Incentives and discoveries

“War and peace” was the Nobel prize lecture by laureate Robert Aumann on game theory, which he delivered also in IISc Bangalore in 2006, of course among other places. I was in the auditorium that day, as a clueless 16 year old who understood nothing. I left the auditorium remembering one statement he made: (paraphrased) “If there is one word that sums up economics, it is incentives“. I did go back and read his lecture much later and found that it was in fact very accessible to non-expert readers and quite insightful. Even today I marvel at game theory — I don’t really understand it, but I know that when I do, it will be profound and life changing.

Why would I believe something will be profound and life changing when I understand it? The answer lies in the universality of it. Physics is universal — if I go out to build a playhouse in my backyard, I am working with the laws of physics. All activities in the real world are under the jurisdiction of the laws of physics. I can do nothing in the real world that isn’t allowed by the laws of physics. I believe that incentives is also universal. For instance, I wouldn’t be building a playhouse if I didn’t have an incentive for it, especially if there are many people involved in building it. No activity can be done unless it is well-incentivised. 

There are things that can be built in principle, but we don’t know how to incentivise people to build them. An old example is the supersonic commercial flight ( see concorde). There were flights, that took just over 3 hours from London to New York. It is allowed within the laws of physics for us to do this, but apparently, it is not incentivisable economically to keep such a service available.  There is an idea called the space elevator. It is an elevator to outer space; just a very, very long cable (100s of 1000s of km) that extends from the earth to outer space and stays upright due to the centrifugal force balancing out the gravity. The laws of physics and the limits of technology perhaps allow this. It can be built in principle, but we don’t know to incentive work towards it. It’s even awkward to say “can be built in principle” (it can’t be built if you can’t incentivise it!).

This brings us to the question of fundamentals: Are economic limitations as fundamental as physical ones? That is, if building some device or accomplishing a task would have to violate a law of physics, we say ‘it can’t be done even in theory‘. Should we say the same if it violates a law of economics? There are a couple of hairs to split here. First, if a task doesn’t violate any law of physics, but we don’t have the necessary technology, we say it could be done in theory, only to me limited by technology, and technology evolves with time. This is different from saying that it cant be done even in theory. However, one has to note that theory evolves in time, too. Similarly, among economic impossibilities, we want to distinguish between tasks that can’t be incentivised in practice today and those that cant be incentivised even in theory. So now back to the question, are the economic impossibilities as fundamental (or more) as the physical impossibilities?

Something about this question makes me uncomfortable. It is the following: There are theoretical physicists who think about what would happen in the universe if a particular law were to be violated. So indeed, the work in this space is not constrained by the laws of physics. However, even these theoretical physicists are working under the laws of economics. For instance, such a field of study wouldn’t exist, if there wasn’t enough incentive for it. So it seems we maybe able to escape from the laws of physics, by sitting inside a theory room and theorizing as we wish, but even that activity can’t practically escape the laws of economics — it can be done only if it is has a sufficient incentive, in the form of intellectual interest, for example. This would lead us to conclude that the economic impossibilities are more fundamental. And that’s uncomfortable for me because, one of the reasons why I chose to do physics is that it represents the reality that cant be escaped from.

There are some amusing consequences of the difficult-to-incentivise tasks. When faced with a hard technological challenge, it’s a common practice to start with a simpler problem. However, it may not be easy to incentivise efforts towards the simpler problem (some would say it’s not really simpler in that case). For instance consider the case of colonising the moon. It is perhaps easier than colonising mars, but apparently, it is easier to incentivise large scale efforts towards the latter. It is probably related to the psychological impact of colonising a place that feels like it is outside the realm of earth.

A couple of years ago, I wrote a post about Finding a stone in the darkness, which was also on this topic. It was about the events surrounding the discovery of the famous Gaussian (i.e., Normal) distribution, particularly focusing on the incentives that led to this discovery. In short, the Normal distribution was develop by Gauss in order to predict the position and therefore discover a new “planet”, Ceres. At that time, the Normal distribution was nothing more than a mathematical jargon that not many people cared about; they cared about Ceres. It was Ceres that captured the imagination of people, ( astronomy has been very captivating to the public and continues to be so) and therefore generated incentives for work in this direction. A by-product of this work was the Normal distribution, which later tuned out to be vastly more impactful than the discovery of Ceres itself.

When you spot a Stag

When WhatsApp released the blue-tick read-receipt feature, it was followed by jokes about seemingly ridiculous extensions of the feature. If I send a message to you, the blue tick tells me that you saw the message. In its next feature, WhatsApp can perhaps send another message to you saying that I saw the blue tick so that you know that I know that you received the message. And then it can send another message to me letting me know that you know that I know that you received the message. And this can continue ad infinitum. This apparently funny and benign hierarchy of information is actually at the heart of an unsolvable problem known as the coordinated attack problem. It has a few strong consequences;  it partially explains why we have open secrets that everyone knows yet no one acts on it; and why people tolerate bad authorities for a long time even after everyone knows that the authority is bad and should be ousted, and why do sociological changes seem to take a long time. At a deeper level, this information hierarchy is central to how values are formed and destroyed in human societies. The blue-eyed islander puzzle illustrates how complex this hierarchy can be.

