Friday, December 27, 2013

Jacob Bernoulli: a birthday

Welcome back after the Christmas – thanks for your wishes and even gifts. I hope you enjoyed the Saturnalia.

Jacob Bernoulli was born on December 27th, 1654, as the first big hero of the Bernoulli family that turned out to be rather remarkable. It included 3 great sons of Niklaus Bernoulli (who was a descendant of some doctors and spice traders etc.) in the 17th century; and 6 additional mathematicians and physicists in the 18th century.

I would still say that Jacob Bernoulli (from the group of the 3 sons) was the most important mathematician among these 9 men (Bernoulli numbers, differential equations, \(e\), and other things below); and Daniel Bernoulli (from the later group of 6) was the most important physicist (Bernoulli's principle of fluid mechanics, a manifestation of the conservation of energy, and St Petersburg paradox in game theory). I will only discuss these two men and their contributions.

Jacob Bernoulli was born in Basel, Switzerland. Jacob's father Niklaus wanted his son to study theology – which he did – but he also wanted Jacob to avoid mathematics and astronomy – which he didn't. In his 20s, Jacob would travel all over Europe for 6 years and interacted with some of the greatest minds. Only when he returned to Basel, he would start his comfortable academic job and, more importantly, a remarkable discovery spree in his 30s.

Bernoulli would be among the first ones who would decode Leibniz's incomprehensible writings – and he became a big fan of Leibniz. In fact, he was such a fan that he entered the Newton-Leibniz calculus war on the evil side. Jacob Bernoulli also contributed something to astronomy – namely a wrong theory of comets.

But you may be more interested in his contributions that were not wrong. Well, he was really the father of \(e\approx 2.718\) that he defined as\[

e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n

\] tbe amount of money you get after you add \(100\%/N\) interest to the base \(1\) \(N\) times. If you add \(50\%\) twice, you will have \(1.5\times 1.5=2.25\) dollars. If you increase the number of additions of the interests but reduce the interest to \(1/N\), your total final wealth will converge to \(e\). I believe that this simple definition of \(e\) – and why it's the most natural base of the exponentials and logarithms – is one of the most underestimated insights of basic enough mathematics.

He would place an image that was supposed to be a logarithmic spiral, \[

r = a\exp(b\varphi),

\] on his tomb that also contains a Latin inscription written by his two kids (they missed him, the greatest mathematician, etc.). But let's return back to his life. Jacob Bernoulli would play with the differential equation\[

y'+ P(x)y = Q(x)y^n

\] which is nonlinear but may be solved by a power-law substitution. Perhaps more importantly, he would invent the Bernoulli numbers,\[

B_n = -n\zeta(1-n),

\] which I wrote using perhaps the "more famous" Riemann zeta function (of negative arguments). These rational numbers appear in millions of contexts of number theory, in the Taylor expansion of \(\tan\) and \(\tanh\), in formulae for the sums of finitely many integer powers (it is not a coincidence that the related zeta function of negative integers also expresses the regulated values of the divergent sums of the integer powers). They are relevant for combinatorial problems involving alternating permutations, Faulhaber's formula, and lots of other things, including the Bernoulli polynomials (polynomial approximations of \(\sin x\)).

In string theory, they play a prominent role in the formulae for the Schnabl's exact vacuum solution to the cubic string field theory. I believe that due to this work, Martin Schnabl – and his followers – are among the world's greatest experts in the Bernoulli numbers.

Jacob Bernoulli would also coin the Bernoulli map, a generalization of the "multiple of a real number mod one", and the inequality \((1 + x)^r \geq 1 + rx\). The following curve, the lemniscate of Bernoulli, is defined by\[

P\in {\rm Lemniscate}\,\Leftrightarrow \, d(P,F_1)d(P,F_2) = \frac 14 d(F_1,F_2)^2


In the context of the probability calculus, he would study the binomial distribution ("Bernoulli trial", "Bernoulli process", "Bernoulli sampling", "Bernoulli distribution" which only takes values \(0\) or \(1\) with probabilities \(p\) and \(1-p\)) and its generalizations to more than two outcomes ("Bernoulli scheme"). Especially in the context of the binomial distribution, he would notice the law of large numbers, at least through some examples. The transfer operator is also sometimes named after him. Well, many very modern insights invented recently also carry his name because of some mathematical links (hidden Bernoulli model in speech recognition, for example).

He died in Basel at age of 50.

