You expect the physical world around you to behave a certain way. If you knock a glass off the table, it will fall to the floor and likely shatter. How do you know this? Primarily from everyday experience, which your brain distills into your intuition. Once you’ve seen enough
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Feynman
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Beloved late physicist Richard P. Feynman (1918–1988) first met his hero Paul Dirac (1902–1984) during Princeton University’s Bicentennial Celebration in 1946 and then again at least twice, in 1948 and 1962.
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Schrödinger’s Equation
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What is Wave-Particle Duality?
Why we need quantum physics.
Computer Science
About Scientific Theories
What amount of effort really goes towards formulation?
Before we begin, be aware that talking about control theory in any meaningful way means talking about linear algebra. Please don’t be intimidated, even if you don’t have the first idea what any of the symbols or terms mean. Linear Algebra is hard but its core ideas are intuitive and I will explain everything as we go.
In the first article, we explored the world of control systems. We saw how control systems are present all around us for ages and how it makes our life safe and easy. The problem “control of an inverted pendulum” was proposed to understand various control strategies and concepts. Refer this article for insight into the inverted pendulum.
For this study, I will be placing the inverted pendulum on a cart with a friction-less base. The connection between the pendulum rod of length L (assumed weightless) and the cart has damping (d). Damping is the resistance towards change in angular speed of the rod. The figure below shows the setup:
Here, u is the control force to balance it, M is mass of cart, m is the mass of pendulum ball.
I will not go through the derivations of the equation. That is something we can find anywhere easily. It is also well understood with a video rather a textbook derivation. On the other hand, to understand the analysis of the equation one needs the understanding of state variables and equilibrium points. State variables are the minimum number of variables with which we can completely define a system. Too formal? Don’t worry. Take, for example, the train for which you are waiting on the platform. You look into your train tracking app to get an idea when it will arrive. Which information is needed by the app to correctly predict the arrival time of the train? Well, its current position, its speed and the traffic on the track will be used to correctly predict the arrival time. So, look back on the definition and you will find that the current position, the current speed and the traffic are the state variables here.
Real life example of state variables
If you express this in terms of equations and mathematical model then the whole expression is called state-space expression.
Now, the equilibrium points. Put all the derivative of the state variables to zero you will get the equilibrium point. Basically, in the equilibrium point, there is no change in the state variables with respect to time. Hence, the rate of change is zero. So, when the train arrives at the platforms and stops, you get the equilibrium point.
The train has arrived. There is no change in the position, speed and the traffic on the track.
Mathematical Modelling
The equation for the inverted pendulum is given below. You can see how the equation are written in terms of state variables, which are, the position of the cart {x}, its speed {v}, the angle which the ball pendulum makes with the vertical {θ} and its angular velocity {ω}. So, the state vector X = [x, v, θ, ω]’, where “ ‘ ” denotes the transpose.
The dynamic equation of the inverted pendulum on cart.
These are the differential equations (equations which describe a rule for the rate of change of a function with respect to one or more of its input variables). The dot “ . ” on the top of state variables means their rate of change with respect to time.
Mathematical models can be well coded in MATLAB. MATLAB is an engineering-specific programing language, helps us to write the mathematical models easily. Check the simple code of the above equations below. To view all the MATLAB codes related to this series, go to this link of GitHub.
This is how the dynamic equations look like in MATLAB code
Above, dx is the time derivative of the state vector X.
If you find the equilibrium point here by equating all the above equations to zero, you will get two points. These two points, corresponding to either the pendulum down (θ = 0) or pendulum up (θ = π) configuration; in both cases, v = ω = 0. The upwards position is an unstable equilibrium point. The downward position is stable. We will later see how we can identify different the equilibrium points using simple math. A disturbed pendulum will always go to the stable position unless a control force is applied (see the videos below).
From here onward, the complete setup of the pendulum and the cart will be called the system.
Equilibrium points of an inverted pendulum
Developing the linear model
Looking at the equation above, we see that terms like sin(θ), cos(θ), ω² make the system non-linear. They pose a problem in designing the block-diagram or representing in a neat constant matrix of a state-space format or transfer function format. Hence, modelling and analysis become difficult. Also, many linear control techniques are available like PID, LQR (we will discuss them when we design the controller) and they are easy and tried-and-tested methods to deals with linear systems.
So, we try to develop a linear model of this system using a process called linearization. For linearization, we need an operating point as explained in the appendix. The equilibrium point is an operating point. But we have two equilibrium point. Hence, the linearized expression depends upon which point is chosen. The linearized matrices look like this as shown below
The linearized state-space expression
where b=1 for the pendulum upward equilibrium point, and b=-1 for the pendulum downward equilibrium point. The above expression can be simply written as
State-space model of the linearised system
We call A as the state matrix, B as the input matrix, C as the output matrix and D as the direct output matrix. Y is the output matrix. Look how clean and easy the above expression is when compared to the non-linear dynamics. The uncontrolled system performance depends upon A matrix. The code looks like this
Analysis of the linear equations
Before proceeding, let us revisit the stick on the finger analogy from the second article. As mentioned there, to move the stick forward, you first need to pull the finger back, giving a forward tilt and then move forwards. This type of system where the physics of the system forces it to dip/move in the wrong direction initially to achieve its correct direction is called “non-minimum phase system”. Other examples include the altitude manoeuvre in aircraft and parallel parking.