In game theory, if you and I know something, it is called mutual knowledge (of first order) between us. If I know that you know and you know that I know, it is called mutual knowledge of the second order. And If I know that you know that I know and you know that I know that you know :D, it is mutual knowledge of the third order. This can be confusing, but think of it in the context of a card game, assuming that we are talking about knowing what card I have. Each layer of the information Hierarchy has a different implication on the game — they are all different. At this point if you are reminded of a sequence from the TV series F.R.I.E.N.D.S, yes, that is exactly what I am talking about. The infinite-order mutual knowledge is called common knowledge. If I have a piece of news X, and I call up 10 of my friends individually and give them the news, then it is a first-order mutual knowledge of these 10 people. Each of them know, but none of them are sure if the remaining 9 know. Instead, if I bring them all in a room and announce it, then X is a common knowledge of these 10 people. In this case, they all heard it together, so they all know, they know that everyone else knows, they know that everyone knows that everyone else knows.. you get the point. If you have attended conferences, you might think about the difference between a talk and a poster in this context.

Consider a village with 100 people and a village head. Let us suppose that the village head is bad in some way and needs to be ousted. Let us also assume that this requires all 100 people to revolt at the same time and if any one of them doesn’t revolt, the revolt will be suppressed. Practically, it is more likely that only about 50 of them need to revolt, but we can go with this for the purpose of argument. What level of mutual knowledge of the fact that the village head is bad do we need here? If everyone individually knows that the head is bad, that is not really sufficient for a revolt — no one wants to be the only one revolting. Therefore no one revolts, unless they are certain that everyone else will revolt too. Therefore, everyone needs to know that everyone else knows that the head is bad. And, to be sure that everyone else revolts, everyone needs to know that everyone else knows that everyone knows that the head is bad and ad infinitum. Strictly speaking, no finite order of mutual knowledge is sufficient for a revolt. This is the coordinated attack problem. Practically, revolts don’t happen until the mutual knowledge has reached a sufficiently high order. The coordinated attack problem is often presented with the two-party example: two military convoys coordinating an attack from different places. This will need the information of the attack time to be a common knowledge between the two. These examples are interesting, but the principle is far more general — it applies to any action that requires a coordinated effort. Information wouldn’t be actionable unless it has reached a high order of mutual knowledge. Indeed, an effective way to control riots is to prevent public gatherings — which only prevents information from being shared at higher levels of the hierarchy, while allowing it to spread at lower levels.

One can imagine that the villagers not revolting is an equilibrium, that keeps things stable; all villagers revolting is also an equilibrium, perhaps a more desirable one. But only some of them revolting is a non-equilibrium state, worse than both the equilibria. This is a system with two local equilibria, much like the Stag hunt problem in game theory. In the stag hunt problem, two hunters, with an arrow each are approaching a pair of rabbits from opposite directions. They can’t communicate directly. If they both target the rabbits, each of them will get a rabbit. However, one of them spots a stag, for which, they need 2 arrows, i.e., both of them need to shoot simultaneously. A coordinated attack on the stag is obviously better than carrying on with targeting the rabbits; however, this needs a common knowledge of the presence of the stag. It is not sufficient even if both the hunters spot the stag; each of them needs to know that the other has seen the stag; and that the other knows that they have spotted the stag and so on. If only one of the hunters attacks the stag, then it runs away and the hunter gets nothing. 

We can generalize the stag hunt problem to n hunters with one arrow each targeting n rabbits, and a stag walks in where one needs to shoot m arrows. This will mean that shooting at the stag is desirable only when there is a subset of at-least m hunters for whom the presence of the stag is common knowledge.  The above two examples appear to be about revolts and stag hunting; however, the former can be thought of as a representation of a society destroying an old value and the latter, as a society changing a value. 

Finding a stone in the darkness

In January 1801, an elated astronomer, Joseph Piazzi announced that he had discovered a piece of stone, possibly a new planet in the night sky. He had tracked its location for 41 days after which he said it disappeared from the night sky; perhaps it had moved along its orbit to the other side of the sun so that it is now in the day sky, not to be seen.

The response from astronomers, none of whom other than Piazzi had seen this planet, was an admixture of excitement, skepticism and anticipation. From the observations for 41 days by Piazzi, which included only 21 points in the sky, it could be concluded that the planet lives in between Mars and Jupiter and that it may visit our night sky again in about an year. But where exactly in the night sky and when exactly after an year? When and where must an astronomer point his telescope in order to see this planet again? These questions had no answer. Astronomers feared that the piece of stone may disappear into darkness for eternity.

Intense attempts to calculate the path of the planet followed. The fundamental problem was to find a path, a curve of some sort, like an ellipse that passes through the 21 points. The difficulty, however, was that those known points covered a mere 3 degree arc and were subject to significant errors. No one knew how to solve this problem.

The excitement around this problem infected onto a 24 year old mathematician. This young man set aside most other commitments to solve this problem. Almost an year later, on December 7, 1801, an astronomer pointing his telescope where the young man told him to, spotted a faint object. It was later confirmed that this object is indeed the planet they were looking for.  They had found the piece of stone in the darkness.

This piece of stone is now known as Ceres, the largest among the asteroids, not a planet. The young man was Carl Friedrich Gauss. 

Along with the stone, two other bright objects were discovered. One is the now-famous Gaussian distribution, which Gauss developed to model the errors in the observed data points. The other is the least-squares curve fitting technique, which Gauss derived from the Gaussian distribution of error.  One curious question here is, is it an accident that the endeavor of finding a stone in the darkness led to revelation of such universal mathematical concepts? Perhaps the difficulty in finding Ceres in the sky was indicative of a deficiency of the day’s mathematics. The problem was only superficially about finding a stone in the darkness; lurking underneath, was a fundamental mathematical problem that we didn’t know how to solve. More generally, lurking beneath every natural event that we can’t predict, there is either a  fundamental physical law that we don’t know or a mathematical problem that we can’t solve.

