## Archive for the ‘Thoughts’ Category

### Freeman Dyson on Wittgenstein

October 21, 2012

[ source ]

Wittgenstein, unlike Heidegger, did not establish an ism. He wrote very little, and everything that he wrote was simple and clear. The only book that he published during his lifetime was Tractatus Logico-Philosophicus, written in Vienna in 1918 and published in England with a long introduction by Bertrand Russell in 1922. It fills less than two hundred small pages, even though the original German and the English translation are printed side by side. I was lucky to be given a copy of the Tractatus as a prize when I was in high school. I read it through in one night, in an ecstasy of adolescent enthusiasm. Most of it is about mathematical logic. Only the last five pages deal with human problems. The text is divided into numbered sections, each consisting of one or two sentences. For example, section 6.521 says: “The solution of the problem of life is seen in the vanishing of this problem. Is not this the reason why men, to whom after long doubting the sense of life became clear, could not then say wherein this sense consisted?” The most famous sentence in the book is the final section 7: “Wherof one cannot speak, thereof one must be silent.”

I found the book enlightening and liberating. It said that philosophy is simple and has limited scope. Philosophy is concerned with logic and the correct use of language. All speculations outside this limited area are mysticism. Section 6.522 says: “There is indeed the inexpressible. This shows itself. It is the mystical.” Since the mystical is inexpressible, there is nothing more to be said. Holt summarizes the difference between Heidegger and Wittgenstein in nine words: “Wittgenstein was brave and ascetic, Heidegger treacherous and vain.” These words apply equally to their characters as human beings and to their intellectual output.

Wittgenstein’s intellectual asceticism had a great influence on the philosophers of the English-speaking world. It narrowed the scope of philosophy by excluding ethics and aesthetics. At the same time, his personal asceticism enhanced his credibility. During World War II, he wanted to serve his adopted country in a practical way. Being too old for military service, he took a leave of absence from his academic position in Cambridge and served in a menial job, as a hospital orderly taking care of patients. When I arrived at Cambridge University in 1946, Wittgenstein had just returned from his six years of duty at the hospital. I held him in the highest respect and was delighted to find him living in a room above mine on the same staircase. I frequently met him walking up or down the stairs, but I was too shy to start a conversation. Several times I heard him muttering to himself: “I get stupider and stupider every day.”

Finally, toward the end of my time in Cambridge, I ventured to speak to him. I told him I had enjoyed reading the Tractatus, and I asked him whether he still held the same views that he had expressed twenty-eight years earlier. He remained silent for a long time and then said, “Which newspaper do you represent?” I told him I was a student and not a journalist, but he never answered my question.

Wittgenstein’s response to me was humiliating, and his response to female students who tried to attend his lectures was even worse. If a woman appeared in the audience, he would remain standing silent until she left the room. I decided that he was a charlatan using outrageous behavior to attract attention. I hated him for his rudeness. Fifty years later, walking through a churchyard on the outskirts of Cambridge on a sunny morning in winter, I came by chance upon his tombstone, a massive block of stone lightly covered with fresh snow. On the stone was written the single word, “WITTGENSTEIN.” To my surprise, I found that the old hatred was gone, replaced by a deeper understanding. He was at peace, and I was at peace too, in the white silence. He was no longer an ill-tempered charlatan. He was a tortured soul, the last survivor of a family with a tragic history, living a lonely life among strangers, trying until the end to express the inexpressible.

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Source:

Freeman Dyson, What can you really know?, The New York Review of Books, November 8, 2012.

### Gell-Mann on learning

September 27, 2012

Murray Gell-Mann on why being in awe inhibits learning:

I said I’d rather be poor or die than be an engineer because I would be no good at it. If I designed something it would fall down. When I was admitted to Yale, I took an aptitude test, and when the counselor gave me the results of the exam, he said: “You could be lots of different things. But don’t be an engineer.”

After my father gave up on engineering, he said, ‘How about we compromise and go with physics? General relativity, quantum mechanics, you will love it.’ I thought I would give my father’s advice a try. I don’t know why. I never took his advice on anything else. He told me how beautiful physics would be if I stuck with it, and that notion of beauty impressed me. My father studied those things. He was a great admirer of Einstein. He would lock himself in his room and study general relativity. He never really understood it. My opinion is that you have to despise something like that to get good at it.

