A Mathematical Gangster Strikes

Normally we think of natural sciences as being empirical sciences first and foremost, but even so, the experimental falsification or verification of hypotheses is still preceded by rational conception.

This is especially true in physics, where theory is way ahead of the ability to experimentally verify all the concepts that have been created. Even Einstein had to wait four years before the first empirical proof of general relativity could be obtained in 1919 by observing the position of stars during a solar eclipse (which showed that the mass of the sun bent the light rays coming from said stars, altering their relative position in the sky). This was notwithstanding the fact that general relativity could explain the anomalous rate of precession of the perihelion of Mercury's orbit, which was first recognized as posing a problem for Newtonian mechanics in 1859. Even though the 1919 observations were reconfirmed in 1922 as well as by later observations, sufficiently precise measurements to remove all remaining doubt could only be performed in the 1960s. 


Leonhard Euler by Handmann“Mathematical gangster” Leonhard Euler (1707-1783), in a famous painting showing him wearing rather odd Prussian headgear made from silk. While Napoleon became a cognac, Euler eventually became a Swiss banknote.

(Pastel by Jakob Emanuel Handmann)


Extensive precision-testing of general relativity effectively began in 1959, with the observation of the gravitational red-shift (which Einstein had first predicted in 1907) in the Pound-Rebka experiment. Similarly, several of the effects predicted by special relativity could only be empirically verified a long time after the theory had been conceived. For instance, the first empirical observation of time dilation and length contraction was made in the Ives-Stillwell experiment in 1938.

Since Einstein's time, the gap between the advances in theory and the ability to confirm theoretical concepts by experiment has arguably grown even larger in physics. For instance, many of the ideas of quantum physics require sufficiently strong high energy particle accelerators for experimental confirmation (see the hunt for the Higgs boson – from theoretical prediction to experimental proof, four decades passed).

Highly advanced theories like string theory may never be fully confirmed by experiments, simply because some of their elements are literally beyond our three-dimensional world. That theory is so far ahead of the possibility to conduct experiments falsifying or confirming it is not least due to the fact that a purely aprioristic science – namely mathematics – is the main tool employed by theoretical physicists.


CERN / LHC / Teilchenbeschleuniger

The compact muon solenoid detector of the Large Hadron Collider (a 27 kilometer/17 mile ring of superconducting magnets 100 meters/330 feet underground) at CERN. A 5- sigma signal at around 125 giga-electron volts was detected in 2012 and in June this year it was finally confirmed that it indeed represented the Higgs boson.

(Photo credit: CERN)


In this context, we have recent come across a video by Numberphile, in which the work of mathematician Leonhard Euler on infinite series is discussed. Professor Edward Frenkel explains that Euler was so to speak a “mathematical gangster” by employing highly unconventional (and hence “not allowed”) thought processes to certain problems. Euler came to the heretical conclusion that the divergent series 1+2+3+4+5+…., which at first glance simply appears to expand toward infinity, can actually be assigned a finite number.


Counter-Intuitive Result, Practical Applications

So what is the sum of this infinite series of all natural numbers? Rather counter-intuitively, Euler found out it is actually – 1/12. It is perhaps better to state that the result “isolates the finite part” of the series, as Frenkel puts it. This result was only confirmed a century later by Riemann. But how can that be? As Frenkel explains in the video below, although it is a seemingly counter-intuitive result, it (and analogous results) finds countless practical applications in physics.

He also explains why it is possible to “break the rules” in an aprioristic science. As long as a theory is logically consistent, it is possible to describe things that appear “impossible” on the surface. As an example he cites the square root of -1.  This is a number that makes no sense at first glance. How can there be a square root of a negative number? However, the theory of complex numbers which includes the square roots of negative numbers, is a logically consistent theory which is simply based on different axioms. Importantly, it is actually a quite useful theory that can help in elucidating other mathematical problems. In this sense such “unrealistic” modeling is comparable to models employed in economics (such as e.g. the evenly rotating economy). In economics, such models describe a world that doesn't exist in reality, but they are nevertheless indispensable for explaining economic laws and are providing us with the wherewithal to understand the real world of change.


