Essay Abstract

Physics and mathematics are grounded in man's evolved ability to freely create mental models and use them to manage interactions with the natural world.

Author Bio

David Hestenes is Emeritus Professor of Physics at Arizona State University. He is a Fellow of the APS and Overseas Fellow of Churchill College, Cambridge. He has also been UCLA University Fellow, NSF Postdoctoral Fellow, NASA Faculty Fellow and Senior Fulbright Research Fellow. His scientific research has focused on development and application of Geometric Algebra as a unified mathematical language for physics and engineering. For contributions to physics education he received the 2002 Oersted Medal from the American Association of Physics Teachers and the 2014 Excellence in Physics Education Award from the American Physical Society.

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Dear Professor Hestenes,

Very enjoyable read. Since I agree completely with your statement "The primary cognitive activities in science and mathematics involve making, validating and applying conceptual models!", I knew the rest of your essay was going to be interesting.

I enjoyed your definition of models "A model is a representation of structure in a given system", allowing for very complex modelling that includes change over time and complex objects that interactions with multiple objects.

My essay involves modelling the particles of the standard model. The particle models specifically contain properties you talk about to make them complex enough to be proper models of the real particles. I hope you get a chance to have a look.

Good luck with your essay and you deserve a good rating.

Regards,

Ed Unverricht

    This is by far the best essay of all that I have read.

    Have you personally found some utility a priori thinking along these lines when throughout your significant career you thought about the value of say GA as a foundation on which to model physical reality, or is this more in reflection, looking back?

    One more question, GA or Octonion Algebra, which in your opinion possesses more promise as the foundation on which to build mathematical models of our physical reality? Why?

    Thank you so much for participating here,

    Rick Lockyer

      Dear David Hestenes,

      First, I speak for many here when I say thank you, thank you, thank you for Geometric Algebra. I and many others agree with Doran and Lasenby that it is "the most powerful available mathematical system developed to date."

      I fully agree that modeling is most characteristic of man as you point out in detail. I earlier developed a theory of theory-making based on a robot (to minimize 'baggage') designed to extract structure from measurement data to model 'features' [both static and dynamic] of reality, as touched on in my essay. I hope you will find time to look at it and would welcome any feedback.

      With highest regards,

      Edwin Eugene Klingman

        Dear Professor Hestenes,

        We have read your excellent essay with great interest, and also browsed through your Ref. [21] (review on Geometric Algebra).

        We have a few queries. In your essay you write:

        "Modeling theory asserts that physical and mathematical intuitions are merely two different ways to relate products of imagination to the external world. Physical intuition matches structure in mental models with structure in physical systems. Mathematical intuition matches mental structure with symbolic structure. Thus, structure in imagination is common ground for both physical and mathematical intuition."

        Is it already apparent from this statement as to why `mathematics is so highly effective in description of physical laws' or does one need to add an intermediate cognitive step in going from the physical world to its mathematical description? [something like the physical intuition searching for a mathematical intuition it can fit into].

        How does your modelling compare with the work of Lakoff and Nunez, which you cite (their book). In their book, they spell out in some detail how cognitive mechanisms lead to the mathematics of various number systems, algebra, calculus, set theory, and there is this nice example of the Euler identity as understood through metaphors. In your modelling, is it possible to set up a detailed description of this nature, using cognition to recover different aspects of mathematics? [Our apologies if this is already there in your listed work and we have not caught up with it].

        We admit to not being aware of your extraordinary program of Geometric Algebra. Would you say there is any way in which mental modelling picks out GA as a more natural description, as compared to conventional mathematics? Or is this a conscious judgement based choice one makes?

        In your AJP article on GA, you discuss quantum mechanics, and the limited validity of the Schrodinger equation (no spin). It is not clear to us if GA can in any way address the deep problems of quantum theory, already existent at the level of the Schrodinger equation, such as the quantum measurement problem. We will appreciate your thoughts on this.

        Lastly we cannot resist noting that we also have an essay in this contest, suggesting cognitive science as a basis for the connection between physics and mathematics.