If you admire it sufficiently, you’ll be in awe of it, so you’ll never learn it. My father thought it must be very hard, and it will take years to understand it, and only a few people understand it, and so on. But I had a wonderful teacher at Yale, Henry Margenau, who took the opposite attitude. He thought relativity was for everybody. Just learn the math. He’d say, “We’ll prepare the math on Tuesday and Thursday, and we’ll cover general relativity on Saturday and next Tuesday.” And he was right. It isn’t that bad.

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Source:

Susan Kruglinski, The Man Who Found Quarks and Made Sense of the Universe, DISCOVER Magazine, April 2009.

### Born under the second law of thermodynamics

August 26, 2012

[ source ]

Neil Armstrong once portrayed himself as follows [1, 2]:

I am, and ever will be, a white-socks, pocket-protector, nerdy engineer—born under the second law of thermodynamics, steeped in the steam tables, in love with free-body diagrams, transformed by Laplace, and propelled by compressible flow. As an engineer, I take a substantial amount of pride in the accomplishments of my profession.

Neil “slipped the surly bonds of earth” and “trod the high untrespassed sanctity of space” [3]. He died yesterday [4, 5].

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Sources

[1] Neil A. Armstrong, Greatest Engineering Achievements of the 20th Century (transcript of speech), National Press Club, February 22, 2000.

[2] Neil A. Armstrong, Greatest Engineering Achievements of the 20th Century (audio of speech – Neil’s address starts at approximately 5:30), National Press Club, February 22, 2000.

[3] John Gillespie Magee, Jr., High Flight, England, 1941.

[4] John Noble Wilford, Neil Armstrong, first man on Moon, dies at 82, The New York Times, August 25, 2012.

[5] Craig Nelson, Neil Armstrong, hero with a slide rule, The Wall Street Journal, August 25, 2012.

### From classical deduction to Bayesian probability

August 15, 2012

Terence Tao on classical deduction and Bayesian probability:

In classical logic, one can represent one’s information about a system as a set of possible states that the system could be in, based on the information at hand. With each new measurement of the system, some possibilities could be eliminated, leading to an updated posterior set of information that is an improvement over the prior set of information. A good example of this type of updating occurs when solving a Sudoku puzzle; each new cell value that one learns about constrains the possible values of the remaining cells. Other examples can be found in the classic detective stories of Arthur Conan Doyle featuring Sherlock Holmes. Proof by contradiction can also be viewed as an instance of this type of deduction.

A modern refinement of classical deduction is that of Bayesian probability. Here, one’s information about a system is not merely represented as a set of possible states, but by a probability distribution on the space of all states, indicating one’s current beliefs on the likelihood of each particular state actually being the true state. Each new measurement of the system then updates a prior probability distribution to a posterior probability distribution, using Bayes’ formula

$P (A \mid B) = \displaystyle \frac{P (B \mid A) P(A)}{ P(B) }$.

Bayesian probability is widely used in statistics, in machine learning, and in the sciences.

To relate Bayesian probability to classical deduction, recall that every probability distribution has a support, which (in the case when the space of states is discrete) is the set of all states that occur with non-zero probability. When performing a Bayesian update on a discrete space, any state which is inconsistent with the new piece of information will have its posterior probability set to zero, and thus be removed from the support. Thus we see that whilst the probability distribution evolves by Bayesian updating, the support evolves by classical deductive logic. Thus one can view classical logic as the qualitative projection of Bayesian probability, or equivalently, one can view Bayesian probability as a quantitative refinement of classical logic.

Alternatively, one can view Bayesian probability as a special case of classical logic by taking a frequentist interpretation. In this interpretation, one views the actual universe (or at least the actual system) as just one of a large number of possible universes (or systems). In each of these universes, the system is in one of the possible states; the probability assigned to each state is then the proportion of the possible universes in which that state is attained. Each new measurement eliminates some fraction of the universes in a given state, depending on how likely or unlikely that state was to actually produce that measurement; the surviving universes then have a new posterior probability distribution, which is related to the prior distribution by Bayes’ formula.

It is instructive to interpret Sherlock Holmes‘ famous quote, “When you have eliminated all which is impossible, then whatever remains, however improbable, must be the truth,” from a Bayesian viewpoint. The statement is technically correct; however, when performing this type of elimination to an (a priori) improbable conclusion, the denominator in Bayes’ formula is extremely small, and so the deduction is unstable if it later turns out that some of the possibilities thought to have been completely eliminated, were in fact only incompletely eliminated. (See also the mantra “extraordinary claims require extraordinary evidence”, which can be viewed as the Bayesian counterpoint to Holmes’ classical remark.)

