Identity and Equivalence

One of the conditions of being human – in some ways an advantage, in others a limitation – is that we formulate our thoughts in terms of entities – people, places and things. It’s the best way we know to relate to our surroundings. Simply put, we like to think in terms of distinct, separable, independent entities. And we love to create distinctions – to identify and delineate one thing from another. Scientists especially love to categorize, and to identify fundamental properties of entities. From phylogeny, to the periodic table of elements to the standard model of particle physics, it is a large part of how we make sense of the world.

Sometimes, these divisions are reasonable. One particularly sensible instance is the delineation of places. Within America, there are 50 states. When I’m in California, there’s no sense in which I’m in Alaska or Wisconsin, or even in bordering Nevada. If I were to walk eastward, I would continue to be in California until one particular, well-defined moment when I cross over into a neighboring state. In other words, there are agreed upon, sharp boundaries between one state and the next. And while geographic location may not be the only unique characteristic of a state, it certainly can be used to unambiguously discriminate one from the next.

Other entities are harder to define. Take colors for instance: on its face, color seems fairly tame. Early on we learn about the three primary colors, red, yellow and blue. We are taught that by combining these three, we can arrive at any color we would like. We learn that the rainbow is composed of 7 colors, and we are given a handy acronym, ROYGBIV, for remembering these colors. In most everyday situations, these naive definitions hold up, and there is no cause for concern. Armed with our experience and rules of thumb, we typically agree on what color we see reflecting off of an object.

However, the problem is just that – we almost always agree. Lexicographer Kory Stamper of Merriam Webster wrote a fascinating article in Slate about the exceedingly unhelpful definitions of certain colors that can be found in the historical literature.

One example Stamper gives is Vermillion: “a variable color averaging a vivid reddish orange that is redder, darker, and slightly stronger than the chrome orange, redder and darker than golden poppy, and redder and lighter than international orange.”

If you were given this definition, could you go to a color wheel and pick out a precise location that realizes vermillion? Sure. And so could I. And so could everyone and their grandmother. But there is absolutely no guarantee the “colors” that we pick out would be the same. In other words, while the quote provides a few reference points of colors that are redder, lighter and darker etc., it does nothing to quantify how much so. It tells us nothing about where golden poppy ends and vermillion begins.

And if it’s this difficult to pin down colors, how difficult must it be – or is it even possible at all – to develop consistent and complete definitions for more complicated concepts? And how are we to think about our own identity, personhood, and sense of self?

This is a question that has consumed philosophers for millennia. In fact, it’s really not just a question but an entire sub-discipline of philosophy. Some of the sharpest minds in history have taken a crack at this problem, among them Descartes, Locke, Hume and Kant. The goal here is not to replicate their arguments or try to come up with some tidy all-encompassing definition for what it means to be something rather than something else. Rather, I’m going to draw on a few examples from physics and mathematics that may provide some insight into this problem. It is left to the reader to decide how convincing these arguments are, and to what extent they bear consequences in our lives.

Mathematics, in its ultimate generality, is the study of structure. There are many different branches of pure math, from algebra and calculus, to topology, logic, and number theory. The methods of proof vary by topic, but the common thread is the imposition of a certain “structure” on a set, and the importance of “maps” between sets, which preserve that structure. Forgive the complete lack of rigor in the paragraphs that follow.

Let’s bring this back down to Earth with an example: In the study of logic, the “structure” is informally called “interpretation”. Two statements are “equivalent” in the logical sense if they are true under the exact same models, or interpretations. Take the following two statements:

If Bill Gates is in Palo Alto, then he is in California.
If Bill Gates is not in California, then he is not in Palo Alto.

I and II are not exactly the same in the sense that the words in each statement are arranged in different orders. Yet for any possible situation, the truth or falsity of I and II will always be the same. (II is the contrapositive of I, so I can be derived from II and II can be derived from I. These derivations are related to the map between the statements.) They are logically equivalent. In the context of Logic, this equivalence is what matters. In other words, their logical content is their essence.