        Thanks for putting up such an insightful essay, and with regards,

        Anshu, Tejinder

          Dear Prof. David Hestenes,

          Your essay is non-rhetoric, 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 Prof. Hestenes

            Do you think that Algebras of Grassmann,Hamilton and Clifford were written down under the motivation to "model" physical world.I strongly disagree with that and so will Grassmann,Hanilton and Clifford.All of these algebras have deep metaphysical underpinings and cant be subsumed to a mechanics of modelling.I would like you to refer to David Finkelestein,Basil Hiley and Louis Kauffman to see how

            Clifford algebras are not only a natural language for physics but they also reveal deeper physics and mathematics.

              Dear Professor Hestenes,

              There are many interesting observations about human psychology and intuition in your essay, and they can certainly help to improve the pedagogy of physics. But the invocation of Kant in this context is rather puzzling, and exactly for the reasons you dismiss as a "red herring". Kant's entire motivation in his philosophy (in the First Critique and the Prologomena) was to somehow account for the existence of a body of priori synthetic knowledge, including Euclidean geometry. Kant recognized (correctly) that Euclidean geometry is not analytically true (i.e. the denial of the axioms does not yield a contradiction) but insisted that it was nonetheless necessarily true of physical space. And the only way he could figure out to reconcile these claims is to make Euclidean space the form of outer intuition. But Euclidean geometry just is not true of physical space, and space (or space-time) is not the form of any human intuition. In fact, in every detail Kant's predictions about science have been refuted. He thought that science would have to postulate deterministic laws (refuted by quantum theory), that parts of Newtonian dynamics would never be rejected by scientific theory (refuted by both quantum theory and Relativity) and so on. No doubt, humans come pre-equipped with methods of reasoning, but scientific progress has largely been about seeing the limitations and errors built into those methods. Kant thought that the outcomes of this reasoning must be true, and so could not be about any mind-independent world. To save the village of physics, Kant had to destroy it.

              No doubt, all reasoning is "rule-governed", just because the drawing of consequences in an apparently random way from premises would not be regarded as reasoning at all. But whatever rules were provided by evolution are not sacrosanct and above correction (as your essay illustrates). To serve Kant's purposes, the specific rules had to be sacrosanct and above correction, to form the foundation of a priori knowledge. So Kant is not a good place to draw inspiration for an account of scientific method.

              Regards,

              Tim Maudlin

                David,

                You conclude your essay by noting that there is a very powerful mathematics available that few people know of ... specifically, that of geometric algebra.

                For whatever it is worth to you, I can state with complete certainty that the word is spreading. How do I know this? Because I am a chemical engineer. My formal education did not include Hamilton or Clifford. I have only recently learned of you. But my post-education search has brought me to these ideas.

                You have my most sincere thanks.

                Best Regards and Good Luck,

                Gary Simpson

                  I occasionally tutor high school math and college algebra students. Some have great trouble with the subject. After reading your paper, I have the idea that the troubled student does not build models to represent math objects and processes. Then, they cannot use the characteristics of models you list in your essay to build understanding.

                  I have published a book titled, "The Theory and Practice of Magic Deception," sold on Amazon dot com. It is intended for magicians in the performing arts to improve their ability to entertain with magic. The book is selling well to the magic industry and a significant mid America university bought a case full for a special class on understanding how the mind works. Using the language of your essay, the book posits that the mind consists of a wide variety of models to represent daily actions and events. The point of the book is to suggest that a skilled performer study these common models and trigger them with rehearsed real world actions to support deception. The result is that the skilled performer can create a reality that does not exist. The effect is astounding and can be captured with a camera. One example from the book was used in a video documentary about how the human mind works presented by Brian Green. To me, the concept you have offered has far reaching consequences.

                  I have submitted an essay, "Modeling Reality with Mathematics," to this contest purporting that models in the physics community are supported by opinion leaders, not necessarily a comparison to reality. An example would include the ether model that existed for over 2000 years to explain light phenomena. The point of the essay is that opinion leaders are forcing physicists to use only mathematical models and avoid models depending on imagery.