To gain altitude, the tail ailerons bend downwards to increases the angle of attack and hence, the lift. But that downwards force on the ailerons forces the airplane to dip down initially. On right is the altitude graph depicting the initial dip.
Non-minimum phase systems can easily be identified from their transfer function. The presence of a right-hand side (RHS) zero makes it a non-minimum phase system. The transfer function (output/input) of the system is shown below:
The transfer function with poles and zeros with b=1
One of the roots of the numerator of G(s) i.e. zeros is positive (+2.9458) is a result of the system being a non-minimum phase system. Zeros only chance the amplitude (except the RHS zeros) of the response, not the overall characteristic. Here we can see that the system has an RHS pole. RHS pole leads to positive exponential terms like e^t, which leads to an unstable response.
When the model is solved with b=-1, we see that all the poles of the system are negative (LHS) and the response is stable. Hence, when simulated with b=-1, we get stable dynamics as shown below in the simulation section.
“Most mathematicians prove what they can, von Neumann proves what he wants”
It is indeed supremely difficult to effectively refute the claim that John von Neumann is likely the most intelligent person who has ever lived. By the time of his death in 1957 at the modest age of 53, the Hungarian polymath had not only revolutionized several subfields of mathematics and physics but also made foundational contributions to pure economics and statistics and taken key parts in the invention of the atomic bomb, nuclear energy and digital computing.
Known now as “the last representative of the great mathematicians”, von Neumann’s genius was legendary even in his own lifetime. The sheer breadth of stories and anecdotes about his brilliance, from Nobel Prize-winning physicists to world-class mathematicians abound:
”You know, Herb, Johnny can do calculations in his head ten times as fast as I can. And I can do them ten times as fast as you can, so you can see how impressive Johnny is” — Enrico Fermi (Nobel Prize in Physics, 1938)
“One had the impression of a perfect instrument whose gears were machined to mesh accurately to a thousandth of an inch.” — Eugene Wigner (Nobel Prize in Physics, 1963)
“I have sometimes wondered whether a brain like von Neumann’s does not indicate a species superior to that of man” — Hans Bethe (Nobel Prize in Physics, 1967)
And indeed, von Neumann both worked alongside and collaborated with some of the foremost figures of twentieth century science. He went to high school with Eugene Wigner, collaborated with Hermann Weyl at ETH, attended lectures by Albert Einstein in Berlin, worked under David Hilbert at Göttingen, with Alan Turing and Oskar Morgenstern in Princeton, with Niels Bohr in Copenhagen and was close with both Richard Feynman and J. Robert Oppenheimer at Los Alamos.
An émigré to America in 1933, von Neumann’s life was one famously dedicated to cognitive and creative pursuits, but also the enjoyments of life. Twice married and wealthy, he loved expensive clothes, hard liquor, fast cars and dirty jokes, according to his friend Stanisław Ulam. Almost involuntarily, his posthumous biographer Norman Macrae recounts, people took a liking to von Neumann, even those who disagreed with his conservative politics (Regis, 1992).
This essay aims to highlight some of the unbelievable feats of “Johnny” von Neumann’s mind.
Early years (1903–1921)
Neumann János Lajos (John Louis Neumann in English) was born (some say “arrived”) on December 28th 1903 in Budapest, Hungary. Born to wealthy non-observant Jewish bankers, his upbringing can be described as privileged. His father held a doctorate in law, and he grew up in an 18-room apartment on the top floor above the Kann-Heller offices at 62 Bajcsy-Zsilinszky Street in Budapest (Macrae, 1992).
John von Neumann at age 7 (1910)
Child Prodigy
“Johnny” von Neumann was a child prodigy. Even from a young age, there were strange stories of little John Louis’ abilities: dividing two eight-digit numbers in his head and conversing in Ancient Greek at age six (Henderson, 2007), proficient in calculus at age eight (Nasar, 1998) and reading Emile Borel’s Théorie des Fonctions (“On some points in the theory of functions” ) at age twelve (Leonard, 2010). Reportedly, von Neumann possessed an eidetic memory, and so was able to recall complete novels and pages of the phone directory on command. This enabled him to accumulate an almost encyclopedic knowledge of what ever he read, such as the history of the Peloponnesian Wars, the Trial Joan of Arc and Byzantine history (Leonard, 2010). A Princeton professor of the latter topic once stated that by the time he was in his thirties, Johnny had greater expertise in Byzantine history than he did (Blair, 1957).
Left: John von Neumann at age 11 (1915) with his cousin Katalin Alcsuti. (Photo: Nicholas Vonneumann). Right: The Neumann brothers Miklós (1911–2011), Mihály (1907–1989) and János Lajos (1903–1957)
“One of his remarkable abilities was his power of absolute recall. As far as I could tell, von Neumann was able on once reading a book or article to quote it back verbatim; moreover, he could do it years later without hesitation. He could also translate it at no diminution in speed from its original language into English. On one occasion I tested his ability by asking him to tell me how A Tale of Two Cities started. Whereupon, without any pause, he immediately began to recite the first chapter and continued until asked to stop after about ten or fifteen minutes.”
