Since the start of the lockdown almost three months ago, I have noticed a significant increase in the number of selfies that decorate the virtual walls welcoming me on Facebook and Instagram. Gone are the pictures of large groups of people having dinner, partying at clubs, or celebrating someone’s birthday, and in their place, there have cropped up images showing the happy and optimistic faces of lonely photographers, secluded in the comforts of their own homes.

The disappearance of the multitude in these pictures accentuates the individual, making more prominent those facets that make us unique. Despite the restrictive nature of the confinement, we find ways of setting us apart from the rest, making the point that, though we might be similar, we are far from identical.

At every step of our journey, we find bifurcations on the road that take us all deeper down the forest of reality, but apart one from the other. Of course, this is not surprising. The eruption of knowledge, skills and ideals that resulted from the development of our civilisation has created a world where we can craft as filigree our existence, up to the point where we must claim “Surely there is nobody else like me!”

There are trivial ways of establishing you are different from the rest: You are the only person on Earth who has your exact fingerprint pattern, or the only person that has your national I.D. or passport number. But being different is not the same as being unique. While you are different from others purely because of the forces of randomness, you are unique because there is an element of value that makes you exceptional: Perhaps you were the first woman to travel to space, the first African to win a Nobel Prize in Literature or the first athlete of your country ever to win a medal in the Paralympic Games.

Unable to claim to be the first one accomplishing something spectacular, I devised a little game that helped me thinking about my uniqueness. It goes like this: What is the smallest number of large social groups to which you belong but whose intersection has you as its only member? For example, you might realise you are the only Chinese painter who lives in Canada and practices capoeira (four groups) or the only Polish dentist in Spain that rides a motorbike and goes to church every Sunday (five groups). The fewer the groups, and the bigger they are, the more special is the conclusion, as it is more difficult that you pass the intersection criteria: You might be the only lawyer of your family, but surely you are not the only lawyer of your city!

In my case, I seriously suspect I am the only person who belongs simultaneously to the following five groups (in parenthesis, the guesstimated population of each of them):

• Latino (626 million)
• Permanent resident of the United Kingdom (67 million)
• Mathematician (~2 million with MSc/PhD in the world?)
• Blogger (~30 million in the world?)
• Woodcarver (~2 million in the world?)

(For those with competitive minds, you may turn the result of this exercise in a metric measuring your uniqueness by calculating something like the average of those populations, divided by the number of groups times one million. This would give a score between 0 and 7600, mine being just 29.)

Destiny can falsify my rationale just by presenting me another Latin-American mathematician living in the U.K. that writes a blog and enjoys the occasional carving of wooden objects. Still, something tells me that, if there were more people with that peculiar profile, I would have met them by now.

Having absolutely no basis for what I am about to say, I conjecture that every person in this planet can solve my challenge with no more than half a dozen groups, each of them consisting of at least a million people. In other words, human experiences may be chaotic and unfathomable, but perhaps there is a way to compress them by projecting them to low-dimensional manifolds, each one holding one and only one of us.

If this statement were true, it would be an acknowledgement to our liberal societies, which extol the virtues of individualism. Perhaps it should be a cause of celebration that the richness of possible tastes, occupations and values allow each person to be a special snowflake, the sole master of a unique bucket.

You are unique. Great! Now what?

Establishing that you are unique, regardless of the method you use, is an arbitrary fabrication of your mind, a specific way of organising the information of who you are and how you relate with the others. The fact that it is an arbitrary mental state, however, does not mean that it is irrelevant.

Surely you have experienced the boost of confidence that comes from perceiving yourself as someone special in a positive way. Such elated estate is called narcissism, a behaviour that we demonise, but one that we may need at least in a small dose, to stay more or less sane on this planet. The danger with narcissism is that it is terribly addictive, and you might find yourself hunting for the next fix of adulation – coming from others or yourself – just so the rush of feeling special does not fade. Narcissists are the junkies of narcissism, and if you have ever crossed paths with one of them, you probably know that they would not struggle a bit answering the question of why they are unique.

My little game of looking for groups of people of which you are the only person in their intersection does not preclude you from using it to single you out as a unique person but in a negative way. After all, perhaps you are indeed the only person in Birmingham, going through a divorce, who got a lousy year-end review from his boss, and whose car just broke down in the middle of the highway (four groups). From what I have seen in others and myself, this is a pervasive way of rationalising those stages in life when everything goes wrong, but a particularly unhelpful one. It is a way of rewiring your brain to think that your reality is so particular that nobody gets you, that you are irredeemable alone on this planet.