                  Your essay discusses the use of models to understand the world around us. I am curious of your point of view about social pressures controlling the use of models for our understanding.

                  Al Schneider

                    Dear Professor Hestenes,

                    You have concluded that five types of structure suffice to characterize any scientific model.

                    Links among the parts is the main subject of my article. I would be very grateful if you find any inconsistency or wrong concept among the structures in my essay.

                    Best Regards,

                    Branko Zivlak

                      I agree with your argument. Interestingly, I mentioned both Copernican revolution and Kant too in my essay. My conclusion is that the realist view of theories is untenable.

                        Thank you Gary,

                        Yes geometric algebra is spreading rapidly, and I no longer feel the need to promote it.

                        But it has not yet been blessed by the high priests of physics and mathematics.

                        I do not see how you gathered from my essay that I "think that Algebras of Grassmann,Hamilton and Clifford were written down under the motivation to "model" physical world." Rather, I claim that they aimed to create rules to express geometric in accord with their intuition (The second category in my list of universal structures), that were then used to "model the physical world.

                        Hamilton was quite explicit in how he did it. He began with rules for generating rotations in the plane by multiplication with the unit imaginary i. Then he looked for rules connecting similar generators j and k in orthogonal planes. The result was the famous rules for quaternion multiplication. Grassmann took his cues directly from Euclid. Based on the intuition that a moving point sweeps out a line, a moving line sweeps out an area and a moving area sweeps out a volume, he created his "Algebra of Extension," about which I have written a lot. Clifford amalgamated insights of both Grassmann and Hamilton.

                        I do not think that Finkelstein and Hiley have made such fundamental contributions. However, Kauffman's work is a great example of the interplay of topological intuition with its mathematical representation.

                        You raise many points about Kant that are worth discussing, but I do not see them as contravening anything in my essay. You seem to cast him as the ultimate dogmatist, whereas I see him as striving to make sharp distinctions in a very muddy subject.

                        My aim is not to defend Kant, but to take advantage of his best insights. His views on Euclidean geometry were the best available at the time. But I claim that his argument about the role of rules in justifying those views transcended mistakes in those views and applies to current views on non-Euclidean geometry.

                        Also I do not claim that evolution provided any rules, but only that the rules adopted by science and mathematics must be articulated with intuition to enable understanding of the physical world.

                        Dear Rick,

                        Absolutely. I say so quite explicitly in my book. "New Foundations for Classical Mechanics." Especially in the last chapter in the first edition. Although it was replaced in the second edition, it is available on my Geometric Calculus Website.

                        Every algebra including Octonians can be expressed as a subalgebra of Geometric Algebra.

                        In your essay I think you are arguing against the position of "naïve realism". Your view corresponds to what is commonly called "scientific realism," which I regard as the view of my essay.

                        Dear Ed,

                        It was a joy to read your excellent essay.

                        Keep up the good work!

                        .........David H

                        Dear Ed,

                        In case you haven't met, let me introduce you to the Robot designed to do probabilistic reasoning by E.T. Jaynes in his book "Probability Theory, the Logic of Science," which I regard as one of the greatest books of the twentieth century.

                        He might help with the reasoning in your essay, which I find refreshing, though I am still not ready to make a final judgment on Bell's theorem.

                        .........David H

                        Dear David,

                        "You mentioned that in context of Copernican revolution in science in Newton's Principia (1687),Kant shifted the focus of epistemology from structure of the external world to structure of mind. His revolutionary insight was that our perceptions and thoughts are shaped by inherent structure of our minds. Kant's primary question: What does the structure of science and mathematics tell us about how

                        the human mind works? "What, precisely, is thinking?" -- Einstein

                        Modeling theory asserts that physical and mathematical intuitions are merely two different ways to relate products of imagination to the external world.Thus, structure in imagination is common ground for both

                        physical and mathematical intuition.Kant began by identifying construction in intuition as a means for acquiring certain geometrical knowledge. Kant's notion of geometrical proof is by construction of figures, and he argues that such proofs have universal

                        validity as long as the figures are "determined by certain universal conditions of construction. Kant's argument is often dismissed because it led him to conclude that Euclidean geometry is certain a priori.Because we now know that non-Euclidean geometry can be associated with the same intuitive construction simply by changing the rules

                        assigned to it."