Daniel Bernoulli (1700-1782) used to be an important fluid mechanic. He would be a forefather of the carburetor and the airplane wing. You may imagine that centuries before the age of cars and airplanes, such a contribution had to be a bit theoretical. Indeed, what I mean is purely his discovery of the Bernoulli principle\[

{v^2 \over 2}+gz+{p\over\rho}=\text{constant}

\] which simply expresses the conservation of the "total energy per unit mass of the fluid". The first term is the kinetic energy, the second one is the potential one, the third one is from the pressure, and their sum has to be constant because the particles of the fluid may only gain or lose energy given by the first two terms when the pressure changes and does work which is accounted for by the pressure term.

He would lay the foundations of the kinetic theory of gases and thermodynamics – although I don't claim that he is the only guy who may appear as the subject of a similar sentence.

But Daniel Bernoulli would also invent a "paradox" called the "St Petersburg Paradox" in the probability calculus or game theory that would become the main topic of countless papers by economists – the avalanche is still continuing these days. The idea is simple and provocative. You may buy a "lottery ticket" that will pay you \(2^n\) dollars if the head appears \(n\) times in a row. So you will get nothing if you get "tails" immediately, you get one dollar if the first one is "head" and the second one is "tail", two dollars if you get "two heads" stopped by "one tail", four dollars for "three heads followed by a tail", and so on. How much such a ticket is worth?

The expectation value of the bounty is\[

E = \sum_{n=1}^\infty \frac{1}{2^{n+1}} \cdot 2^n = \sum_{n=1}^\infty \frac 12 = \infty.

\] So you should buy the ticket for any price – the average wealth you will win is infinite. Most people wouldn't even pay $25 for such a ticket. Why?

The number of solutions and "talking points" people – starting from Daniel Bernoulli himself and continuing with folks like Keynes, Samuelson, and others – is huge. First, people don't really think about the expectation value of the money but the utility that increases "sublinearly" with the money. If you're too rich already, one dollar is "nothing". So if you replace the factor \(2^n\) by a quantity that increases less quickly with \(n\), you may get a finite sum. But if the ticket promises you a bigger payoff than \(2^n\), you may restore the divergence in the sum so one might think that this "utility trick" doesn't solve the basic paradox.

Alternatively, you might say that the divergence arises from terms with very high values of \(n\) which correspond to tiny probabilities. Effects with too tiny probabilities may be said to be "impossible" and if you just truncate the terms with \(n\gt n_0\) with some large \(n_0\), for example forty, the sum becomes convergent again. Many economists claim that this explanation is problematic because people, on the contrary, tend to overestimate events with very tiny probabilities (the interest of some brainwashed people in the hypothetically possible apocalypse caused by global warming – whose probability is zero for all practical purposes and for most of the impractical purposes as well – is an example of this irrational bias).

At this point, I would say that the economists seem to assume that the people's thinking is utterly rational and it maximizes "something". However, people may be victims of illusions. Too difficult math problems may encourage them to think irrationally. In particular, people tend to think that "exponentially large numbers are just huge" (think of anti-growth überlunatics like Alexander Ač for whom "the exponential function" is a synonym of "apocalypse"). In the St Petersburg lottery ticket case, it's "manifest" that the probabilities to get too many heads are "exponentially tiny" which is why people tend to approximate them by zero – even though many of the terms are actually weighted by probabilities that are larger (less insanely small) than the probabilities relevant for ordinary lottery tickets they normally buy. So one should distinguish the maximization of a well-defined utility function from the research of a real-world person's brain which is simply not an infallible calculator or a superb mathematician!

One may also make many other points. The exponentially large bounty cannot be paid by any bank in the world if \(n\) is sufficiently large – a few dozens – which should also lead you to cutoff the sum, reducing the divergent expectation value to a finite one (a few dozens of dollars if the lottery is organized by Bill Gates). Alternatively, people tend to rely on the fact that the institution offering the lottery ticket can't behave insanely so if you would win an infinite amount of dollars in average, they would lose an infinite amount and they can't be stupid enough to offer you such conditions. Again, this conclusion is wrong because the average payoff is demonstrably infinite – and the sociological argument why a rich bank wouldn't be crazy enough to offer you insanely good conditions (for you) is simply another example of the imperfect reasoning of the potential players in the lottery.


  1. Civil engineers write Bernouli's principle in terms of unit weight (N, lbf) instead of unit mass, so

    v^2/2g + z +p/gamma

    While an affront to physics purists, it is a great convenience in engineering design. All energies are expressed in terms of elevation (m, ft), and starting from, say, the water surface in a reservoir you can easily calculate what parts of a city you can serve without pumping.

