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

    Dear Jayakar,

    If I understand you correctly, I agree. The Brain is the Noumena from which the Phenomena of Mind emerges. In other words, the Mind (conscious or unconscious) is a state of matter, the Brain. To explain how is, perhaps, the ultimate challenge of science.

    I believe that will require some version of statistical mechanics, to explain how "macro" states of Mind emerge from "micro" Brain states. It is clear that there are many more degrees of freedom in Brain states than in Mind states, so Mind states must emerge as some sort of cooperative phenomena. The ultimate objective of my Modeling Theory of Mind is to describe the structure of Mind (states) with sufficient precision so we know what we are trying to explain with models of Brain states.

    The link between physics and mathematics in nature boils down to the fact that there are regularities (patterns) in nature discovered by physics, and mathematics has been developed as a science of patterns.

    Thanks for your comment......David

    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

    6 days later

    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

      Thanks for your input, Chris.

      You may not know that Rob MacDuff and I started the Cognitive Instruction in Mathematical Modeling (CIMM) program together. But I just gave it the name and contributed ideas about Modeling Instruction for physics students, while Rob has persisted in developing and applying his innovative techniques to teach mathematics to young children. He has succeeded brilliantly, and I have personally witnessed the delight and productive engagement of third-graders in his math activities.

      While Robb and I have many friendly disagreements, you are mistaken to think that we diverge "at a fundamental philosophical level." Rather, we disagree about specific details, most recently about time and motion.

      Robb's problems in getting CIMM recognized by the math education community stem partly from its revolutionary approach and partly from his uncompromising critique of standard practice. To be effective in persuasion, one must recognize value in your opponent's point of view.

      Rick,

      For construction of the Octonion product in GA with some deep analysis, see the book by Pertti Lounesto: Clifford Algebras and Spinors (Cambridge U Press, 1997).

      In fact, every algebra has a GA representation, just as it has a matrix representation.

      .....David

      8 days later

      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.

      Addenda

      Two embedded hyperlinks in my comment failed to transfer:

      "... clearly stated by Isaac Newton (p. 2, 1st paragraph" refers to the page and paragraph in which he is quoted in Robert MacDuff's essay "Mathematics of Science" in this contest,

      The "CIMM program" referred to in the last paragraph of my comment was intended to link to https://trueddotorg.wordpress.com/tag/algebra/.

      David, I was going to let your response go without comment in deference to you, but could not do it. Sure, basis element products must be closed for the set for any possible algebra, making every possible basis element product a linear combination of the full set in the general description of any algebra. So one can represent the product of any two algebraic elements a*b using an n x n matrix for "a" and an n x 1 matrix for "b" and thinking of the result as simply an ordered set of coefficients over the vector space R^n. But one cannot represent any algebra with an equivalent matrix representation for both "a" and "b" redefining "*" then as matrix multiplication. Non-associative algebras cannot be represented in this manner, since matrix multiplication is associative.

      As for the author's GA representation of the octonion product, there is no basis (pun intended) for the use of "=" in the expression. On the left is the product of two octonion algebraic elements, which is not simply an ordered set of coefficients over R^8, each coefficient is attached to an octonion basis element. The non-scalar octonion basis elements have no equivalents on the GA side of the expression. There can't be because every GA is associative for multiplication, meaning every triple of GA basis elements is associative. This is not the case for octonion algebra, only the seven quaternion subalgebra triples are associative. A subalgebra is formed from a subset of basis elements of the larger algebra, so octonion algebra cannot be a subalgebra of any associative algebra. One should not turn a blind eye to the basis elements. To do so is to conflate vector spaces with algebras.

      Rick

      5 days later

      Dear Professor Hestenes,

      Because I am not familiar with modeling theory, your essay introduced me to some new concepts. I thank you for this. My question is about the relation of models to real things and events. As I understand it, models belong in worlds 2 and 3, while real things and events are in world 1. How do we move from worlds 2 and 3 to world 1? Perhaps I should ask, do we move from worlds 2 and 3 to world 1? According to cognitive semantics, language does not refer directly to world 1, but to mental models and their components in world 2. I take it that mathematics is part of world 3. So, when a person interprets a mathematical structure, the person is attempting to establish a morphism between the mathematical structure in world 3 and a mental model which is in world 2. I am not sure how world 1 fits into this picture. Perhaps it is not supposed to fit into the picture, but is really the realm of what Kant thought of as the noumena or things-in-themselves, which, in Kant's view, fall outside the proper use of human understanding. I would appreciate it if you would clarify this for me. Thank you.