Another interesting place where one can contrast classical deduction with Bayesian deduction is with regard to taking converses. In classical logic, if one knows that $A$ implies $B$, one cannot then deduce that $B$ implies $A$. However, in Bayesian probability, if one knows that the presence of $A$ elevates the likelihood that $B$ is true, then an observation of $B$ will conversely elevate the prior probability that $A$ is true, thanks to Bayes’ formula: if $P(B \mid A) > P(B)$, then $P(A \mid B) > P(A)$. This may help explain why taking converses is an intuitive operation to those who have not yet been thoroughly exposed to classical logic. (It is also instructive to understand why this disparity between the two types of deduction is not in conflict with the previously mentioned links between the two. This disparity is roughly analogous to the disparity between worst-case analysis and average-case analysis.)

Bayesian probability can be generalised further; for instance, quantum mechanics (with the Copenhagen interpretation) can be viewed as a noncommutative generalisation of Bayesian probability, though the connection to classical logic is then lost when one is dealing with observables that do not commute. But this is another story…

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Please do note that Terence Tao’s original post contains neither links nor boldface highlighting. I took the liberty of adding those for convenience and emphasis. To improve legibility I also wrote the mathematical expressions in $\LaTeX$.

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Source:

Terence Tao, “A modern refinement of classical deduction is that of Bayesian probability”, Google Buzz, April 4, 2010.

### Vladimir Arnold on mathematical models

July 3, 2012

Vladimir Igorevich Arnold (1937-2010) on mathematical models [1]:

The scheme of construction of a mathematical theory is exactly the same as that in any other natural science. First we consider some objects and make some observations in special cases. Then we try and find the limits of application of our observations, look for counter-examples which would prevent unjustified extension of our observations onto a too wide range of events.

As a result we formulate the empirical discovery that we made (for example, the Fermat conjecture or Poincaré conjecture) as clearly as possible. After this there comes the difficult period of checking as to how reliable are the conclusions.

At this point a special technique has been developed in mathematics. This technique, when applied to the real world, is sometimes useful, but can sometimes also lead to self-deception. This technique is called modelling. When constructing a model, the following idealisation is made: certain facts which are only known with a certain degree of probability or with a certain degree of accuracy, are considered to be “absolutely” correct and are accepted as “axioms”. The sense of this “absoluteness” lies precisely in the fact that we allow ourselves to use these “facts” according to the rules of formal logic, in the process declaring as “theorems” all that we can derive from them.

It is obvious that in any real-life activity it is impossible to wholly rely on such deductions. The reason is at least that the parameters of the studied phenomena are never known absolutely exactly and a small change in parameters (for example, the initial conditions of a process) can totally change the result. Say, for this reason a reliable long-term weather forecast is impossible and will remain impossible, no matter how much we develop computers and devices which record initial conditions.

In exactly the same way a small change in axioms (of which we cannot be completely sure) is capable, generally speaking, of leading to completely different conclusions than those that are obtained from theorems which have been deduced from the accepted axioms. The longer and fancier is the chain of deductions (“proofs”), the less reliable is the final result.

Complex models are rarely useful (unless for those writing their dissertations).

The mathematical technique of modelling consists of ignoring this trouble and speaking about your deductive model in such a way as if it coincided with reality. The fact that this path, which is obviously incorrect from the point of view of natural science, often leads to useful results in physics is called “the inconceivable effectiveness of mathematics in natural sciences” (or “the Wigner principle”).

Here we can add a remark by I.M. Gel’fand: there exists yet another phenomenon which is comparable in its inconceivability with the inconceivable effectiveness of mathematics in physics noted by Wigner – this is the equally inconceivable ineffectiveness of mathematics in biology.

“The subtle poison of mathematical education” (in F. Klein‘s words) for a physicist consists precisely in that the absolutised model separates from the reality and is no longer compared with it.

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I did remove one comment in parentheses to shorten the passage. Hence, this is not an exact reproduction of the original passage in [1]. The “Wigner Principle” that Arnold refers to is the thesis of Eugene Wigner’s famous article [2]. Lastly, I do wonder how many theoretical chemists, theoretical physicists, and systems biologists out there were angered by Arnold’s “heretical” thoughts.

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References

[1] Vladimir Igorevich Arnold, A.V. Goryunov (translator), On teaching mathematics, 1998.

[2] Eugene Paul Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, February 1960.