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.