Richard's Paradox

This is one of the really cool paradox I came across. Find the fallacy in this paradox.

Consider the properties of natural numbers that can be described in, say, the english language. For instance, the property of a number being a prime can be described as "a number x that is not divisible by any other number other than 1 and itself". As an another example, the property of a number that has an integer square root can be described as "a number x that is a product of an integer by itself", etc. 

Now these descriptions can be listed one by one, based on the number of letters present in them. For example, the description "a number x that is not divisible by any other number other than 1 and itself" has 61 letters and the description "a number x that is a product of an integer by itself" has 41 letters and so the latter description will be listed before the former description. If two descriptions have the same number of letters then they can be arranged alphabetically in the order of appearance. 

Call a number x as Richardian if x does not satisfy the description listed in the x^th row. Thus, 41 is Richardian whereas 61 is not Richardian. 41 is Richardian since 41 does not have an integer square root and 61 is not Richardian since 61 is a prime number. Now this property of a natural number being Richardian that is described can then be listed in a row. Let the row number corresponding to this description be n.

The question is then "is n Richardian?"

If n is Richardian, then by definition it should not satisfy the description in the n^th row. But, the n^th row description is the definition of a Richardian number and so n will not be Richardian. On the other hand if n is not Richardian, then it does not satisfy the description on the n^th row and so it will be Richardian. Thus, n is Richardian if and only if it is not Richardian.

Isn't this cool? So what is the fallacy here?

Immanuel Kant Series - Part I

Immanuel Kant (1724-1804) is the foremost and central figure in western philosophy. His greatest work Critique of Pure Reason stands out as one of the most influential philosophical treatise of all times. In the coming series, I am planning to record notes which I find important (and interesting) when trying to understand Kantian philosophy. 

A small note on Kant's view on his days which I thought was interesting. It appears Kant did not enjoy his youthful days. He says:

Many people imagine that the years of their youth are the pleasantest and best of their lives; but it is not really so. They are the most troublesome; for we are then under strict discipline, can seldom choose our friends, and still more seldom have our freedom.

Now on to philosophy. There were at least two outstanding questions related to metaphysics which led Kant to formulate his philosophy. One was Descartes famous cogito ergo sum (I think, therefore I am) and the other was the antagonism between Leibniz's rationalism and Hume's empiricism. 

cogito ergo sum

According to Descartes it is senseless to doubt our existence. The fact that we can think of our existence means that we exist, for sure. Roger Scruton¹ remarks "here doubt only confirms what is doubted". However, critical analysis of cogito ergo sum reveals our existence is not vouched by that statement, but rather a thought exists about our existence. cogito acknowledges there is a thought and ergo sum acknowledges our existence. But even when taken together it doesn't mean "I" exists but rather a thought that "I exist". Thus, when Kant was framing his thesis, there was no argument that was readily available to affirm or deny the existence of "I".

Leibniz's Rationalism 

Rationalism is a philosophy that emphasizes that all knowledge is derived primarily from reason and claims to provide an absolute description of the world that is independent of observers experience. Leibniz belonged to rationalist school of thought in which he described the world consisting of infinitely many individual monads each of which living eternally outside space and time. Each monad has a view of reality but ultimate reality is inaccessible to any monad through experience although it can only be realized through reason. Such a reality is more like a surface defined by function of several variables, where each variable is a monad. And experience of reality for a particular monad can be imagined as a restriction of other variables to a constant, during which a section of reality, and not the whole, can be described.

Hume's Empiricism

In contrast to Leibniz, Hume was a empiricist. Empiricism is a philosophy that believes all forms of knowledge are a product of experience and not reason. We gather information only through the senses and so there is no reason to believe reason can alone derive any knowledge. Far from it, it is only through experience all knowledge are gathered. There is no such thing as monad or soul, because experience doesn't tell us if there are any. Also rationalists claims of ultimate description of the world accessible only through pure reason are false since they run contrary to sense perceptions. There could be no single description of the objective universe. Everyone has their view of the world which gets superposed thorough relations. On other words, relationships of experiences only define the objective world and nothing else.

