Continuation of a Conversation with a Mathematician on the Seeming Applicability of Mathematics and Reason to Physical Reality

This is a continuation of a Twitter discussion I am having with wrf3 on the applicability of mathematics and reason to physical reality. We have moved it here for a more convenient forum for lengthy responses. Wrf3 is a mathematician and software engineer.

I’ll briefly restate my position on this question and then respond to certain specific points and questions from wrf3’s last reply.

I recognize two distinct innate modes of human thought: rational objectification of events in the world; and esthetic experience of Being.

Borrowing from Kant’s epistemology, I understand objective rational thought to be a subjective ordering of chaotic and manifold sense data by means of innate reason working through the capacities of understanding and drawn in the imaginary sensibilities of space and time. The resulting representations do not exist as such in the external world, but are also not purely arbitrary constructions. Unique aspects of events in physical reality condition the stimulus on our senses in unique ways, and these unique stimuli provide unique content for our forms of representation. This creates a degree of correspondence between representation and thing-in-itself, without which we would have no success in manipulating our environment, and evolved reason would have had no adaptative benefit as we would not have been able to distinguish predators from prey.

This objectification is a sort of vastly simplified sketch of an imagined representation of a manifold reality existing outside time and space and not directly knowable. This simplified sketch reduces our representation to what we predict to be crucial for our immediate attention, and from it we plan our actions. Over time we expanded this faculty to contemplation of the cosmos and our place in it. Philosophy and science are emergent extensions of this basic capacity.

This capacity has succeeded within certain limitations. We can only imagine (or conceive of) the world as in space and time obeying causal law. That in no way implies the world actually is that way, but only that as conditions of our understanding we have no choice but to reduce the world to those elements. Beginning in the late 19th century and greatly expanding since is a perceived conflict between deeper aspects of reality and our limited understanding. This understanding evolved within the very narrow band of reality we call Newtonian or classical physics, and it works reasonably well within its home boundary. When we roam too far afield, however, the world becomes stubbornly counterintuitive as in the case of relativity and quantum physics. We have approached the depths where our limited representational faculty is confounded by events that defy space, time and causality. To describe these events, we can only resort to purely abstract models rather than sense data, and almost unnoticed, physics reverts back to metaphysics.

This brings us to Poincare and Wigner. Poincare demonstrated the hermetic nature of rational constructs through the examples of Euclidean, Lobachevskian, Riemannian and his own fourth type of geometry, where each is perfectly internally consistent but each with its own assumptions contrary to the other three. In physics the analog is the contradictions among Newtonian physics, Relativity and quantum theory. In his Empirical Law of Epistemology, Wigner demonstrates the applicability of mathematics to physical reality as an epistemological issue rather than an ontological question, thus demystifying any apparent applicability. Building from Poincare, he shows natural laws to be mathematical descriptions valid within limited boundaries of space, time and chosen events to include in the description. Other chosen events, or space/time boundaries will yield different laws. He describes this as the inherent epistemological limit of the invariability principles.

Wigner shows the primordial connection between mathematics and physical reality as a practical application of reason-derived mathematics. Practical mathematics evolved from a purely esthetic pursuit about 5000 years ago when the world “suggested” events to our understanding which we reduced to our idea of number. We abstract from multiple events, focusing on likeness and de-emphasizing difference in order to count things, such as grain shipments. The further we abstract from events, however, the more complex the mathematics leading to a hermetic rational construct. As we further abstract and expand the scope of applicability, we reach a point where the bands of invariability principles snap, and another construct is necessary. The application of purely subjective reason to the universe is thereby demystified as approximate and provisional epistemological constructs. Wigner concludes that we have been fortunate up to now that the correspondence between mathematics and the universe has been beneficial but we have no right to expect that to continue forever.

Our older primordial mode of understanding is esthetic. It was our primary mode of knowledge in the West until the time of Socrates. Pythagoras saw mathematics as a purely esthetic pursuit connected to music and the rite of Dionysus. For him, mathematics gave us understanding of music as vibration, and vibration as the fundamental cosmic element. An idea that eventually reappears with quantum field theory.

This pre-Socratic world was essentially pre-metaphysical. Reality was understood as unmediated sensed experience (Heidegger’s “Ereignis”) of the cosmos. It was experienced musically or understood as Logos; i.e. the fullness of poetic language as originating language from the experience of Being still dripping with music. It was our sympathetic voicings of the cosmic vibrations.

This mode of knowledge is our intuition of the profound mystery and power of Being, of which we are a part. Most importantly, it is not a metaphysical construction, but dwells purely within the sensed experience. This is in contrast to Socratic reason, where Logos attenuates to logic – thought and language with the music wrung out of it. A dry rational abstraction of the world. Heidegger describes this as the beginning of metaphysics, where the Logos of A is A becomes the logic of A=A. The uniqueness of A as its own being is denied for the sake of practical representation, as we saw above with Wigner’s origin of practical mathematics. The “is”, which once had expressed the fullness of Being itself and was the most urgent element becomes reduced to a mere copula.

Each of the two modes of thought has its own purpose and is beneficial within its limits. I retain Heidegger’s distinction between them as “Truth” arising from esthetic experience, and “Correctness” inhering in objectification. Unlike some, I do not diminish the importance of scientific objective discovery when seen within its proper boundary. It is, however, a superficial correctness revealing measurement and imputed causality. It gives us critically important information from correspondence without which we might not have survived at all, and certainly would not have achieved our current level of existence. I am equally aware of the threat inherent in the common contemporary assumption that it fully describes Being. It can never tell us what something is, but merely superficial dimensionality. Again, Being is reduced to copula.