— Excerpt, The Computer from Pascal to von Neumann by Herman Goldstein (1980)
An unconventional parent, von Neumann’s father Max would reportedly bring his workaday banking decisions home to the family and ask his children how they would have reacted to particular investment possibilities and balance-sheet risks (Macrae, 1992). He was home-schooled until 1914, as was the custom in Hungary at the time. Starting at the age of 11, he was enrolled in the German-speaking Lutheran Gymnasium in Budapest. He would attend the high school until 1921, famously overlapping the high school years of three other “Martians” of Hungary:
Leo Szilard (att. 1908–16 at Real Gymnasium), the physicist who conceived of the nuclear chain reaction and in late 1939 wrote the famous Einstein-Szilard letter for Franklin D. Roosevelt that resulted in the formation of the Manhattan Project that built the first atomic bomb
Eugene Wigner (att. 1913–21 at Lutheran Gymnasium), the 1963 Nobel Prize laureate in Physics who worked on the Manhattan Project, including the theory of the atomic nucleus, elementary particles and Wigner’s Theorem in quantum mechanics
Edward Teller (att. 1918–26 at Minta School), the “father of the hydrogen bomb”, an early member of the Manhattan Project and contributor to nuclear and molecular physics, spectroscopy and surface physics
Although all of similar ages and interests, as Macrae (1992) writes:
“The four Budapesters were as different as four men from similar backgrounds could be. They resembled one another only in the power of the intellects and in the nature of their professional careers. Wigner […] is shy, painfully modest, quiet. Teller, after a lifetime of successful controversy, is emotional, extroverted and not one to hide his candle. Szilard was passionate, oblique, engagé, and infuriating. Johnny […] was none of these. Johnny’s most usual motivation was to try to make the next minute the most productive one for whatever intellectual business he had in mind.”
— Excerpt, John von Neumann by Norman Macrae (1992)
Yet still, the four would work together off and on as they all emigrated to America and got involved in the Manhattan Project.
By the time von Neumann enrolled in university in 1921, he had already written a paper with one of his tutors, Mikhail Fekete on “A generalization of Fejér’s theorem on the location of the roots of a certain kind of polynomial” (Ulam, 1958). Fekete had along with Laszló Rátz reportedly taken a notice to von Neumann and begun tutoring him in university-level mathematics. According to Ulam, even at the age of 18, von Neumann was already recognized as a full-fledged mathematician. Of an early set theory paper written by a 16 year old von Neumann, Abraham Fraenkel (of Zermelo-Fraenkel set theory fame) himself later stated (Ulam, 1958):
Letter from Abraham Fraenkel to Stanislaw Ulam Around 1922-23, being then professor at Marburg University, I received from Professor Erhard Schmidt, Berlin […] a long manuscript of an author unknown to me, Johann von Neumann, with the title Die Axiomatisierung der Mengerlehre, this being his eventual doctor dissertation which appeared in the Zeitschrift only in 1928 […] I asked to express my view since it seemed incomprehensible. I don’t maintain that I understood anything, but enough to see that this was an outstanding work, and to recognize ex ungue leonem [the claws of the lion]. While answering in this sense, I invited the young scholar to visit me in Marburg, and discussed things with him, strongly advising him to prepare the ground for the understanding of so technical an essay by a more informal essay which could stress the new access to the problem and its fundamental consequences. He wrote such an essay under the title Eine Axiomatisierung der Mengerlehre and I published it in 1925.
In University (1921–1926)
As Macrae (1992) writes, there was never much doubt that Johnny would one day be attending university. Johnny’s father, Max, initially wanted him to follow in his footsteps and become a well-paid financier, worrying about the financial stability of a career in mathematics. However, with the help of the encouragement from Hungarian mathematicians such as Lipót Fejér and Rudolf Ortvay, his father eventually acquiesced and decided to let von Neumann pursue his passions, financing his studies abroad.
Johnny, apparently in agreement with his father, decided initially to pursue a career in chemical engineering. As he didn’t have any knowledge of chemistry, it was arranged that he could take a two-year non-degree course in chemistry at the University of Berlin. He did, from 1921 to 1923, afterwards sitting for and passing the entrance exam to the prestigious ETH Zurich. Still interested in pursuing mathematics, he also simultaneously entered University Pázmány Péter (now Eötvös Loránd University) in Budapest as a Ph.D. candidate in mathematics. His Ph.D. thesis, officially written under the supervision of Fejér, regarded the axiomatization of Cantor’s set theory. As he was officially in Berlin studying chemistry, he completed his Ph.D. largely in absentia, only appearing at the University in Budapest at the end of each term for exams. While in Berlin, he collaborated with Erhard Schmidt on set theory and also attended courses in physics, including statistical mechanics taught by Albert Einstein. At ETH, starting in 1923, he continued both his studies in chemistry and his research in mathematics.