How about something a bit more abstract. In graph theory, a graph is defined by its vertices (a set of points), and edges (a set of pairs of vertices), the latter of which describe the connections between vertices.

Take for example the following simple graph:






In this graph, which we can call G, the vertices are a, b, c, d, and e, or V = {a, b, c, d, e}. The edges are the lines drawn between these vertices, E = {(a, b), (b, c), (c, d), (d, a), and (a, e)}. We can write G as G = (V, E) – this contains all of the defining information about the graph.

For graphs, the manifestation of equivalence is in mappings called isomorphisms.






At first glance the two graphs above – a pentagon and a star, seem quite different. On the page (or computer screen), the lines in the star cross each other, while those of the pentagon clearly do not. However, this difference is inconsequential from a graph-theoretic perspective. The two graphs are actually isomorphic – i.e. they have the same underlying structure.

One way to see this is to label the vertices of the pentagon a through e, and then write down all of the edges in the graph in terms of these vertices:

E1 = {(a, b), (b, c), (c, d), (d, e), and (e, a)}.

If we do the same for the star graph, labeling the vertices 1 through 5, we find that the edges are

E2 = {(1, 3), (3, 5), (5, 2), (2, 4), and (4, 1)}.

Looking at these two edge sets, it is obvious that if we make the substitution

a → 1, b → 3, c → 5, d → 2, and e → 4,

then the graphs have the same edge sets. Despite different vertex labels and a different appearance, the graphs behave in the same way – just as in the Logic example, these two have the same essence.

Sure, mathematics may exist in its own world, where it is not confined to the laws of physical reality. Obviously humans are far more complex and nuanced than can be mathematized. But these types of equivalences, which pervade every branch of pure math, beg deep philosophical questions.

What gives an entity its essence?

What make it it rather than something else?

What is essential to my own identity?

In some sense, I am different than the Jacob I was 18 years ago singing the ABC’s and playing with Sesame Street action figures. And I am different from the Jacob that I will be when I have grown bald and hard of hearing. I have matured. I’ve gained life experience. Even the cells in my body, comprising my physical self, die and are replaced every few weeks. Yet through it all I have – and will continue to – retain the fundamental quality of Jacob-ness.

Just like relabeling vertices on a graph, I could legally change my name without changing who I am. I could shave my head, or put on a top hat to disguise myself, but I will just be an equivalent Jacob, as the second logical statement above was merely the first statement in different guise.

John Locke famously believed that one’s material body had no bearing on their personal identity. In his Essay Concerning Human Understanding, he presents an example of a prince who wakes up in the body of a cobbler. Locke claims that after the transfer of consciousness, the person now corresponding to the cobbler is one and the same with the person who previously corresponded to the prince.

Okay so physical presence is not essential to one’s self identity. Then what is? What of opinions, thoughts, beliefs, and even actions? Perhaps quantum mechanics can provide some insight.

In quantum physics, a particle is represented by a mathematical object called a wave-function. One of the simplest quantum systems is the two-level atom. Like a classical analog, there is a ground state and an excited state. The quantum-ness comes into play in that the atom does not have to be in just the ground or the excited state at any given moment – instead it can be in a superposition of the two. The mathematical object used to represent such a system is a two-dimensional complex valued Hilbert space, – the state of the system being any normalized vector in that space.

Things get more interesting when multiple quantum systems interact: such systems can become entangled. When this happens, the state of the combined systems cannot be described individually. For example, when two two-level atoms are entangled, the state of the combined system cannot be described by two vectors in . Rather, they must be described by a vector in . When one atom is measured, the state of the second atom changes instantaneously.

In some sense, the state of a person is their worldview – their thoughts and tendencies. And if only figuratively, we are all entangled with those around us. It is impossible to disentangle our beliefs from the influence of society. Stark changes in the world around us, which we may call formative events, impact our internal state in undeniable ways. Our state – feelings, beliefs and desires – are distinct from our fundamental nature in the same way as the state of a particle is distinct from the mathematical object – the structure – which defines it.