  2. You know from experience that more than a few heads is exceedingly unlikely, so you base your price on what has a reasonable likelihood of taking place.

  3. It's one of the loopholes I wrote.

  4. Engineers or the other people may face trouble when the fluid is flowing to a region with a different g - because then at most one of the forms of the principle may be right. ;-)

  5. Excepcional essay Lucretius, it's a compendium of my own views on the subject after lots of discussions with those atheists who think that religion is in some way disabling for a scientist. I'd add a simple amendment: What would a world without religion...? definitely an ugly world without the contributions of religious people to art (Velazquez, Raphael, Caravaggio, Rubens ...), to music (Mozart, Bach, Verdi ...), Philosophy (Augustine of Hippo, Thomas Aquinas St. Augustine, Locke, Calvin...), and of course, something that many people forget, to science itself:

    - List of Catholic scientists (

    - List of Roman Catholic cleric–scientists (

    - List of Jesuit scientists (

    - List of Jewish scientists and philosophers (

    - List of Muslim scientists (

  6. Man, did I hit a nerve for you to conflate my comment into an all out new atheist attack and to set up a whole brigade of straw men to attack with imo quite ridiculous arguments and non-sequiturs. I was obviously NOT laying out a formal argument. The Devils of Loudun was a spur of the moment example that popped into my mind.
    Klari von Neumann said that John was very childish. His deathbed conversion stunned many of his friends and was obviously induced by fear, a usual tool of most religions. Pascal's wager is craven and contemptible. The sort of "God" who would punish someone for not believing is certainly a small god, a puny god.
    The idea that an atheist must not care about others--be altruistic is total bs as is the trotting out of Stalin etc. Unfricking believable that someone as intelligent as you could be so brainwashed. If you think that my brief comment was my argument against religion, you need a new mental CPU.

  7. Don't you think that altruism--helping others-- is better motivated by decency rather than fear of punishment or promise of reward? I find that accusations by the religiously-impaired that atheists or agnostics must be immoral, or selfish to be infuriating. The idea that you need religion to be decent...gad, what a cringe-worthy thought. Wolves are altruistic to their pack members, as are elephants. Unless I am wrong, they are not worshipping a god in their image, and claiming that unbeliever elephants must be rogue animals, potential Stalinesque pachyderms. Pardon the rant (and it is a rant, so don't bother assuming it is an argument. )

  8. Your second reply is no improvement on your first one.

    “The Devils of Loudun was a spur of the moment example that popped into my mind.”

    OK, but the fact that the first “spur of the moment” example you come up with is so bad speaks for itself.

    “Klari von Neumann said that John was very childish. His deathbed conversion stunned many of his friends and was obviously induced by fear, a usual tool of most religions.

    Well, if the only thing religion can do is cure childish people of fear, that is already much more than what you can offer, isn’t it?

    “Pascal's wager is craven and contemptible. The sort of "God" who would punish someone for not believing is certainly a small god, a puny god.”

    There is no need at all to interpret Pascal’s wager in terms of “belief”. In fact, Pascal speaks only about “behaving as if one believed” and this can refer to keeping the ethical principles. While it is true that Christians and in particular Catholics put a big stress on belief, the Jews don’t. In fact, they require the Gentiles only to keep the seven Noahide Laws and belief is not one of them. Pascal himself admitted that a man may not be able to believe and should not be punished for this.

    “The idea that an atheist must not care about others--be altruistic is total bs as is the trotting out of Stalin etc.”

    This is absurd, who said “must not care”? My point was only that he cannot justify why one should care. I can’t believe that you don’t understand such a simple difference - obviously you are pretty desperate.

    “If you think that my brief comment was my argument against religion, you need a new mental CPU.”

    I am sure it was not you better argument but so far I have not seen that you are capable of anything worth of being called “an argument”. This last remark strongly reinforces that impression.

  9. Clearly you are replying to post that you have not bothered to read. How else could you imagine that I worte that “atheists or agnostics must be immoral, or selfish” while at the same time repeatedly stating that I was myself an agnostic (or even an atheist)?

    What I wrote was that, in my opinion, there is no argument based on atheistic grounds that justifies not just simply mild altruism but risking or sacrificing of one’s own life (and that of one's own family ) for a complete stranger or even less for someone who you view as an enemy. (This is the kind of example that I had in mind ).

