Athens, Digital Signatures, and Epistemology
Plus artificial borders, conjectures & refutations, notation...
I was recently in Rome celebrating Christmas. It is a haven of ancient architecture, sculptures, and art. But out of all the paintings I glimpsed, this fresco — Raphael’s The School of Athens — captivated me.
In it are the who’s who of the ancient world: Pythagoras, Socrates, Plato, Aristotle, Ptolemy, Euclid and more, debating and discussing ideas in a mythical university.
Immediately, my mind began teeming with questions. How did Athens become such an intellectual juggernaut? How did Athens produce a stream of legendary philosophers and mathematicians?
To find out, join me in a journey of discovery — a journey that will cover digital signatures, Karl Popper, artificial borders, and much more.
A few months ago, I found myself reading about digital signatures and cryptography. I marvelled at the sheer efficiency of the concept: the digital signature is mathematically linked to the message being signed. Here’s how it works: suppose Mr Spock wants to sign the message ‘live long and prosper’ (to an exceptionally fortunate recipient, needless to say!) He will apply a hash function like SHA256 to generate a digest, then use his private key to encrypt the digest, creating a digital signature. The recipient can decrypt the signature using Mr Spock’s public key for verification (see below).
Note that since the digital signature is a one-to-one mathematical function of the message, the signature itself changes for a different message. This makes it difficult to tamper. Contrast this with conventional ink-on-paper signatures, which are the same (invariant) for all messages.
Turns out, this insight is crucial to understanding the seemingly unrelated prosperity of ancient Athens. How so? Let’s find out!
[R2-D2 and C-3PO are droids, engaged in an eternal battle of wits.]
C-3PO: R2, I’m afraid I don’t see how Athens has anything to do with digital signatures! Are you sure you are not malfunctioning?
R2-D2: Oh, ‘3PO. Be patient, my dear friend. Good things take time. A journey of a light-year begins with…
C-3PO (cuts him off): Wise aphorisms, you have begun speaking. Trying to imitate Master Yoda, are you?
R2-D2 (ignores accusation): Say, do you see how digital signatures are closely tied to the message? Do you see how ink-on-paper signatures are the same regardless of the message being signed?
C-3PO: I do.
R2-D2: Excellent. Now, make a similar distinction in explanations. ‘Sound’ explanations are ones that are closely tied to the underlying phenomenon they seek to explain. Au contraire, ‘unsound’ explanations are decoupled from the phenomenon in question.
C-3PO: Wh-what?! You know I have trouble comprehending highbrow comments like that one.
R2-D2 (chuckles in Morse code): Oh, ‘3PO. You know I can’t resist making them! Anyway…do you ever wonder why seasons occur? Some people once suggested the sun’s anger causes summer. Now, such a proposition can quickly be taken apart by simply asking ‘why anger and not fear?’ The sun’s emotions, if it has any, have nothing to do with why we experience summer!
C-3PO: Hmmm. I see. In fact, those who promote ‘unsound’ explanations can easily re-package their statement once refuted. They can change ‘fear’ to ‘envy’, then ‘hunger’ and so on. But what if I suggest that Earth’s axis of rotation is at a tilt with respect to our plane of rotation around the sun? Then, if proven false, I have nowhere to hide! My reputation is on the line. I have skin in the game; or, in our case, metal in the game. Am I right?
R2-D2: You are! I must admit, ‘3PO — conversing with me over the years has made you smart!
[In a cruel irony, the summer heat causes sweat to trickle down the droids’ metal surfaces. The impurities in sweat being electrical conductors, both droids experience mild shocks and must rest before they can speak again.]
Now, it is crucial to understand that unsound explanations, being decoupled from the phenomenon, can apply to anything. They are one-size-fits-all (OSFA).
The Perils of OSFA: Black Wednesday
Let’s rewind the clock to 1992. Europe has adopted the exchange rate mechanism: a pegging of European currencies at fixed values, within a certain band. Post the fall of the Berlin Wall, Germany’s Bundesbank is maintaining high interest rates in anticipation of inflationary pressures. As a result, the deutsche mark is appreciating against other currencies.
The United Kingdom, on the other hand, is in a recession. The British central bank wants to cut interest rates to revive the economy, but cannot do so because the pound sterling is near the lower end of the permissible band against the German deutsche mark. The Bank of England is forced to intervene and purchase pounds from the open market, depleting their forex reserves in the process.
