ideas) that I present there. If you are interested, I invite you to take a look at those papers and contact me with any questions. I would be delighted to discuss them. But in short, I do believe that forces and motion are derivable from these very basic symmetries. I believe that a great deal of physics will be derivable this way now that we understand how mathematics is related to physics.

Tejinder Singh

Thanks Kevin, for your clear and precise reply, which I agree with. I do believe this is path breaking work and I hope to make time to read and understand the papers you have referred to.

This is probably already a direction contained in some of your works: it will be great to understand how quantum mechanics might originate from symmetries, in the sense in which you approach the problem. I believe that a successful application of the symmetry idea ought to yield not quantum theory, but a more general theory to which quantum mechanics is an approximation, because by itself quantum theory suffers from various shortcomings including the measurement problem. [I am reminded of Stephen Adler's work on Trace Dynamics, described in his book `Quantum theory as an emergent phenomenon', where quantum theory is approximately emergent as an equilibrium statistical thermodynamics of an underlying mechanical theory of non-commuting matrices. Fluctuations about equilibrium modify quantum theory in a desirable way]. I will be much interested in knowing your views on the symmetries - quantum theory connection.

Regards,

Tejinder

Edwin Klingman

Dear Kevin Knuth,

Thanks for your essay heavily focused on symmetry. May I recommend another that you might like: Aldo Filomeno's essay focused on gauge symmetry.

You look at symmetry and additivity. I begin with logic structures and quickly lead to addition structures, which leads to the concept of distance and identity.

And yes, it was somewhat surprising that one can derive the Feynman rules for combining quantum amplitudes by relying on symmetries. One learns a lot in these FQXi contests!

Early in your essay you state that "as a physicist interested in the foundations, assumptions cause me concern."

They cause me concern too, and I have been concerned for example that Bell's assumption of a constant field leads to a null experimental result, an obvious contradiction before one even gets into any other analysis. My conclusion is that Bell relied on an oversimplified assumption. The interesting thing is that a local model based on a more realistic assumption actually delivers the quantum correlation unless one throws away measurement information. I invite you to read my essay and I welcome your feedback.

Best,

Edwin Eugene Klingman

Kevin Knuth

Dear Edwin,

Thank you for your kind comments, as well as for pointing me both to your essay and the essay of our fellow essay submitter Aldo Filomeno. I will certainly focus my attention on them.

Thank you

Kevin

Kevin Knuth

Dear Joe

Thank you for your comments. Much of your thinking seems to be currently focused on the concepts of abstractness and uniqueness, which admittedly has not been my focus. I do not have further thoughts at the moment, but will be happy to respond if any come to me.

Ted Erikson

And how would you classify the symmetry of activities of a regular tetrahedron with an inscribed sphere? Seems to me to be a "duality" with connections. see topic #2408.

Kevin Knuth

Dear Ted

Thanks for your comment.

I am not sure how this question relates to my essay. Perhaps it is more related to topic #2408 as you note.

Sincerely

Kevin Knuth

Jacek Safuta

Dear Kevin,

Your essay is really fantastic and inspiring. You ask very important questions that I have not asked myself. Below I try to comment main aspects of your essay.

The aspect related to quantification.

I start from your example: "...2+1 defines 3 [...] this is an axiom of measure theory, which means that it is assumed." Obviously we can easily find examples where 1+1+...=1 or defines 1. The examples are delivered by every superposition of waves creating a wavepacket. We perceive an electron as a countable object (1 piece), eternal and indestructible unit of matter. In reality it is a wavepacket. We perceive (with measuring instruments) its envelope and not single waves creating the packet. We perceive e.g. an apple and not elementary particles and interactions that create the wavepacket called an apple. So adding apples or electrons we add wavepackets. Adding many fermions and bosons we get 1 apple. After consumption this wavepacket is decomposed into smaller wavepackets up to vacuum. Laughlin said that "The modern concept of the vacuum of space, confirmed every day by experiment, is a relativistic aether . But we do not call it this because it is taboo." So I call this elastic medium spacetime (as Einstein did).

The aspect related to ordering.

"...this should not be surprising since it has been generally believed that the laws of physics reflect an underlying order in the universe. It is explicitly demonstrated that some laws of physics not only reflect such order, but in fact can be derived directly from it." I propose to find that underlying order in the Thurston geometrization conjecture, proved by Perelman. It states that every closed 3-manifold can be decomposed into pieces that each have one of eight types of geometric structure, resulting in an emergence of some attributes that we can observe. We can assign every fundamental interaction and matter to the proper Thurston geometry with metric.