“Evidently, a Ph.D. thesis and examinations did not constitute an appreciable effort” — Eugene Wigner
Two portraits of John von Neumann (1920s)
In mathematics, he first studied Hilbert’s theory of consistency with German mathematician Hermann Weyl. He eventually graduated both as a chemical engineer from ETH and with Ph.D. in mathematics, summa cum laude from the University of Budapest in 1926 at 24 years old.
“There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. von Neumann didn’t say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann” — George Pólya
From von Neumann’s Fellowship application to the International Education Board (1926)
His application to the Rockefeller-financed International Education Board (above) for a six-month fellowship to continue his research at the University of Göttingen mentions Hungarian, German, English, French and Italian as spoken languages, and was accompanied by letters of recommendation from Richard Courant, Hermann Weyl and David Hilbert, three of the world’s foremost mathematicians at the time (Leonard, 2010).
In Göttingen (1926–1930)
The Auditorium Maximum at the University of Göttingen, 1935
Johnny traveled to Göttingen in the fall of 1926 to continue his work in mathematics under David Hilbert, likely the world’s foremost mathematician of that time. Reportedly, according to Leonard (2010), von Neumann was initially attracted to Hilbert’s stance in the debate over so-called metamathematics, also known as formalism and that this is what drove him to study under Hilbert. In particular, in his fellowship application, he wrote of his wish to conduct (Leonard, 2010)
“Research over the bases of mathematics and of the general theory of sets, especially Hilbert’s theory of uncontradictoriness […], [investigations which] have the purpose of clearing up the nature of antinomies of the general theory of sets, and thereby to securely establish the classical foundations of mathematics. Such research render it possible to explain critically the doubts which have arisen in mathematics”
Very much both in the vein and language of Hilbert, von Neumann was likely referring to the fundamental questions posed by Georg Cantor regarding the nature of infinite sets starting in the 1880s. von Neumann, along with Wilhelm Ackermann and Paul Bernays would eventually become Hilbert’s key assistants in the elaboration of his Entscheidungsproblem (“decision problem”) initiated in 1918. By the time he arrived in Göttingen, von Neumann was already well acquainted with the topic, in addition to his Ph.D. dissertation having already published two related papers while at ETH.
Set theory
John von Neumann wrote a cluster of papers on set theory and logic while in his twenties:
von Neumann (1923). His first set theory paper is entitled Zur Einführung der transfiniten Zahlen (“On the introduction of transfinite numbers”) and regards Cantor’s 1897 definition of ordinal numbers as order types of well-ordered sets. In the paper, von Neumann introduces a new theory of ordinal numbers, which regards an ordinal as the set of the preceding ordinals (Van Heijenoort, 1970).
von Neumann (1925). His second set theory paper is entitled Eine Axiomatisierung der Mengenlehre (“An axiomatization of set theory”). It is the first paper that introduces what would later be known as the von Neumann-Bernays-Gödel set theory (NBG) and includes the first introduction of the concept of a class, defined using the primitive notions of functions and arguments. In the paper, von Neumann takes a stance in the foundations of mathematics debate, objecting to Brouwer and Weyl’s willingness to ‘sacrifice much of mathematics and set theory’, and logicists’ ‘attempts to build mathematics on the axiom of reducibility’. Instead, he argued for the axiomatic approach of Zermelo and Fraenkel, which, in von Neumann’s view, replaced vagueness with rigor (Leonard, 2010).
von Neumann (1926). His third paper Az általános halmazelmélet axiomatikus felépitése, his doctoral dissertation, which contains the main points which would be published in German for the first time in his fifth paper.
von Neumann (1928). In his fourth set theory paper, entitled Die Axiomatisierung der Mengenlehre (“The Axiomatization of Set Theory”), von Neumann formally lays out his own axiomatic system. With its single page of axioms, it was the most succinct set theory axioms developed at the time, and formed the basis for the system later developed by Gödel and Berneys.
von Neumann (1928). His fifth paper on set theory, “Über die Definition durch transfinite Induktion und verwandte Fragen der allgemeinen Mengenlehre” (“On the Definition by Transfinite Induction and related questions of General Set Theory”) proves the possibility of definition by transfinite induction. That is, in the paper von Neumann demonstrates the significance of axioms for the elimination of the paradoxes of set theory, proving that a set does not lead to contradictions if and only if its cardinality is not the same as the cardinality of all sets, which implies the axiom of choice (Leonard, 2010).
von Neumann (1929). In his sixth set theory paper, Über eine Widerspruchsfreiheitsfrage in der axiomatischen Mengenlehre, von Neumann discusses the questions of relative consistency in set theory (Van Heijenoort, 1970).
Summarized, von Neumann’s main contribution to set theory is what would become the von Neumann-Bernays-Gödel set theory (NBG), an axiomatic set theory that is considered a conservative extension of the accepted Zermelo-Fraenkel set theory (ZFC). It introduced the notion of class (a collection of sets defined by a formula whose quantifiers range only over sets) and defines classes that are larger than sets, such as the class of all sets and the class of all ordinal numbers.
Left: John von Neumann in the 1920s. Right: von Neumann, J (1923). Zur Einführung der transfiniten Zahlen (“On the introduction of transfinite numbers”). Acta Litterarum ac Scientiarum Regiae Universitatis Hungaricae Francisco-Josephinae, sectio scientiarum mathematicarum, 1, pp. 199–208.