There is a second interesting feature of this event. What was the value of this endeavor from the perspective of Gauss and the astronomers? A success in this endeavor would mean finding the stone in the darkness; a failure would mean losing the stone. What does any of this mean? What do we gain if we find the stone and what do we lose if we don’t? From the perspective of Piazzi and many other astronomers, it was of profound value to make new friends in the sky, who can help them navigate the sky in the future. In this case, the end product of this endeavor was  to rediscover Ceres in the sky and track her location, so that they can keep in touch with her. From the perspective of Johann Bode, another astronomer, the hopeful end product of this endeavor was to show that God indeed didn’t leave a large gap between Mars and Jupiter; he placed Ceres appropriately in between, in accordance with Bode’s law. However, as Gauss mentions in his book, from his perspective, the value of this endeavor lies in establishing the importance of the mathematical problem that lies beneath. The end product was to solve this mathematical problem. Ceres was a piece of stone to some; she was a friend in the sky to others and to Gauss, she was a messenger of a mathematical treasure.

Interestingly enough, Gauss had considered the problem of predicting the orbit of a planet starting with very few points known, as a mathematical curiosity even before he became aware of the case of Ceres! It was usual for astronomers to find orbits of comets with very few points known. However, they could make a variety of intuitive assumptions about the orbit, making the problem simple. Gauss, as any mathematician would, naturally sought to solve the most general version of this problem, with no assumptions about the orbit. However, he anticipated that astronomers would overlook this problem considering it to be a mathematician’s fantasy with no practical value. The case of Ceres, as Gauss himself notes, was exciting to him because it inflated the value of this problem among astronomers.

Of the discoveries made during this episode, Ceres is perhaps the least remembered today. However, she assumed the primal position in 1801, during the event. The endeavor was powered by the value attached to finding Ceres. Natural sciences are about knowing our home, the universe, better. I believe that mathematics, in an unusual way, is also a part of the physical universe. The more we explore the latter, the more we will learn about the former.

Before concluding, I want to remark on the role of risk and trust in availing the value of exploration. Scientific research is an exploration. Like all explorations, it involves, at-least, the inevitable risk of not finding anything. By its nature, the outcome of a scientific exploration can’t be evaluated by someone who is not an expert. In other words, the question of whether a researcher has been honest in doing what they should do can only be answered by another researcher sharing the same expertise, and perhaps the same source of income ;). Therefore, such explorations can be carried out only when the explorers are trusted. Scientists are trustees. The story of the discovery of Ceres, however, was remarkable. With no knowledge of its details, Gauss’ methods were validated by the astronomers. In fact both the fields of astronomy and mathematics were mutually validated by each other.

Possibility and Reality

I remember curious conversation about ghosts with my little cousin, a few years ago. I was telling her that ghosts are not real. So she asked me, “why is there the word ‘ghost’, if there are none?” That was a smart question from a seven year old kid. I didn’t want to discredit her intellectual curiosity by giving a silly/funny answer; I wanted to give an actual answer. So I tried to articulate what I knew in a simple way so that she could understand. I do not remember what I said, but I think I didn’t succeed in communicating it to her.

In hind sight, I can think of one way I could have answered that question. In Sanskrit, there is a word called Bhūta. This word has been adapted in many Indian languages. It has two meanings: ghost and past. So I could have said, some people have had fearful experiences in the past, and when they recollect the terror from the past, they represent it by a scary Bhūta. I am not aware of an etymological connection between the two meanings of the term, but it is at least a convenient coincidence. In hind-hind sight, I can think of a better way :D. I could have said, in a majority of the stories and movies that feature a ghost, the ghost is a person who is dead, and had suffered injustice in the past. The ghost appears in order to avenge the suffering; it is a personification of terror.

This question is in fact a very broad one. The vocabulary of human languages are much bigger than what is necessary to talk and think about objects and events that are real. Why do we want to talk and think about objects and events that are not real, and further treat them as if they were real? An offshoot of this question is the one concerning god, which was a topic for a different blog  post.

A quick answer is that imaginary situations are a necessary part of thinking, in general. Events or situations that are not real, often appear in logical arguments. For instance,  “If X was alive today, Y would have happened” is a common example of what is known as a counterfactual proposition. Such propositions attempt to analyze our understanding of reality, based on an admittedly counterfactual premise. The conclusions are often unverifiable — there is no objective way to test if Y would have really happened if X was alive today. This makes such arguments vulnerable to abuse; for instance, in a political discourse, such arguments are merely another form of rhetoric. However, from a more philosophical perspective, the thinking that goes behind every decision making process does involve a counter factual proposition of some form. Indeed, the process of thinking, fundamentally involves considering events that may not be real. While this argument validates the need for considering such counterfactual propositions, it does not fully answer the  question at hand, for it leaves out another kind of counterfactual propositions — those with a surreal premise.

This is a stronger version of the kind of counterfactual propositions that I discussed above. The example I gave in the previous paragraph started with a premise that is  not factually accurate, while respecting all the laws of thought and all the natural laws— it could have been real. However, one can start with a premise that violates certain natural laws; for instance, assuming that fairies or demons or something supernatural exists, and carrying out an analysis while still respecting the laws of thought. While such arguments can be logical, their conclusions can not be real. They can not be connected to elements of reality. Indeed, one can build a theory that disregards the laws of physics, but nevertheless it is perfectly logical, i.e., it respects the laws of thought. Most of mathematics is constructed this way — with no regards for natural laws and a good part of theoretical physics is also constructed this way exploring possibilities that are not realities.  We may term situations that are illogical , i.e., that violate the laws of thought as impossible; those that violate the laws of physics, but respect the laws of thought as possible and those that respect the laws of thought and the laws of physics as real.

