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RECENT POSTS IN THIS TOPIC

**Kevin Knuth**: *on* 5/5/15 at 5:02am UTC, wrote Dear David, I very much enjoyed your essay, and despite the fact that we...

**Alma Ionescu**: *on* 4/22/15 at 14:32pm UTC, wrote Dear Professor Hestenes, It was greatly interesting to read an account of...

**John Wsol**: *on* 4/22/15 at 13:20pm UTC, wrote My laptop keyboard is failing-now I have an intermittent 's'. (I've had to...

**John Wsol**: *on* 4/22/15 at 3:39am UTC, wrote **Dear Professor David Hestenes, ** Astonishing! Finally I meet...

**Sylvia Wenmackers**: *on* 4/21/15 at 20:34pm UTC, wrote Dear David Hestenes, This is a great essay! For a general audience,...

**Cristinel Stoica**: *on* 4/21/15 at 14:26pm UTC, wrote Dear Professor Hestenes, Your essay is brilliant, and is no wonder, coming...

**David Hestenes**: *on* 4/13/15 at 6:13am UTC, wrote Dear Larry, I will try to answer you as briefly as I can. World 1 (the...

**Laurence Hitterdale**: *on* 4/12/15 at 18:50pm UTC, wrote Dear Professor Hestenes, Because I am not familiar with modeling theory,...

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Philosophers, physicists and neuroscientists discuss how our sense of time’s flow might arise through our interactions with external stimuli—despite suggestions from Einstein's relativity that our perception of the passage of time is an illusion.

A devilish new framework of thermodynamics that focuses on how we observe information could help illuminate our understanding of probability and rewrite quantum theory.

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Objective reality, and the laws of physics themselves, emerge from our observations, according to a new framework that turns what we think of as fundamental on its head.

FQXi FORUM

October 14, 2019

CATEGORY:
Trick or Truth Essay Contest (2015)
[back]

TOPIC: Modeling the Physical World with Common Sense and Mathematics by David Hestenes [refresh]

TOPIC: Modeling the Physical World with Common Sense and Mathematics by David Hestenes [refresh]

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.

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.

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

report post as inappropriate

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

report post as inappropriate

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

report post as inappropriate

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

report post as inappropriate

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.

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.

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

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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

report post as inappropriate

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

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

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

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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 "

I fully agree that modeling is most characteristic of man as you point out in detail. I earlier developed a

With highest regards,

Edwin Eugene Klingman

report post as inappropriate

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

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 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

report post as inappropriate

Thank you for your kind words on my essay. I do not expect to convince many people that Bell

Thank you for introducing ET Jaynes

"…

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

report post as inappropriate

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...

view entire post

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...

view entire post

report post as inappropriate

Thanks for your probing queries, Tejinder.

I will not repeat them in my answers, but you can correlate them by paragraph.

You have identified my basic claim about the common ground for physics and mathematics in intuition. Beyond that there are many details, and one should look at specific cases. I suggest you look at my reply to rukhsan ul haq wani for examples. There is more about Grassmann in my reference [29].

Lakoff is a major figure in cognitive linguistics, especially for his work on metaphors.

He demonstrates how metaphors play a crucial role in cognition. Unfortunately, I had to eliminate comments on that to meet the character limit on my essay. But many details are given in my reference [3].

GA is a tool –– deliberately designed to integrate algebra and geometry, and thereby facilitate geometric intuition. The published literature on GA testifies to its effectiveness in this domain.

Evidently you have not looked deeply enough at my papers to see that I make strong and unique claims about the relevance of GA to the foundations of quantum mechanics. Look at my ref. [22], which has the same web address as [21]. I demonstrate that GA reveals hidden geometric structure in the Dirac equation that relates electron spin to complex numbers in quantum mechanics in an essential way. I believe this shows you will never get to the bottom of quantum mechanics from the Schroedinger equation. You might also like to look at my essay “Electron time, mass and zitter,” which got second prize in a previous FQXi contest.

Now I must take a look at your essay.

………David H.

I will not repeat them in my answers, but you can correlate them by paragraph.

You have identified my basic claim about the common ground for physics and mathematics in intuition. Beyond that there are many details, and one should look at specific cases. I suggest you look at my reply to rukhsan ul haq wani for examples. There is more about Grassmann in my reference [29].

Lakoff is a major figure in cognitive linguistics, especially for his work on metaphors.

He demonstrates how metaphors play a crucial role in cognition. Unfortunately, I had to eliminate comments on that to meet the character limit on my essay. But many details are given in my reference [3].

GA is a tool –– deliberately designed to integrate algebra and geometry, and thereby facilitate geometric intuition. The published literature on GA testifies to its effectiveness in this domain.