"...here it is explicitly demonstrated that some laws of physics not only reflect such order, but in fact can be derived directly from it. This has enormous implications for the direction and progress of foundational physics in the sense that it enables one to see that common mathematical assumptions, such as additivity, linearity, Hilbert spaces, etc., while familiar, are most likely not fundamental..." That is right! In my opinion additivity, linearity, Hilbert spaces, etc. show the effectiveness of geometry rather than mathematics as a whole set of abstract structures. The additivity is what we perceive in macro or micro world (as I noticed earlier) and it presents our language rather than reality. Without that humans' perception baggage there are only wavepackets and their superpositions that create constantly evolving and dynamic geometric picture. The geometry devoid of human language is universal language itself, comprehensive probably even for aliens or future supercomputers.

The aspect related to symmetries.

You claim...I believe that the answer lies in the deeper symmetries that various problems exhibit... and you quote Jaynes: "the essential content ... does not lie in the equations; it lies in the ideas that lead to those equations." Please note that only geometric structures (objects) can show symmetry in transformation (technically an isometry or affine map) that maps the object onto itself. Thurston introduced his version of symmetries in geometry. That is too extensive to describe here. It would deserve a separate essay.

If you are interested you can find details in my essay.

I would appreciate your comments. Thank you.

Jacek

Kevin Knuth

Dear Jacek

Thank you for your questions and comments.

Your first question relates to closure, which I addressed in the essay. In my example of combining a cup of water, a napkin and a lighted match, the order in which you combine them matters if you choose a description where you count the things: water, napkin, match. However, on combination, these things are not closed. You get other things such as ashes and gas products. Alternatively by choosing to consider the constituent atoms to comprise the system, the system is now closed and addition works.

Second, you make a proposal to find underlying order. You describe a three manifold. But I am not sure why you would imagine such a thing to be fundamental. Moreover, one would at least want to consider a 4-manifold. But then again, the concept of what one considers to be fundamental is then again at issue.

Third, you discuss additivity and note that "The additivity is what we perceive in macro or micro world (as I noticed earlier) and it presents our language rather than reality." This is precisely why in my essay I decided to present a careful analysis of how additivity arises. It is clearly not simply our language. It is a necessity given our desire to consistently quantify systems exhibiting very simple properties: closure, order, commutativity, and associativity. Any potential for speculation as to the reason for additivity has been removed.

Last, you make a note that such symmetries are only exhibited by geometric structures. This is simply untrue as given by the explicit examples in the essay. Symmetries such as commutativity, associativity and distributivity are critical and they have gone relatively unnoticed in physics, which focuses on symmetries resulting from conservation laws (mass, energy, charge, etc). None of these are geometric per se. And in fact, in our work on emergent spacetime (link below), we have shown that geometric concepts are derivable from order-theoretic concepts thus challenging the fundamental status that geometry has enjoyed in so many minds for so long.

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

Thanks again for your comments.

I look forward to reading your essay.

Sincerely

Kevin Knuth

Ed Unverricht

Dear Kevin,

Thank you for your clear and well thought out essay. I liked your perspective, "..there are two distinct aspects to the role that mathematics plays. The first aspect is related to ordering and associated symmetries, and the second aspect is related to quantification and the equations that enable one to quantify things." and I think it provides a basis for considerable thought and discussion.

Your suggestion that we "step back and release ourselves from familiarity and consider order and symmetry to be fundamental, then we see these equations as rules to constrain our artificial quantifications in accordance with the underlying order and symmetries of our chosen descriptions." highlights the importance of underlying order and symmetries. I especially agree with the comment "quantum mechanics is not a generalized probability theory any more than information theory, geometry, and number theory are generalized probability theories"

I have used groups and symmetries as a way of visualizing the fundamental particles on the standard model in my essay here. I feel this in many ways is an example of your first aspect of order and symmetries. The second aspect of quantification and equations confirm the proper ordering and symmetries and allow for the predictive power so important to physics.

All the best regarding your essay, it made an enjoyable and thoughtful read.

Regards,

Ed Unverricht

Kevin Knuth

Dear Ed

Thank you for your kind comments and for highlighting the thoughts that resonated with your own.

I look forward to reading your essay.

Sincerely

Kevin Knuth

Pankaj Mani

Dear Kevin,

You have focused on how mathematics, which appears to be the result of both human creativity and human discovery, can possibly exhibit the degree of success and seemingly-universal applicability to quantifying the physical world as exemplified by the laws of physics.

You have also taken your thought in line with Hamming.

You are trying to explain which one is more fundamental mathematics or physics?

Its indeed the clue what governs the mathematical equations,operators e.g. addition,multiplication etc. Its certain laws of invariance,symmetry order.Its not mathematics describing physics but the laws of invariance,symmetry,order behind those mathematical structures explaining that of physical reality. If we look at the algorithm of formation of numbers and operators e.g. ,-,*,/, we will find that the particular invariance, symmetry,order is drawn from the physical world reality itself e.g quantum structure of energy levels of atoms.

It is possible that the invariance,symmetry,order of classical mathematical structures e.g.hypotheses of geometry don't resemble the those characteristics of quantum world.This raises the deeper question in context of Riemann's geometry,that hypotheses of geometry don't at conform at quantum level.