Edward Frenkel on Leonhard Euler and the series 1+2+3+4+5+…which equals -1/12


So how exactly did Euler arrive at this astonishing result? The method is actually quite simple, and it is explained below by physicist Tony Padilla. Padilla also points out that Euler's work on divergent series is indeed quite useful in physics (including, as it were, string theory).


Tony Padilla on the method employed by Euler to calculate the result


If you want to know why exactly the sum of Grandi's series 1-1+1-1+1-1+1… equals ½, that is shown here by Dr. James Grime:


Why does Grandi's series equal ½?


And lastly, here is Eduard Frenkel again, talking about the Riemann hypothesis and Riemann's zeta function. Once again, “gangster” Euler's work is mentioned as a path-breaking:


Eduard Frenkel on Riemann's zeta function and the Riemann hypothesis


This is not only an example for an aprioristic science arriving at results that can be usefully applied in a natural science. You also have the chance to win $1 million if you manage to prove (or disprove) Riemann's hypothesis.

If you were laboring away on the Poincare conjecture, we regret to inform you that you are too late. After mathematicians had tried for almost a century to solve it, Grigory Perelman finally succeeded in 2002. By 2006 his proof had successfully withstood review by several teams of mathematicians. He was offered a Fields medal and was awarded the Millennium Prize of $1 million – and promptly declined both. Incidentally, the Poincare conjecture has important implications for understanding the universe shortly after the big bang.



Grigori Perelman explains his proof of the Poincare conjecture in a lecture delivered in 2003.

(Photo credit: Frances M. Roberts)



Control room at CERN

(Photo credit: CERN)




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10 Responses to “Apriorism and the Natural Sciences”

  • VB:

    This is a bunch of nonsense. The Russian on the first video is just full of hot air – he is just talking without explaining or proving anything. The guy on the second video has one valid sentence in the whole talk – “this is a fake sum”. That’s it. You can’t sum a divergent series.

    The analogies with square root of 2 and of -1 one do not apply. Just as you CANNOT come up with a rational number that is a square root of 2 and a non-complex number that is a square root of -1, you cannot come up with a ratio and say that it is the sum of a divergent series. You *can* come up with a new TYPE of number and say that it marks the sum of a divergent series and base a whole mathematical system on it – but this isn’t what these guys are doing.

    The sum of a divergent series is infinity. You can’t multiply or divide infinity. Even if 1 = 4S, where S is infinity, you can’t conclude from that that S is 1/4. The *proper* way to deal with the different kinds of infinities was introduced by Cantor with the aleph series of numbers.

  • worldend666:

    It seems if you read a bit of background the proofs above that the sum of all integers =1/12 are illegal although there is a conventional proof which I don’t dare tackle. If you haven’t watched at least one of the videos then stop reading now :)

    Specifically the part timer’s proof above relies on 1-1+1-1+1…equalling a half when in fact it does no such thing. Worse, if you just reorder the numbers to start with -1instead of 1 then following the convoluted logic would lead you to conclude the sum is in fact equal to minus a half. Clearly it cannot equal both a half and negative a half.

    • 23571113:

      look, these are divergent series. when someone says it is equal to 1/12 or whatever, they don’t mean it converges to 1/12 in the usual sense. they have a precise definition of what is means for a (divergent) series to equal to a number, and that is not the usual definition (for a convergent series). for example, one can say: “Definition: If a series is divergent, then we define its (special) value to be equal to 10.” There is nothing wrong with making this definition and then one can go tell people, hey actually every divergent series has value 10. Well, yes, by definition! You can come up with other useless definitions, and may even be able to play around with the definitions.

      The point is, they not only have a well-defined notion of what it means to say the given series is equal to 1/12, but actually has some useful and non-trivial applications for this concept.

