Modeling the Physical World with Common Sense and Mathematics by David Hestenes

Dear Prof. David Hestenes,

Your essay is factual, and educative. From 'commonsense' to 'thinking' to 'modeling' it portrays a clear path of evolution. Quoting from your essay, "CS concepts should be regarded as alternative hypotheses about the physical world that, when clearly formulated, can be tested empirically." "Thinking is a hardwired human ability to freely create mental models and use them for planning and controlling interactions with the physical world." "the transition from common sense to scientific thinking is not a replacement of CS concepts with scientific concepts, but rather a realignment of intuition with experience."

I completely agree with your view, "Likewise the tools of mathematics were invented, not discovered; though it may be said that theorems derived from structures built with those tools are discovered." In my opinion, there indeed need be just one law in mathematics, the law of addition; it is eternal. The structures are based on this fundamental law and are invented; the theorems derived from the structures follow the fundamental law, and are discovered.

You ask the question, "What accounts for the ubiquitous applicability of mathematics to science? You suggest co-evolution of physics and mathematics as the possible reason." I think it is more fundamental than mere co-evolution: A static world does not have any 'laws'. The only role of law is governing changes. Changes can happen by way of 'motion' only. Motion follows mathematical laws. Thus, all the changes in the physical world follow mathematical laws. That is why mathematics is applicable to science, the study of the physical world. The co-evolution is thus predetermined.

I would like to draw your attention to my essay: A physicalist interpretation of the relation between Physics and Mathematics, and my site: finitenesstheory.com.

Dear Professor Hestenes,

Many thanks for your detailed response. We will study your Refs. [22], [29] as also your 2008 FQXi essay.

Tejinder

Dear Professor Hestenes,

Could you please explain to me why you thought that my comment about the real Universe was inappropriate?

You are I hope aware that suppression of the truth is unethical.

Eagerly awaiting your answer,

Joe Fisher

Dear Professor Hestenes,

You wrote: "As we grow and learn through everyday experience, each of us develops a system of common sense (CS) concepts about how the world works."

This is my single unified theorem of how the real Universe is occurring: Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING. Proof exists that every real astronomer looking through a real telescope has failed to notice that each of the real galaxies he has observed is unique as to its structure and its perceived distance from all other real galaxies. Each real star is unique as to its structure and its perceived distance apart from all other real stars. Every real scientist who has peered at real snowflakes through a real microscope has concluded that each real snowflake is unique as to its structure. Real structure is unique, once. Unique, once does not consist of abstract amounts of abstract quanta. Based on one's normal observation, one must conclude that all of the stars, all of the planets, all of the asteroids, all of the comets, all of the meteors, all of the specks of astral dust and all real objects have only one real thing in common. Each real object has a real material surface that seems to be attached to a material sub-surface. All surfaces, no matter the apparent degree of separation, must travel at the same constant speed. No matter in which direction one looks, one will only ever see a plethora of real surfaces and those surfaces must all be traveling at the same constant speed or else it would be physically impossible for one to observe them instantly and simultaneously. Real surfaces are easy to spot because they are well lighted. Real light does not travel far from its source as can be confirmed by looking at the real stars, or a real lightning bolt. Reflected light needs to adhere to a surface in order for it to be observed, which means that real light cannot have a surface of its own. Real light must be the only stationary substance in the real Universe. The stars remain in place due to astral radiation. The planets orbit because of atmospheric accumulation. There is no space.

Warm regards,

Joe Fisher

Dear Prof. David Hestenes,

Quoting You, "There is no such thing as direct observation, only indirect perception". This indicates our difference in approach or perception regarding the nature of the physical world. I think the physical world is real and we are also real, and our sensory organs are designed to observe this reality to the extent required, and so the physical structures we observe directly are real and not merely a perception.

Given the basic properties, it is mathematical laws that decide the emergent structures. The emergent properties of the structures depend both on the structure and the basic properties. Mathematical laws again decide the next level structures and so on. All these can happen only if motion is one of the basic properties; otherwise there will not be any changes, and hence no laws. This is what I mean by saying, 'Mathematics governs the changes in the physical world'. And, that is the reason why "mathematics can be applied to model changes in the physical world".