I want to point out that it may appear unreasonable that we want to call those situations that disregard the laws of physics but respect the laws of thought as possible. Shouldn’t we call them impossible simply because they disregard the laws of physics? This is the perspective taken commonly in experimental physics. Mathematics and experimental physics, in general, do not agree on the definition of the possible. In mathematics, possible is anything that can be conceived without logical contradictions. In experimental physics, possible is anything that can be conceived without logical contradictions, and conforming to the known laws of physics. The classic example, that puts this conflict of definitions in the spotlight is the notion of virtual displacement in Lagrangian dynamics, in classical mechanics. It has the reputation of being an elusive concept among students, for reasons that are perhaps related to the conflict of the two perspectives mentioned above. The Lagrangian is a functional defined over the phase space coordinates and its integral along a path in the phase space is known as the action corresponding to the path. The notion of virtual displacement appears when we minimize the action among all paths connecting two fixed endpoints in the phase space. In mathematics, this is done by the stationary conditions — the path that minimizes the action has the property that when it is perturbed, keeping the Lagrangian the same, it results in no first order correction to the action. This perturbation, in physics, is known as a virtual displacement. It is a mathematical possibility, however, under no circumstance can it be a physical reality — the path of a physical system in the phase space can not be changed without changing the Lagrangian. In other words, because this perturbed path violates Newton’s laws it can not be real. The name ‘virtual displacement’ is perhaps a representation of this problem, and possibly a consequence of a debate when the term was coined. There is no physical interpretation for the concept of virtual displacement, which is often a characteristic of ideas from mathematics. Mathematics studies the possibilities; experimental physics studies the reality.

What is the value of thinking about possibilities that have no relation to physical reality? Well, that is the definition of abstraction. For instance, in mathematics, a set is sometimes defined by collecting together different possibilities, most of which are not real.  Abstraction is the precursor of creation; anything that is created existed in the creators’ mind in an abstract form well before it was created. An abstract idea that has no representative a physical object, can guide us to create one. This is perhaps a vague statement, but I will give one simple example. mathematics often has a general solution to a class of problems. The class of problems is an abstract set of problems that belong to the same family. The general solution is an algorithm that works on any member of the class of problems it purports to solve. This general solution, albeit abstract, is indeed a machine; at least the ghost of a machine. Charles Babbage, who was also a mathematician, came up with the idea of the analytical engine, which was a mechanical representation of an abstract algorithmic solution to a class of problems.

The relation between abstraction and creation, I think, is very broad. In this regard, I want to end by remarking that every human creation has an abstract ghost, to be elaborated  in a possible future blog post.

Truth and trust

“Guess what, I have 101325 hair strands in my head” said a friend when I was in elementary school. I looked at him in disbelief and he said “look, if you don’t believe me, count it for yourself” :D. I said I trust him and actually, I still believe he has 101325 hair strands :D. That was a joke. Now let’s get to some serious people who I very much trust and serious claims which I believe in. I am a physicist and almost all of what I know in physics are beliefs, supported by my trust for other physicists of the present and the past. I believe that LIGO observed gravitational waves — I wasn’t a witness when the data was taken, nor did I verify each element of their technical setup. In fact, I don’t even have the technical knowledge to go in and verify an entire setup that big. Indeed, even someone who does would still take an impractical amount of time. The different parts of the LIGO team, sure trust each other. What if, one of the thousand computers they use was programmed to putout any desired data? This of course, is a conspiracy theory, trying to survive upon a Russel’s teapot argument. But nonetheless, the burden of verification, so to speak, is so large that one just has to give up on the verification.  I haven’t verified Young’s double slit experiment, Michelson Morley experiment, etc etc. Even if I did, some of these experiments are too complicated — involve too many components that I didn’t build myself (or watched them built), and therefore trusting another human being is inevitable.  These experiments were done by physicists in the past. I trust them. I believe that they did it. I will argue soon, that these are not “silly” concerns or ones that only promote conspiracy theorists.

Can’t I avoid basing my truth upon trust?. Can’t I do an ab initio verification of every claim that is important to me? Actually I can. Take for example, the Pythagoras theorem. I know a few proofs, and I can decide the validity of these proofs without trusting another human being. More generally, if I come across any mathematical claim, I can, in many cases do an independent examination and decide for myself whether it is true or not. That is the nature of mathematics, the one that differs from experimental sciences. However, mathematicians do base their beliefs on trust, when there is no time to verify each and every claim.  Nevertheless, if necessary, it is not an impossible deal to do an ab initio verification of mathematical claims.

In contrast, in experimental sciences, every generation of scientists will lose their entire life to rediscovering what was already known, if they decide to base their knowledge upon pure evidence and not trust. This is an inevitable consequence of the burden of verification.  So our notion of truth about the physical world appears to be linked to trust between people at a very fundamental level. There is one issue even with mathematical claims; non-mathematicians are mostly untrained to decide the validity or invalidity of a mathematical proof. In fact, even a silly trick proving 1=2 can be hard for a non-mathematician to invalidate. Simply believing that there is something wrong just because the end result is outrageous is not a logical invalidation!. The silly tricks used to prove 1=2 can also be used to prove some less-obviously-wrong, but nevertheless wrong statements and a vast majority of the people would fall for it, if they didn’t trust a real mathematician. This holds for statements regarding anything, including those that I am no expert at. Therefore, contrary to what is apparent, I would become incredibly gullible, if I were to decide the validity of everything I am told, on my own without trusting anyone.