                        Let me cite Riemann's lecture excerpt :how Riemann geometry(On the hypothesis which underlie geometry) connects external world to human mind which I have explained in my essay using Mathematical Structure Hypothesis.

                        Riemann's presentation demonstrate an important fact of mental life: human beings do not sit outside the universe, investigating it from a fixed, stable location - rather, creative mental activity is itself a universal power, and must be itself considered by anyone seeking a unified physical view of the world. The structure of external world and human mind are basically the same.This is the matter of deep consciousness. We all have hypotheses about the nature of space itself, and we have preconceptions about constructions in space,which faker Euclid didn't question his assumptions.Riemann's examination of curved spaces. Triply extended manifolds (such as space) can be curved! Riemann's conflict with discrete versus continuous manifold leads us into the domain of another science, the realm of physics beyond mathematics.He changed the notion of geometry unlike Euclidean. In context of Einstein's relativity,they exist as action-spaces, not geometric spaces.Vladimir Vernadsky's passionate search for understanding the nature of life and cognition led him to hunt for geometries capable of expressing activities of life that he knew simply could not exist in a Euclidean space.Vernadsky also wrote much about the different kind of living time distinct from abiotic time. In evolutionary living time, for example, before and after are not merely distinguished chronologically, as before being not-after and after being the opposite of before, but rather after is fundamentally different than before, being a time in which higher developments of new life processes exist. This is seen much more strongly in human time. In our economic time, the power of the human species - and we are ourselves a physical force - changes categorically with new discoveries of principle. Economic times differ qualitatively, not quantitatively. And such human time doesn't just "happen" like the ticking of a clock, it has to be created through discovery and driven by passion! This is the spacetime of economic development.With Riemann, "geometry" itself completely changes its meaning - it isn't the stage upon which events unfold, it's the shape of action itself!the most powerful of physical forces: the human mind. Creative thought is a physical force: it has physical effects just like electromagnetism, plasma, biological processes. A true Riemannian geometry, based firmly on the principles that lie behind perceived appearances, must take creative mind into account.There is only one world to discover and act on. Mind discovers, mind acts, mind creates. Riemann brings us into reality, and shows that the principles underlying reality cohere with the mind. While he concluded his lecture with the need to abandon mathematics for physics, to truly achieve Riemann's program, we must go beyond physics to economics; we must include the progressing development of the powers of the human mind.

                        It is known that geometry assumes, both the notion of space and the first principles of constructions in space, as given in advance. She gives definitions of them which are merely nominal, while the true determinations appear in the form of axioms. The relation of these assumptions remains consequently in darkness; we perceive neither whether and how far their connection is necessary, nor a priori, whether it is possible.

                        From Euclid to Legendre (to name the most famous of modern reforming geometers) this darkness was cleared up neither by mathematicians nor by such philosophers as concerned themselves with it. The reason of this is doubtless that the general notion of multiply extended magnitudes (in which space-magnitudes are included) remained entirely unworked.

                        The question of the validity of the hypotheses of geometry in the infinitely small is bound up with the question of the ground of the metric relations of space. Either therefore the reality which underlies space must form a discrete manifold, or we must seek the ground of its metric relations outside it, in binding forces which act upon it.This leads us into the domain of another science, that of physics, into which the object of today's proceedings does not allow us to enter.

                        Thus , if Kant shifts from the structure of external world to the structure of mind.This is what Riemann's geometry true meaning is. How human mind and the external world both exists within each other because of Vibration and they are infact same at higher level of consciousness.This peculiar geometry which is linked to paradox of consciousness and Russell's paradox geometry, you may refer to my essay.

                        Anyway wonderful essay.

                        Regards,

                        Pankaj Mani