So, this coherency and compatibility is the key why mathematics has been so effective in Physics. because my Mathematical Structure Hypothesis states that mathematical abstraction and physical reality both originate from Vibration. This is why sometimes mathematics explains physics and other times physical theories solve the mathematical problems.its a two-way interdependence but the different manifestations of the same Vibration.The mechanism of perception of Integers,addition all are its products deeply to be discovered ,how? Thats why e.g.entropy which was considered to be physical phenomenon also governs the mathematical structures as in Poincare,Geometrization conjectures.

Thats why the clue is to match the laws of invariance,symmetry,order of the two,otherwise it leads to mutual friction. We need to match the physical characteristics of mathematical abstractness with that of physical reality. Riemann Hypothesis is all about this laws of invariance behind functioning of these operators, complex analytical continuation.

In context of Skolem paradox,a particular model fails to accurately capture every feature of the reality of which it is a model. A mathematical model of a physical theory, for instance, may contain only real numbers and sets of real numbers, even though the theory itself concerns, say, subatomic particles and regions of space-time. Similarly, a tabletop model of the solar system will get some things right about the solar system while getting other things quite wrong.

This is because that when mutual laws of invariance matches each other,it gives right result and otherwise wrong because of mutual friction.

Its equally important that as Wigner said that physical laws are conditional statements. What mathematics tries to do is superficially tries to find out the quantity relations between observables rather than why something happens.

What should be field of further research is as Richard Feyman said- The next great era of human awakening would come ,today we dont see the content of equations.

So,the nest era of research is to discover those laws of invariance which governs the mathematical equations itself and then match it with physics.

Anyway, you have written a great essay .

Regards,

Pankaj Mani

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!

Jose Koshy

Dear Kevin H. Knuth,

I found your essay interesting. Your remark about addition is thought provoking. 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. The rest of the mathematical structures are axiomatic, where axioms can be regarded as properties that we arbitrarily assign. I have been thinking in that direction for some time, but not reached a final conclusion.

In my opinion there are no physical laws; the physical world has no laws of its own. It has only some basic properties. The rules applicable to it are that of mathematics. Or simply, 'the properties are physical' and 'the laws are mathematical'. Why is it so? The changes in the physical world happen by way of motion, the rules of which are mathematical. So a changing world follows mathematical rules. I request you to go through my essay: A physicalist interpretation of the relation between Physics and Mathematics

If it seems interesting, kindly visit my site: finitenesstheory.com, where I propose a theory of everything based on fundamental properties and consequent emergent strucures.

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.

Jayakar Joseph

Dear Kevin,

It's true that Mathematics has deeper roles in Physical Laws, while we consider that Symmetry and Asymmetry are the basic kind of abstracts for both Physical and Mathematical entities, whereas quantization limits the value with 1 and returns 0 on absence of a quanta, while not an abstract.

Thus, Mysterious connection between Physics and Mathematics begins with Three-dimensional Structure formation on the Universe, by the transformation from pre-existence.

With best wishes,

Jayakar

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.

Allen Francom

Hi Kevin,

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

From the ground up, such as cellular automata do not seem to present the idea of "selection criteria" so that there is therein an "observer" to make choices as you might describe.

Either something has to "evolve" in that soup to "see" what you're saying, or it has to be totally inherent from the get-go, by definition.

Would you see his efforts as yet another expression of a "measure" still standing on these same principles ? Or as something yet more fundamental and there is some "gap" between his bits flipping and the ability for set theory stuff to exist at all - "what observer/criteria" - in that universe... when do the eyes happen... How...

Did that make sense ? This is a hard one to frame, I've never asked it before.

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?

Robert MacDuff

Kevin, we meet again. I do like your essay, however, I am not certain I understand what you mean by this: "One can now see that ordering, commutativity and associativity underlie a class of universal phenomena. I will next discuss how this leads to mathematics which gives rise to physical laws with a degree of universal applicability."

Do these underlie physics or are they simply a means of encoding what "Laws Describe"? An empirically familiar regularity is that some collections of objects (pencils, pennies, rocks, etc.) are invariant under regrouping: a law. What is meant is that regrouping creates things that are the same but not the same: the law of the included middle. They are different in that the grouping is different but quantitatively they are the same. Consider these three principles: 1/ a + b = b + a, 2/ (a +b) + c = a + (b + c) & 3/ a(b + c) = ab + ac. 2 & 3 state regrouping principles. However 1 is a definition of identity as orientation is not encoded symbolically. In other words it doesn't make any difference which group is label a or b. A mathematical interpretation of these symbols is entirely different as the intent is to reason about symbol construction with a serial ordered set of symbols; a + b = c.

It seems to me that set theory is when objects may not be invariant under regrouping but there is a way of determining the variance. Thus I found it odd that you would start out with set theory. Am I missing something, as it seems to me that you started out assuming the law: invariance under regrouping, with your reference to the union of sets. Have you not assumed that arbitrary members are not lost upon being joined?

Cheers

Rob MacDuff

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