Hume's empiricism and especially this extreme skepticism was unacceptable to Kant and he noted Hume woke him up from "dogmatic slumbers". Roger Scruton¹ notes:

Kant did indeed have a lasting quarrel with Leibniz and with the Leibnizian system.  But it was the sense that the problems of objectivity and that of causal necessity are ultimately connected that led him towards the outlook of the Critique of Pure Reason. It was only then that he perceived what was really wrong with Leibniz, through his attempt to show what was really wrong with Hume.

References

1. Roger Scruton, Kant: A Very Short Introduction, Oxford University Press; Revised edition December 6, 2001.

Modern Science and Collaborations

Three months back there was an article in Nature (Policy: Free Indian science dated Apr. 02, 2014) discussing the status of science in India and how it should be free from government and bureaucratic interference. This is a good article and points towards the right direction. At the end of the day, science like religion should be de-coupled from governance. This is just basics. This article also brings to the fore the legacy of good science pursued under prominent Indian scientists decades back which leaves us wondering whether the era of monumental ingenuity by lone scientists, are over. Contemporary science often presents itself as a collaborative enterprise devoid of towering geniuses. This is an upsetting view, especially for Indian science because there has not been a single Nobel prize or Fields medal winner after our independence, despite our success in collaborative endeavors such as space and pharmaceuticals. Here I attempt to argue that in science, the era of individual geniuses are not over so that we can hope to reclaim the past legacy (provided we take much needed reforms). In fact, far from it, I claim that in disciplines requiring extensive collaborations, towering geniuses are inevitable outcomes as much as in the disciplines of minimal collaboration. To see this, we need to analyze the structure and the philosophy of science. 

Science is not one single entity. There are at least two main divisions: formal sciences and experimental sciences. Formal sciences can be typically characterized by analytical propositions, i.e., those propositions that are true by their inherent meaning and not how they relate to the real world. In formal sciences like mathematics, theoretical physics/computer science and logic, the progress is largely determined by individual efforts with little scope for collaboration. This is simply because of the structure of these sciences. In these sciences, a new progress cannot merely shelter known propositions. Any new theory should subsume old theories: not just reconciliation of known ideas but proper inclusion of known facts. It is not mere accommodation of known doctrines in the new framework but explanation of available actualities within the new framework. Special relativity includes Newtonian mechanics and not merely accommodates it. Any result that holds true for a vector space should hold true for a Hilbert space. On the other hand, collaboration involves exchange of ideas, although can aid in the formation of a grandeur theory, often works on the level of ideas than on intuition. Encompassing theories are invented purely on the levels of intuition where collaboration has minimal influence. Therefore, there are ample scope for geniuses, who can stand out from the ordinary, by building a better bigger theory. This doesn't mean the progress is linear and a great theory, as a giant leap in our understanding, can take its time to arrive. In these systems of thought, history determines who is great and who is not. Blessed are those scientists who were recognized during their lifetimes. Such recognition need not always happen and certainly we cannot expect them to happen during the short window of our times. Events that leave footprints in the history of science are separated in time far greater than the average human life. For instance, it took more than 160 years for special relativity at appear (it was proposed in 1905) after Newton passed away in 1727. In my opinion, Paul Erdos, the hungarian mathematician could easily be one of the great historical figures who died in the recent past. Currently, Terrence Tao is considered an extraordinary mathematician and history has the potential to elevate him to a genius like David Hilbert. Therefore, at least in the case of formal sciences, where collaboration is of limited importance, we can expect history to determine superiority. And often history repeats itself.

