Category Archives: Philosophy

Life, the Universe, and Everything

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(Click here for the PDF version of this presentation.)

Math is everywhere, hidden in places where we don’t even expect to see it. For example, take a look at the following image:

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What do you see?

Most people say “music.” People who have studied the piano might recognize this as a piano score. And a true enthusiast might recognize it as the third movement of Beethoven’s Moonlight Sonata.

What you’ve probably never thought of before, though, is that a musical score is actually a form of graph. It tells the performer what combination of notes to play at a given moment in time. In other words, it shows sound as a function of time.

In the image below, I’ve added labeled axes to draw attention to this:moonlight_sonata_graph

Now consider a photograph. Below is one of the most spectacular images I found when Googling “photograph.” (Thanks to whoever posted it!) I love how it shows the strings of mucus frozen in time.

tiger_photo

Anyway, a photograph itself is also just a type of graph — and not just metaphorically. In fact, even the way images are produced in our brains is just a way of numerically graphing the intensity and frequency of light that falls on different portions of our retinas. In essence, your retina is the x-y plane and the light is the quantity being graphed.

Below is what the photograph looks like when graphed in three dimensions from different angles, with the colors changed to a different color scale:

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Now here is the same graph when viewed from directly above, so that the tiger is easier to make out:
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Here’s another example of a great photo:
frog_photo

And here it is with the same procedure applied to it. This one works a little better than the tiger because it isn’t filled with little white spots that end up looking like noisy spikes in the graph.frog_photo_graphs_1

Below is the graph when viewed from directly above, just as I did for the tiger. Pretty cool, huh?frog_photo_graphs_2

Now consider something that really seems to have nothing to do with math: a piece of literature. Below is the first paragraph from A Tale of Two Cities, by Charles Dickens.
Tale_of_2_cities

It, too, can be considered as a type of graph. It’s a graph that tells the reader what words to speak or think as a function of time:Tale_of_2_cities_graphThere are, of course, many other examples of graphs:

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What I’m saying is that anything can be thought of as a kind of graph. Really, though, it’s not just graphs that are so powerful, but numbers themselves. This is because numbers encode information. For example, an entire song can be encoded in a single number. So can a photograph, or even a movie.

What’s particularly fascinating is that physicists now believe that physical reality itself is composed of information. In fact, the universe might even be digital. And since numbers encode information, it is possible that the entire universe could be represented by a single number.

Take a minute to meditate on that.

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If that’s true, then there’s only one thing we can conclude…

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This post is based on a PowerPoint presentation I made for my math students in an attempt to inspire them. Here it is in PDF form:

Math Is Everything (PDF version)

Trespassing on Einstein’s Lawn

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Trespassing on Einstein’s Lawn, by Amanda Gefter, is a memoir of a girl on two simultaneous quests. One is to find the answer to the ultimate question about life, the universe, and everything—namely, why is there something rather than nothing? The other is to publish a book about it and become a legitimate, big-time author. She appears to have succeeded to a significant degree on both counts and does a good job telling the story about how she did it.

Gefter’s treatment of the physics is an excellent demonstration of why popular science writing shouldn’t be left to scientists alone. Whereas scientists tend to be wrapped up in their own particular theories (e.g., string theory), a good journalist is better positioned to take a step back and make an unbiased assessment of what all the different theories out there are saying (even if the lack of bias is partly due to a lack of technical understanding). Gefter has attempted to do this, both by studying cosmology extensively on her own and by interviewing the big players in the field.

By taking this approach, she has done a more convincing job than anyone else has yet done—as far as I know—of presenting a satisfying explanation of how everything that we experience (i.e., life, the universe, and everything) can truly come from nothing (and actually be nothing), even though we perceive it to be something. Using qualitative conceptual arguments, she presents a compelling case for how the universe arises from nothing, without requiring any external laws of physics, such as quantum mechanics or general relativity, to exist a priori to govern the behavior of the nothingness.

Her central thesis, as I interpret it, is that something can’t be fundamentally “real” unless it is invariant—that is, unless it exists in all reference frames. In other words, if a reference frame can be found in which a thing doesn’t exist, then that thing is not “real.” She begins with a list of candidate components of reality, such as space-time, particles, fields, and forces; and in the course of her interviews with the most respected physicists of our time, crosses each item off the list.

In the end, with everything crossed off of her list, she concludes that the universe is ultimately made of nothing—which is the only philosophically satisfying conclusion anyway. It is very important to note here that what Gefter means by “nothing” is not space, but actual nothingness—devoid of all properties whatsoever, including any set of governing mathematical rules. With the help of a certain physicist, she even goes so far as to suggest how the mere imposition of a boundary—which is itself nothing—creates information from nothing. This emergence of information evidently initiates a cascade from which everything emerges.

It is reminiscent, Gefter notes, of how, in the field of mathematics, the entire set of real numbers can be constructed from the empty set alone—i.e., from nothing. She also mentions other pleasing analogies with mathematics that can be drawn, not the least of which is the possibility that, in the spirit of Gödel’s incompleteness theorem, no set of physical laws that can be constructed within physical reality could ever give a complete, external description of physical reality.