      Best wishes,

      Laurence Hitterdale

      Dear Larry,

      I will try to answer you as briefly as I can. World 1 (the physical World) includes all there is. It includes humans with brains that generate a World 2 (a Mind) for each individual. World 2 is a world of human experience, including perception that generates mental models of World 1 and action that modifies objects in in World 1. This is the origin of "common sense" knowledge, which is sufficient to navigate and survive in World 1. Common sense takes World 2 as the given world, so it does not recognize World 1 as something different. Science begins with the recognition that there is a world of "noumena" that cannot be directly perceived, but can only be known indirectly by constructing models. Thus science explores World 1. The exploration is facilitated by the invention of World 3 (human artifacts including language and, especially mathematical symbols). Mathematical symbols are meaningless in themselves, but acquire meaning by morphisms with mental models; such a symbol-model pair constitutes a mathematical concept.

      8 days later

      Dear Professor Hestenes,

      Your essay is brilliant, and is no wonder, coming from you (I am familiar with your works in Geometric Algebra and I am following them for 20 years). Indeed, it was about time that someone takes Kant to the next level, and cognitive science is pretty much the home place for this. You said "my working hypothesis will be: The primary cognitive activities in science and mathematics involve making, validating and applying conceptual models!". Not only I agree, but I think these may be the primary cognitive activities in most our daily activities, maybe in a less rigorous, more approximative and pragmatic manner. This would explain why our brain is so good at doing science! Human mind seems to me to be a shape shifter, able to take the shape of the things you put in it. Although this domain is so different than your writings with which you used me, I find here the same quest for universality that permeates your mathematical physics and pedagogical works. Thanks for the excellent reading!

      Best wishes,

      Cristi Stoica

      Dear David Hestenes,

      This is a great essay! For a general audience, maybe some of the lists could be dispensed with and maybe the name of 'worlds' for domains is a bit misleading, but I loved to read it. In my own contribution, I talked about mathematics as a form of 'constrained imagination'. Now, I just wish I had read more of your work earlier, so I could have cited some portions of it!

      My vote is a 9/10.

      Best wishes,

      Sylvia Wenmackers - Essay Children of the Cosmos

      Dear Professor David Hestenes,

      Astonishing! Finally I meet someone who understands the contributions Cognitive Science has to offer the rest of the sciences. As I read your clearly written & thought provoking essay I'm excited by the many parallel areas of mental cognitions gained by watching my own mind recursively: perceive, pattern-match, relate, abstract and construct mental models. Here are some perspectives and insights that support or augment some of your many valuable insights:

      You say Kant argued "fundamental laws of nature, like the truths of mathematics, are knowable precisely because they do not describe the world as it really is but rather prescribe the structure of the world as we experience it." Please allow me to point out the distinction between "experiences" vs. how we "perceive" the world that we experience. In my essay I say "Higher Perspective is the Key to Understanding Anything." meaning that in our mind we MUST step above our sensory perceptions and see the system (that we hope to comprehend) from a dimensional perspective ABOVE that system. Example, watching planets move in retrograde motion we might say the planets change direction. But when we project our mental perspective ABOVE the solar system (perpendicular to the ecliptic) then we see that the planets are uniformly moving in the same clockwise direction, only at different speeds relative to each other.

      I go on to describe a mental perspective above Space~Time and describe what a non-inertial reference frame would look like. The subsequent model resolves the Dark Matter, Dark Energy and Dark Flow mysteries and even describes, step-by-step, how quantum gravity works.

      Geometric Algebra/Calculus is new to me - perhaps you can suggest what areas of GA I will find most applicable to my idea of Combinatorial Quantum-wave Mechanics. Where the 4D geometry of my Cosmic Onion Model (holographic 4D Hyperspherical standing-waves) is the context in which the ever-expanding 3D surface of the "Now-Manifold" experiences Space~Time expansion. "Particles" are the manifestations of wave-icles which are double-twisted springs (aligned along the time dimension) which manifest mass & charge as a consequence of their frequency (M = hf/c[up]2[/up]) and 4D-geometry.

      As I read your "RULES AND TOOLS FOR THINKING AND DOING" they remind me of information hierarchy: Data, Information, Knowledge, Understanding and Wisdom. Where Data is raw perceptions, Information adds to perceptions the context from which data derives meaning, Knowledge adds an awareness as to the degree of certainty that we hold a datum's value and/or "know" its meaning. Understanding brings in the mental framework that grasps the context of the problem domain and the boundary conditions that enclose that domain and the rules/laws that govern it. Mainstream science seems to stop at "Knowledge" and fails to ask the most important epidemiological question "Do we really know what we think we know?" thus falling short of this lofty ideal of achieving a True Understanding. Wisdom applies understanding to purposefully steer activities in the present, to bring about a preferred future. Hopefully, one which transcends short-sighted, selfish interests, by serving the sustainable "Greater Good of All."

      Thanks for introducing me to Geometric Algebra/Calculus I look forward to adding this to my mental toolbox.

      -- Cosmologically yours,

      -- John Wsol