Inspired by the works of Georg Cantor, Ernst Zermelo’s 1908 axioms for set theory and the 1922 critiques of Zermelo’s set theory by Fraenkel and Skolem, von Neumann’s work provided solutions to some of the problems of Zermelo set theory, leading to the eventual development of Zermelo-Fraenkel set theory (ZFC). The problems he helped resolve include:
The problem of developing Cantor’s theory of ordinal numbers in Zermelo set theory. von Neumann redefined ordinals using sets that are well-ordered using the so-called ∈-relation.
The problem of finding a criterion identifying classes that are too large to be sets. von Neumann introduced the criterion that a class is too large to be a set if and only if it can be mapped onto the class of all sets.
Zermelo’s somewhat imprecise concept of a ‘definite propositional function’ in his axiom of separation. von Neumann formalized the concept with his functions, whose construction requires only finitely many axioms.
The problem of Zermelo’s foundations of the empty set and an infinite set, and iterating the axioms of pairing, union, power set, separation and choice to generate new sets. Fraenkel introduced an axiom to exclude sets. von Neumann revised Fraenkel’s formulation in his axiom of regularity to exclude non-well-founded sets.
Of course, following the critiques and further revisions of Zermelo’s set theory by Fraenkel, Skolem, Hilbert and von Neumann, a young mathematician by the name of Kurt Gödel in 1930 published a paper which would effectively end von Neumann’s efforts in formalist set theory, and indeed Hilbert’s formalist program altogether, his theorem of incompleteness. von Neumann happened to be in the audience when Gödel first presented it:
“At a mathematical conference preceding Hilbert’s address, a quiet, obscure young man, Kurt Gödel, only a year beyond his PhD, announced a result which would forever change the foundations of mathematics. He formalized the liar paradox, “This statement is false” to prove roughly that for any effectively axiomatized consistent extension T of number theory (Peano arithmetic) there is a sentence σ which asserts its own unprovability in T.
John von Neumann, who was in the audience immediately understood the importance of Gödel’s incompleteness theorem. He was at the conference representing Hilbert’s proof theory program and recognized that Hilbert’s program was over.
In the next few weeks von Neumann realized that by arithmetizing the proof of Gödel’s first theorem, one could prove an even better one, that no such formal system T could prove its own consistency. A few weeks later he brought his proof to Gödel, who thanked him and informed him politely that he had already submitted the second incompleteness theorem for publication.”
— Excerpt, Computability. Turing, Gödel, Church and Beyond by Copeland et al. (2015)
One of Gödel’s lifelong supporters, von Neumann later stated that
“Gödel is absolutely irreplaceable. In a class by himself.”
By the end of 1927, von Neumann had published twelve major papers in mathematics. His habilitation (qualification to conduct independent university teaching) was completed in December of 1927, and he began lecturing as a Privatdozent at the University of Berlin in 1928 at the age of 25, the youngest Privatdozent ever elected in the university’s history in any subject.
“By the middle of 1927 it was clearly desirable for the young eagle Johnny to soar from Hilbert’s nest. Johnny had spent his undergraduate years explaining what Hilbert had got magnificently right but was now into his postgraduate years where had to explain what Hilbert had got wrong”
— Excerpt, John von Neumann by Norman Macrae (1992)
Game Theory
Around the same time he was making contributions to set theory, von Neumann also proved a theorem known as the minimax theorem for zero-sum games, which would later lay the foundation for the new field of game theory as a mathematical discipline. The minimax theorem may be summarized as follows:
The Minimax Theorem (von Neumann, 1928). The minimax theorem provides the conditions that guarantee that the max-min inequality is also an equality, i.e. that every finite, zero-sum, two-person game has optimal mixed strategies.
The proof was published in Zur Theorie der Gesellschaftsspiele (“On the Theory of Games of Strategy”) in 1928. In collaboration with economist Oskar Morgenstern, von Neumann later published the definitive book on such cooperative, zero-sum games, Theory of Games and Economic Behavior (1944).
Left: von Neumann, J. (1928). Zur Theorie der Gesellschaftsspiele (“On the Theory of Games of Strategy”). Right: First edition copy of Theory of Games and Economic Behavior (1944) by John von Neumann and Oskar Morgenstern (Photo: Whitmore Rare Books).
By the end of 1929, von Neumann’s number of published major papers had risen to 32, averaging almost one major paper per month. In 1929 he briefly became a Privatdozent at the University of Hamburg, where he found the prospects of becoming a professor to be better.