Gefter doesn’t spend much time talking about Zen, but she does mention it, and one cannot help but think that she has given a rather compelling case for the truth of Zen from an actual physics perspective. Nothing is everything, and everything is nothing. And no system can observe itself, for it would then cease to exist upon making the observation.

My own analogy is that a blank canvas, which contains nothing, also contains everything—in more than one sense. While still blank, the canvas retains the potential to become any of the infinitely many possible paintings that could conceivably be painted on it. Moreover, if you actually do paint every last one of the infinitely many possible paintings on the canvas, it will become blank again (i.e., pure white or black, depending on the medium). Perhaps our universe is just one of the infinitely many possibilities that exist simultaneously on the blank canvas of nothingness.

Though Gefter’s case is conceptually compelling, it is neither academically rigorous nor airtight, and she acknowledges this. Should some component of the universe be shown to be truly invariant, her core thesis would go out the window. And there are still a few points about which I don’t feel satisfied. For example, why can observers (which don’t necessarily have to be conscious) exist within nothing? (For supposedly it is the observer who creates the boundary that gives rise to information.) Can we really talk about “observers” and “boundaries” within nothingness without invoking some sort of governing system of definitions and rules? Are we really talking about nothing then?

The other part of Gefter’s quest—to become a bona fide science writer—is interesting and inspiring in itself. It’s not a rags-to-riches story, but it’s an excellent example of how it can be possible to attain seemingly unattainable goals. That’s what the title is really about. In the beginning, she was neither a scientist nor a writer. And so she was not just literally trespassing on Einstein’s lawn when she visited his old house in Princeton; she was, in carrying out her quest, venturing into territory where she didn’t rightly belong. But in the end, she earned her spot on the lawn.

Gefter’s success comes from four components that I have been meditating on recently: passion, discipline, assertiveness, and luck. Throughout her quest, she maintained a very intense passion, largely instilled by her father, for the ultimate questions about physical reality. She also demonstrated the discipline to stick to that quest over a period of several years—attending conferences, writing articles one at a time, interviewing physicists, keeping a detailed journal, and ultimately sitting down to write out the book itself. Assertiveness played an important role when, in situations where most people would have thought there was no hope trying, she nevertheless called up high-profile scientists and publishers to try to get her foot in the door—and it worked. Finally, of course, she had plenty of luck. More than anything else, she was lucky to have a father who planted and cultivated the seed of her passion and then provided the financial, intellectual, and emotional support necessary for her to set off on her quest.

Throughout the book, Gefter draws many clever parallels between the mysterious physical phenomena that she is investigating and things that are going on in her personal life. For example, physicists’ conclusion that it is inherently impossible to construct a consistent description of the universe that takes into account more than one point of view at a time—i.e., you can only have one observer—was a nice parallel with her reluctant decision to write the book on her own after the publisher rejected her proposal for a book by a father-daughter duo.

I did grow a bit tired of Gefter’s self-deprecating refrain about how she was an impostor, a fake who didn’t deserve to be present at the conferences she attended or at the magazines where she worked. I also grew a bit tired of the “Oh my God, I’m in the presence of one of the greatest physicists who ever lived!” exclamation that seemed to accompany every single interview she did. And I didn’t appreciate the dig she took—which seemed rather mean-spirited to me—at a waiter who said that he had also majored in philosophy of science. She was making an attempt to draw another clever parallel, contrasting the success she hoped to achieve with the waiter’s apparent lack of success; but the net effect was to shatter the image of humility she seemed to be working so hard to create throughout the rest of the text.

Nevertheless, Gefter did a fantastic job documenting her quest, and the overall picture that she paints of the present status of cosmology is far more satisfying than any other I’ve read—precisely because she takes into account many different physicists’ points of view. She begins by saying:

Reality is a huge mystery, and you have a choice to make. You can run from it, you can placate yourself with fairy tales, you can just pretend everything’s normal, or you can stare that mystery in the eye and try to solve it. If you are one of the brave ones to choose the latter, welcome to science. Science is the quest to solve the eternal riddle.

Then, at the end, she is able to say, “Physics isn’t the machinery behind the workings of the world; physics is the machinery behind the illusion that there is a world.” Before reading the book, I would have dismissed this statement as pseudo-philosophical mumbo-jumbo based on feelings that surely have no grounding in an actual understanding of physics. But I am now convinced that it may be a deep and legitimate conclusion indeed—and I am very glad that I took the time to read her story.

(Thanks to my father for giving me the book.)

The Physics of Free Will

bluePillRedPillThe French mathematician and physicist Pierre-Simon Laplace famously boasted that if he knew the exact position and velocity of every particle in the universe at a given instant in time, he could predict with perfect precision the state of the universe at any time in the future (and, presumably, the past as well). Such was his faith in the universality, immutability, and sovereignty of the classical laws of motion, that he believed no particle, out to the very farthest reaches of space and back to the earliest moment in time, could escape the path predetermined for it by all of the interactions it was destined to have with the rest of the universe since creation. It is a nice-sounding boast, as it paints a picture of an unshakable, perfectly ordered world; and yet, it is terribly disturbing, as it utterly disallows any concept of free will whatsoever.