    Using a mild word like “decency” in this context is absurd, but even more absurd is to think that this has anything to do with “fear”.

    I dispair at the possibility of explaining this to someone like you so I won’t even bother to attempt it.

  10. You could look at Kenneth Ring 1997 paper “Near-Death and Out-of-Body Experiences in the Blind: A Study of Apparent Eyeless Vision“

    However, we don’t know whether the congenitally blind experient’s in Ring’s paper are actually ‘seeing’ in the way the sighted ‘see’, indeed there is a great deal of doubt about that, as Ring himself admits.

    However, their particular condition makes them an interesting group to study, as you've pointed out, it’s quite well documented that the congenitally blind don’t appear to dream ‘visually’.

    So, whether during the OBE NDE, the congenitally blind are ‘seeing’ in the way the sighted ‘see’, or not, remains an open question. However it is interesting that these experient’s seem very sure that whatever they experienced during their OBE NDE, it contained perceptions which were distinctly different, from their non-visual dreams.

    Hidden away in Rings paper is an interesting quote from perhaps his best congenitally blind subject, 'Vicki', who said she was never able to discriminate colours during her OBE NDE, but only “…different shades of brightness…”.

    If you know anything about Edwin Land’s work on human colour perception, he showed that this is exactly how the retina sees – in different shades of brightness. Land’s experimental work clearly demonstrates that secondary processing must be taking place in the cortex to create our perception of colour, and that colour itself arises solely out of our brain.

  11. Listen Lucretius---I think your arguments are simple and in fact, stupid, since you are stooping to ad hominem stuff. IMO the Devils is a great example, as is David MacKay's "Extraordinary Popular Delusions and the Madness of Crowds". You might want to read Huxley's "Grey Eminence" about Joseph Surin and Richelieu.

    "He cannot justify why one should care" Really? How about altruism, decency, empathy, etc. If you think religion is required for these things, you are farther gone than I thought.
    If you can't see that "behaving as if one believed" is contemptible, then I feel sorry for you.
    If you can pretend to believe something to avoid punishment, that is craven.
    Spinoza had ethical principles, and by all accounts, was a good human being. So, for that matter was Nietzsche.

    As with Laplace's comment to Napoleon that wrt God that "I have no need for that hypothesis". Just as I do know the difference between "must

    not care" and justifying why one should care, I also know the difference between someone using false arguments, condescension, and straw man tactics. My comment to Anne about Seth seems to have triggered your version of the Spanish Inquisition. I prefer not to pretend to believe things that I believe to be false--
    You believe what you want---"Nothing is neither good nor bad but thinking makes it so."
    I expect this to be our last entanglement. You can blather on if you wish.

  12. I also take no pleasure in this pseudo-argument because you behave as if you were as dumb as your spiritual and intellectual ally Cynthia, although in the past you did not always seem so. So it must be something special about this topic. But on this topic you appear incapable of understanding the simplest things so the natural conclusion is that, like the people who refuse to try to understad quantum mechanics, you are also refusing to use your brain.

    So I will keep it short. Your bringing up “fear” etc, shows that you completely failed to understand the most basic point, so basic in fact that I have so far not bothered to make it explicit. The key point is the difference between MORALITY BASED ON ABSOLUTE MORAL PRINCIPLES and RELATIVISTIC MORALITY, which is all that you can offer as an alternative. Your statement that altrusim “is better motivated by decency rather than fear of punishment or promise of reward” is a perfect example. I did not bother to answer it but now will. As an atheist, which is what I am, as I have pointed many times, I can say with perfect sincerity that I don’t give a damn about your “decency”. What you call decent or indecent is of no importance to me at all, it is just your opinion and you can keep it. If a situation arose that for some reason I could save your life with, say, some risk to mine, or alternatively felt the need to kill you, the fact that you and other people who I don’t care about will call it decent or indecent would not matter to me. What would matter are two things: what people who I care about think about it, and my own conscience. But there is not guarantee that the former (the people who I care about) will actually mind my killing you, in case of war. My conscience is more of a problem because it is not entirely known to me, but it can reasonably be considered as something related to “acting as if one believed” because I have no doubt at all is that it is rooted in Christian and Jewish morality. In my case, anyway, whenever I deal with a moral issue, I find myself acting as if I believed in the Jewish and Christian God, with the emphasis on Jewish because there are things in Christian morality (such as “turning the other cheek”) that I do not accept.