Quantum Mechanics
In a shortlist von Neumann himself submitted to the National Academy of Sciences later in his life, he listed his work on quantum mechanics in Göttingen (1926) and Berlin (1927–29) as the “most essential”. The term quantum mechanics, largely devised by Göttingen’s own twenty-three year old wunderkind Werner Heisenberg the year before was still hotly debated, and in the same year von Neumann arrived, Erwin Schrödinger, then working from Switzerland, had rejected Heisenberg’s formulation as completely wrong (Macrae, 1992). As the story goes:
“In Johnny’s early weeks at Göttingen in 1926, Heisenberg lectured on the difference between his and Schrödinger’s theories. The aging Hilbert, professor of mathematics, asked his physics assistant, Lothar Nordheim, what on earth this young man Heisenberg was talking about. Nordheim sent to the professor a paper that Hilbert still did not understand. To quote Nordheim himself, as recorded in Heims’s book: “When von Neumann saw this, he cast it in a few days into elegant axiomatic form, much to the liking of Hilbert.” To Hilbert’s delight, Johnny’s mathematical exposition made much use of Hilbert’s own concept of Hilbert space.” – Excerpt, John von Neumann by Norman Macrae (1992)
Starting with the incident above, in the following years, von Neumann published a set of papers which would establish a rigorous mathematical framework for quantum mechanics, now known as the Dirac-von Neumann axioms. As Van Hove (1958) writes,
“By the time von Neumann started his investigations on the formal framework of quantum mechanics this theory was known in two different mathematical formulations: the “matrix mechanics” of Heisenberg, Born and Jordan, and the “wave mechanics” of Schrödinger. The mathematical equivalence of these formulations had been established by Schrödinger, and they had both been embedded as special cases in a general formalism, often called “transformation theory”, developed by Dirac and Jordan.
This formalism, however, was rather clumsy and it was hampered by its reliance upon ill-defined mathematical objects, the famous delta-functions of Dirac and their derivatives. [..] [von Neumann] soon realized that a much more natural framework was provided by the abstract, axiomatic theory of Hilbert spaces and their linear operators.”
— Excerpt, Von Neumann’s Contributions to Quantum Theory by Léon Van Hove (1958)
In the period from 1927–31, von Neumann published five highly influential papers relating to quantum mechanics:
von Neumann (1927). Mathematische Begründung der Quantenmechanik (“Mathematical Foundation of Quantum Mechanics”) in Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse pp. 1–57.
von Neumann (1927). Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik (“Probabilistic Theory of Quantum Mechanics”) in Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse pp. 245–272.
von Neumann (1927). Thermodynamik quantenmechanischer Gesamtheiten (“Thermodynamics of Quantum Mechanical Quantities”) in Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse. pp. 273–291.
von Neumann (1930). Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren (“General Eigenvalue Theory of Hermitian Functional Operators”) in Mathematische Annalen 102 (1) pp 49–131.
von Neumann (1931). Die Eindeutigkeit der Schrödingerschen Operatoren (“The uniqueness of Schrödinger operators”) in Mathematische Annalsen 104 pp 570–578.
His basic insight, which neither Heisenberg, Bohr or Schrödinger had, in the words of Paul Halmos was“that the geometry of the vectors in a Hilbert space have the same formal properties as the structure of the states of a quantum mechanical system” (Macrae, 1992). That is, von Neumann realized that the state of a quantum system could be represented by the point of a complex Hilbert space, that in general, could be infinite-dimensional even for a single particle. In such a formal view of quantum mechanics, observable quantities such as position or momentum are represented as linear operators acting on the Hilbert space associated with the quantum system (Macrae, 1992). The uncertainty principle, for instance, in von Neumann’s system is translated into the non-commutativity of two corresponding operators.
Summarized, von Neumann’s contributions to quantum mechanics can be said to broadly be two-fold, consisting of:
The mathematical framework of quantum theory, where states of the physical system are described by Hilbert space vectors and measurable quantities (such as position, momentum and energy) by unbounded hermitian operators acting upon them; and
The statistical aspects of quantum theory. In the course of his formulation of quantum mechanics in terms of vectors and operators in Hilbert spaces, von Neumann also gave the basic rule for how the theory should be understood statistically (Van Hove, 1958). That is, as the result of the measurement of a given physical quantity on a system in a given quantum state, its probability distribution should be expressed by means of a vector representing the state and the spectral resolution of the operator representing the physical quantity.
First edition copy of Mathematische Grundlagen der Quantenmechanik (1932)
His work on quantum mechanics was eventually collected in the influential 1932 book Mathematische Grundlagen der Quantenmechanik (“Mathematical Foundations for Quantum Mechanics”), considered the first rigorous and complete mathematical formulation of quantum mechanics.
Quantum mechanics was very fortunate indeed to attract, in the very first years after its discovery in 1925, the interest of a mathematical genius of von Neumann’s stature. As a result, the mathematical framework of the theory was developed and the formal aspects of its entirely novel rules of interpretation were analysed by one single man in two years (1927–1929). — Van Hove (1958)
Operator theory
Following his work in set theory and quantum mechanics, while still in Berlin, von Neumann next turned his attention to algebra, in particular operator theory which concerns the study of linear operators on function spaces. The most trivial examples are the differential and integral operators we all remember from calculus. von Neumann introduced the study of rings of operators through his invention of what is now known as von Neumann algebras, defined as:
Definition of a von Neumann algebra. A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identify operator
The work was published in the paper Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren (“On the Algebra of Functional Operations and Theory of Normal Operators”) in 1930.