It is a particularly easy trap to fall in, but one that is difficult to escape from. As a corollary of my statement before, I conjecture that every person in this planet can solve my challenge with less than half a dozen groups of at least a million people, but in such a way that your uniqueness stems from being inferior to everybody else. Or, in other words, that everybody can use the argument of being unique to feel miserable.

I’m not saying I’m Batman I’m just saying nobody has ever seen me and Batman in a room together

But, how did we end up thinking about being unique in the first place? The quality of being “the only one of a kind” does not seem to arise in the natural world: everything that surrounds us comes in multiples, not in unique realisations. Sure, any hunter-gatherer in the plains of East Africa 50,000 years ago could already notice that the patterns of spots on leopards are all different. Still, it takes an extraordinary leap of imagination to conceive that each leopard is an unrepeatable entity. The Sun and the Moon might be the only objects in the domain of the primitive man that he could point to as being completely exceptional, and perhaps our reverence to them ignited our curiosity for other samples exhibiting such characteristic. But who knows.

However, we don’t need to go to the outside world to notice examples of exceptionality. Our subjective experience is blatantly unique, as it becomes evident when we interact with others. The same events, conveying the same information, can be (and often are) interpreted in as many ways as the number of people experiencing them. Following this argument, we get very quickly in murky terrain, as we suddenly find ourselves in the company of the idealists.

Christian Wolff, in his Psicologia Rationalis of 1740, calls idealists “those who not only admit the ideal existence of the bodies in our soul but that in fact deny the real existence of the world and the bodies”. In other words, idealism is a philosophy that asserts that “reality” is in some way indistinguishable or inseparable from human understanding and perception; that it is in some sense mentally constituted, or otherwise closely connected to ideas. The conclusion of some philosophers that there is no external world but just inner machinations of the mind, involved a long and complicated debate, with various factions disputing the essence of reality. However, I do find fascinating to track the thoughts on this topic of a radical idealist that happens to be my favourite writer: Jorge Luis Borges.

Borges professed admiration for Berkeley, Schopenhauer, Swedenborg, and de Quincey, key figures of the idealist doctrine. Some scholars see in Borges’ literary work some clumsy attempts to follow their steps, but my reading is somehow different, as what I see are ludic explorations of these philosophical questions. The fantastic worlds that Borges portrayed in his short stories, essays and poems are vehicles where he plays with the idealist assumptions, taking them to their wildest consequences.

The wonderful “Tlön, Uqbar, Orbius Tertius” is such a perfect example. A long story by Borges’ standards, it describes a fictional world called Tlön, where the people hold an extreme form of idealism, giving way to a society so different from ours. Without an external world to reference, the inhabitants of Tlön do not have the concept of things, which means that in their language there is no need for nouns, leading to a conception of the world that pretty much rules out all Western Philosophy. In Tlön, there is no History, no Ontology, not even the possibility of a priori deductive reasoning from first principles.

In some of his other works, Borges’ aggressive pursuit of idealism takes him to exotic lands. Like Hume, he finds straightforward the refutation of space, but he marvels as of why stopping there. Time as well is impossible, as he reflects in “A New Refutation of Time”. The ultimate consequence of his views, and the most dramatic, in my opinion, is his refutation of plurality, which takes us back to the original topic of this entry. The sense of human uniqueness that I described above pales as insignificant and inconsequential next to Borges own views on the matter. In the poem “Descartes” he already writes:

“I am the only man on Earth, but perhaps there is neither Earth nor man.”

But it is in the poem entitled “You”, rumbling as if giving a passionate sermon, where he tells us that perhaps we are unique, just not in the way we think we are:

“In all the world, one man has been born, one man has died.

To insist otherwise is nothing more than statistics, an impossible extension.

No less impossible than bracketing the smell of rain with you dream of two nights ago.    

The man is Ulysses, Abel, Cain, the first to make constellations of the stars, to build the first pyramid, the man who contrived the hexagrams of the Book of Changes, the smith who engraved runes on the sword of Hengist, Einar Tamberskelver the archer, Luis de León, the bookseller who fathered Samuel Johnson, Voltaire’s gardener, Darwin aboard the Beagle, a Jew in the death chamber, and, in time, you and I.

One man alone has died at Troy, at Metaurus, at Hastings, at Austerlitz, at Trafalgar, at Gettysburg.

One man alone has died in hospitals, in boats, in painful solitude,  in the rooms of habit and of love.

One man alone has looked on the enormity of dawn.