Long story short: havoc ensues, hedge funds make a killing, and the UK is forced to exit the ERM after raising interest rates by 5 percentage points in a single day.
Margaret Thatcher had recognised the instability inherent in the ERM. As she says in this parliamentary speech from 1990-
…there is no way in which there can be a single currency until all economies are at the same state of development, the same state of prosperity…
We learn that sound macroeconomic policies tend to be tailored to each country’s particular conditions (demographics, cultural aspects, development stage etc.) rather than OSFA.
The Perils of OSFA: Artificial Borders
For a fascinating example (my favourite) of the unsoundness of OSFA, we must transcend disciplines and enter the domain of geography.
What exactly is the Middle East? As Tim Marshall writes in Prisoners of Geography-
The Middle of what? East of where? The region’s very name is based on a European view of the world…
During the First World War, the Ottoman Empire — comprising most of today’s Middle East — was fighting against France and Britain. In 1916, British diplomat Mark Sykes and his French counterpart François Georges-Picot pulled out a map and did something that continues to cause havoc more than a century later: drew a line.
To be precise, they drew a line to the split the land, should they win the war. When European officers sitting in a posh office somewhere arbitrarily draw lines on a map to create national borders, you don’t need to be a genius to figure out what happens.
What happens when people from different communities and religions are forced to live together? What happens when they need a passport to meet a relative, just because apparently they are crossing a freshly-invented border?
It’s not that hard. But guess what? That’s kind of — at least in part — the story of Africa too. The Europeans overlooked the fundamental truth that sound, natural borders must consider local intricacies like mountains, rivers and demographic factors.
By the way, this is why many artificial borders tend to be straight lines. The concept of fractal dimensionality can be used to quantitatively analyse the soundness of borders, but more on fractals in a separate post. In the meantime, here is a wonderful paper for those interested.
Our discussion so far has revolved around explanations and their soundness. But let’s take a step back. Surely explanation is not the only form of knowledge, right?
Let’s take another look at the School of Athens, shall we? In the centre, under the arches, stand two iconic figures: Plato on the left (in pink) and Aristotle on the right (in blue).
Take a closer look at their hands. Plato points upwards — some say this symbolises his preference for theory and pure mathematics — while Aristotle faces his palm downwards, indicating his emphasis on the practical.
A key dichotomy presents itself whenever one thinks of knowledge, namely the distinction between ‘pure’ and ‘applied’.
Pure knowledge (‘P’) relates to the discovery of an underlying law of nature — say, the laws of thermodynamics — which in turn governs what transformations are and aren’t possible (for instance, the ruling out of perpetual motion machines.) Applied knowledge (‘A’), on the other hand, consists of a set of instructions for carrying out these transformations (for example, how one can use the laws to design a diesel engine).
Understanding the interconnectedness between P and A is an illuminating experience. Keen readers of Gödel, Escher, Bach will point out that the said relation is an exotic isomorphism. Not to worry, since R2-D2 and C-3PO are back to the rescue!
C-3PO: Exotic isomorphism. What an exotic phrase! R2, would you care to impart your knowledge on this matter?
R2-D2: Sure. Before I do so, can I interest you in some snacks? Perhaps we can enjoy some Wookie Cookies? (C-3PO nods affirmatively. R2-D2 opens the cookbook and places the recipe in front of a scanning device, connected to a robot that presents ready cookies in precisely 101 seconds.) Now, you see, our inanimate robot friend here has transformed the recipe instructions into these delicious cookies. Such a transformation or mapping, one that conserves information, is called an isomorphism.
C-3PO: I see. So the mapping of P onto A is an isomorphism?
R2-D2: Spot on! You must recognise, ‘3PO, that P serves as the epistemic base for A. Suppose you wish to construct a lightsaber someday…
C-3PO (interrupts excitedly): As a matter of fact, I do!
R2-D2: All the more reason to investigate! As I was saying, to build a lightsaber, you must first acquire the required knowledge of electromagnetism, lasers and so on. You see, the probability that robots like our cookie-machine are created in a society without knowledge of mechanics is negligible. Such inventions, if any, would be based on chance and accident alone, and would be one-off rather than sustained.
C-3PO: I must confess R2, now that you enumerate everything I need to learn, my hopes of building a lightsaber are dashed. I see your point, though. A wide knowledge base of P is a prerequisite for sustained technological innovation. But I have an observation to make. Surely, once I educate myself about the concepts you suggested, a lightsaber is not the only thing I can make, right?