  • Mark Humphrey:

    You mention the Big Bang theory. I doubt the validity of the Big Bang, on the basis of philosophy. The theory posits that the universe, meaning all the exists, came into existence via the Big Bang. Before the Big Bang there was, supposedly, nothing; after the bang, existence came into being.

    But the idea of “before and after” presupposes “time”; time measures the motion of things that exist. So there is no time, “before” existence; there is only time within existence. This means that there can be no beginning and no ending of existence. If there is no beginning or end, then existence is infinite: it has always been and will always be. This idea is similar to the vague concept of “God” that the religious hold to, except there is no question that existence exists.

    The idea of infinite regression is congruent with the idea that something cannot come from nothing. The characteristics and aspects of whatever exists change, but various entities have always and will forever have being.

    • No6:

      The Big Bang theory suggests that this universe came about 13.7 billion years ago. It says nothing about other possible universes or the existence of a greater Cosmos.

    • SavvyGuy:

      @Mark Humphrey: “I doubt the validity of the Big Bang, on the basis of philosophy”

      Well, if you doubt that all of existence started out from a singularity, then please ponder this…why is gravity an attracting force between masses, and not a repelling force?

      My pea-sized brain tells me that the attracting nature of gravity is a remnant of the Big Bang, and therefore at some point in the future the Universe will stop expanding and start contracting due to gravity. Eventually, all of existence will collapse back into the singularity whence it came from, and it will be Big Bang redux all over again.

      Not only is space fractal, but so is time!

  • gigi:

    Assume S = 1+2+3+4+5+…
    S = S
    => 0 = S – S
    => 0 = (1+2+3+4+5…) – (1+2+3+4+5…)

    => 0 = 1+((2+3+4+5+6…) – (1+2+3+4+5…))
    => 0 = 1+(2-1)+(3-2)+(4-3)+(5-4)+(6-5)+…
    => 0 = 1+1+1+1+1+…

  • woodsbp:

    Pater, are you sure about this?

    “In this sense such “unrealistic” modeling is comparable to models employed in economics (such as e.g. the evenly rotating economy). In economics, such models describe a world that doesn’t exist in reality, but they are nevertheless indispensable for explaining economic laws and are providing us with the wherewithal to understand the real world of change.”

    Unrealistic modelling, as you term it, is I believe not appropriate to economics – since economics is hardly an aprioristic science. Economics is in reality a social science – its proper name being Political Economy, so it cannot have ‘laws’ in the sense of the immutable natural laws of physics and chemistry.

    However, you can certainly adduce logical premises, and time and factor-bounded correlates which may be mistakenly described as causal, hence the overwhelming temptation to call them as ‘laws’. It would indeed be wonderful if there were predictable causes-and-effects in Political Economy. Alas, there are none. But this will not prevent the furious faith of fervent folk believing otherwise. And there being none so ‘queer’ as folk.

    Now whilst we are on the subjects of ‘laws’ and ‘mathematical rules’, there are two such entities which are of some real significance to Political Economy – but rarely get mentioned. Perhaps economists are actually ignorant of them, or perhaps they do know about them but consider them to be quaint and irrelevant. Whatever.

    The first are the 3 No. Thermodynamic Laws. The second are the mathematical rules of exponential (growth) functions. It would be wonderful if economists (and those who are economically inclined) would give more care and attention to these two entities. They, with some certitude, may be deployed to explain why we are enduring such a prolonged economic ‘downturn’. Not everything mind, but enough to cause concern.

    Oh, one last thing. You can never prove anything ‘positive’, ie. confirm it. But you can adduce statistically reliable evidence, that at a particular level of statistical confidence your Null Hypothesis may be rejected, and you accept an alternate hypothesis.


    • Economics is definitely an aprioristic science, and modeling is a necessary mental tool to explain economic laws. I will write more on this, and discuss your objections in the next article on the topic. Obviously I have not explained these issues in detail in this article, but I will flesh out these arguments next time.
      This article was only meant to show that apriorism also has its place, to some extent, in the natural sciences.

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