L'affaire Socks: Sometime ago, I thought I would cheat the universe that plays tricks with my always missing second sock, by purchasing all socks of the same kind - this way all I had to do was find any two socks and I was good. And to my utter consternation this past weekend, I could not find ANY socks. I then had no choice but to propitiate the sock gods by paying a visit to the temple (Target).
It is always puzzling to me how to be modern, yet timelessly grounded and relevant. As a child of modern science, I cannot help but thing that science offers such a facility. However, one might run the risk of slipping into mechanistic experimentation and measurement if science were the only focus. But, 'the philosophy of science' offers a very tempting sobriquet that allows one to experience the wonder of our lives and this world, and yet be tethered to all the advantages that science brings and is yet to bring us.
The two questions I try and think through are:
i) Are equations an accurate representation of reality? Or is a heuristic (i.e. a rules based, intuitive leap) better?
ii) Is the field of quantum physics, the best effort of modern scientific man at apprehending reality?
John Wheeler, the Princeton Physicist and mentor to three Nobel Laureates (including Richard Feynman) once said, "IT is BIT" - meaning that the ultimate reality we are trying to describe is actually information (bit/byte). A wondrous saying, indeed!
Part I: Philosophy of Science - Equations vs Heuristics
The write-up in this link below started my rumination on this whole topic, and by the way, sent me into paroxysms of delight like nothing has in a while - I dreamily wished I were a philosopher of science - what better pointy-headeness to aspire to in this modern age? I suppose it doesn't pay?
Hail Mary Pass - I remembered my advance math classes in engineering in college and later in graduate school in Philadelphia. We would be solving second and third order differential equations - which was particularly relevant in control theory. The difference between undergraduate and post-graduate studies was this - in the undergraduate studies, we had a manual with tables. So if we had to find the right size bolt, and the right type of material for a particular application (fastening a robot to the concrete shop-floor would be an example), we would open our reference book, and run down the criteria, and pick the right solution. What we were taught was application, not how the answer was derived. The answers were always in ranges. It did not quite occur to me why the answers were so.
In graduate school, much to my consternation, the curriculum completely abandoned application. Gone were the reference books. Instead, we had to work out by solving fairly complex equations the answer that was in the manual. That was when I had my first insight into the nature of the problem we were trying to solve. Linear and first order equations were easy to solve. The output was proportional to input, varied by slope.
But the non-linear equations became tricky right of the bat. Output was not proportional to input. The simplest example of this is an asymptotic curve. And then when you got to second order differential equations (such as when laminar liquid flow transitioned to turbulence), it was not easy, and most often impossible to accurately predict the outcome. Meaning it was never easy to say exactly when a laminar flow would switch to turbulence (in fact, it is not possible). However, you could introduce a 'constant' that would help predict with accuracy the range within which laminar flow would become turbulent. And it was that 'constant' that found its way into the reference book that we used in the undergraduate school.
Whenever you hear the word 'constant' in science, you can be safe in assuming it is a bit of a hail mary pass. Meaning, that it was plugged in there into the equation to best predict the outcome. What the constant is not generally easy to explain. In the case of laminar flow changing to turbulent flow, a similar constant is used based on the type of liquid. In astronomy we have read of the 'cosmological constant'
Instead, I decided to bail on this course and head towards Wall Street/Consulting and try my hand at making non-linear compensation.
Chaos - In the early 90s, Chaos Theory broke loose into popular consciousness, and given my old studies, I found this very fascinating. In chaos theory, you vault past building and solving non-linear equations - and look at describing phenomena through fractals. I was so mesmerized by this that I spent hours programming and visualizing chaos phenomena.
One example. I programmed a triangle and labeled each corner a,b - c,d and e,f. And then I rolled a virtual dice inside the triangle and for each roll of the dice I placed a dot inside the triangle. Every roll of the dice would be another dot half the distance to one of the corners. If the dice rolled 1 or 2 it would go to one corner. 3 or 4 to another. 5 or 6 to the third. Since it was on the computer, I let it roll millions of times overnight. In the morning, I was astonished to see the result. It was millions and millions of little triangles. Here was the kicker. No matter how much I zoomed in, it was millions and millions of triangles AT THE SAME FIDELITY! In other words I had created a fractal world of triangles. In philosophical terms, it was turtles all the way down . . .
A solid foundation - This brings us to the debate between algebraic equations or heuristics as best explaining natural phenomena. Equations have a strong logical foundation, and can credibly build the story. But often seem to fall short of fully explaining the phenomena, or worse, from a science and engineering perspective, fail to predict the outcome.