If, someone makes an extraordinary claim (like Einstein was somehow wrong) and bears the burden of proof by providing a 3000 page document, how would I bear the burden of verification? It is easy to disregard a claim and call it cranky simply because it defies something widely accepted, but that is not justifiable by any principle — after all, a popular knowledge isn’t necessary correct. What if the 3000 page document consists of very well framed prolific arguments, but there is a tiny tiny flaw in page 2598 which makes the whole argument collapse?  I have to read through carefully to find such a flaw. Moreover, the overall burden of verification, as I argued before is impossible for one man to bear. So the natural way out is to trust someone who deals with the concerned subject for a living — an astrophysicist, if the claim is astrophysical and a Biologist if the claim is biological etc. This is popularly known as accepting the ‘scientific consensus‘. Not to forget, such a consensus is not fool proof — neither in principle, nor in practice.   In principle, it is very much an act of trusting someone. In practice, the moment a scientific consensus is forming public opinion, political forces will attempt to tamper with it. One can never be sure which one is closer to the truth; is it the existing consensus, or is it an opposition to it?

Political tampering of scientific consensus is perhaps as old as civilizations, science and politics. It opens a set of interesting questions —  how does an individual decide who to trust? how does someone become trustworthy? and how does a society design itself so that the majority always trust the most trustworthy? Here I am not concerned with them; rather, I want to promote these questions up to the philosophical one arising from the inevitable reliance of truth on trust :

Is truth mostly inaccessible to an individual minus the society?

The problem with the word trust is, it requires more than one person. If I was the only one living on this planet, no one else to ask anything, no one to trust, what would “truth” look like? Based on the understanding that truth is only that which I have verified, the above considerations imply that the volume of my truth is limited. There is a price I have to pay to know something is true — the burden of verification. Indeed, this volume of truth is so small that it is almost fair to say most of the truth is inaccessible to one individual minus the society.

One of the somewhat disappointing implications of this embedding of trust in truth is that the advise “don’t accept without questioning” is really, “rethink who you want to trust” 😀 — something much less cooler than the former, but nevertheless a good thing once in a while.  But as I indicated in the first paragraph, the biggest implication to me is the narrowing of the gap between knowledge and belief.

The truth accessible to the individual minus the society is different from the truth we know of — it is primitive and pure. It is pure because it doesn’t involve trust; it is that and only that, which I am sure of, even if I don’t trust anyone else. Of course, one can dissect the “I” further and ask, to what extent can I trust my own sense organs?, but that is a question for a separate blog post; one that addresses the self minus the sense, can be very interesting :D. Back to the truth of the individual minus the society. It is primitive because, its volume is limited by what can be verified by one person — that would include only elementary facts like “I am” etc. :D.

The truth that we know of is sophisticated but impure. It includes answers to very complex questions — ones about a distant galaxy, ones about the minute atoms, ones about the deep sea, all of which have an inherent trust involved because of which, I call it impure. That is not too bad, because, it is possible for such a truth to be actually very close to reality, in the case where everyone is honest.

I will end with a remark on the question mentioned before, of how does a society make sure that what its subjects believe as true are indeed close to the reality. To the least, we can classify the situation into three cases — first, everyone is honest, in which case, what people believe is indeed very close to reality. Second, many people are dishonest, but with different motives; so the resulting contradictions would spread mistrust and therefore decrease the volume of truth that people believe they know. Third, many people are dishonest, but with the same motive — perpetrating a specific myth. That’s the hardest situation because it is indistinguishable from the first case!.

The Seeds of Mathematics

Mathematics is an introspective science, as opposed to experimental sciences(Physics, Chemistry etc). Unlike experimental sciences, progress in mathematics doesn’t require getting data from deep sky, probing into an atom or colliding elementary particles at high speeds. Numbers and other concepts are in our head and all open problems are in our head. This suggests a possibility, that even if we locked ourselves in a cave, cutting off all communications with the nature, we may still be able to continue our progress in mathematics. The subject of this blog post is to examine if  this is really possible.

The converse of this question has a clear answer: Halting mathematical progress would halt progress in all experimental sciences. Mathematics develops ways of thinking, that are employed in understanding the nature. It appears that mathematics is on its own; it doesn’t depend on any of the other sciences. It can continue progressing without any other sciences. This view is expressed in this xkcd comic. But I am going to contradict this view (and I am going to contradict other aspects of that cartoon in my next blog post :D).

Although it is not apparent, progress in mathematics requires us to experimentally probe into nature. The ideas involved in mathematical research are seeded by our experimental probes. Exploration of nature is a key not only to develop physical sciences, but also to develop mathematics, although it is a purely logical subject. A problem is deemed solved, only after a logically consistent solution is found. However, the question “what is an interesting problem to solve?” doesn’t quite have a logical answer.  G. H. Hardy has described  this question to be akin to asking where do poets, writers and other artists get ideas for their work. Conceiving new problems is a work of imagination. And imagination is always seeded by reality.