In the case of experimental sciences, it can be argued that today's science has increasingly become collaborative. Several reasons could be attributed to it, including availability of information, increased networking, abundance of scientists etc. However, it is important to recognize that the very nature of these sciences allow exchange of ideas at the fundamental level. This is because experimental facts cannot be derived from a well-defined and agreed set of axioms. Whatever be the reasons for collaboration in these sciences, it is not straight-forward to see why they should result in geniuses. Thomas Kuhn, in his classic book, The Structure of Scientific Revolutions talks about normal science and paradigm shifts. Basically, progress in science proceeds in three distinct phases. Prescience that lacks a central paradigm comes first. In this phase scientists scramble to define a central defining principle. As an example, in the beginning of 20th century there was struggle to define a unit of inheritance until Gregor Mendel established its laws. This prescience phase is usually followed by normal science. Normal science is a puzzle-solving exercise proceeds by development-by-accumulation and attempts to refine and strengthen a central paradigm. This phase is extremely productive and keeps scientists in business. In this developmental phase,  the works of professional scientists who conform to the central paradigm are acknowledged and honored. Scientific results that doesn't reconcile with the central paradigm are not considered as refuting the paradigm but viewed as a mistake of the researcher. For example, in modern times, any result that do not conform to the central dogma of molecular biology would be considered as a blunder of the researcher in designing an experiment. As results anomalous to the central paradigm appear in increasing frequency, there comes a crisis point in science when a new paradigm that accommodates the original framework as well as the anomalous results, emerges. This is a period of revolutionary science when contradictory theories and ideas are reconciled and merged into a broader framework. There have been several revolutionary periods in the history of science. In the recent times, negative experiments of Michelson-Morley in 1887 (that attempted to validate the theory of aether) and eventual emergence of special relativity in 1905, marks one such period in experimental physics. This phase also brings in towering researchers. These pioneering scientists are stalwarts who question the central and well established principles, just to come up with an inclusive framework.

Thus, regardless of whether science is formal or experimental, pioneers are bound to arise and the need of the hour for Indian science is to create an environment that works to this advantage. The rise of such glorious scientists whenever contradictions overwhelm normal science, reminds of the classic verse in Bhagavad Gita:

paritrāṇāya sādhūnāḿ

vināśāya ca duṣkṛtām

dharma-saḿsthāpanārthāya
sambhavāmi yuge yuge

Experimental and Formal Sciences

The difference between experimental sciences and formal sciences is the difference between Reductionism and Reductio ad absurdum. 

Reductio ad absurdum is simply not possible in any experimental framework and is the hallmark of sciences based on formal logic.

India's Secularism

For some, it is real wonder how India continues to exist in spite of multitude of faiths and millions of gods. When chaos and confusion should be the norm with such diversity, she is a more or less a very peaceful country. Communal tensions and killings, although avoidable, are only exceptions than examples. They are statistically insignificant and bound to exist as long as India is secular. To understand and appreciate the true spirit of Indian secularism, we need to move beyond its mundane definition and our routine desire to be secular.

To be truly secular is to be truly scientific. One may wonder why? The simple reason is this: if one is inquisitive enough to genuinely understand the functioning of this universe, the baggage of religious predispositions would automatically vanish at some point. Assumptions and quest for knowledge are often tangential and at crossroads. A scientist often confronts beliefs and faiths and soon learns to abandon them. To a scientist, science becomes the religion and nature, the God. A genuine scientist would be interested in knowing why things happen than merely accepting how things happen. He would question why India is secular than observing it as being secular. For a scientific mind, the wonders of this world are not because of how they are, but why they are. These wonders can be explained to a certain degree and this knowledge gives, not only the power to discern, but to view them in its extraordinary glory. A scientist appreciates desirable wonders as facts and not merely aspires them as known. In Indian context, a genuine social scientist would appreciate India's secularism not as an aspiration, but as a fact. 