Heuristics on the other hand get you to the outcome. However, they may not be universal. In other words a given heuristic applies to a given phenomenon, and accepts a range within which the prediction can occur.
Perhaps, equations are the foundation from which one can make a intuitive/heuristic leap to observe and predict natural phenomena. And with the Universe a potentially infinitely complex place, we are always going to be in the business of using both.
An initial conclusion - Science and Engineering typically content themselves with predicting observed phenomena and try to eschew questions around meaning. The why in science is more around why is that happening? Philosophy it seems to me is concerned with meaning. What is the meaning of that particular phenomenon. Ages ago, science and philosophy where all rolled into one. And since enlightenment, the two split paths. I think scientists at the peak of their work find philosophical questions irksome. Philosophers potentially look at scientists as robotic if they don't ask the 'meaningful' question.
Many physicists like Schroedinger, Oppenheimer, Heisenberg, Einstein, Feynman, Wheeler, Millikan, Rutherford all speculated on broader questions much later in the careers. Schroedinger even published a celebrated book titled 'What is Life?' based on a series of lectures he gave.
People eventually seem to get around to what philosophers have been asking all along - what is the meaning of it all.
However, let me say this from purely my vantage - in the last several hundred years, there has been definitive and tangible advances in our understanding in the sciences. But the question raised in philosophy still has us running in circles (chasing our tails).
Part II: Philosophy of Science / Re: Quantum Physics
To get a sense for either how small we really are - or to get a sense for how big we really are - it is worth watching this YouTube video titled the Powers of Ten: http://www.youtube.com/watch?v=0fKBhvDjuy0
Turtles all the way down - We are mostly reductionists at heart. Meaning, that if we are confronted with a problem, we try to break it down and get to the root of it. And in Physics, in particular, scientists over the ages and in the last century have gone all out to dig deep and get to the root of it all. The Large Hadron Collider in Switzerland is a great example of the level of investment and the extent to which we are willing to go to do so.
But at least to my popular science reading, it feels like it is 'turtles all the way down.' What I am referring to is the opening joke in the very funny book 'Plato and Platypus walk into a bar.' It goes like this:
Dimitri: If Atlas holds up the world, what holds up Atlas?
Tasso: Atlas stands on the back of a turtle.
Dimitri: But what does the turtle stand on?
Tasso: Another turtle.
Dimitri: And what does that turtle stand on?
Tasso: My dear Dimitri, it’s turtles all the way down!
So one is tempted with the realization that 'it is turtles all the way down,' and just give up and wonder, boy, what is the meaning of it all.
Richard Feynman, who burst into popular consciousness when he put the obfuscating NASA bureaucrats in place in the post-Challenger investigation, said the following of meaning seekers: “Philosophy of science is about as useful to scientists as ornithology is to birds.”
To which a wag replied, “it is likely that ornithological knowledge would be of great benefit to birds, were it possible for them to possess it.”
A metaphor - Here is a quote shared by a friend that I like: “Classical electrodynamics, special relativity, general relativity - have simple structures so that once one grasps them everything else becomes mechanical and assured. They are great trees whose overall structure can be seen from the distance and the climb planned ahead, branch by branch. Quantum field theory, however, is like a jungle. At any place there is luxuriant foliage whose admiration can absorb a career. But there seems no place to stand from which one sees anything other than a tangle upon a tangle upon a tangle.”
The meat of the debate in the link revolves around: can we only hope to get a true glimpse of physical reality through the ‘language’ of mathematics? However in the field of Quantum Physics, things have turned out interestingly and differently.
Is the cat alive or dead? One of the great strikers of imagination in popular science is Werner Heisenberg. He is the one known for the 'uncertainty principle.' I suspect only trained physicists truly understand what he said. However, in popular science it is sufficient to understand that when it come to actually describing an electron, it behaves as BOTH as a particle and a wave.
In any case, Heisenberg relates this very interesting story about Neils Bohr, who is considered the father of quantum mechanics.
“At the end of the lecture, Bohr came over and asked me to join him that afternoon for a walk over the Hainberg Mountain. This walk was to have profound repercussions on my scientific career. Perhaps it is more correct to say that my real scientific career only started that afternoon when Bohr told me … atoms were not things! We talked for about three hours. And for the first time I saw that one of the founders of quantum theory was deeply worried by its difficulties. Bohr had immense insight, a result not of mathematical analysis but of observation of the actual phenomena. He could sense a relationship intuitively [heuristically] rather than derive it formally [mathematically].”
But Heisenberg did not want to treat the atom like a little solar system, but rather like a virtual oscillator which could produce all the frequencies of the spectrum. He created a system called matrix mechanics to show how the atoms worked. The new approach came with no visual aids - this was a purely mathematical formalism, difficult to use and impossible to visualize. It simply gave the right answers.