Evidently you have not looked deeply enough at my papers to see that I make strong and unique claims about the relevance of GA to the foundations of quantum mechanics. Look at my ref. [22], which has the same web address as [21]. I demonstrate that GA reveals hidden geometric structure in the Dirac equation that relates electron spin to complex numbers in quantum mechanics in an essential way. I believe this shows you will never get to the bottom of quantum mechanics from the Schroedinger equation. You might also like to look at my essay “Electron time, mass and zitter,” which got second prize in a previous FQXi contest.

Now I must take a look at your essay.

………David H.

Dear Professor Hestenes,

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

Tejinder

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

Tejinder

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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...

view entire post

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...

view entire post

post approved

Thanks for your probing queries, Tejinder.

I will not repeat them in my answers, but you can correlate them by paragraph.

You have identified my basic claim about the common ground for physics and mathematics in intuition. Beyond that there are many details, and one should look at specific cases. I suggest you look at my reply to rukhsan ul haq wani for examples. There is more about Grassmann in my reference [29].

Lakoff is a major figure in cognitive linguistics, especially for his work on metaphors.

He demonstrates how metaphors play a crucial role in cognition. Unfortunately, I had to eliminate comments on that to meet the character limit on my essay. But many details are given in my reference [3].

GA is a tool –– deliberately designed to integrate algebra and geometry, and thereby facilitate geometric intuition. The published literature on GA testifies to its effectiveness in this domain.

Evidently you have not looked deeply enough at my papers to see that I make strong and unique claims about the relevance of GA to the foundations of quantum mechanics. Look at my ref. [22], which has the same web address as [21]. I demonstrate that GA reveals hidden geometric structure in the Dirac equation that relates electron spin to complex numbers in quantum mechanics in an essential way. I believe this shows you will never get to the bottom of quantum mechanics from the Schroedinger equation. You might also like to look at my essay “Electron time, mass and zitter,” which got second prize in a previous FQXi contest.

Now I must take a look at your essay.

………David H.

I will not repeat them in my answers, but you can correlate them by paragraph.

You have identified my basic claim about the common ground for physics and mathematics in intuition. Beyond that there are many details, and one should look at specific cases. I suggest you look at my reply to rukhsan ul haq wani for examples. There is more about Grassmann in my reference [29].

Lakoff is a major figure in cognitive linguistics, especially for his work on metaphors.

He demonstrates how metaphors play a crucial role in cognition. Unfortunately, I had to eliminate comments on that to meet the character limit on my essay. But many details are given in my reference [3].

GA is a tool –– deliberately designed to integrate algebra and geometry, and thereby facilitate geometric intuition. The published literature on GA testifies to its effectiveness in this domain.

Evidently you have not looked deeply enough at my papers to see that I make strong and unique claims about the relevance of GA to the foundations of quantum mechanics. Look at my ref. [22], which has the same web address as [21]. I demonstrate that GA reveals hidden geometric structure in the Dirac equation that relates electron spin to complex numbers in quantum mechanics in an essential way. I believe this shows you will never get to the bottom of quantum mechanics from the Schroedinger equation. You might also like to look at my essay “Electron time, mass and zitter,” which got second prize in a previous FQXi contest.

Now I must take a look at your essay.

………David H.

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.

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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.

report post as inappropriate

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.

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.

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)...

view entire post

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)...

view entire post

report post as inappropriate

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.

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.

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

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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

report post as inappropriate

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

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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

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Good to hear about your work, Al.

From your note, it looks to be in complete accord with Modeling Theory,

and I believe it can be a boon to teaching that is enlightening as well as entertaining.

As you know there is a huge psychology literature on how our expectations shape what we see. As you know, there is no psychology in the K-12 curriculum, and I have long been wondering how to squeeze it in. Maybe magic is the way to go!.

Modeling Theory has generated what is arguably to most effective and widely used approach to STEM education. To learn more about it, check out: modelinginstruction.org

Actually, I don’t agree that physicists are forced by opinion leaders to “use only mathematical models and avoid models depending on imagery.” I suppose you are referring to opinions that “the mechanisms of quantum mechanics cannot be visualized.” While many competent physicists take that point of view, I assure you that visualization plays a crucial role in their practice of physics. The debate on the foundations of quantum mechanics is by no means settled.

……….David H.

From your note, it looks to be in complete accord with Modeling Theory,

and I believe it can be a boon to teaching that is enlightening as well as entertaining.

As you know there is a huge psychology literature on how our expectations shape what we see. As you know, there is no psychology in the K-12 curriculum, and I have long been wondering how to squeeze it in. Maybe magic is the way to go!.

Modeling Theory has generated what is arguably to most effective and widely used approach to STEM education. To learn more about it, check out: modelinginstruction.org

Actually, I don’t agree that physicists are forced by opinion leaders to “use only mathematical models and avoid models depending on imagery.” I suppose you are referring to opinions that “the mechanisms of quantum mechanics cannot be visualized.” While many competent physicists take that point of view, I assure you that visualization plays a crucial role in their practice of physics. The debate on the foundations of quantum mechanics is by no means settled.