The two major thrusts of Western thought since the Enlightenment are the overcoming of metaphysics and the primacy of esthetic knowledge over reason; or restoring the “is”. The former began with Francis Bacon, who set the trend of methodically moving areas of knowledge from the realm of metaphysics to objective physical science. The Enlightenment solidified the claim that only knowledge through the senses can ground valid judgments. With knowledge now grounded in the senses, the Romantic period began the process of re-establishing esthetics as the mode of profound truth of the universe as experience. It could be said that there is more profound truth in a great work of Beethoven than in all of Einstein’s writing. Both of these trends come together through the two great 20th Century thinkers, Heidegger, and the later Wittgenstein.

Discussion Points from wrf3

1. 1/ How do you know it (Reason) is literally nothing like that (sensation)? Do you not experience your thoughts? Rain is the motion of atoms in ways that you can feel on your skin. Reason is the motion of atoms in other, more constrained, ways. If you don’t experience that motion, you can’t reason.

1/ It shouldn’t be hard. First, you make a distinction between internal/external for thoughts vs. experiences. “Inside your head” vs. “out in the world”. But it’s the same nature in both places, the same atoms, same principles.

The above are responses to my claim that reason is essentially different from reality as an a priori condition of objective thought projected onto the world, and therefore no direct ontological applicability exists. We do seem to start with a surprising basic assumption that consciousness itself is a feature of the physical universe and not some metaphysical ideal existence. Thankfully, we have no need to go through another tedious debate about duality.

To the first response, I will repeat my original answer to that question: while we have dedicated receptors and neural paths for each sensation, no such thing exists for reason. I cannot experience reason the way I do light. It is a pure idea with which I can construct an understanding of the world from the light that impinged upon my retinas. The only way I can perceive reason is to transfer it to the a priori imaginary sensibilities of space and time, where I can create mathematics or logical forms, but this is entirely without external sense data. Without converting to the imaginings of space and time, I have no intuition of reason at all.

In your second response you seem to be making the same claim for reason that I make for esthetic experience: the removal of subject/object metaphysics for unmediated connection between consciousness and reality. Consciousness can be measured physically as various types of brain waves. Coincidentally, quantum theory describes all of existence as wave. In addition, both quantum events and the consciousness seem to exist beyond mechanistic causation. The new subject of quantum mind is attracting top physicists and neuroscientists and perhaps offers the path to understanding. I can see a possible physical connection between quantum wave fields and brain waves that eliminate any separation of subject and object. That is, in fact, what I claim for poetry and music as unmediated experience of truth.

You would make that same claim for reason and mathematics to justify the applicability of mathematics to the physical universe. Your model, however, centers on atoms, not waves, and leaves the exact principles unspecified. Where I have sympathetic vibration, you have an assertion that a certain atomic arrangement creates reason, and since atomic arrangements are part of nature, that implies an exact connection and description of truth. That would imply, however, that all ideas would be equally true since they are all atomic arrangements. It would also leave unexplained the limitations of Wigner’s invariability principles, which seem to demonstrate the inability of reason to grasp anything larger than a very limited set of events within limited space and time.

At this point I won’t go into your other responses, but would rather save them for later as the above is the crucial difference between our views. Instead, I would ask you to demonstrate why reason is an atomic arrangement, and why it being a part of nature would imply truth; and along with that how you would explain erroneous ideas and the limits of the invariability principles. I would like to hear an analogous process to my notion of immediate sensation of vibration.

8 thoughts on “Continuation of a Conversation with a Mathematician on the Seeming Applicability of Mathematics and Reason to Physical Reality

  1. Jeff, I’ve put an introduction to my reply on my website here. It does not directly answer your questions, but does provide some background and motivation for what I’m going to say. Hopefully of interest is a link to a text file containing our Twitter conversation to date. It isn’t pretty, but it is searchable and I find that to be an immense help.

    I note that you’re asking me to provide the answer to Hofstadter’s question in Gödel, Escher, Bach: “What is a self, and how can a self come out of the stuff that is as selfless as a stone or a puddle?” Hofstadter wrote hundreds of pages to say “I don’t know.” Trying to answer that convincingly in a blog post is going to be difficult — but not impossible (I hope!). The saving grace is that I’m not writing anything that isn’t already established. Only the arrangement of the pieces into a coherent narrative is “new”. I will note that my response will cause vehement disagreement with the Thomists’, but I disagree with them as much, if not more, than I disagree with you.

    Give me a few days to write up and semi-polish my answers to your questions. I’ll post again here when I’m ready. Note that I’m working on this material for a book, so I might as well get something out into the wild now.

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  2. Part IIa, “The Road To Logic”, is up. “The Road To Truth” is being written… All of this stuff is easy in that the ideas are simple. But the presentation is hard, because it has to be built from the ground up. Literally. All I have is the “dust of the ground” and making complicated objects from dust requires pictures. And I have to check that my exposition is correct, which means I want to use machine checking. Which means code… But enough of my troubles. Back to writing.

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  3. Part IIb, “The Road to Logic” is up. I’m sorry it took me longer than planned. Even worse, I’ve decided to add another part to my response: “Reason and Logic” which follows “The Road To Logic.” It’s absolutely critical that I explain things such that the criticism that I’m sneaking something in a side door won’t survive examination, and that takes multiple revisions for clarity. At least, for me. The words on paper never cooperate with the ideas in my head.

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  4. I’ve posted part III and an “intermission”. The intermission contains a Table of Contents. I think I’ve answered the bulk of your questions; at least enough, IMO, to give you an opportunity to respond. At the very least, I’m interested in seeing a response so I can, if necessary, clarify or expand things.

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