What then is our fundamental essence? Is it an immaterial soul? A naked, unencumbered nature? Or something else entirely? Perhaps it’s a combination of qualities or, as Locke believed, continuity of consciousness. Maybe it is impossible to define precisely, and we must embrace the vagueness, as in our discussion of color. As a scientist, I would like to believe that there is something. But as a human being, I’m torn. Whatever it is, it’s not my place to say what that essence is. It is up to each of us to decide what makes us who we are.

The Question of Science

When I was 12, a friend gave me The Ultimate Hitchhiker’s Guide as a birthday present – 5 books bound together with an ornately engraved gold and black cover. It was a behemoth of a book, chronicling the adventures of Englishman Arthur Dent and his extraterrestrial friend Ford Prefect as they traveled through the galaxy. It seemed like the Bible of Science Fiction, and I absolutely couldn’t put it down. I think I read the entire series over a Spring Break one year, opting out of swimming, instead immersing myself in science fiction lore.

Through Arthur and Ford, I vicariously ventured into all corners of Douglas Adams’ universe. I encountered a cow that wanted to be eaten, a Paranoid Android older than the Universe, and a babel fish, which, when placed in your ear, allows you to understand any language.

In the first installment, The Hitchhiker’s Guide to the Galaxy, a group of hyper-intelligent beings builds a supercomputer called Deep Thought, in order to help them solve the mysteries of the universe. They ask Deep Thought the ‘Answer to the Ultimate Question of Life, the Universe, and Everything’, which turns out to be 42. But, as the computer aptly points out, the Beings know the answer but not the question. In the sequels, armed with an answer, the extraterrestrials backtrack through possible equations and formulas, and build an even more powerful supercomputer to discover the question.

One of the reasons I love the series is that every instance of science or adventure is so incredibly fantastical. But recently I’ve found that some of the ideas in the Hitchhiker’s Guide hit close to home.

When I first thought about writing this blog post, I wanted to write about physical intuition, the ability to qualitatively explain complex processes by thinking about them in terms of everyday experiences. I’ve since come to realize that “everyday experiences” doesn’t quite do justice to the way most people perceive science. Science is about asking questions and, hopefully, getting answers. However, in many ways, we don’t even know which questions we should be asking.

Science asks how, not why. It’s a subtle difference, one that is easy to ignore or overlook. But it lies at the heart of the way we see the world. How looks for an explanation, a sequence of events that brings a system from point A to point B. It is the search for processes that accurately describe the world around us. Even cause and effect lie in the realm of how. Why, on the other hand, begs a purpose, an intention.

These two questions seem so genuinely similar that we group them together, using them interchangeably. The lack of distinction is so ingrained in our society that it has become a part of our language. My roommate here at CERN, Emil Öhman is originally from

Sweden, so English is his second language. Every night over dinner we talk about philosophy and physics, and even though he is fluent in English, he always asks me to clarify what I mean when I use the word reason.

Emil brought to my attention the fact that reason has two completely separate meanings. Sure enough, as attests, we use the word as a synonym for ‘motive’, the why, and ‘cause’, the how. The former connotes purpose, which is inherently human, while the latter seeks to explain natural occurrences. Consciously or not, we often use reason ambiguously, hinting at both meanings simultaneously.

Syntactical quirks aside, this inability to distinguish between how and why is an integral part of the way we interact with science. Through high school and my freshman year of college, I’ve heard the phrase ‘physical intuition’ thrown around in physics and math classes in the likes of “from our physical intuition, we can see that…” and “it’s obvious if you use your physical intuition, that this problem can be distilled down to…”

In many cases it is the most powerful problem solving technique that we have. Sometimes the math required to solve a problem is incredibly involved, and you have to think back on all of the physical processes you’ve observed in your life, pick out a few that share a semblance of similarity, and compare and contrast them in the hopes of developing a hunch. It allows us to run thought experiments in our heads, taking as an axiom that the physical laws that govern all processes are the same.

The classic examples of using physical intuition tend to fit this mold fairly well. Richard Feynman famously asked his students to find the weakest point on an infinitely round table with four evenly spaced legs. In the absence of physical intuition, this problem would require applied physics and intensive mathematical calculation. Using personal experiences, it is easy to guess that the table is weakest between any two of its legs.