One man alone has felt on his tongue the fresh quenching of water, the flavour of fruit and flesh.

I speak of the unique, the single man, he who is always alone.”

If there is no external reality, our subjective experience right at this moment is all there is, but with the disturbing consequence that our loneliness is deeper in nature. The heroes and monsters of history books; our relatives and neighbours; the people that we love and we hate; they all exist only as constructs of our minds. The only person that ever existed and will ever exist is whoever is writing these lines, who happens to be the same person that is reading them.

This is Fight Club but at a cosmic scale.

Existence and Uniqueness

Borges’ poem is of haunting beauty, one of my favourites for sure. But the image he portrays is an aberration that clashes with my belief of an objective world (the same belief professed by the heresiarchs of Tlön). This feeling led me to the observation – perhaps the only one I am really trying to make in this text – is that there is something contra naturam in the credence that something can be unique; that it is an idea so strange and counterintuitive that it rarely appears in any argument and only with great effort. In the zoo of bizarre abstractions that were created by the human intellect (like “continuum”, or “equilibrium”, or “justice”) the consideration that “there is only one of a kind” deserves a special place for our appreciation and bemusement. Other than our fervour for being unique creatures, and Borges’ universe of only one man, I am aware only of three different examples where humans have toyed with this preposterous but seductive idea.

The first one is in Mathematics, where one of the essential skills of the trade is being able to prove by a sequence of logical steps the existence and uniqueness of a given abstract object. Take any book of any field in Mathematics (Algebra, Topology, Analysis, whatever). The chances are that you will find early in the discussion of a new topic the insistence of the author to establish that that thing which is about to consume her attention for the next hundred pages is real (as real as anything can be in Mathematics) and unambiguous.

In my corner of the field, the object in question is typically the solution of a specific problem: a differential equation, an interpolant over a set of nodes, a rational function whose singularities behave in a certain way. I was surprised when I looked in my PhD dissertation and found that I relied 14 times on the argument of something being unique. That’s quite a lot! And that was Applied Mathematics, so I can only imagine how much more often this argument is used among my intellectual cousins in Pure Mathematics.

A preferred manoeuvre to attack a uniqueness problem in Mathematics is to set up a logical ambush and postulate the existence of two such objects. Similar in the spirit of Borges’ refutations, what you do is to follow to the last consequences such hypothesis until you run into something impossible. It is not so much that you affirm the uniqueness of something as much as rejecting the possibility of plurality.

The law of noncontradiction, a metaphysical principle already postulated by Aristotle, is the foundation of this way of reasoning. I have not checked it yet, but I presume that already Euclid must have used these proofs by contradiction to show in The Elements that this triangle, that circle, or such polygon are unique.

I am not sure what happened next, but somehow here we are, 25 centuries later, still thinking about the uniqueness of some phantasmagoric entities that appear in the realm of Mathematics. I have not found a book that tells this story, so I think it would be a great project to write one, tracing the evolution of uniqueness in Mathematics, disentangling the importance of its role through the ages, and illustrating these discussions with dozens of exemplary theorems. Perhaps one day.

A ping-pong, bouncing on the walls of eternity

The second example of uniqueness arises in Physics. In the spring of 1940, Richard Feynman, a young doctoral student at Princeton University, got a phone call from John Wheeler, his advisor. The two men had been discussing for weeks why all electrons ever detected have identical charges, masses, and other physical properties, but they had reached no conclusion. Now, Wheeler was calling his student, excited to share with him a dazzling vision he just had. Feynman would remember the call years later when receiving the Nobel Prize of Physics:

I received a telephone call one day at the graduate college at Princeton from Professor Wheeler, in which he said, “Feynman, I know why all electrons have the same charge and the same mass” “Why?” “Because they are all the same electron!”

It is an understatement to say that there are many electrons in the universe: its order of magnitude is 10 to the 80 (that is, a “1” followed by 80 “zeros”). But what Wheeler suggested that day was that all those electrons were just the manifestation of one single electron that was travelling back and forth in time, impacting the universe at the temporal slice that we call “the present” at different points in space. Theoretical physicists describe the possible events experienced by a particle through what is known as “world lines”, threads in the 4-dimensional space that constitutes the space-time. The traditional (i.e., normal, intuitive, not crazy) way of thinking these world lines is that each particle has its own, independent from the others. However, Wheeler saw that perhaps all of them were really just one single thread, making 10 to the 80 turns at both ends of eternity, giving us the illusion that there were so many electrons right now.

