Dear Professor Hestenes,
You have concluded that five types of structure suffice to characterize any scientific model.
Links among the parts is the main subject of my article. I would be very grateful if you find any inconsistency or wrong concept among the structures in my essay.
Best Regards,
Branko Zivlak
I agree with your argument. Interestingly, I mentioned both Copernican revolution and Kant too in my essay. My conclusion is that the realist view of theories is untenable.
Thank you Gary,
Yes geometric algebra is spreading rapidly, and I no longer feel the need to promote it.
But it has not yet been blessed by the high priests of physics and mathematics.
I do not see how you gathered from my essay that I "think that Algebras of Grassmann,Hamilton and Clifford were written down under the motivation to "model" physical world." Rather, I claim that they aimed to create rules to express geometric in accord with their intuition (The second category in my list of universal structures), that were then used to "model the physical world.
Hamilton was quite explicit in how he did it. He began with rules for generating rotations in the plane by multiplication with the unit imaginary i. Then he looked for rules connecting similar generators j and k in orthogonal planes. The result was the famous rules for quaternion multiplication. Grassmann took his cues directly from Euclid. Based on the intuition that a moving point sweeps out a line, a moving line sweeps out an area and a moving area sweeps out a volume, he created his "Algebra of Extension," about which I have written a lot. Clifford amalgamated insights of both Grassmann and Hamilton.
I do not think that Finkelstein and Hiley have made such fundamental contributions. However, Kauffman's work is a great example of the interplay of topological intuition with its mathematical representation.
You raise many points about Kant that are worth discussing, but I do not see them as contravening anything in my essay. You seem to cast him as the ultimate dogmatist, whereas I see him as striving to make sharp distinctions in a very muddy subject.
My aim is not to defend Kant, but to take advantage of his best insights. His views on Euclidean geometry were the best available at the time. But I claim that his argument about the role of rules in justifying those views transcended mistakes in those views and applies to current views on non-Euclidean geometry.
Also I do not claim that evolution provided any rules, but only that the rules adopted by science and mathematics must be articulated with intuition to enable understanding of the physical world.
Dear Rick,
Absolutely. I say so quite explicitly in my book. "New Foundations for Classical Mechanics." Especially in the last chapter in the first edition. Although it was replaced in the second edition, it is available on my Geometric Calculus Website.
Every algebra including Octonians can be expressed as a subalgebra of Geometric Algebra.
In your essay I think you are arguing against the position of "naïve realism". Your view corresponds to what is commonly called "scientific realism," which I regard as the view of my essay.
Dear Ed,
It was a joy to read your excellent essay.
Keep up the good work!
.........David H
Dear Ed,
In case you haven't met, let me introduce you to the Robot designed to do probabilistic reasoning by E.T. Jaynes in his book "Probability Theory, the Logic of Science," which I regard as one of the greatest books of the twentieth century.
He might help with the reasoning in your essay, which I find refreshing, though I am still not ready to make a final judgment on Bell's theorem.
.........David H
Dear David,
"You mentioned that in context of Copernican revolution in science in Newton's Principia (1687),Kant shifted the focus of epistemology from structure of the external world to structure of mind. His revolutionary insight was that our perceptions and thoughts are shaped by inherent structure of our minds. Kant's primary question: What does the structure of science and mathematics tell us about how
the human mind works? "What, precisely, is thinking?" -- Einstein
Modeling theory asserts that physical and mathematical intuitions are merely two different ways to relate products of imagination to the external world.Thus, structure in imagination is common ground for both
physical and mathematical intuition.Kant began by identifying construction in intuition as a means for acquiring certain geometrical knowledge. Kant's notion of geometrical proof is by construction of figures, and he argues that such proofs have universal
validity as long as the figures are "determined by certain universal conditions of construction. Kant's argument is often dismissed because it led him to conclude that Euclidean geometry is certain a priori.Because we now know that non-Euclidean geometry can be associated with the same intuitive construction simply by changing the rules
assigned to it."
Let me cite Riemann's lecture excerpt :how Riemann geometry(On the hypothesis which underlie geometry) connects external world to human mind which I have explained in my essay using Mathematical Structure Hypothesis.