Indian political systems that harp on secularism, at best do so only as an aspiration. The main-stream visual media that has little to do with fact checking yearns for secularism. It is then no wonder we see hue and cry about maintaining secularism when hope and sensationalism dominate over understanding and facts. While one can genuinely appreciate the plurality in India and aspire for its peaceful maintenance, few really understand why it bound to be so. They question, if millennium years of conquest has not crushed the inclusivity in our society, can decades destroy its diversity? This marks the difference between being ordinary and being scientific. Between hope and reason. For some, choosing between the Congress and the BJP.

The prevalent secularity of India cannot be attributed only to the tolerance levels of people towards other faiths. The fundamental reason for India's secularity is the constitutional authority of the Vedas upon which different schools of thoughts are founded. The canonical triad consisting of Upanishads, Brahma Sutra, and the Gita act as the Supreme Court interpreting and guarding this constitution. The predominant religious schools are lower level courts trying to resolve the conflicts to their convenience and logic. Even schools that don't agree with the constitutional authority of the Vedas, like Buddhism and Jainism, are allowed to flourish. Existence of God is at best only an inference, a corollary, rather than an axiom. God exists because Vedas say so. Since Vedas are written by men and women, they are subject to interpretation and as a consequence, the notion of God is subject to interpretation as well. This gives ample scope for anybody to choose any faith or even be an atheist. In fact, several religious schools like the Samkhya founded by Kapila, are atheistic in nature.

Throughout the history of India, religious schools have verbally battled each other on the interpretation of the Vedas. Samkara vehemently argued against (based on the canonical triad) sunyavada to establish monotheism. Ramanuja again using the canonical texts disagreed with Samkara to establish qualified monotheism. Arguments and opposition on religious doctrines are embedded in the very fabric of Indian society that makes us multicultural with tolerant ethos. Since duality is a feature embedded in the Vedas, it extends beyond the individuals and flows into the society. This is the rational explanation to India's secularism and diversity we see today and why it is bound to exist. And this is precisely the reason why the Supreme Court of India described hinduism as a way of life than as a set of religious beliefs. Hinduism is more or less like Unix, an open-source which can be customized to one's liking, Vedas being the kernel.

Unfortunately, politicians, media and ordinary people neither have the time nor the inclination to understand the nuances of philosophies and logic and hence chaos and commotions are an inevitable outcome, especially during an election season. The responsibility is on the individuals to understand and reconcile these philosophical aspects of hinduism with their experience. Doing so they will be able to appreciate India's secularism in a deeper sense which will be intellectually and morally satisfying. Indian philosophy by Dr. S. Radhakrishnan is a good starting point. Lastly, I would like to quote a passage from Nasadiya Sukta (Rg. Veda - 10:129) which reads as follows:

Whence all creation had its origin,
he, whether he fashioned it or whether he did not,
he, who surveys it all from highest heaven,
he knows - or maybe even he does not know

These classic verses and especially the last sentence puts forth an agnostic view of the universe with the sense of scientific wonder. This is the spirit we need to entertain and encourage.

Natural Selection

I should probably confess that at the time of writing this post, I did not have any clue of what natural selection is (or for that matter, evolution). Looking back, I just see how silly I was. Six years of experience in biological and medical institutes has taught me to understand and appreciate science in its broad and pristine perspective. Let me explain the idea behind natural selection.

Natural selection is one of the mechanisms through which evolution can happen. Evolution can happen if any of these five key factors are present in a population system:

1. Mutation

2. Natural selection

3. Non-random mating

4. Genetic drift &

5. Gene flow

If none of these are present, then at least for diploids, the allelic and genotypic frequencies will remain constant (the famous Hardy-Weinberg equilibrium) through generations and no evolution will take place. At the end of the day, evolution is just a gradual change in the allele frequencies across generations and it is not synonymous with natural selection. Although selection need not be the only factor in evolution, it is a key mechanism when the population is sufficiently large. Selection requires at least three necessary and sufficient conditions in order to maintain evolution:

1. Variation of traits within a population
2. Heritability of traits
3. The traits should confer fitness (reproductive) or survival advantage.