The classic example for an interesting problem is the one that led to Fermat’s last theorem: ‘are there integers a, b and c such that aⁿ+bⁿ = cⁿ for some integer n?’. Surely, the complete answer to this problem was profoundly useful, only because large number of people worked on it and it developed a great deal of understanding of numbers. However, the problem was presumably seeded by the Pythagoras theorem. We have integers, like (3,4,5) such that 3²+4²=5², so  a curious question like ‘what if we change the exponent to a number other than 2?’, would have been the origin of Fermat’s last theorem.

Another question in mathematical research that doesn’t have a logical answer is “what set of axioms should we choose?”. Axioms, like problems, are chosen by taking a cue from previous mathematical theories. The resulting structural similarity between different mathematical theories has been capitalized in a theory called category theory.

Where do these ‘previous mathematical theories’ get their axioms and problems from? There must be a starting point, a seed for every mathematical idea. These seeds come from outside- from our interaction with nature. Cutting off interaction with nature will cut off the supply of new seeds. But that doesn’t entirely stop mathematical progress; instead, ideas for new mathematical theories will be entirely dependent on the old mathematical theories. Over a timescale of several hundred years, this is a significant setback to mathematical progress. Seeding of mathematics by interaction with nature is a slow process. In fact, we are still benefiting from the seeds of Pythagoras theorem.

The seeds of Pythagoras theorem

‘Geometry’ stands for measurement of the earth. ‘Earth’ here doesn’t mean the planet earth or the globe; it means land; real estate. Geometrical ideas were developed as a result of extensive measurement of land, during early human settlements. The most influential of these was the Pythagoras theorem. Let me go through its development, in its three chronological stages: the content, the statement and the proof, to identify its seeds.

Given two sides of a right triangle, knowing how to calculate the third side is the essential content of Pythagoras theorem. This may be done using a formula, or using tabulated data or using similarity of triangles. All these methods carry the basic wisdom– “the third side of a right triangle is not an independent variable”. Any civilization that built large planned settlements knew the content of Pythagoras theorem.

Explicit statements probably came several thousand years after the content. Early statements of Pythagoras theorem were in terms of areas. There are records of statements in Babylonian scriptures(2000 BC), in the Vedas(Shulva Sutras, 800 BC) and Chinese scriptures. An explicit statement  could’t have brought any change in the applicability. Perhaps, the room/house in which the statements were written was constructed using the content of Pythagoras theorem :D. However, it brought big changes in theory. Geometrical shapes were understood by cutting them into triangles. Triangles were now understood by cutting them in to right triangles. Right triangles took precedence over other triangles, leading to a new branch- Trigonometry.

A partial proof was recorded in 800 BC(Shulva Sutras) and a complete proof in 500 BC(Pythagoras). Presumably, there were unrecorded proofs prior to this. There are some theories that Pythagoreans might have been communicating with Chinese schools of mathematics. The proof was seeded by two pieces of intuition, which were developed when planning settlements. One is that land can be measured in areas, which can be added and subtracted by joining and cutting pieces of land. The second is scaling; a big piece of land can be scaled down and represented on paper or a flat stone. All proofs of the Pythagoras theorem are based on areas of triangles or scaling of small triangles to big ones(A book, The Pythagorean Proposition lists 370 proofs). Scaling is in fact a logical implication of the properties of areas. But it is likely that it was developed independently.

The proof, of course, had no practical implications. Proofs generally store methods of thinking. Even today we feel its impact on our thinking. The proof raised the status of the mere formula, ‘a² + b²=c^2′ to a theorem, resulting in the discovery of irrational numbers. Furthermore, it redefined the whole of geometry in terms of a single quantity- the distance between two points. The more advanced forms of geometry- Riemannian geometry and even Differential geometry contain the germs of Pythagorean distance.

The Pythagoras theorem and all of its intellectual impact on Mathematics are seeded by man’s physical exploration in to measuring land. A writer, within his lifetime travels extensively to gain experiences of reality that can seed his imagination. Mathematics is also seeded by explorations of real world, but this seeding has a longer timescale, much longer than a mathematician’s lifetime.

 

The atheist debate

Debating the existence of God and the relevance of religion is the doorstep to understanding the role of imagination in reality. Imagination is a tool of dynamics of reality- Imagination, shaped by the past of reality, shapes the future of reality. It evolves reality in time.

To explain the above statement with an example, consider a chess game. The board, the pieces and the players are real. The game setup and rules are imaginary. In the imagination, the board is a war-field, each piece is a certain type of warrior, and so on. The future of reality, i.e,  the next move to be made by the players is entirely guided by this imagination.

The sense of loss or win is also determined by the imagination. Losing a pawn is a much smaller loss than losing the queen- although in reality, they are both just pieces of plastic or wood.

God is an imaginary entity. So are the rules of religion and the associated wins and losses, rights and wrongs. In what way does it impact the reality? What is the magnitude of this impact? Is it possible for a civilization to exist without religion?

A civilization without a religion is likely to collapse internally or remain primitive. We could have seen why is this true, if we had a chance to watch the formation of a civilization, and observe how they came up with God and religion.  We can do so, but such an experiment will take several thousands of years, and so, it better be a thought experiment. 😀

A thought experiment

Let us choose an inhabitable, but uninhabited island, far off from the rest of the world as the site of our experiment. Let us then initiate a civilization, with young children. For a few generations, we have to silently protect them, making sure that they survive safe. Later on, we can cut off all contacts with that island. A a few generations later, the people in the island will forget about us, and it will grow just like any natural civilization; no civilization remembers a time when they didn’t have a language of communication and a system of documentation. They will eventually find us, after they invent ships and start sailing, but this will take a very long time.