"The laws change as we learn more". Do you mean that laws 'actually' change? Or, is it that you meant the changes happen in our 'perception' regarding the law?

Dear David,

In a biological system the signal transduction is causal for sensing, processing and retaining the knowledge acquired from its environment; in that, the electro chemical units that is causal for the formation of Mind with sensible Phenomena and the non-sensible Noumena coined by Kant, are the mathematical units only.

In relevant to this science behind Epistemology, I think, whatever the mathematical or physical model we inspire, there is a link between Physics and Mathematics by nature.

With best wishes,

Jayakar

Dear Professor Hestenes,

FQXi.org has labeled my work "OBNOXIOUS SPAM," and removed it from several sites where I had posted it.

Sir, do you have a real complete skin surface? Does every animal, insect, and can of soda have a real complete surface? It is physically impossible to perceive a single real surface. No matter in which direction one looks, one will only ever see a plethora of real surfaces. At each point where surfaces touch, a real sub-surface forms. All surface travels at the same constant speed. Each sub-surface travels at a unique speed that remains less than the constant speed of surface. That is why although everything is traveling at the same speed, each object remains in its own unique position.

Grateful that you read my comment and at least made an answer to it.

Joe Fisher

Thanks so much for responding.

I have a difficult time seeing how a non-associative algebra could ever be a subalgebra of an associative algebra, since selective zero valued coefficients applied within the associative algebra clearing out all but the chosen subalgebra basis elements would necessarily still need to be associative. Certainly one may go the other way where there are 7 associative quaternion subalgebras within the non-associative octonion algebra.

You might find my 2012 FQXi essay The Algebra of Everything interesting. The algebraic structure of octonion algebra gives us clues on how the equations describing nature must look and actually do conform to the idea that there is no preferred definition for octonion algebra. This means observables are invariant to all possible chiral changes after singularly enumerating the quaternion associative triplets. I have demonstrated the the divergence of the stress-energy-momentum forms for electrodynamics are but a subset of the full octonion (non-tensor) representation, mandated to be what they must be by enforcing octonion algebraic invariance, as it must in a more inclusive approach. The Lorentz transformation falls out of restricting the two portions of each field component to transform in kind. Look at the essay, and if you are interested, drop me an email and I will send you a PDF of my in progress book that goes well beyond the limited essay. It is not and can't be your GA since you seem to insist on associative product structure. I think it is superior since it is a successful representational structure that is mandated by the octonion algebra itself and not by hand inserted. I think octonion algebra itself could be considered a "geometric algebra", just not yours.

Thanks and with much respect,

Rick

Dear David Hestenes,

Thank you for your kind words on my essay. I do not expect to convince many people that Bell is wrong with my essay. My hope is that I will convince a number of people that Bell may be wrong, based on my analysis. This would represent a very significant change from today's situation, in which Bell's conclusions are stated as fact. I believe that time and effort spent on understanding my theory will call Bell's physical assumptions into question, and I have faith that once the questioning begins, the right answer will be forthcoming.

Thank you for introducing ET Jaynes Probability Theory: the Logic of Science. Like you I regard his as one of the greatest books of the 20th century, and I treat him in my 2013 FQXi essay, Gravity and the Nature of Information, which you might also find interesting. It is a little more "blue sky" than my current essay but one Jaynes quote from that essay is well suited to my current essay on Bell, to wit:

"... a false premise built into a model which is never questioned cannot be removed by any amount of data."

In closing, I thank you yet again for Geometric Algebra, Space-Time Algebra, and all the rest of your work. You have contributed well to mankind.

Edwin Eugene Klingman

Dear Prof. Hestenes,

Wonderful essay! I agree with your arguments, and enjoyed reading them. I share some of your views in my essay. I would be honored to have your opinion.

Best regards,

Mohammed

Dear Prof. Hestenes,

Congratulations! You've outdone himself with this paper! Never have you explained your tremendously influential and successful Modeling Theory with greater clarity and power or reached so deeply into the realm of the mind, exploring the relationship between mental and formal conceptual models!