……….David H.

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

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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

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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.

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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...

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"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...

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Dear Panjak,

I agree with your assertion that "Mathematical structures have no independent existence without physical reality.”

Your assertion that “Everything in Universe including mathematical structures and physical reality is Vibration" may be an interesting hypothesis, but it lacks adequate scientific support.

I agree with your assertion that "Mathematical structures have no independent existence without physical reality.”

Your assertion that “Everything in Universe including mathematical structures and physical reality is Vibration" may be an interesting hypothesis, but it lacks adequate scientific support.

Dear David,

Let me quote scientifically the role of Vibration.That which we call matter and mind are one and the same substance. The only difference is in the degree of vibration. Mind at a very low rate of vibration is what is known as matter. Matter at a high rate of vibration is what is known as mind. Both are the same substance; and therefore, as matter is bound by time and space and causation, mind which is matter at a high rate of vibration is bound by the same law. Mind becomes matter, and matter in its turn becomes mind, it is simply a question of vibration.

It's the hypothesis that requires to be explored further more scientifically.

Regards,

Pankaj Mani

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Let me quote scientifically the role of Vibration.That which we call matter and mind are one and the same substance. The only difference is in the degree of vibration. Mind at a very low rate of vibration is what is known as matter. Matter at a high rate of vibration is what is known as mind. Both are the same substance; and therefore, as matter is bound by time and space and causation, mind which is matter at a high rate of vibration is bound by the same law. Mind becomes matter, and matter in its turn becomes mind, it is simply a question of vibration.

It's the hypothesis that requires to be explored further more scientifically.

Regards,

Pankaj Mani

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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.

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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.

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Hello Jose,

Here are a few comments on your essay from the point of view of mine.

“Any explanation of the physical world should start from the description of the background . . . The background description is based entirely on direct observation”

There is no such thing as direct observation, only indirect perception.

“So the fact that the physical world changes with time indicates that motion is a property of the physical world.”

I agree that motion is fundamental. For more see my “Modeling Games” reference [5] and compare it with what you say about Newton.

“Thus, in explaining the physical world, there should be a clear distinction between properties and laws: properties should be physical

and laws should be mathematical.”

I agree with this, but think you slip up when you say:

'Mathematics governs the changes in the physical world'.

Rather, mathematics can be applied to model changes in the physical world.

The relation of properties to laws is an analogy. The laws change as we learn more, but the physical properties remain what they are.

Here are a few comments on your essay from the point of view of mine.

“Any explanation of the physical world should start from the description of the background . . . The background description is based entirely on direct observation”

There is no such thing as direct observation, only indirect perception.

“So the fact that the physical world changes with time indicates that motion is a property of the physical world.”

I agree that motion is fundamental. For more see my “Modeling Games” reference [5] and compare it with what you say about Newton.

“Thus, in explaining the physical world, there should be a clear distinction between properties and laws: properties should be physical

and laws should be mathematical.”

I agree with this, but think you slip up when you say:

'Mathematics governs the changes in the physical world'.

Rather, mathematics can be applied to model changes in the physical world.

The relation of properties to laws is an analogy. The laws change as we learn more, but the physical properties remain what they are.

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?

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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?

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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

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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

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Sorry Joe,

It was my post that was inappropriate, because it was intended for someone else and I pasted it in the wrong box. When I tried to remove it, your post was expunged and I lost your contact.

When I get your original post back I will reply.

.......David

It was my post that was inappropriate, because it was intended for someone else and I pasted it in the wrong box. When I tried to remove it, your post was expunged and I lost your contact.

When I get your original post back I will reply.

.......David

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...

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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...

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You do indeed nicely articulate a coherent common sense view of how the world works.

I part company with you when you say:

“Each real object has a real material surface that seems to be attached to a material sub-surface.”

I would say “perceived object” instead of “real object.” The rest of your account is coherent from a CS point of view, but it is inconsistent with the science of light and vision.

Respectfully......David H.

I part company with you when you say:

“Each real object has a real material surface that seems to be attached to a material sub-surface.”

I would say “perceived object” instead of “real object.” The rest of your account is coherent from a CS point of view, but it is inconsistent with the science of light and vision.

Respectfully......David H.

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

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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

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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

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

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

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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

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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...

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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...

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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.

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.

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...

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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...

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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

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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

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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...

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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...

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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/.

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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/.

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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

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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

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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.

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.