Riemann's presentation demonstrate an important fact of mental life: human beings do not sit outside the universe, investigating it from a fixed, stable location - rather, creative mental activity is itself a universal power, and must be itself considered by anyone seeking a unified physical view of the world. The structure of external world and human mind are basically the same.This is the matter of deep consciousness. We all have hypotheses about the nature of space itself, and we have preconceptions about constructions in space,which faker Euclid didn't question his assumptions.Riemann's examination of curved spaces. Triply extended manifolds (such as space) can be curved! Riemann's conflict with discrete versus continuous manifold leads us into the domain of another science, the realm of physics beyond mathematics.He changed the notion of geometry unlike Euclidean. In context of Einstein's relativity,they exist as action-spaces, not geometric spaces.Vladimir Vernadsky's passionate search for understanding the nature of life and cognition led him to hunt for geometries capable of expressing activities of life that he knew simply could not exist in a Euclidean space.Vernadsky also wrote much about the different kind of living time distinct from abiotic time. In evolutionary living time, for example, before and after are not merely distinguished chronologically, as before being not-after and after being the opposite of before, but rather after is fundamentally different than before, being a time in which higher developments of new life processes exist. This is seen much more strongly in human time. In our economic time, the power of the human species - and we are ourselves a physical force - changes categorically with new discoveries of principle. Economic times differ qualitatively, not quantitatively. And such human time doesn't just "happen" like the ticking of a clock, it has to be created through discovery and driven by passion! This is the spacetime of economic development.With Riemann, "geometry" itself completely changes its meaning - it isn't the stage upon which events unfold, it's the shape of action itself!the most powerful of physical forces: the human mind. Creative thought is a physical force: it has physical effects just like electromagnetism, plasma, biological processes. A true Riemannian geometry, based firmly on the principles that lie behind perceived appearances, must take creative mind into account.There is only one world to discover and act on. Mind discovers, mind acts, mind creates. Riemann brings us into reality, and shows that the principles underlying reality cohere with the mind. While he concluded his lecture with the need to abandon mathematics for physics, to truly achieve Riemann's program, we must go beyond physics to economics; we must include the progressing development of the powers of the human mind.
It is known that geometry assumes, both the notion of space and the first principles of constructions in space, as given in advance. She gives definitions of them which are merely nominal, while the true determinations appear in the form of axioms. The relation of these assumptions remains consequently in darkness; we perceive neither whether and how far their connection is necessary, nor a priori, whether it is possible.
From Euclid to Legendre (to name the most famous of modern reforming geometers) this darkness was cleared up neither by mathematicians nor by such philosophers as concerned themselves with it. The reason of this is doubtless that the general notion of multiply extended magnitudes (in which space-magnitudes are included) remained entirely unworked.
The question of the validity of the hypotheses of geometry in the infinitely small is bound up with the question of the ground of the metric relations of space. Either therefore the reality which underlies space must form a discrete manifold, or we must seek the ground of its metric relations outside it, in binding forces which act upon it.This leads us into the domain of another science, that of physics, into which the object of today's proceedings does not allow us to enter.
Thus , if Kant shifts from the structure of external world to the structure of mind.This is what Riemann's geometry true meaning is. How human mind and the external world both exists within each other because of Vibration and they are infact same at higher level of consciousness.This peculiar geometry which is linked to paradox of consciousness and Russell's paradox geometry, you may refer to my essay.
Anyway wonderful essay.
Regards,
Pankaj Mani
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.
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.
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.
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 do not see any inconsistency in your relations among physical constants.
But I am reminded of Arthur Eddington's comment:
"I won't believe the experiment until it is confirmed by theory."
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
Dear Prof. David Hestenes,
Your essay is factual, and educative. From 'commonsense' to 'thinking' to 'modeling' it portrays a clear path of evolution. Quoting from your essay, "CS concepts should be regarded as alternative hypotheses about the physical world that, when clearly formulated, can be tested empirically." "Thinking is a hardwired human ability to freely create mental models and use them for planning and controlling interactions with the physical world." "the transition from common sense to scientific thinking is not a replacement of CS concepts with scientific concepts, but rather a realignment of intuition with experience."
I completely agree with your view, "Likewise the tools of mathematics were invented, not discovered; though it may be said that theorems derived from structures built with those tools are discovered." In my opinion, there indeed need be just one law in mathematics, the law of addition; it is eternal. The structures are based on this fundamental law and are invented; the theorems derived from the structures follow the fundamental law, and are discovered.
You ask the question, "What accounts for the ubiquitous applicability of mathematics to science? You suggest co-evolution of physics and mathematics as the possible reason." I think it is more fundamental than mere co-evolution: A static world does not have any 'laws'. The only role of law is governing changes. Changes can happen by way of 'motion' only. Motion follows mathematical laws. Thus, all the changes in the physical world follow mathematical laws. That is why mathematics is applicable to science, the study of the physical world. The co-evolution is thus predetermined.
I would like to draw your attention to my essay: A physicalist interpretation of the relation between Physics and Mathematics, and my site: finitenesstheory.com.
Dear Professor Hestenes,
Many thanks for your detailed response. We will study your Refs. [22], [29] as also your 2008 FQXi essay.
Tejinder
Dear Professor Hestenes,
Could you please explain to me why you thought that my comment about the real Universe was inappropriate?
You are I hope aware that suppression of the truth is unethical.
Eagerly awaiting your answer,
Joe Fisher
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