I know it’s round, but it is possible that the Earth is flat

I start with ten brazenly provocative statements that may irritate or surprise a frequent reader of this blog but that in a strict sense are all true:

  1. It is possible that the Earth may be flat.
  2. It is possible that man has never set foot on the Moon.
  3. It is possible that the Earth may be only six thousand years old.
  4. It is possible that the theory of evolution by natural selection is wrong.
  5. It is possible that there is no such thing as global warming.
  6. It is possible that aliens built the Egyptian pyramids.
  7. It is possible that the 5G technology caused the coronavirus pandemic.
  8. It is possible that vaccines do not help to prevent diseases.
  9. It is possible that homeopathy can cure illnesses.
  10. It is possible that the Democratic Party may have stolen the 2020 US election.

Let us pause for a moment on that first statement.

In my geography classes at school, I learned about a dozen experiments that proved the Earth was round: there was the one of the ship that disappears in the distance at sea, the one involving lunar eclipses and another one with time zones. Also, someone surely made me realize that the Sun and the Moon certainly look spherical and made me question how strange it would be if our Earth did not have the same shape. And of course, there were the beautiful photos taken from space, showing a cheerful blue ball floating on a black background.

And even with all this, the Earth may not be round, right? I mean, can’t this all be a gigantic conspiracy? A silly misunderstanding? I confess I have never bothered to do any of the experiments myself that claim to prove that the Earth is spherical, and instead, my sensory experiences of when I have been in a desert or on the beach in front of the sea make me think that we live on a table of formidable dimensions. I did not take the photos of the Earth from space; they could well be the product of some skilful artist at some Rosicrucian lodge’s service that has taken complete control of our planet. And what the stars do in the sky with their own shapes only tells us that they are the round ones, not us.

But although I can entertain these questions, the truth is that I know that the Earth is round. If you had asked me a while ago why I knew this, I would have given you an elaborate and confusing explanation, a long and tedious intellectual harangue with way too many invocations to the scientific method. But it was Matthew Broderick, the wonderful Hollywood actor famous for that 1980s masterpiece Ferris Bueller’s Day Off, who made me realize a couple of years ago why is it that I really know that the Earth is round.

Professor Williams and Mrs Pysner

Fall 1996. The Hayden Planetarium in New York offers its regular Astronomy course for adults, a series of informal talks on planets, stars, and galaxies for cosmos enthusiasts. As has been customary for many years, the professor is Mark Williams, a man in his forties, professionally trained in astronomy, but whose research career never really took off. Mark is courteous and respectful, maybe a little shy, and although he puts effort into his classes, they are a bit boring – something that perhaps he feels himself. Mark is no Carl Sagan; his students know it, his colleagues know it, his family knows it, and he knows it.

One day Mark comes to the classroom deeply overwhelmed; you know, men in their forties have this kind of mid-life crisis, his teenage son despises him, his marriage drifts without emotion, teaching the secrets of space to some skill-less adults is too frustrating. Halfway through the class, Mark gets into a heated argument with one of his students, Mrs Pysner, a woman in her fifties who has not the slightest idea of ​​Astronomy, or Physics, or Science, and who has been bothering Mark since the start of the course with her silly questions:

Mrs Pysner [Raising her hand]: I have a question.

Mark: Yes?

Mrs Pysner: How do you know what those galaxies look like at all?

Mark: Well, that we know from –

Mrs Pysner [Interrupts]: I mean, how do we know any of this?

Mark: Well, that’s actually a perf-

Mrs Pysner [Interrupts again]: How do we even know the Earth is really round?

Mark: All right.

Mrs Pysner: I mean, unless it’s not. Because it sure looks flat to me. Anyway, isn’t it all pretty much just a theory?

Mark [Eyes wide open, visibly irritated, can’t believe this bullshit he’s hearing but manages to calm himself down]: No, it’s round.

Mrs Pysner: But how do we know?

Mark: Well-

Mrs Pysner [Shrewedley, interrupts once more]: But do you see what I mean?

Mark [About to explode, God knows how this man is controlling himself so as not to become violent in the face of such imbecility]: Yes, I do know what you mean, and I’d like-

Mrs Pysner [Interrupts again]: Do you know what I mean? I’m sorry. Go ahead. 

Mark [Finally pissed off, he starts yelling, he can’t stop himself anymore]: All right. There are easily a dozen experiments you could do to prove that the Earth is round. But the real reason most of us know that the Earth is round is because somebody found out before us, AND TOLD US SO!