Albert Einstein used physical intuition to formulate his theory of relativity. He realized that if he were in a metal box isolated from the outside world, and he felt a downward acceleration, he wouldn’t be able to tell if the box was a rocket accelerating upward, or if a gravitational body was pulling him closer. Einstein didn’t need to run experiments or create a perfectly isolated system out in space in order to come to his conclusion. Instead, the Equivalence Principle, as it became known (for equating inertial and gravitational acceleration) required physical intuition and a bit of creativity.

But it seems like the analogies are now in large part based on social structures. Physical intuition has grown to view particles as people, and subatomic phenomena as social interactions. I remember my Chemistry teacher in high school explaining the force between protons and electrons as “opposites attract”, as if it were an obvious corollary to a similar phenomenon in dating. We colloquially referred to certain elements as “wanting” to have a complete valence shell of electrons, when we really meant that atoms of that element lose or gain electrons because they are unstable.

We even personify Natural Selection, the process by which life evolves to handle environmental and social conditions. I’ve heard over and over again that Evolution “favors” those that are fit. Evolution itself has no wants or desires. It just happens to be true that on average organisms that are more fit to survive, survive.

Whereas physical intuition used to ask how, our personification of science has added an element of why. Instead of asking, “how did such biodiversity come to exist?” we ask, “why were these organisms chosen to survive?” By bringing inanimate objects and abstract processes to life, we give them thoughts, feelings, and even emotions! We impose upon them the human constructs of intent and purpose.

Indeed, it’s such a satisfying way to view the world. Emil thinks of particles as people that push each other away when they get too close and invade personal space, and pull each other closer when they grow too distant.

I only recently realized that my own conception of particles and elements relies heavily on social interaction. On the whole, the more fundamental a particle, the more stable it is, and the larger and more complex it becomes, the more prone it is to decay. Some, like the proton, have lifetimes longer than the existence of the Universe, while others live for fleeting fractions of a second. The same logic applies to elements. Those with more atoms tend to be less stable than elements containing only a few. I think about these particles and elements in terms of social groups; the larger the group, the greater the likelihood that members will disagree, and someone will want to leave. Complexity spurs on collapse.

There’s something poetic in our relationship with Science. We are composed of particles, but we view particles as if they were innately human. We personify elements and assign rationale to mechanisms that have no inherent purpose. Our social, as well as physical experiences, point us to the answer…

Which brings us back to the ‘Answer to the Ultimate Question of Life, the Universe, and Everything’. Is it wrong to incorporate why into our understanding of Science? Does why complement the empiricism of how, or refute it? Clearly viewing particle physics through the lens of social structures has limitations. It could, however, lend new approaches to old problems. After all, why appeals to something deep down, which how alone can never hope to satisfy. Could there be two fundamental questions, two ‘Ultimate Questions of Life, the Universe, and Everything’?

Inside the Heart of a Glacier

Right now it’s 3:30 in the afternoon and I’m sitting on the side of the road with my friend Nick. Along with Manjari, who is out exploring, Nick and I are in the small French town of Chamonix, situated in a valley in the Alps, about two hours outside of Geneva.

Chamonix is a beautifully odd mixture of idyllic European facades and touristy trinket shops. Cafes and chapels are interspersed with ATMs and hotels. The Tourist Center has a line dedicated to Japanese visitors, although far more come from China. And the music in restaurants is entirely in English, even when none of the waiters speaks the language: The Beatles are especially popular.

The unmistakable golden arches of the fast-food empire are easily visible from the bus stop – which is where we are right now. But the McDonalds logo is dwarfed by Mont Blanc, rising regally in the background.

Nick and I are waiting for the 5:00 bus back the Geneva after a weekend full of sightseeing. Since yesterday morning, we’ve hiked along the Alps, walked on a glacier, and ridden cable cars up to Aiguille de Midi, “Needle of the Noon”.