We can observe how the civilization develops, from a remote sensing satellite.  Of course, this will take several generations of observation in reality, and that is the reason why this is a thought experiment.

This setting can be used to analyze many things. Our question here is of relevance of religion and God: Will the civilization in the island necessarily develop a religion and a God?  Let us refer to our history. We know of a large number of civilizations that existed sometime in the past, somewhere in the world. How many of these didn’t have a god or a religion? Turns out, most of the known civilizations have a religion and god(s), with extremely sparse exceptions. Pirahã people is one such example. They don’t believe in any deity, but they do believe in spirits. However, they are not an independently grown civilization; they are a subtribe of a bigger tribe. So, this doesn’t really tell us how to evolve the civilization in our island without a religion.

Does this mean that no civilization can exist without religion and God? There are two possibilities: One, religion is a part of the growth of a civilization, or two, all those civilizations that didn’t develop a religion collapsed too soon to leave any footprints of their existence, and so we don’t know about them. Perhaps, they collapsed because of not having a religion.

For one thing, the civilization in our island should say something about death; something nice like, dead people become stars in the sky, or they become spirits or, they go to heaven/hell. Otherwise, the civilization will collapse internally. People are glued in to a society by an emotional attachment(relation, friends,, etc). This attachment also has a bad facet – it causes anguish, particularly over death, which is certain. If it is not dissipated, it can potentially crush the civilization. So, a strong civilization needs a strong attachment and a robust way of dissipating destructive emotions. Evidently, rituals associated with death and afterlife are a big chapter in every religion.

Moving ahead, the most prominent feature of a religion is, it creates God, as a protector of all :D. Is it really necessary to have an imaginary protector? Will the civilization in our island develop such an imaginary protector?. Well, if it doesn’t, it will never explore outside the island, and so, it will make a very slow progress in science!. Let us see why:

A civilization will attach value to life of a person(and many more things), not only that a person values his own life, but also, others value his life. Any prospect of loss of life will therefore induce an emotion called fear. It prevents the civilization from exploring too far away from their safe home. An imagination of a protector, can create a counter emotion to fear and therefore make it possible to explore. Knowing that this protector is not real does not alter anything!; Imagination can create real emotions.  One example where this method of evading fear is employed is, explorations in the ocean. Sailors are known to be superstitious, in order to evade the fear due to risks in their sailing. (Sailor’s superstitions. Why aren’t there similar superstitions with today’s astronauts? This has a simple answer  😀 ). Therefore, the people in our island may never find us, if they don’t imagine a protector!

Exploration is the key for scientific progress. Scientific progress is not a process carried out by scientists alone. It is carried out by the entire society. Scientists need a strong support from all sections of the society. As an example, let us consider the big revolution brought by Newton’s laws of motion(they partly caused the industrial revolution). What does it take for the civilization in our island to make this breakthrough?. It takes three things, in order of decreasing importance:

1. A thorough documented knowledge of the objects in the sky. This is accumulated by a thousand years of night sky observers
2.  A thorough knowledge of the surface of the earth, and how the sky looks when viewed from different locations on the earth. This is gathered by exploratory sailors.
3.  A genius like Isaac Newton

The people in our island will never get to this without being able to explore. As paradoxical as it is, science has gained a  little from some superstitions too!. :P. 

So, the civilization in our island should have a method of dissipating destructive emotions, in particular, it should have something nice to say about death. And it should also have a protector(or a means to evade fear). Do these two complete a religion? I don’t think so. I have considered only those aspects that affect the stability and growth of the civilization. Religion also has another kind of value that is shared by the arts- music, dance, stories etc. In societies where religion is strong, it appears to influence the way people think(something I don’t understand). That is a subject of another blog post. I will conclude now by saying, man created God, and then God created man!. 

Action and Expression

Why should I be rational?- Part III

<< Part-I, << Part -II

Imagine, there is table in front of you. You placed sugar at one end of the table, honey at the other end and an ant at the center. The ant can make its personal choice of whether to go for sugar or for honey.

Suppose, in addition, the ant has moods- three possible moods- happy, sad and neutral. It’s mood changes randomly between these three at irregular intervals. Like we have an urge to express our feelings, let us assume that this ant too has an urge to express its mood. It has no means of verbal communication to do so; but there is still a way: It is walking towards its choice, honey/sugar. When it feels happy, it stops where ever it is, and takes a few steps towards the honey. When it feels sad, it stops and takes a few steps towards the sugar. When it is neutral, it has nothing to express; it continues walking towards its choice. This is a way to using the walking as a channel of communication to express its mood- just because it has an urge to do so.

Now, as an observer, you don’t know the ants choice- sugar or honey. Neither do you know of its changing moods. There no way to find out the ant’s choice or its mood just by watching it. If at some point, the ant is moving towards the honey, it could either be an action: i.e., it has chosen to get the honey or, it could be an expression of its mood.

So far we have assumed that the ant itself knows which of its movement is an action and which is expression. If it didn’t consciously know, and it tried to judge from its own motion, it would be impossible to do so. The ant doesn’t know what it wants!.
This is an analogy I used to define action and expression. We too use every available channel to express what we feel. And, like the ant, sometimes we ourselves are unable to identify an expression or an action.

This urge to express is one reason behind non trivialities in a language- figures of speech. Many metaphors are born because the mind uses multiple channels to express itself. Abusive terminologies(that claim the untrue 😛 ) are an expression of anger.