Your anchoring of your ideas in the work of Emmanuel Kant and adaption of the "cognitive linguistics" explanation of how models relate to the "real world" have more firmly relocated the objects referenced by language to mental models inside the mind. Yet the goal remains clear: to build our ability to construct models that can successfully map the world as it exists, though that world remain forever beyond the ability of our minds to directly know it.

This work, and the Modeling Method of Physics Instruction - as taught to me by Dwain Desbien who was then a graduate student in your group at Arizona State University and communicated through your lectures and many papers - transformed my teaching and touched the lives of many of my students. That system fell short however when I attempted to apply its methods to lower-level math classes, or even to physical science classes for deeply mathematically-challenged students. The Models of Mathematics that you identified, the models the course "Integrating Mathematics and Physics" in your group's Modeling Instruction program focused on, were evidently beyond the reach of these students, not sufficiently fundamental to address their difficulties. Some 12 years later, I see no sign here that you have taken your models of mathematics to the new and more fundamental level I was seeking.

Comparing this paper with "Mathematics of Science" by your sparring-partner Rob MacDuff (elsewhere on this FQXi contest board), your influence on each other is evident; yet at a fundamental philosophical level you diverge. For while you seem resigned to mathematics remaining a self-referential body of rules (which it certainly is now), MacDuff holds out the tantalizing prospect of freeing it, or at least freeing science in its use, from this constraint.

You refer to mathematics as "the science of structure," where "mathematical intuition matches mental structures with symbolic structures," but distinguish it from the rest of science with "mathematical models have as referents only themselves," You quote Saunders MacLane approvingly, "... mathematics is not concerned with reality but with rule," and sanctify this universal doctrine with your closing words: "Kant put his finger on the source of this stunning revolution: the use of rules to harness the powers of human intuition."

MacDuff has challenged this doctrine that math must be a purely self-referential set of rules, that it can have only itself for a referent. His critical break with you and with all of mathematics of the recent past is the introduction of a set of axioms and models of science that have as their referent observable and inferable properties and structure of objects in the real world, a system that has sufficient internal consistency to construct a mathematics from it. He thus places mathematics firmly within the realm of science rather than as an exception, as an embedded and integral part of the structure of science rather than as a supporting discipline adapted to its purposes.

I look forward to seeing the fruits of your ongoing dialogue with MacDuff. I am however concerned that he gets so little acknowledgment from anyone within the Academy. Given the widespread evidence from teachers that MacDuff's ideas and the CIMM curriculum developed from them have solved the problem of communicating mathematics to the vast numbers of students who are currently excluded or marginalized by its difficulty, it behooves the academic community to overcome our instinctive but natural resistance to heresy coming from outsiders and give his work a closer look.

Professor Hestenes,

There's no higher compliment I can give: You are the thinker that I wanted to be.

Being in personal agreement with Popper's philosophy of science called critical rationalism, I see in your short essay all the elements of his correspondence theory; i.e., the correspondence of mathematical model to independent physical result.

I am not the critical rationalist I wanted to be, because my mathematical soul still craves the freedom of rational idealism. About 20 years ago, I was deeply affected by one of the best short pieces of fiction that I have ever read, then or since -- Jim Cowan's "The Spade of Reason."* The narrator says early on:

" ... some minds create weird models and those minds may be mad. I don't know about that. But I do know that one kind of madness is not knowing that the model is all we will ever know."

And ends with:

"I am sane, and I am ready to leave.

"Look outside. Like lightning drawn to a lightning rod, the night's dew has condensed on the very tip of each blade of grass. The laws of physics are written so that the dew must collect as tiny globes of water on each blade's tip, not as a film of moisture smeared over the whole lawn. Each drop will scatter the low light of the rising sun. While I am walking to the gate the lawn will look as if someone has thrown away a million diamonds. Why should it be so beautiful? I can't explain it, and that's how I know I'm sane. Reason is only the sixth sense, the silent sixth sense, no more reliable than the other five. As imperfect , and as capable of causing pain or ecstasy.

"So maybe the message was truly random. Or maybe there is a God who takes an interest in things, dazzling us with the morning dew and sending us messages. But don't forget that some religions have trickster gods."