R2-D2: I’m impressed, ‘3PO. You catch on quickly. You are absolutely right: a single idea within P can serve as the epistemic base for multiple A. With your soon-to-be-acquired knowledge, you can host an interplanetary light display, among almost-countless other things. On the other hand, a single A is likely to draw on multiple P. Knowledge of a single subject matter will not suffice in your quest to build a lightsaber.
C-3PO: Eureka! This intricate connection means that greater P creates exponentially more opportunities for A. Or what our friend from the neighbouring galaxy, Droid Econ, might call ‘increasing marginal returns’.
R2-D2: Well done my friend!
C-3PO: Oh, I couldn’t have done it without you…But I just realised, you didn’t clarify the ‘exotic’ part. Would you do so now?
R2-D2: Very well. You see, the earlier mapping from recipe to cookie was a rather dull one. It merely involved the robot doing some mechanical processes without any creative input of its own. This might be called a prosaic isomorphism. On the other hand, mapping P onto A requires a creative exploration of how theories can be applied. Which is why it is called invention!
C-3PO: Hmmm…So while P contains the knowledge required for A, one still needs to extract this knowledge.
R2-D2: Exactly. While a large epistemic base of P creates the potential for A, it does not, in and of itself, guarantee that inventions will occur.
C-3PO: Then what does?!
R2-D2: External changes. More precisely, changes that improve the ease of access to P. Changes that lower the costs of accessing P. Incentives play a key role, as do institutions that encourage and enable enterprise.
[At this point, unable to resist their temptations, the droids went off to watch Star Trek: Enterprise!]
Of the various changes that facilitate P —> A, notation is particularly interesting. Notation comprises methods to represent P, for instance coordinate systems. Even though P is notation-invariant, well-chosen notation can greatly improve ease of access to P, e.g., how assembly and high-level languages have helped popularise computer programming.
I like to categorise these into prosaic and exotic notation. Prosaic notation is mechanical, often a mere ‘gimmick’ for abbreviation, to avoid writing convoluted or long expressions repeatedly; e.g., writing (cos θ + i sin θ) as (cis θ).
Exotic notation is creative, a different way of expressing P altogether. For instance, complex numbers can be written in the form a + bi in Cartesian coordinates, but can equivalently be written as [r(cos θ + i sin θ)] in polar coordinates, and [r(e^iθ)] in exponential form. Likewise, one can think of Newtonian, Lagrangian and Hamiltonian formulations of classical mechanics as exotic notations.
Under some conditions, it seems like prosaic notation can somehow extend beyond mere abbreviation. Consider Poisson brackets; meant to be a simplification to avoid writing a long expression of partial derivatives, they end up being unusually useful. To fully understand what I mean, I would highly recommend watching this lecture by Leonard Susskind. How and why exactly this occurs is somewhat unclear to me: perhaps by writing convoluted expressions in a simpler form, the saving of labour enables us to see ideas we might not have seen otherwise?
While we are on notation, it is worth mentioning the differences in ‘obviousness’. For instance, De Moivre’s theorem says that a complex number (r, θ) raised to the power n is equal to (r^n, nθ), i.e., radius gets exponentiated, and angle multiplies. Leonhard Euler recognised that this is similar to the behaviour of exponential functions.
Enter the magic of notation: when expressed in exponential form, e^iθ = cos θ + i sin θ (this follows from a Taylor/Maclaurin series expansion of sin, cos, and e^x, which is beyond our current purposes.) De Moivre’s theorem, which can be slightly tricky to prove otherwise, is plainly obvious in exponential form (I encourage you to give it a try!)
The following excerpt from GEB seems uncannily similar-
Programming in different languages is like composing pieces in different keys, particularly if you work at the keyboard … certain kinds of figurations “lie in the hand” in one key but are awkward in another … This shows how a notational system can play a significant role in shaping the final product.
Likewise, The Selfish Gene has this apt quote-
Often the most important contribution a scientist can make is to discover a new way of seeing old theories or facts … a change of vision can, at its best, achieve something loftier than a theory. It can usher in a whole climate of thinking, in which many exciting and testable theories are born, and unimagined facts laid bare.
I have long been fascinated by questions like ‘how is knowledge created?’