However, all these are instances where we consciously know that it is an expression. There are examples where we are unaware of this. A day before every exam, we “decide” to be better prepared next time. But this “decision” never gets acted. :D. This is an instance where the mind uses the process of deciding, as a channel to express that it is repenting. This is often a case where we don’t consciously know that it was an expression and not an action. An action too, is quite common. A decision to turn left while driving, for instance, is an action. It gets enacted without any hassle.

Many of our decisions are mere expressions. Being unable to identify actions and expressions is quite common. While both action and expression are essential, it is important to know which is an action and which is an expression. Before, this, we need to understand all the differences between them.

An action is a thought that is born in the mind and flows out through the body. It can’t be stopped in between, by a purely internal force. The thought is not complete until it is acted. It is a single piece- it can’t be broken in to thinking part, and enacting part. Action belongs to the deterministic part* of the future. Like all other deterministic parts of the future, it already exists in the present as a thought, but is invisible. It becomes visible in the future. Therefore, action is a part of reality. Therefore, questioning an action or an inaction is questioning the existence/non existence of a part of reality- it is an existential question.

An expression is always preceded by a strong emotion, which is to be expressed. It is not a part of reality. So, an obvious way to distinguish between action and expression: if a chain logical deductions leads to visibly absurdity, it is an expression. Expression is itself absurd, but its absurdity is usually invisible. So,, many times, we don’t identify expressions as expressions(like the confused ant). This is a state of illusion. The illusion is broken by a chain of logical deductions starting from it, that reaches an absurdity. We are a part of reality, and everything that we want to call reality must be deducible from it. Expression/ surreal objects can’t replace the reality.

* “the sun will rise up tomorrow” is a deterministic part of the future to our present knowledge

We are Incomplete

Why Should I Be rational?- Part II

<< Part-I

Can man survive all by himself without even the knowledge of the existence of others somewhere? It seems, he can. He can look for food himself. He can fight for his life against predators himself. Our body has a process to fight every challenge to its survival. And such a process has a closed end within the body- it does not involve any other member of the species. In this sense, such processes are complete. We can therefore say we are individuals.

However, there are some processes that are not complete. E.g weeping. Tears are not like a digestive juice, which is produced as a part a complete process- digestion. Another example is screaming.

When a man meets with an accident, and is wounded badly, he screams uncontrollably. This screaming is an involuntary reaction to pain. It does not contribute to healing of the wound. A complete process to heal the wound is initiated separately. It may take days, or may not succeed at all. But screaming is not a part of it. It is an open ended process and not a part of a complete process. It is incomplete.

Incomplete processes are a call for help, to other members of the species who could be around. The human mind is equipped to initiate incomplete processes, which means, it knows that it is not alone. Also, we are tuned to respond, on hearing a call for help from another member of the species. The incomplete process is then completed by a second individual, who receives it. So such a process initiated in us is to be completed by others. Therefore, we are incomplete individuals.

What is the mechanism of the response? I believe, an incomplete process produces the same emotion in the second individual as that of the first, in a much weaker form. For instance, when a man dries in pain, the cry produces the same emotion- pain in a very weak form, in the listener and prompts him to attend for help. When a musician plays on stage, people enjoy by resonating with what he expresses through his music. Thinking is also a sequence of well controlled emotions. When someone lectures a proof in mathematics, he is expressing this sequence. Anyone who understands it essentially resonates with it.

Incomplete processes form a weak link between people. In short, we are wired to both seek empathy from others and to show empathy to others. Resonating with others’ emotion is the most fundamental form of communication. It is the reason why we developed languages, common beliefs, common hope, religion, and finally, civilizations. It is the origin of all surreal objects.

An incomplete process is an expression of one’s feeling to others. I have carefully chosen the term expression here. It is chosen in opposition to action, where in we execute a decision. Expression is born from the urge to communicate what we feel, and ends with communicating it. Following this urge, the human mind attempts to use every available channel of communication as a mode of expression. There are several channels of communication, other than verbal. Two people playing chess, for instance, are intensely communicating with each other through the chess board, even though they don’t speak to or even look at each other. Making a decision can also be used as a channel to communicate. Our mind, by nature, attempts to utilize every such channel to express what it feels.

Part-III >>

Why should I be rational?

Part I: Truth and logic

Trying to be rational is placing restrictions on oneself. If I don’t want to be rational, I can be sometimes rational and sometimes irrational :D. Like playing a game without observing its rules, not being rational is easier. So why should anyone try to be rational? One observation is: we are more peaceful when we are rational. This is just an observation, not an answer. Moreover, it could be that it is the other way round: we are rational when we are peaceful :P.

Being rational is a way of accepting the reality. Reality is anything that is either verified through senses, or deduced from another reality(that is verified through senses). A rational argument is a link connecting two realities. Reality is interconnected within, through logical deductions. In other words, the set of realities is closed under deductions. Therefore, no untrue statements can be deduced starting from a true statement. Also, starting from an untrue premises, some deductions will be untrue. Some of them might be visibly absurd, and this way, deduction can be used to identify untrue statements. Thus, logic is used to keep our self within reality.

Therefore, the question really is, why should I restrict myself to reality?. Is it possible to live totally in an imagination, by believing it is true?. If not, what is the role of imagination? Also, why do we feel more peaceful when rational?

To answer these questions, we need to understand the origin and nature of all surreal objects that we can think of. I have broken my thoughts on this in to two other posts, due to its length :D.

Part -II >> , Part-III >>