On the other hand, another great Physicist in this field Erwin Schroedinger used a different approach. He is famous for the example of the 'Schroedinger's cat' - where funnily and intriguingly he talks about the state of a cat being dead or alive in a closed box not being known till we actually open the box.
He developed an equation known as Schroedinger's equation that was a wave that described in some magical way the quantum aspects of the system. A wave description in Physics is a very classical approach that has been visualized and understood for decades if not centuries.
Shall the twain meet ? Schroedinger wondered if there was any relationship between his own theory and Heisenberg’s matrix mechanics. In 1926, he found a remarkable result of his own analysis. To his surprise, he showed that the two theories were completely equivalent from a mathematical point of view.
One was based on a clear conceptual wave model of atomic structure and the other claimed that such a model was meaningless. Yet both gave the same result.
On a related note, my father who is a Physicist says, that Feynman diagrams are heuristic(al) rather than driven by equations – which also significantly advanced knowledge in this field.
Summary - what this entire discussion adds up to it seems to me, is the following. We use different tools to apprehend the world we live in. Mathematics is one way. Rules based heuristics is another way. But honestly, we can use all the help we can get! As the Powers of Ten video clip from YouTube shows, we are suspended between the worlds of unimaginably big and unimaginably small.
Somehow we are endowed with a interconnected set of neurons in our brains that are able to manipulate symbols and make sense of this world. We can ascribe meanings and patterns to everything we see. But perhaps it is more critical that we see beyond the patterns with every tool in our toolkit, and make sense of this infinitely complex, and infinitely wondrous universe we occupy.
8 comments:
Terrific stuff. Very interesting perspective.
1. You say, “Equations have a strong logical foundation, and can credibly build the story. But often seem to fall short of fully explaining the phenomena, or worse, from a science and engineering perspective, fail to predict the outcome.” I agree that something like a ‘mere’ table of equations, such as was in your undergrad ‘manual with tables,’ or, in this context, quantum mechanics which is also a matrix or table of equations, by their nature will not fully explain the underlying phenomena. However, I’d question your next statement, “Heuristics on the other hand get you to the outcome. “ Because, in the case of quantum mechanics at least, these mathematical formalisms are incredibly accurate predictors of outcomes. And every attempt at heuristic explanations, they’ve failed.
But I also agree with the author of the subject piece who said both formalisms and heuristics can be used to make progress. And I guess you’d agree too as you write, “Perhaps, equations are the foundation from which one can make an intuitive/heuristic leap to observe and predict natural phenomena. And with the Universe a potentially infinitely complex place, we are always going to be in the business of using both.”
After all, Einstein used the heuristic of a thought experiment in which he rode on a photon to imagine what that perspective would look like and what that perspective must therefore mean about the nature of space and time and gravity.
2. Also, you write, “Science and Engineering typically content themselves with predicting observed phenomena and try to eschew questions around meaning. The why in science is more around why is that happening? Philosophy it seems to me is concerned with meaning. What is the meaning of that particular phenomenon. Ages ago, science and philosophy where all rolled into one. And since enlightenment, the two split paths. I think scientists at the peak of their work find philosophical questions irksome. Philosophers potentially look at scientists as robotic if they don't ask the 'meaningful' question.”
Actually, the sort of philosophy we’re talking about here also eschews meaning. Foundationalism is just as abstract as physics at the foundational level of ontology and epistemology (the latter you might say being about trying to get past our human desire to add meaning). Now when you get to the philosophical disciplines of Aesthetics and Ethics, meaning is unavoidable. But these subjects are to foundationalist what engineering is to the theoretical physicists: as per the sitcom The Big Bang Theory the Sheldons of foundationalism look down there nose at the Wolowitzs of ethics and aesthetics. I mean, who, they sniff, still believes there’s any meaning to be in all of this?
And even in aesthetics and ethics, it’s generally acknowledged that any ‘meaning’ is historical, provisional and socially constructed. That is, they’re pretty much with the physicists on this. But that has meant that philosophy has lost some of its usefulness for the average person who wants to look for meaning. But both physicists and philosophers are likely to tell you that there is no meaning beyond that which you create for yourself. That’s what the end of foundationalism means to the layman. And the folks don’t like it!
3. But I and Lawrence Cahoone agree with your statement “the question raised in philosophy still has us running in circles.” As Cahoone puts it below, “Philosophy moves forward in a spiral, rendering some theories unsupportable, pressing forward with others, and recycling parts of older theories in new projects.”