First, selection requires variation in phenotypes which are amply provided by mutations or polymorphisms. Technically, mutations provide variations in genotypes that account for phenotypes through the principles of molecular biology. They provide the variety (or phenotypic traits) so that nature can pick and retain the traits that are beneficial for survival/reproduction. 

The second key factor for selection to take place is the heritability of traits. Selection can act only on phenotypes and not on genotypes. Therefore, for a trait to evolve in a population, it should be heritable in the first place. If a trait is not heritable, it will not be passed into future generations and the trait will go into an evolutionary dead-end.

Third, the traits should benefit survival or reproduction, otherwise known as fitness advantage. The traits that are not beneficial are weeded out through environmental or genetic machineries.

Natural selection is simple yet a profound concept. Selection works based on a subset of population rather than individual(s). It is a statistical property, like color for instance. Individual atoms/molecules may not have a specific color, but color can emerge due to aggregation. Similarly, when a group of individuals in a population develop a certain trait that is more favorable to survival/reproduction, assuming the trait is heritable, there is a good chance of passing it on to the next generation by sheer probability.

One of the Best Non-constructive Proofs

This is one of the finest proofs in mathematics as it completely exposes the non-constructive philosophy of mathematics as propounded by David Hilbert.

Theorem: There exists irrational numbers p and q such that p^q is a rational number.

Proof: Consider the number a = √2 ^√2 .

Case I:

a is a rational number. We are done. Just take p = √2 and q = √2. Therefore, p^q is a rational number.

Case II:

a is an irrational number. In this case, take p = a and q = √2.

Thus, by the laws of arithmetic, p^q = (√2 ^√2)^√2 = (√2 ) ²  = 2, a rational number. 

QED.

The beauty of this proof is that at the end of the day (or the proof) we really don't know if a is a rational number or an irrational number. Note that this is technically a valid proof without any fallacies. 

Mathematicians who harp on constructivism will frown at this proof and the law of excluded middle is the real culprit. For more details about constructivism (and Brower-Hilbert controversy) look into this link.

Note: It turns out that a is indeed an irrational number as a consequence of Gelfand-Schneider Theorem since √2 is an algebraic number (as it is the solution of the polynomial equation x²-2 = 0).

On Highly Recurrent Cancer Mutations

I hit upon this insight which I thought I should share it with you. This is purely from analytical perspective.

One would wonder why in some cancers, certain residues/mutations (like R132H, V600E etc) are almost always found at high frequencies. There could be a simple game theoretical explanation to it.

In game theory, there is something called mixed and pure strategies. For instance, let us say Charlie and Ruth are playing Rock-Paper-Scissors game. Charlie/Ruth can choose to call “Rock" all the time (or for that matter either “Paper" or “Scissors" all the time). This is called pure strategy – that is, choosing a particular option all the time. On the other hand, Charlie/Ruth can alternate between rock,  paper and scissors in certain frequencies. For instance, calling rock 5/10 times, paper 2/10 times and scissors 3/10 times. This is called mixed strategy. 

There is a nice theorem that says: if Charlie “knows” what Ruth is going to play, then the optimal counter strategy for Charlie is always a pure strategy and not a mixed strategy. The same result holds for Ruth as well. This is a very interesting result because regardless of what the other person plays (could be pure or mixed with any frequency), if a player “knows” what is going to be played, the optimal play for him is to choose a pure strategy. And pure strategy is choosing one option all the time.

Therefore, applying this in a cancer setting: if a gene like IDH or BRAF somehow “senses” the frequency of other events, the optimal strategy for it would be a pure strategy or in other words, mutating at a particular residue all the time. That could just be R132H or V600E. The gene just needs to sense the pattern rather than the frequency of actual events in order to get mutating at a particular residue almost always.

I know this argument devoid of any biology does not say why that particular residue, but it could be a small rationale for why we see what we see.