Broderick, who plays Mark, and Jenny Galloway, the woman who plays Mrs Pysner, continue with the dialogue of the play The Starry Messenger that I am watching at the Wyndham Theater in London. However, for a moment, I disconnect from what is happening on stage. Mark’s last sentence, pronounced with anger and like someone who finally admits a terrible secret, falls on me like a profound epistemological revelation.

Almost anything we know, we know it because someone else told us so. Our direct experience only lets us glimpse an infinitesimal portion of reality. The rest is gossip.

I have never set foot in the ruins of Persepolis, but many people have told me stories of that place, so I know that Persepolis exists.

I never met Artaxerxes IV, but many people have told me that the Persian nobleman was for a short time King of Kings, so I know Artaxerxes lived.

Once, I saw a bottle of cyanide. Until that moment, I did not know anyone who had drunk that liquid, but many people had warned me that if I did, I would die, so I knew that the content of that bottle was deadly poison, and I did not drink it.

My knowledge about Persepolis, Artaxerxes, and cyanide is widely shared around the world. However, many issues in our times are deeply controversial, while our understanding of them is built on what we have been told. The efficacy of vaccines, the arrival of man on the Moon, the theory of evolution, global warming, aliens’ presence on our planet, the United States elections. In all these cases, there are strong voices that try to convince us of one side or the other. In all these cases, I agreed with one alternative and not the other, as I trusted the former and not the later.

A postmodern reader might argue that the discussion ends here, that each side builds upon a series of stories, narratives, and traditions, and that both are equally valid because they are both human constructions – everything is relative, nobody can claim some cognitive supremacy over reality. Who knows, maybe.

However, I believe that there is an essential difference in the argumentative core that each field uses to defend its positions: one side emphasizes what is probable, and the other emphasizes what is possible. I think this difference puts the former on higher ground than the latter. And I feel the confusion around both concepts – what is probable and what is possible – are at the heart of many of the ideological crusades that we wage relentlessly in our times.

Probably to be or probably not to be, that is the question

Our knowledge is not a bag of trivia data. The dates of battles, the atomic weights of the elements in the periodic table, or the specific metric used in romantic poetry are just the pieces of information we use to build what we know. Just as we cannot say that a pile of bricks, concrete and glass is equivalent to a house, we cannot say that a series of facts about history, chemistry, or literature is equivalent to knowledge. What is fundamental are the connections we form between these cognitive pieces, and the more extensive they are, the deeper our understanding of reality.

And yet, the interpolations that we do to build our knowledge are subject to uncertainty, forcing a probabilistic nature in what we know. With an important caveat: Probability in this context should not be interpreted from a frequentist perspective (if an event is observed x times in a hundred independent experiments, the probability of that event occurring is x per cent), but as a formal system logic that extends the Aristotelian model in which reality is binary, just true or false.

Probability is a theory that manages to represent and manipulate the concept of uncertainty. Establishing such a link is not straightforward, and the formalization between probability and uncertainty was only done until the middle of the 20th century, mainly around Richard Cox’s work. Under a series of axioms that characterize uncertainty, Cox’s Theorem achieves an isomorphic representation between all binary logics subject to uncertainty with the conditional probability theory. To put it in other words: the operations with which the probability functions are manipulated also serve to manipulate uncertain premises. Among these manipulations, the one given by Bayes’ Theorem is perhaps the most relevant to our discussion.

Bayes’ Theorem is a mathematical machine that transforms the certainty of what we know by incorporating new evidence. For example, let’s go back to the issue about our planet’s shape, one in which way may have some degree of certainty but perhaps not the truth. Suppose we hear of a new experiment that provides further evidence pointing in favour of the Earth’s roundness. Bayes’ Theorem then brings together the following four elements:

(A) The probability that the Earth is round before knowing the experiment’s result (the prior probability).

(B) The probability of observing the new experiment’s result assuming the Earth is really round (the likelihood).

(C) The probability of observing the new experiment’s result, independent of whether the Earth is round or not (the marginal probability)

(D) The probability that the Earth is round after knowing the experiment’s result (the posterior probability).

Bayes’ Theorem says that A x B / C = D, or in the case of our example, that we can update our certainty about the shape of the Earth, from before we know the new evidence, (A), to the one after it, (D), simply by multiplying the first one by the expression B / C.

