Sunday, November 11, 2018

Sunday Night Ramble…

"If my mental processes are determined wholly by the motion of atoms in my brain, I have no reason to believe that my beliefs are true... and hence I have no reason for supposing my brain to be composed of atoms."--- J.B.S. Haldane, "Possible Worlds" (1927)
The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.— Eugene V. Wigner
(longish ramble ahead....)
Last Friday was one of those oddly serendipitous days in some ways — and that’s despite the fact that I was stewing over missing Ben Orlin’s night-before presentation at our local University. ARRRRRRRGH!! — had planned for weeks to attend, but for a whole series of reasons didn’t make it. Luckily, someone had posted his hour-talk from a previous stop (which I assume was the same as here), so I went online later and viewed that.

A couple days prior, a mathematician/blogger had sent along something to read for any comments, and parts of it reminded me of a favorite quote I’ve used here previously from provocateur David Berlinski (for those who’ve seen me employ it multiple previous times I beg forgiveness, and indeed apologies for all the long quotes coming below):
"Like any other mathematician, Euclid took a good deal for granted that he never noticed.  In order to say anything at all, we must suppose the world stable enough so that some things stay the same, even as other things change. This idea of general stability is self-referential. In order to express what it says, one must assume what it means. Euclid expressed himself in Greek; I am writing in English. Neither Euclid's Greek nor my English says of itself that it is Greek or English. It is hardly helpful to be told that a book is written in English if one must also be told that written in English is written in English. Whatever the language, its identification is a part of the background. This particular background must necessarily remain in the back, any effort to move it forward leading to an infinite regress, assurances requiring assurances in turn. These examples suggest what is at work in any attempt to describe once and for all the beliefs 'on which all men base their proofs.' It suggests something about the ever-receding landscape of demonstration and so ratifies the fact that even the most impeccable of proofs is an artifact."-- D. Berlinski (from "The King of Infinite Space")
The interplay of language, meaning, abstraction, perception… and, mathematics/science is an ongoing interest of mine. Another quote I’ve used elsewhere is from venerable Bertrand Russell toward the end of a frustrating career (1957) trying to formalize all of mathematics:
"I have come to believe, though very reluctantly, that it [mathematics] consists of tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-legged animal is an animal."  

Again, I think this reflects on the complex interplay of language and human thought — hmmm, all of mathematics as tautologies?… maybe all of meaning is tautological, just substituting one set of human scratchings or sounds for another… is all of knowledge just one gigantic Thesaurus? ;)) No, I don’t believe that, but a lot of “knowledge” does seem illusory or mirage-like and certainly changing. Often meanings, metaphors, analogies and the like simply refer back on themselves within an enclosed bubble.

Later on Friday afternoon I stumbled across this new essay at the Scientific American site, “Proofs and Guarantees” (actually reprinted from “The Mathematical Intelligencer”), which seemed to hit some of the same notes, questioning assumptions. It ends accepting the “fallibility of the initial axioms or other first principles” of math, and thus acknowledging an evolutionary nature to mathematics, as something that is not necessarily fixed over time (just as science is not static and self-corrects over time):