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

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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:

Best wishes,

Cristi Stoica

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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

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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

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My laptop keyboard is failing-now I have an intermittent 's'. (I've had to switch to an external Bluetooth keyboard.) Spelling corrector, which usually helps, turned 'epi_{s}temological' into 'epidemiological'. ;-)

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Dear Professor Hestenes,

It was greatly interesting to read an account of the art of teaching science from the expert himself. Your section about mental models reminded me that there are studies comparing physical intuition with mathematical intuition from a neurological perspective and usually people are better at one category than they are in the other. What are your thoughts on this? What kind of wiring must your students have had to be better in a category? Which wiring is best for mathematical intuition and which for physical?

Your essay covers successfully a lot of information and presents it with extraordinary clarity. It would be very interesting if this is the seed of a book, because I think that many neurologists and psychologists would be very interested in your point of view. Your point of view is all the more special because you had access to a lot of young people just when their brains were undergoing the last transformations of the adolescence and their cognitive structures were settling into place. It is obvious that your method of teaching was very good for their needs, which means there are many interesting facts to learn from your experience, some of them perhaps surprising and new.

Thank you for a spectacular read and wish you all the best!

Alma

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It was greatly interesting to read an account of the art of teaching science from the expert himself. Your section about mental models reminded me that there are studies comparing physical intuition with mathematical intuition from a neurological perspective and usually people are better at one category than they are in the other. What are your thoughts on this? What kind of wiring must your students have had to be better in a category? Which wiring is best for mathematical intuition and which for physical?

Your essay covers successfully a lot of information and presents it with extraordinary clarity. It would be very interesting if this is the seed of a book, because I think that many neurologists and psychologists would be very interested in your point of view. Your point of view is all the more special because you had access to a lot of young people just when their brains were undergoing the last transformations of the adolescence and their cognitive structures were settling into place. It is obvious that your method of teaching was very good for their needs, which means there are many interesting facts to learn from your experience, some of them perhaps surprising and new.

Thank you for a spectacular read and wish you all the best!

Alma

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Dear David,

I very much enjoyed your essay, and despite the fact that we had the chance to discuss this in great detail last week, I thought that I should still leave some comments.

I sincerely appreciate your focus on modeling. This is what we do in science. We make models. Its easy to forget that. Much confusion arises when we confuse the model with that which is being modeled. I believe that this is what Ed Jaynes called the Mind Projection Fallacy, and I believe that this fallacy lies at the root of some of the ideas that mathematics is real in the same sense that the physical world is real.

As you noted, I now see that the approach that I took in my essay is well-founded in modeling. You put it very nicely when you wrote:

"Note that a conceptual model establishes an analogy between a mental model and its symbolic representation. Mathematical models are symbolic structures, and to understand one is to create a mental model with analogous structure."

I was also surprised to learn that Kant was the one who first formulated the abstract associative and commutative rules for addition. You know, of course, that these properties along with closure and ordering are all that is necessary to establish additivity. The basic idea is that if you want to quantify pencils, and since pencils when grouped (I like that word better. Thanks for the suggestion) obey closure, associativity and commutativity, and groups of pencils can be ordered, one is then constrained to quantify the grouping of pencils using an invertible transform of additivity.

Here is where our essays meet (or rather join):

1. Symmetries form the basis of analogy

2. Symmetries constrain any attempt at consistent quantification (map to a total order).

3. These symmetry-based constraints act on quantifications and thus form the basis of quantitative laws.

4. Thus the laws originate from the analogy.

I think that we have attained a deeper understanding here!

Thank you!

Kevin

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I very much enjoyed your essay, and despite the fact that we had the chance to discuss this in great detail last week, I thought that I should still leave some comments.

I sincerely appreciate your focus on modeling. This is what we do in science. We make models. Its easy to forget that. Much confusion arises when we confuse the model with that which is being modeled. I believe that this is what Ed Jaynes called the Mind Projection Fallacy, and I believe that this fallacy lies at the root of some of the ideas that mathematics is real in the same sense that the physical world is real.

As you noted, I now see that the approach that I took in my essay is well-founded in modeling. You put it very nicely when you wrote:

"Note that a conceptual model establishes an analogy between a mental model and its symbolic representation. Mathematical models are symbolic structures, and to understand one is to create a mental model with analogous structure."

I was also surprised to learn that Kant was the one who first formulated the abstract associative and commutative rules for addition. You know, of course, that these properties along with closure and ordering are all that is necessary to establish additivity. The basic idea is that if you want to quantify pencils, and since pencils when grouped (I like that word better. Thanks for the suggestion) obey closure, associativity and commutativity, and groups of pencils can be ordered, one is then constrained to quantify the grouping of pencils using an invertible transform of additivity.

Here is where our essays meet (or rather join):

1. Symmetries form the basis of analogy

2. Symmetries constrain any attempt at consistent quantification (map to a total order).

3. These symmetry-based constraints act on quantifications and thus form the basis of quantitative laws.

4. Thus the laws originate from the analogy.

I think that we have attained a deeper understanding here!

Thank you!

Kevin

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