The arithmetic is quite simple. For example, if we had a 50% certainty that the Earth was round, and we calculate a 75% probability of observing the new evidence assuming that the Earth is round, and a 62.5% probability of observing it regardless of the shape of the Earth, then, the probability that Earth is round has risen to 60%:

50% x (75% / 62.5%) = 60%.

Bayes’ Theorem’s power arises when one iterates it many times, modifying an a priori probability with the results of new evidence and thus producing a new a posteriori probability, which will become the new a priori of the next iteration. Mathematicians proved that regardless of which was the very first a priori probability assigned to a theory (say, some distant ancestor thousands of years ago gave 0.01% to the probability that the Earth is round), the sequence of posterior probabilities converges to the true certainty that the theory we seek to test.

Of course, the challenge of using Bayes’ Theorem is to quantify the expressions that appear in its formula. However, at least as a metaphor, Bayes provides a program for building knowledge based on new evidence. We can incorporate satisfactorily the evidence that we obtain about the phenomena of nature, of society, and ourselves into the model we have of reality, even in the presence of uncertainty. 

We have not escaped Professor’s Williams predicament about knowledge. In most cases, the evidence fueling Bayes’ Theorem will consist of stories that we read in newspapers, books, or specialized journals. We may not trust the sources completely, but at least we know we have a way to cope with their uncertainty. At the end of the day, we should not end up being so baffled as Mrs Pysner was in that classroom.

From my shore, the probability that the Earth is round is very high. And very high is the probability that man has reached the Moon. And that the Earth is four billion years old. And that evolution by natural selection is a valid theory. And that global warming is happening. And that the pharaohs Khufu, Khafre and Menkaure ordered their subjects to build the pyramids. And that 5G technology has nothing to do with the coronavirus. And that vaccines help prevent disease. And that homoeopathic remedies are no better than a placebo. And that Trump lost the 2020 elections.

For all these examples, the accumulation process of evidence transformed my priors into very high posterior probabilities. My certainty about them is sufficiently high that I can firmly say, “I know it”.

Those who like to entertain conspiracy theories tend to overplay a handful of bits of trivia that they read in some online forum (“How can the flag that Americans put on the moon fly if there is no air there?”, “Why we never went back to the Moon?”). For them, it becomes incontestable evidence, regina probatoriumprobatio superlativa, major proven quam omnes aliae probationes, that should throw down the house of cards they think our knowledge is supposed to be. However, they do not know that the process of changing our paradigms requires providing evidence systematically, one after another and after another.

The shape of my knowledge

And although the probabilistic logical system offers a framework to deal with the uncertainty that prevails in our understanding of reality, I must admit that the description I made of it in the previous section creates a somewhat sanitized image of our knowledge. It would seem that through experimentation, we can build clean and well-separated chains that are linking new elements of evidence until we achieve a lucid image of the phenomenon we want to analyze.

This representation is entirely inadequate to me. Knowledge is not a bag full of pieces of information, nor is it a house of cards, nor is it a collection of disjunct chains that connect ignorance with the truth through evidence links.

To me, knowledge rather looks like a beautiful tapestry, weaved somehow haphazardly with the stories we have been told and our personal experiences. One of the messy threads in my own tapestry of knowledge, for example, connects the roundness of the Earth with the gravitational fields, these with the Christoffel symbols, and these in turn with Strasbourg, the European Parliament, the western front of the Second World War, the uranium mines of Namibia, the Bushmen of Africa, the Y-A chromosome haplogroup, The Eagle pub in Cambridge, the rowing regattas in the River Thames, the love I once had for a woman in a distant past. An infinity more threads similar to this run through everything I know, everything I have experienced and everything I have felt.

A new piece of information can alter the tapestry’s appearance, not only in a single specific point but in all those with which it connects. That is what makes it fragile, and that is what makes it strong.

But, if our knowledge brings together these fibers surrounded by uncertainty and probability, what is the role played by the concept of possibility?

Entering the universe of possibilities must be done with the care of an explorer visiting dangerous lands. However, we humans do not heed those dangers, and we routinely let our beliefs wander in that strange space. Unlike what is probable, the universe of the possibilities is a gargantuan, aberrant, and monstrous place, where reality – and realities – are just wandering wisps floating without direction.

Let’s briefly enter that universe.

To be continued.

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