Oh, and loved the about your heading “towards Wall Street/Consulting and try my had [sic] at making non-linear compensation.”
One might even extend the metaphor by saying that you’re seeking to make non-linear compensation through opportunities to arbitrage non-linear market swings!
;-)
1. how did you happen to come across this link?
2. the guy on the program committee - bob batterman - is one of my oldest friends, known him since we were teenagers, went to college and graduate school together. his own work is very interesting but not something most people even in professional philosophy would have any idea about.
Pretty funny that you should send this out. this is a very small club.
Actually, the applicability of mathematical models to this type of (non-euclidean, non-deterministic, non-bivalent) phenomena is something that long interested me and still does. one of the few parts of modern philosophy that still interests me. why middle-sized animals that evolved in a euclidean (for all intents and purposes), seemingly deterministic world of facts should have the ability to create and apply models that are none of the above to non-observable phenomena and get the predictions remarkable accurate is surely a mystery that is hard to fathom.
OK Got it. Socks at Target.
The philosophers seem to have answered these questions for themselves in every age, even from the dawn of civilization.
But for each one of us to assimilate into our own lives... well a lifetime. (no pun intended)
Well, as the wise men say, every man is doomed to follow his line of thought.
Those who are spared, do not argue.
The rest, like me, would sit up at 3 am to debate a point.
The video is simply mind boggling. one of the best videos I had seen. We are able to visualize the extremes in matter of minutes.
While Quantum Field Theory is indeed a tangle, it is the most accurately verified theory ever.
For the finest discussion of waves and particles, read the first chapter of Volume 3 of the famous "Feynman Lectures on Physics". By the way, you don't need to know any physics to read that chapter.
Feynman diagrams are a book-keeping trick that Feynman invented. Much simpler to account for the various terms in a lengthy expansion of fundamental quantities.
I think that this fits into the Quantum Physics we are discussing. Talking about non linear equations and constants, the situation becomes still worse when what we are differentiating itself is not known as in the case of 'probability distribution function'. At that point, it will be a hailmary pass of an invisible football!
I'd suggest that one of the tools for understanding the universe is meditation. Direct experience of stillness, expansiveness and connectedness can defy language - but provide deep insights.
Love the Feynman story... going to post the quote to Facebook. :-)
‘Mathematics is one way. Rules based heuristics is another way’: This profound question posed by you is perhaps is the most difficult and unresolved question.
At the beginning of the twentieth century when the atom started revealing itself, the Physists were stumped. They did not know what to do! The mathematics they knew neatly fit into all the physical phenomena they knew till that time.
The first monkey wrench was thrown by two persons. Max Plank and Rutherford. The first by trying to explain Blackbody radiation through coupled oscillators and the later explaining the atomic structure by the Gamma ray scattering. The first was a mathematical approach and the second can be called a heuristic approach and both were supported by experiments.
When Bohr postulated his four principles, to explain atomic structure, hell broke loose. Because his ideas were so radical and opposed to Max Plank’s fundamental rules of radiation, the only way scientists would accept it was if there is a ‘Scientific and Physical ‘proof’.
In its simplest form, it is to reduce the unmanifest into one of the five senses! If we say, it is hot, we have to see a thermometer or feel it by our body. So, Rydberg had to show that by applying Bohr’s ideas, he could explain the absorption lines in the solar spectrum! That was the only proof Bohr had.
But even Bohr had to use the classical solar system model to explain the atomic structure.
Schrodinger equation started with a simple classical oscillator equation, he substituted the Ψ function for the displacement. Several people used this to explain many experimental facts. But there is always the nagging question ‘what exactly are we doing ?’. For example, if one asks the question what exactly happens when we differentiate a Ψ function twice, we cannot give an answer.
We know that we can explain how the band structure in solids can be explained if we solve that equation. Even here we have to make several statements along the way which are gibberish and un understandable! For example how can one ‘explain’ the entire band structure, (which is not a physical reality) purely based on a mathematical statement that a cosine function cannot be more than ± 1?
Again the answer is that we have to accept the ‘black box’ to give an explanation to the experimental observation.
The real truth about Quantum Physics is that no one knows exactly what is happening.
Concluding on the original question with which we have started, neither mathematical formulation nor heuristic exploration of the submicroscopic world is complete. Each has its own limitation. However,
in my opinion, mathematics has more limitations than the intuitive mind.
But sometimes it can surprise the mind. It is like the hare and tortoise story. The hare is like the mind and the mathematics is like the tortoise. But the hare is so confident that it goes to sleep once in a while and the tortoise over takes it. Though both have the same goal and race to reach it, both do not know what really the goal looks like or even what it is!
Post a Comment