Today we woke up early and took the first train to the glacier, Mer de Glace. In French, Mer de Glace translates as “Sea of ice”, but the name is a vestige of the past. In the 1800s, the glacier was clear white, and easily visible from Chamonix. It was regarded as one of the world’s foremost natural wonders, and was the subject of many paintings and photographs. Since then, the Sea of Ice has receded hundreds of meters and lost a third of its thickness. Deposits of stone and earth, called moraines, have also diluted the crystal color of the glacier.

The first thing you see when you arrive is a vast cavern; previously filled with ice; now almost empty. As you walk down into the depths of the trench, you pass signs marking the levels of the glacier in past years -1820,  1890,        1920,                   1937… the distances between signs grow longer while the years grow closer together.

At the bottom lies the main attraction – an ice cave dug into what is left of the glacier. In the past, Mer De Glace inspired paintings. Now the glacier itself is the canvas. The artwork is framed only by the Earth’s crust. It moves with the glacier at a rate of a centimeter per hour, so every summer it must be entirely re-sculpted.

Sadly, when we got to Mer de Glace, the ice cave wasn’t open. We couldn’t bear to miss such an opportunity, so the three of us snuck in! Descending the last few flights of stairs, we walked along a blue carpet into the mouth of the cave.

The cave itself is an incredible sight! It is about 10 feet across, and penetrates deep into the glacier. The cavern is roughly cylindrical, with smoothed ridges along the walls, like the inside of the tunnel of a wave. Lights line the walls, each one striking the ice slightly differently, creating its own shades of translucency.

More surreal even than the cave are the sculptures inside it. The cave is composed of multiple cavities, each representing a room in an archetypal house. One cavity contains ice sculpted to look like beds. Another mimics a living room, with chairs, couch, and coffee table. A third contains a giant faucet, representing a bathroom.

Walking through the cave, I was struck by the strangeness of the juxtaposition. Why was such a beautiful natural canvas covered with the quintessentially human? Why did the artists model their creation after a house? Did the sculptures change each year? Out of all that is left of the glacier, the cave is the closest we can come to experiencing and appreciating the Sea of Glass. It seems like adding artificial sculptures would detract from that experience.

A part of me thinks that the sculptures are meant to symbolize the age of industrialism overtaking Earth. Maybe the sculptors are saying that humanity has clawed out the heart of Mother Nature, the same way they carved out the core of the glacier. The rest of me feels like it is just a product of proximity to the often incongruent Chamonix.

At its current rate of shrinkage, Mer de Glace could soon be gone. Living in Saint Genis for the summer, which is in the same country, I had to take two buses and a train, and sneak past tour guides to gain entry into the ice cave. In a few weeks, I’ll be back in America, a continent away from Chamonix and the Alps.

I may never again get to walk in the tunnel of translucent ice under France’s largest glacier. I hope to enter the cave once more before it vanishes. But in the meantime, I’ll be imagining a true sanctuary, an ice cave cleared of sculptures.

So Much for Symmetry

One of the first ideas that drew me to science was symmetry – the concept that everything has inherent balance, proportionality, and temporal harmony. I know it’s romantic to think that everything, from electrons to elephants, obeys the same universal laws; but for such a simple idea, it holds surprisingly true.

In Biology, plants and animals adhere to many different types of symmetry. Sunflowers and Nautilus shells spiral outward according to the golden ratio; trees and leaves display fractal geometry, in which the same pattern appears on many scales; and animals show bilateral symmetry, where left mirrors right.

Chemical bonding, too, is governed by symmetry – elements that have the same number of electrons in their outer (valence) shell, bond in the same way. The whole periodic table is based on this structure!

Much of Physics revolves around symmetry in the form of invariance, or lack of change based on coordinate system. It only seems right that in space up is no different than down! In fact, when drawing the diagrams that bear his name, Richard Feynman was so taken by symmetry, he concluded that a particle traveling forward in time is the same as its antiparticle traveling backward in time. In 1972, the Nobel Laureate PW Anderson went so far as to declare, “It is only slightly overstating the case to say that physics is the study of symmetry”!

But every once in a while nature throws a curveball our way. The narwal’s tusk is on its left jaw. And the wrybill is the only bird with its beak always bent to the right.