The process of knowledge creation has twin drivers: conjecture and criticism (C&C), or what Karl Popper might call ‘conjectures and refutations’. Conjecture involves guessing new ideas: this makes knowledge creation a creative process. Criticism involves the judging of these ideas against certain criteria.
Way back in 1880, William James, a Harvard professor — the first to teach a course in psychology — noticed parallels between biological evolution and C&C:
…the new conceptions, emotions, and active tendencies which evolve are originally produced in the shape of random images, fancies, accidental out-births of spontaneous variation in the functional activity of the excessively instable human brain, which the outer environment simply confirms or refutes, adopts or rejects, preserves or destroys, - selects, in short…
Lest one stretch this analogy too far, though, a key distinction is needed: that between random (‘R’) and deliberate (‘D’) processes.
In 1943, Salvador Luria was researching how bacteria developed immunity. At the moment, however, research was far from his mind. He was teasing a colleague — a friend trying his luck at a slot machine during a faculty dance — about his inevitable losses. In that moment, the penny dropped.
At the time, it wasn’t known whether bacteria acquired immunity to viruses and antibiotics through a process within the bacteria, or due to genetic mutation. How might one devise an experiment to distinguish the two possibilites?
Luria realised that if the former was true, the proportion of resistant bacteria would be roughly same in all samples. On the other hand, if mutation was the cause, the presence of resistant bacteria would be highly uneven: like a slot machine’s payouts. This insight earned him the Nobel Prize.
We learn that random mutation causes differences in genetic characteristics within a species: some individuals are now better suited to survive and reproduce (better ‘fitness’.)
Curiously, the difference between R and D extends to their selection stages. D selects ideas on the basis of predetermined criteria like whether or not the idea is a ‘sound’ explanation, or whether or not the idea’s predictions are falsified by experiment. (Being an optimistic fallibilist, I prefer the phrase ‘not falsified’ to ‘validated’ or ‘confirmed’.)
On the other hand, R selects on the basis of an idea’s ability to replicate itself. Often — though not always — the ideas that will best replicate themselves will be those that also pass certain criteria. For instance, variants of a gene that provide their possessor certain benefits, like greater resistance to a disease, are likely to be more successful replicators (but only over a long time period, if at all).
Interestingly, A/B testing seems to be a deliberate process: a digital equivalent of selective breeding. It involves first creating several different versions of a particular webpage; for example, pages with slightly different pictures, colours etc. Incoming users are then assigned to these various pages, and the differences in metrics monitored to optimise for the website’s criteria. For instance, a charity’s website might test out buttons that say ‘donate’ v/s ‘contribute’, then split visitors into two groups (one for each version), monitor the average donation per visitor for these two groups and select the higher one.
Fascinatingly, the themes of R and D have parallels in portfolio dynamics for investors.
In the 1950s, Robert Kirby of Capital Group had a curious experience with a female client. Her husband, who had passed away recently, had created a portfolio by buying $5,000 worth of every stock that Capital recommended and then leaving it untouched.
When Kirby reviewed this portfolio, he found that “[the gentleman] had an odd-looking portfolio. He owned a number of small holdings with values of less than $2,000. He had several large holdings with values in excess of $100,000. There was one jumbo holding worth over $800,000…”
What is evident from this story is that by virtue of differential share price performance over any given time period (say, 10 years), the winners will automatically become a larger proportion of the portfolio.
It is worth noting that the largest allocation at the end of the time frame will be in those stocks with the greatest price appreciation: namely, the best replicators!
Note too that as is characteristic of replicators, the most successful stocks are likely — though not necessary — to be those with the best underlying business performance. (For those interested, here is a wonderful read.)
All the same, portfolio decisions can also be of the D type. By assembling a set of initial companies, then systematically intervening to ‘cut the weeds and water the roses’, one can iterate over time to build a portfolio filled with leaders.
A question that begs asking is whether the best replicators must necessarily be the most ‘advantageous’ ones, given a long enough time frame? (Of course, the definition of ‘long enough’ will vary according to context.) I don’t know the answer as of the date of this writing, but am curious to find out. Perhaps evolutionary game theory can provide some clues?
Using microeconomics as a toolkit, one can conceive a ‘market for ideas’ inspired by Joel Mokyr.
Now, interestingly, the market for ideas (like all markets) can have disequibilria. Thought experiment: consider a society where it is hypothesised that laziness is a virtue. This idea is ridiculous, but assume it is accepted widely in the said society. People would stop going to work, exercising etc.