In Laplace’s world, the innermost workings of the human mind, down to our apparent ability to make decisions and move according to our will, are in fact governed by the laws of motion as they apply to the tiny particles that make up our brains and the physical processes that constitute thoughts — enormously complex to be sure, but entirely predictable with the right amount of knowledge. If true, Laplace’s boast would seem to be an end not just to free will, but to much of the meaning we find in life. After all, what significance is there in a work of art if the artist’s hands were merely being moved by the inevitable firings of neurons determined by the laws of physics since the beginning of time? Creativity would be only an illusion. There would be no spontaneity of thought or expression, no hope of controlling one’s own fate.

The performance of a symphony, with the musicians playing in harmony under the direction of a conductor and to the enjoyment of their audience, would in fact be nothing more than a fully predetermined combination of motions. The composition itself could no longer be truly ascribed to the composer, as he was predetermined from the beginning of time to write down precisely the notes of the piece being performed; and the husband who falls asleep in the audience could not be faulted for his inattentiveness, because that’s just how things had to play out. Even the guy whose cell phone rings during the adagio would be blameless.

But twentieth century physics has shown Laplace’s view to be wrong. We now know that pure chance plays a fundamental role in the outcome of any process. Quantum mechanics has shown irrefutably that particles actually don’t even have precise locations — it’s not just that we can’t measure them precisely enough to know where they are exactly, but rather there is no exact value to be measured. Nor do they have precise velocities. The highly celebrated but oft-misinterpreted uncertainty principle of Heisenberg describes not just limitations in our knowledge about the position and momentum of a particle, but the fuzziness of the particle’s actual being.

This is a huge leap in thinking that most physics students have trouble making, but it is at the heart of quantum mechanics. And not only are every particle’s actual position and momentum fuzzy in a fundamental way, but there is a very real element of pure chance involved in the particle’s behavior. An electron can disappear from one region of space and reappear in another region without crossing the space in between; and just where it reappears is a matter of chance that even the particle itself cannot know ahead of time. Einstein hated these revelations of quantum mechanics, but he recognized their truth. And so now we know on the basis of science alone — even if our bodies and souls are nothing more than extremely complex physical systems — that our fates are not entirely predetermined by the laws of physics. Certain outcomes are highly likely, to be sure, but never certain beyond all doubt (unless, of course, the “many worlds interpretation” of quantum mechanics is correct, in which case every possibility will, with certainty, come to pass in some universe).

So what does this new understanding of our world imply? Does it restore the free will and meaning that Laplace would have robbed from us? At first, it would seem that the answer must be no. The random events described by quantum mechanics cannot directly result in free will, for if they did, there must be a means for an agent of will to tell electrons (for example) where they should appear and interact with other particles (which is essentially the main physical process in our brains that is relevant to thought); but if this were the case, then the electrons would no longer be obeying the laws of chance that they have, in fact, been shown to obey. So it seems that random processes alone cannot account for free will.

However, our universe is governed not by chance alone, but by a most intriguing combination of deterministic rules and random processes — a continuum that fades from pure randomness at the infinitesimally small scale to pure determinism at the infinitely large scale. Could it be that the combination of these two components allows for the construction of something that amounts to more than the sum of its parts? In plane geometry, a straightedge limits its user to the construction of line segments, and a compass limits its user to the construction of arcs; but when the two tools are used in tandem, a whole new level of complexity becomes possible, allowing the geometer to draw impressive figures. Perhaps the classical, deterministic laws of motion and the more recently discovered quantum mechanical laws of chance are, respectively, the straightedge and compass that, when used together, allow for the construction of high-level phenomena such as consciousness and free will, which would otherwise be inaccessible under classical laws or the laws of chance alone.

If so, we would expect such phenomena to emerge in systems that exist at the boundary between the macroscopic scale, where deterministic laws prevail, and the microscopic scale, where quantum mechanics prevails. And it so happens that the human brain (and any mammalian brain, for that matter) is just such a system. The brain as a whole is a macroscopic system composed of networks that are just at the boundary between macroscopic and microscopic, which are further composed of individual neurons that belong to the microscopic realm. Could there be a more suitable system for emergent phenomena such as consciousness and free will to develop? (Some might suggest that the answer to this question is yes: a computer.)

The point of these speculations is not to demean humanity by reducing the soul to a mere physical construct — the above meanderings certainly prove nothing of the kind, nor are they intended to do so — but to suggest that there might be hidden potential in the physical substance of our universe. Matter, space, and time, together with the rules that govern their interaction, may contain some life and magic that we haven’t yet imagined. In Genesis 1:24, God says, “Let the land produce living creatures” [emphasis mine]. Is it possible that the land itself — the material substance of the universe — has, buried deep within it, the very components not only of life and consciousness, but free will as well?