In America
John von Neumann first travelled to America while still a Privatdozent at the University of Hamburg in October 1929 when he was invited to lecture on quantum theory at Princeton University. The visit led to an invitation to return as a visiting professor, which he did in the years 1930–33. The same year this tenure finished, Adolf Hitler first came to power in Germany, leading von Neumann to abandon his academic posts in Europe altogether, stating about the Nazi regime that
“If these boys continue for two more years, they will ruin German science for a generation — at least”
By most accounts, of course, von Neumann’s prediction turned out true. The following year, when asked by the Nazi minister of education “How mathematics is going at Göttingen, now that it is free from the Jewish influence?” Hilbert is said to have replied:
“There is no mathematics in Göttingen anymore.”
At Princeton University (1930–1933)
The circumstances under which von Neumann (and a plethora of other first-rate mathematicians and physicists) would find themselves in Princeton, New Jersey in the mid-1930s is by now well known.
In the case of von Neumann in particular, he was recruited alongside his Lutheran high school contemporary Eugene Wigner by Princeton University professor Oswald Veblen, on a recommendation from Princeton, according to Wigner (Macrae, 1992) to:
“..invite not a single person but at least two, who already knew each other, who wouldn’t suddenly feel put on an island where they had no intimate contact with anybody. Johnny’s name was of course well known by that time the world over, so they decided to invite Johnny von Neumann. They looked: who wrote articles with John von Neumann? They found: Mr. Wigner. So they sent a telegram to me also.”
— Excerpt, John von Neumann by Norman Macrae (1992)
And so von Neumann first came to Princeton in 1930 as a visiting professor. Regarding his work while there, von Neumann himself later in life especially highlighted his work on ergodic theory.
Ergodic Theory
“If von Neumann had never done anything else, they would have been sufficient to guarantee him mathematical immortality” — Paul Halmos (1958)
Ergodic theory is the branch of mathematics that studies the statistical properties of deterministic dynamical systems. Formally, ergodic theory is concerned with the states of dynamical systems with an invariant measure. Informally, think of how the planets move according to Newtonian mechanics in a solar system: the planets move but the rule governing the planets’ motion remains constant. In two papers published in 1932, von Neumann made foundational contributions to the theory of such systems, including the von Neumann’s mean ergodic theorem, considered the first rigorous mathematical basis for the statistical mechanics of liquids and gases. The two papers are titled Proof of the Quasi-ergodic Hypothesis (1932) and Physical Applications of the Ergodic Hypothesis (1932).
Left: von Neumann, J. (1932). Proof of the Quasi-ergodic Hypothesis. Proceedings of the National Academy of Sciences 18 (1) pp. 70–82. Right: von Neumann, J. (1932). Physical Applications of The Ergodic Hypothesis. Proceedings of the National Academy of Sciences 18(3) pp. 263–266.
A subfield of measure theory, ergodic theory in other words concerns the behavior of dynamical systems which are allowed to run for a long time. von Neumann’s ergodic theorem is one of the two most important theorems in the field, the other being by Birkhoff (1931). According to Halmos (1958)
“The profound insight to be gained from [von Neumann’s] paper is that the whole problem is essentially group-theoretic in character, and that, in particular, for the solvability of the problem of measure the ordinary algebraic concept of solvability of a group is relevant. Thus, according to von Neumann, it is the change of group that makes a difference, not the change of space; replacing the group of rigid motions by other perfectly reasonable groups we can produce unsolvable problems in R2 and solvable ones in R3.”
— Excerpt, Von Neumann on Measure and Ergodic Theory by Paul R. Halmos (1958)
At the Institute for Advanced Study
Following his three-year stay as a visiting professor at Princeton in the period 1930–33, von Neumann was offered a lifetime professorship on the faculty of the Institute for Advanced Study (IAS) in 1933. He was 30 years old. The offer came after the the institute’s plan to appoint von Neumann’s former professor Hermann Weyl fell through (Macrae, 1992). Having only been founded three years prior, von Neumann became one of the IAS’ first six professors, the others being J. W. Alexander, Albert Einstein, Marston Morse, Oswald Veblen and eventually, Hermann Weyl.
Institute for Advanced Study in Princeton, New Jersey (Photo: Cliff Compton)
When he joined in 1933, the Institute was still located in the math department of Princeton University’s Fine Hall. Founded in 1930 by Abraham Flexner and funded by philanthropy money from Louis Bamberger and Caroline Bamberger Fuld, the Institute for Advanced Study was and is still a university unlike any other. Inspired by Flexner’s experiences at Heidelberg University, All Souls College, Oxford and the Collège de France, the IAS has been described as:
“A first-rate research institution with no teachers, no students, no classes, but only researchers protected from the vicissitudes and pressures of the outside world.” — Sylvia Nasar
In 1939 moved to its own campus and common room Fuld Hall, the Institute for Advanced Study in a matter of a few years in the early 1930s effectively inherited the University of Göttingen’s throne as the foremost center of the mathematical universe. The dramatic and swift change has since become known as the “Great Purge of 1933”, as a number of top rate academics fled Europe, fearing for their safety. Among them, in addition to von Neumann and Wigner, of course was Einstein (1933), Max Born (1933), fellow Budapestians Leó Szilárd (1938) and Edward Teller (1933), as well as Edmund Landau (1927), James Franck (1933) and Richard Courant (1933), among others.
