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

Professor Hestenes,

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

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

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

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

And ends with:

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

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

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

All best,

Tom

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

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

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

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

My paper Mathematics of Science addresses this to some extent.

David, a brilliant paper!

Cheers

Rob MacDuff

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

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

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

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

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

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

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

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

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

Addenda

Two embedded hyperlinks in my comment failed to transfer:

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

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

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

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

Rick

Dear Professor Hestenes,

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

Best wishes,

Laurence Hitterdale

Dear Professor Hestenes,

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

Best wishes,

Cristi Stoica

Dear David Hestenes,

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

My vote is a 9/10.

Best wishes,

Sylvia Wenmackers - Essay Children of the Cosmos

Dear Professor David Hestenes,

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

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

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

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

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

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

-- Cosmologically yours,

-- John Wsol

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 'epistemological' into 'epidemiological'. ;-)

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

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