Sooner or later, some people would realise that their ‘laziness revolution’ is doing much harm. But perhaps they think it is because they are not being lazy enough! Indeed, they initially turn even lazier, sleeping all day long.
An economist would instantly recognise and label this phenomenon as ‘self-reinforcing’, much like George Soros’s idea of reflexivity. Surely, at some point or the other, people will realise that they were mistaken to begin with, and that laziness is not in fact a virtue. This way, disequilibria are self-correcting in the long run.
Perhaps you notice something: the feedback loop is paramount. Maybe you are thinking: two societies that both adopt the ‘laziness hypothesis’ may return to equilibrium at different times, if one realises the inefficiency quicker. Shorter feedback loops make for greater progress. Spot on!
To understand the factors that influence the feedback loops alluded to above, one must exit the much-cherished comfort of numbers, and enter the realm of intangibles. Especially enduring intangibles, like culture.
The category differences between Athens and Sparta are revealing. While Athens encouraged skepticism and debate, Sparta forbade it. While Athens embraced fallibility and recognised that knowledge is subject to errors, Sparta believed in absolute, unquestionable truths. While Athens distinguished between ideas based on their inherent merit, Sparta did so on the basis of authority.
In many ways, Athens was an early adopter of what Adam Grant calls ‘rethinking’. In his book, Think Again, he writes-
Rethinking is more likely to happen in a learning culture, where growth is the core value and rethinking cycles are routine. In learning cultures, the norm is for people to know what they don’t know, doubt their existing practices, and stay curious about new routines to try out. Evidence shows that in learning cultures, organizations innovate more and make fewer mistakes.
Athens was dynamic: open to change. Sparta was static: opposing change.
Time and again, across various fields, one finds dynamic ideas being ‘sound’ as opposed to ‘unsound’ static ones.
Theories of Intelligence
Carol Dweck is a prominent psychologist at Stanford University. In a groundbreaking 1988 paper with over 16,000 citations, she explains the ‘entity’ and ‘incremental’ theories of intelligence, or what she has more recently called fixed and growth mindset.
In a fixed mindset students believe their basic abilities, their intelligence, their talents, are just fixed traits. They have a certain amount and that's that … In a growth mindset students understand that their talents and abilities can be developed through effort, good teaching and persistence … they believe everyone can get smarter if they work at it.
People who embrace a growth mindset, or incremental theory of intelligence, have been empirically shown to perform better on cognitive tasks.
Questions like ‘why do some nations succeed while others fail?’ come up all the time. A popular suggestion is that of geography: some countries are endowed with beneficial geographic conditions that make them successful. I disagree. While geography is surely an influence, it is not the determining factor.
The suggestion that geography determines a nation’s success (or talent determines an individual’s success) is tantamount to suggesting that the hand one is dealt in poker decides the outcome. Sure, winning with pocket aces is more likely than with 2 and 7, but it is not certain or predetermined.
Let’s relook the earlier question: geography is a static idea. Put differently, it does not account for change. Geography does not explain why ‘world orders’ change every century or so. Surely tectonic plates do not shift so rapidly! Of course, geography is an influence — and an important one too — but, in my opinion, it is not ‘fate’ or ‘destiny’.
Ironically, fallibilism is linked to an optimistic worldview: by admitting that all ideas are prone to mistakes, one creates the possibility of change and progress through error-correction.
Given that correction of errors lies at the heart of progress, it follows that the deadliest mistakes are those that prevent themselves from being corrected: think Russian roulette, prolonged easy money policies, dictatorships…
The single biggest question that remains unanswered to me regarding epistemology is the following. The statement ‘OSFA and static ideas are unsound’ is itself an OSFA. Does this create a contradiction? Or is this a permissible OSFA? Is it the only permissible OSFA? These questions seem remarkably similar to a quote from one of my favourite thinkers, Kurt Gödel-
All generalisations — perhaps except this one — are false.
The ideas in this post have been significantly inspired by The Beginning of Infinity by David Deutsch, and The Gifts of Athena by Joel Mokyr (I highly recommend both.)
Now, this post itself is a conjecture, and I welcome your criticism so that I can improve my knowledge! :)
Feedback and reading recommendations are invited at malhar.manek@gmail.com
Malhar, thank you for spending time to write such a great piece. I felt I was not able to follow multiple analogies connect as they were ranging a wide variety of topics but after reading them agajn -:) I could follow them. Good luck and keep writing.