In arguments between math Platonists and non-Platonists it’s often contended that even (philosophical) non-Platonists ARE Platonists-at-heart when it comes to their livelihood… i.e., a pure mathematician must presume math exists out there to be discovered, in order to carry on their daily work. That may overstate the case, but on-the-other-hand one of my favorite volumes is by non-Platonist, retired mathematician William Byers, “How Mathematicians Think.” The subheading to the book title is: “Using Ambiguity, Contradiction, and Paradox to Create Mathematics” and that succinctly sums up what he argues in the volume, that mathematics is “created” out of the very things that most people presume run counter to it.
Interestingly, a second major influence in Byers’ life (besides mathematics) is Zen Buddhism, known for its mystical focus on contradictions. Here is one passage where he touches upon it:
The second strand in my life was and is a strenuous practice of Zen Buddhism. Zen helped me confront aspects of my life that went beyond the logical and the mathematical. Zen has the reputation for being antilogical, but that is not my experience. My experience is that Zen is not confined to logic; it does not see logic as having the final word. Zen demonstrates that there is a way to work with situations of conflict, situations that are problematic from a normal, rational point of view. The rational, for Zen, is just another point of view. Paradox, in Zen, is used constructively as a way to direct the mind to subverbal levels out of which acts of creativity arise.
Later in the volume he writes:
…every human being lives in a bubble. This bubble contains all their perceptions and cognitions. What exists outside the bubble is not knowable. Radical constructivists 'do not make claims about what exists in itself, that is, without an observer or experiencer.' This is a point that I also made earlier when I claimed that there exists no mathematical knowledge that is completely objective. Mathematical knowledge and truth must be considered as a package with both objective and subjective aspects. The belief in ‘objective mathematical knowledge,’ that is, knowledge that is independent of the beings who know it, is itself a belief and therefore nonobjective. There is no knowledge that is independent of knowing. There is no absolute, objective truth.”
And finally, here’s Byers, at length, in another book, “The Blind Spot,” hitting the same theme:
"It is certainly conceivable that the clarity we perceive in the world is something we bring to the world, not something that is there independent of us. The clarity of the natural world is a metaphysical belief that we unconsciously impose on the situation. We consider it to be obvious that the natural world is something exterior of us and independent of our thoughts and sense impressions; we believe in a mind-independent reality. Paradoxically, we do not recognize that the belief in a mind-independent reality is itself mind-dependent. Logically, we cannot work our way free of the bubble we live in, which consists of all of our sense impression and thoughts. The pristine world of clarity, the natural world independent of the observer, is merely a hypothesis that cannot, in principle, ever be verified. To say that the natural world is ambiguous is to highlight this assumption. It is to emphasize that the feeling that there is a natural world 'out there' that is the same for all people at all times, is an assumption that is not self-evident. This is not to embrace a kind of solipsism and to deny the reality of the world. It is to emphasize that the natural world is intimately intertwined with the world of the mind. In consequence, the natural world is a flow just like the inner world. By stabilizing the inner world through language, logic, mathematics, and science, we simultaneously stabilize the outer world. The result of all this is the recognition that the clarity we assume to be a basic feature of the natural world merely masks a deeper ambiguity. One of the functions of mathematics and science is precisely to deny this ambiguity. This is really the motivation behind the science of certainty." 
Anyway, finally coming back around to Dr. Orlin’s talk, which is all about the relationship between mathematics and science, Ben concludes that they share a symbiotic relationship — two quite DIFFERENT activities feeding off one another (as opposed to the more common conception of math being foundational to science). I couldn’t help but think that perhaps that viewpoint might be broadened out to describe the relationship/interplay between language, thought, and math… entities that are separate but very much feed off one another (though many mathematical aspects of language may not even yet be understood/appreciated).
Give Ben's entertaining, thoughtful talk a watch if you’ve not seen it:

David Chalmers famously talks about consciousness as the “hard problem” of philosophy and psychology, left untouched by resolving the other “easy” or soft problems. How do subjectively-felt  experiences arise out of the conglomeration of matter that is our physical brain? Or, in Thomas Nagel’s famous take, what does it feel like to be a bat?
In recent years there has been a lot discussion and competing theories over “consciousness.” Certainly some of Doug Hofstadter’s past writings touch on these matters, as does Joselle sometimes over at her Mathrising blog… and many many more [including, if you haven't seen it, John Horgan's latest 'free' volume on consciousness, where he speaks to several major thinkers on the topic, HERE].
But then another favorite quote of mine (from Emerson Pugh) is, “If the human brain were so simple that we could understand it, we would be so simple that we couldn’t,” implying that  we will never be able to turn the brain on itself to reveal its own deepest secrets. That’s a sort of “Mysterian” viewpoint (and I’m in the mysterian camp with Colin McGinn, Martin Gardner, Roger Penrose, and others, but plenty of folks oppose it, believing the brain can be fully understood, even duplicated).
Anyway, similarly, I think mathematics has a ‘hard’ problem (philosophically-speaking). It is the one made famous by physicist Eugene Wigner (quoted above). How does one account for the exquisite fit of abstract mathematics with the physical world as we interpret it? How indeed! Max Tegmark’s tempting answer is that fundamentally, mathematics is all there is… mathematics IS the core foundational structure/component of the Universe, or of reality; an intriguing notion, but difficult to flesh out, and not one I see a lot of others gravitating toward. Even if you're a full-out Platonist (like Gödel) and believe mathematics exists in the world, independent of humans, the question remains where did it come from and how are we humans able to access it so successfully? Or has some alien civilization, a million years more advanced than us, recognized mathematics as a truly tautological illusion, and moved on to something else more fundamental by now?
Finally, there's been a lot of emphasis in recent years on "beauty" in mathematics, but now even that view is being drawn into question, with a lot of buzz in particular around Sabine Hossenfelder's recent volume, "Lost In Math" (on modern physics), where she argues the myopic focus on beauty simply leads us astray. Is nothing sacred anymore!... first taking away certainty and truth, and now even our wistful love of beauty. ;)
More and more, I find pieces I'm reading connect somehow back to these interests in language, cognition, consciousness, recursion... maybe some day our grandchildren's grandchildren... or, Ben Orlin... will actually make sense of it all! 

Well, I've let the atoms in my brain (assuming they exist) bounce around a bit too much tonight... time to put them to bed, and perhaps go flying -- something I do quite splendidly in my dreams, but, frustratingly, can't seem to do upon awakening (...though, as Chuang Tzu might ponder, perhaps my flying is real and it's this blog-writing that is just a dream....)

==> Enough of my stream-of-consciousness, IF you want to hear from a real mathematician, I listened to Numberphile's maiden podcast earlier today with Grant Sanderson (of 3Blue1Brown), and it's quite good:

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