All best,

Tom

* (I hope I am not violating any copyright laws by posting the link. I read the story in a book collection of the best science fiction of 1990-something. I was delighted to see it published online. It has a message that I am sure resonates with every visitor to this forum.)

First I would like to say a few words about Dr. Maudlin's rant concerning David's reference to Kant, which is completely unfounded as without Kant's synthetic a priori there would be no perception of structure.

What I love about David's paper is the inclusion of what we know, with how it is that we come to know it. His perception that CS concepts are not misconceptions but rather indicate missing conceptions that require the realignment of intuition with experience is a powerful way to view conceptual change.

David and I completely agree that "A model is a representation of structure in a given system." That is precisely what a model does, however, we differ on how a models encodes structure which brings us back to Kant. I am not enamored by David's taxonomy of structural types as if forces a definition on time based upon how one might measure it. I would rather suggest that there are four different types of structure and they are associated with the different types of mathematical operations, which enable the encoding Kant's synthetic a priori. There may be other types of structure that we can't perceive such as space/time or quantum mechanics but that is another discussion. It is only when you dig down to roots that you begin to realize that we impose mathematical structure on the world so that we can describe "this" in terms of "that", which forms the basis of our understanding.

My paper Mathematics of Science addresses this to some extent.

David, a brilliant paper!

Cheers

Rob MacDuff

David, thank you for your gracious and thoughtful reply, and your kind words about Rob MacDuff's work.

I don't think I have ever publicly acknowledged my debt to you, your writings and the part that you, your colleague Jane Jackson and the many others who have come together around your outstanding Modeling Methods project have played in my development. Your framework for understanding conceptual models and your approach to teaching Newton's physics helped foster a revolution in my own thinking and teaching, one which I could not retreat from if I tried as it has become inextricably a part of how I see the world.

I will grant that some may experience Rob's passionate advocacy for his views and positions as abrasive. However I suspect the deeper reason for the difficulty he's encountering with mathematicians and math educators does indeed lie in an essential philosophical shift embedded in his ideas, one which they perhaps understandably find deeply disturbing.

Galileo, in the title of A Dialogue Concerning the Two Chief World Systems, was of course referring to the Ptolemaic and Copernican systems, but Albert Einstein, in his preface to the Modern Science Library edition, puts his finger on the true essence of the two World Systems: "The leitmotif which I recognize in Galileo's work is the passionate fight against any kind of dogma based on authority."

In our time, the natural sciences have at least nominally abandoned appeals to authority as illegitimate. Mathematics however has over recent centuries gone the other way. Euclidian geometry at least had as its referent the observable properties of phenomena. In my efforts to apply the Modeling Method to helping students construct an understanding of arithmetic and algebra I came to see that math itself stands on the far side of Galileo's Divide, as a "dogma based on authority".

The power that mathematicians have granted themselves to play with the terms of the dogma does not fundamentally alter its condition.

MacDuff's distinction of number as representing a ratio between quantities - an idea which it turns out was clearly stated by Isaac Newton (p. 2, 1st paragraph) but has since been abandoned - is the fulcrum for a fundamental relocation of at least the "scientists' math" to the Science side of Galileo's Divide, as a system where every statement and symbol has as its ultimate referent some observable property or "empirically-familiar regularity" in the realm of real phenomena.

It is not surprising that this proposed paradigm shift sprouted in the soil you and your colleagues prepared. While all of natural science has long been on the Science side of the divide, science education is still largely dominated by a dogmatic, authority-based paradigm. The shift to the model-based paradigm of teaching, learning and constructing knowledge pioneered and supported by your work has brought science education over from the other bank, resulting in a learning experience that is philosophically harmonious with the subject being studied.

The task of designing a Modeling Math is more challenging because not just the pedagogy but the subject matter itself has to be transposed into the Galilean paradigm. I believe that the appropriate pedagogy for this shift is Rob's CIMM program. The newly-christened "structural algebra" on which it now rests owes a huge debt to your work in geometric algebra. Their divergence traces back to that philosophical shift at its heart, which I would invite you to reexamine with an open mind.