In his excellent book “The Quantum Labyrinth: How Richard Feynman and John Wheeler Revolutionized Time and Reality”, Paul Halpern illustrates Wheeler’s idea with the movie “Back to the Future II”. In this movie, Marty McFly has to travel back in time – for a second occasion – to 1955, to the town of Hill Valley. In some hilarious scenes we see two Martys at the same time, and though paradoxical, both manifestations are the same entity, one whose world line got entangle as he drove the DeLorean back and forth in time.

The Dirac equation, which governs the relativistic behaviour of quantum particles, is agnostic to the arrow of time and allows a perfectly reasonable solution of an electron moving in both directions through time. Mathematically, flipping the sign of the charge, of the temporal direction and of the spatial direction of a particle, leads to the same solution. What this means is that a negative charge particle travelling back in time looks like a positive charge particle moving forward in time, what we would know as a positron.

Wheeler’s hypothesis of a one-electron universe was born that day and died that day. Feynman was immediately sceptical of this wild idea and quickly noticed a problem. If we were experiencing multiple manifestations of the same particle, there would need to be the same amount of the ones travelling forward in time as those going back in time. But positrons, like other forms of antimatter, are quite sparse in the universe, adding up to only a fraction of the number of electrons that exist in the universe.

But though Wheeler’s hypothesis was a poetic lucubration rather than a physical model of reality that could be tested thorugh experiments, it gave Feynman the idea of regarding positrons as backwards-travelling electrons. This trick proved to be quite handy for solving equations and became an essential element of the theory of positrons, a work that was later continued by other physicists.

A universe filled only with one single electron seems even lonelier than the one imagined by Borges. Again, collapsing the plurality in something that is unique, results in an interpretation of reality that defies all forms of intuition. My last example is no exception, and though I know next to nothing about it, I presume it is the one that has had the most dramatic and tangible consequences in the world.

Isaiah 46:9

At some point back in the 6th century B.C. or so, a group of men in the region of Judea played with the idea of the unique, reaching the conclusion (or receiving the inspiration, depending on who you ask) that there was only one God. You can find previous attempts here and there of quasi-monotheistic religions, such as Zoroastrianism in the ancient Iranian empires, or the cult of Ra in the New Kingdom of Egypt during the reign of Pharaoh Akhenaten. But what Judaism started was a complete revolution in the way that humans conceive the divine.

Throughout the world, you find examples of civilisations whose cosmology contemplated the existence of not one but many gods, arranged in some hierarchy, which kind of makes sense when you think about it. Humans experience a wide range of problems that not necessarily have an obvious common cause. In the face of the unknown, it makes sense to honour the God and spirits that command the frequency of the rain, which surely are different from those who have power over the flames of fire.

Rome, during the time of the Republic and the first part of the Empire, was a polytheistic civilisation, one that was happy to add new deities of conquered territories to its ever-growing pantheon of gods. Pragmatic as always, Romans did not pay too much attention to the theological conundrums of having under the same roof the gods of Greece, North Africa and Gaul at the same time. As long as everybody was on the same page about their loyalty to Rome, there was no reason to make things complicated.

But of course, things got complicated. Judaism was a first problem to the polytheism of Rome, and in some degree, it played a role in the First Jewish-Roman War of 66-73 CE. However, the causes of that particular revolt – as it happens with most revolts – had to deal more with taxes than with spiritual matters. The real challenge to polytheism came from Christianity, which was not only at odds with the idea of tolerating multiple gods and cults – including worshipping deified Emperors – but that also had the explicit objective of converting the pagans. Christians managed to endure almost three hundred years of anonymity punctuated by the occasional percussion until Emperor Constantine had his miracle vision on the Milvian Bridge, and all we know what happened next.

The tribulations of Judaism, Christianity and Islam have many vertices. Still, it should not escape anyone’s attention the profound consequences that arise when you proclaim the existence and uniqueness of the One True God. As in my other examples, the idea of something being unique creates a peculiar tension with our senses and minds, as it necessarily goes against the acceptance of the plurality.

After writing this very long piece, I end up with more questions than answers. How was the idea of the unique planted in the minds of mathematicians and theologians in the first place? Did it appear any time before? Is there out there something truly unique, with zero copies of itself of any kind, in the physical world? Are there any more examples where we use the argument of uniqueness? Or am I completely wrong, and it is a fundamental idea that illuminates our consciousness and existence?

Sure: One only man; one only solution; one only electron; one only God. Still, the refutation of plurality seems somehow incomplete.