    You must be really stupid if you still believe that I have ever asserted that an atheist cannot act morally or altruistically. What I do assert is that you cannot use atheism to make COMPELLING MORAL STATEMENTS. In other words, there is no statement you can make that I will find morally compelling and your “decency” just proves it. The fact that you call something indecent will only make me shrug my shoulders.

    Compare this with Kossak-Sztucka’s ( ) words:

    We are required by God to protest. ...God who forbids us to kill. We are required by our Christian consciousness. Every human being has the right to be loved by his fellow men. The blood of the defenceless cries to heaven for revenge. Those who oppose our protest, are not Catholics."

    Now replace the word Catholics with atheist and God and heaven with whatever you want. Doesn’t it sound idiotic to you? Don’t you still get the point? No, I guess you don’t.

  13. I don't see why you think that Penrose's addition of (bizarre) musings about consciousness into his science books is compatible with your promoted separation of science and religion.

  14. Quite frankly: I have not paid much attention to these "musings" as they are not the kind of things I am interested in in Penrose's writings, but I have never noticed any direct references to God. Maybe I have missed something "indirect" but Penrose has explicitly denied that sort of thing and I have no reason to believe that he wants to "sneak in" God thorough a back door. Wildly speculative ideas about things about which very little is known are not disqualified by me concept of the "separation".

  15. Dear Lucretius, all science-like comments about God *or* consciousness are wildly speculative. But this extra labeling was done by the two of us, not by Penrose. Penrose didn't separate or label his wildly speculative musings about consciousness from the rest.

    At the end, it's because one can't really separate these two realms. These two realms overlap and fight at many places and one simply has to choose. A promoter of religion or other spiritual ideas about consciousness and similar things is *guaranteed* to soon or later say things that directly clash with established science.

  16. I see a rather big difference between God and consciousness.

    I find God inadmissible in science when used as an explanation. I do not mind science being used to explain God (the most natural scientific explanation of God being that is is a product of human culture and human psychology) even in widely speculative ways as long as no claims are being about having “proved” or “disproved” anything. Such “explanations” will probably forever remain purely speculative.

    On the other hand, the existence of consciousness I consider a fact, explaining of which is a legitimate concern of science, and I do imagine the possibility that one day an adequate explanation could be found. Still, at this time all scientific explanation of consciousness must be considered as “widely speculative”.

    However, doing the opposite: explaining the physical world in terms of “consciousness” would be either religion or metaphysics and I would not consider it science. As far as I know Penrose has not been engaging in the latter.

    “A promoter of religion or other spiritual ideas about consciousness and similar things is *guaranteed* to soon or later say things that directly clash with established science.”

    I am not sure what you mean by a “promoter” of religion. But it is clearly easy to speak of religion which in no way clashes in science. All you have to do is make sure that your views on religion have no empirical consequences at all. One example is the idea of God as kind of “supreme mathematician”, the source and repository of all mathematical truth. Another is God as the ultimate source of all moral values. I very much doubt that either of these ideas is bound to clash with “established science”, except when “established science” attempts to go beyond its proper boundaries. As I have mentioned elsewhere, my views on this matter are very close to those of Freeman Dyson:

  17. It's late and, btw the neurons last used for intense calculus have been petrified for a decade. I was mediocre at it but made up for it in tenacity and sometimes; in the novelty of techniques to get useful (Engineering) results.

    When I asked colleagues to check my work, they ran away; like an Englishman confronted by a hot pepper… when consultants were hired to verify my work, they "proved" it by running numerical simulations for specific dimensions.

    That must've been when the the first Leibnitz-like neurons died of misery in my head.

  18. If g varies enough to be an Engineering problem (and the civil engineers have LARGE factors of safety), then the Engineers will be the last to face trouble. ;-)

    I got into trouble at Uni with a tutor because I'd used g = 10 m/s² in an exercise. He insisted that g = 9.81 (rather foolishly, methinks). The specifications did not state where the objects were situated, so I felt free to use whatever value of physical constant for g was convenient. Engineers read specifications and work out the best/easiest/simplest solution to the problem within the specification.

    Newton's Principia (late-night reading, when I get the chance) has an interesting treatise on the force of gravity, especially wrt the equations for bodies near the surface of a large, distributed mass (i.e. stuff like water on the surface of the planet).

  19. Quantum computer eye candy, care of Google and NASA:

  20. Thanks for your thoughts. I knew they would be different -- and interesting!

  21. Didn't know about Zofia Kossak-Szczucka. Thanks for the link!

  22. Judeo-Christianity has been the real gift giver. The others not so much. Speaking empirically.

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