It’s almost frustrating that out of the millions of species on Earth, all but a few fit symmetries so nicely. What makes them different? What allows left-handed fermions to interact with the weak force when right-handed fermions cannot?

Chaos theory (pictured below), is the study of sensitivity to initial conditions. It says that simple systems, like the double rod pendulum, can have extremely complex results. With just two rods connected at a joint and hung as a pendulum, the double rod pendulum is one of the simplest mechanical systems. Yet it exhibits extremely complex behavior. It never retraces its earlier path, and never repeats a pattern. What’s more, changing the initial conditions even slightly leads to entirely unrecognizable results.

Maybe this sensitivity extends to other systems. In his book, A New Kind of Science, Stephen Wolfram studies cellular automata, or small grids of squares colored either black or white. He identifies 256 simple rules for deciding what color each square would be based on the squares around it, and runs each simulation on his computer. While the vast majority of the simulations produce simple, easily identifiable patterns, and fractals, a small fraction of the tests give different results.

Rule 110, although close to many of the other rules, produces results so complicated that it takes thousands of iterations, and millions of cells, for a pattern to emerge. Until then, it is essentially impossible to predict the behavior of a cell from the prior results.

The mathematician Horace Lamb once said of the chaotic nature of turbulence, “When I die and go to Heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic”.

Maybe chaotic systems can’t be predicted. Maybe their rules can’t be solved for with mathematics or computational power. But I’d like to think that our Universe is governed by simple, elementary laws; and that usually these laws produce simple results, but every once in a while produce an unrecognizable pattern which we perceive as disorder. Narwals and wrybirds break bilateral symmetry. But I’d like to believe that there’s some deeper, truer symmetry that we’re missing. As childish an ideal as symmetry is, it’s soothing to think that there is some underlying order.

Maybe nature isn’t throwing us a curveball; we might just be looking at turbulence the wrong way!

Talking with Titans

In just over a week I’ll be packing my things and getting on a plane to Geneva, Switzerland. I’m going to be working at the ATLAS detector at CERN, the

European Organization for Nuclear Research. It is the world’s largest Physics research complex! Every day for two months I’ll collaborate and talk with people who are just as crazy about Physics as I am, and that is saying a lot! It’s a dream come true for me.

Although I’ll see it with my own eyes shortly, for now, my mental image of CERN consists of dream-like silhouettes of science. There are plenty of pictures of the LHC (Large Hadron Collider), and the ATLAS detector online. But it’s hard to grasp their sheer magnitude.

Before teaching the class Big Science at Sprout, I did some research on CERN, and I came upon a few numbers that hint at CERN’s marvel:

  • 99.999997 – percent of speed of light at which particles travel in LHC
  • 1800 – physicists working at ATLAS detector
  • 210,000 – DVDs worth of data per day analyzed by Data Centre
  • 600 million – particle collisions per second in the LHC
  • 13 billion – estimated dollars spent to find Higgs Boson

But these numbers only scratch the surface of CERN’s scientific prowess and its influence on society. CERN scientists discovered the W and Z bosons, which are carrier particles for the Weak Nuclear force, and the Higgs Boson, whose field gives particles mass. Even the internet, which I am using to write this post, was invented at CERN.

The ATLAS experiment, situated in a cavern dug into the Large Hadron Collider, is one of the biggest sites at CERN. Its detector weighs as much as the eiffel tower, and stands five stories tall. Along with its sister experiment, CMS, the ATLAS experiment was instrumental in discovering the Higgs Boson.

While ATLAS is short for A Toroidal LHC Apparatus, it seems to me an apt acronym. In Greek mythology, Atlas was the Titan who held the world in his hands. As the Titan of navigation and astronomy, Atlas also became associatedwith cartography and maps. I think it’s fitting that in a collosal cavern, a titanic detector is piecing together a different type of map – the map to the fundamental particles.

Right now Geneva seems as far away as Mount Olympus; all I see is a silhouette in the distance. But I can’t wait to talk with the titans of physics, and, hopefully, to hold a little bit of the world in my hands.