Photo of part of the faculty at the Institute for Advanced Study, including its most famous resident Albert Einstein. John von Neumann visible in the background. Julian Bigelow, Herman Goldstine, J. Robert Oppenheimer and John von Neumann in front of MANIAC, the Institute for Advanced Study computer.
Geometry
While at the Institute for Advanced Study, von Neumann founded the field of continuous geometry, an analogue of complex projective geometry where instead of a dimension of a subspace being in a discrete set 0, 1, …, n, it can be an element of the unit interval [0,1].
A continuous geometry is a lattice L with the following properties:- L is modular- L is complete- The lattice operations satisfy a continuity property- Every element in L has a complement- L is irreducible, meaning the only elements with unique complements are 0 and 1.
As with his result in ergodic theory, von Neumann published two papers on continuous geometry, one proving its existence and discussing its properties, and one providing examples:
von Neumann (1936). Continuous geometry. Proceedings of the National Academy of Sciences 22 (2) pp. 92–100.
von Neumann (1936). Examples of continuous geometries. Proceedings of the National Academy of Sciences 22 (2) pp. 101–108;
The Manhattan Project (1937–45)
In addition to his academic pursuits, beginning in the mid to late 30s, von Neumann developed an expertise in the science of explosions, phenomena which are very hard to model mathematically. In particular, von Neumann became a leading authority on the mathematics of shaped charges, explosive charges shaped to focus the effect of the energy of an explosive.
By 1937, according to Macrae, von Neumann had decided for himself that war was clearly coming. Although obviously suited for advanced strategic and operations work, humbly he instead applied to become a lieutenant in the reserve of the ordnance department of the U.S.Army. As a member of the Officers’s Reserve Corps, this would mean that he could get trouble-free access to various sorts of explosion statistics, which he thought would be fascinating (Macrae, 1992).
Left: The photo from von Neumann’s Los Alamos ID badge. Right: John von Neumann talking with Richard Feynman and Stanislaw Ulam in Los Alamos (Photo: )
Needless to say, von Neumann‘s main contributions to the atomic bomb would not be as a lieutenant in the reserve of the ordnance department, but rather in the concept and design of the explosive lenses that were needed to compress the plutonium core of the Fat Man weapon that was later dropped on Nagasaki.
A member of the Manhattan Project in Los Alamos, New Mexico, von Neumann in 1944 showed that the pressure increase from explosion shock wave reflections from solid objects was greater than previously believed, depending on the angle of its incidence. The discovery led to the decision to detonate atomic bombs some kilometers above the target, rather than on impact (Macrae, 1992). von Neumann was present during the first Trinity test on July 16th, 1945 in the Nevada desert as the first atomic bomb test ever successfully detonated.
Work on Philosophy
Macrae (1992) makes the point that in addition to being one of the foremost mathematicians in his lifetime, in many ways, von Neumann should perhaps also be considered one of his era’s most important philosophers. Professor of philosophy John Dorling at the University of Amsterdam, in particular, highlights in particular von Neumann’s contributions to the philosophy of mathematics (including set theory, number theory and Hilbert spaces), physics (especially quantum theory), economics (game theory), biology (cellular automata), computers and artificial intelligence.
von Neumann speaking at the American Philosophical Society in 1957 (Photo: Alfred Eisenstaedt)
His work on the latter two, computers and artificial intelligence (AI) occurred first while he was in Princeton in the mid 1930s meeting with the then-24-year-old Alan Turing when he spent a year at the IAS in 1936–37. Turing began his career by working in the same field as von Neumann had — on set theory, logic and Hilbert’s Entscheidungsproblem. When he finished his Ph.D at Princeton in 1938, Turing had extended the work of von Neumann and Gödel and introduced ordinal logic and the notion of relative computing, augmenting his previously devised Turing machines with so-called oracle machines, allowing the study of problems that lay beyond the capabilities of Turing machines. Although inquired about by von Neumann for a position as a postdoctoral research assistant following his Ph.D., Turing declined and instead travelled back to England (Macrae, 1992).
Work on Computing
“After having been here for a month, I was talking to von Neumann about various kinds of inductive processes and evolutionary processes, and just as an aside he said, “Of course that’s what Turing was talking about.” And I said, “Who’s Turing?”. And he said, “Go look up Proceedings of the London Mathematical Society, 1937”.
The fact that there is a universal machine to imitate all other machines … was understood by von Neumann and few other people. And when he understood it, then he knew what we could do.”
— Julian Bigelow”– Excerpt, Turing’s Cathedral by George Dyson (2012)
Although Turing left, von Neumann continued thinking about computers throughout the end of the 30s and the war. Following his experiences working on the Manhattan Project, he was first drawn into the ENIAC project at the Moore School of Engineering at the University of Pennsylvania during the summer of 1944. Having observed the large amounts of calculation needed to predict blast radii, plan bomb paths and break encryption schemes, von Neumann early saw the need for substantial increases in computing power.
In 1945, von Neumann proposed a description for a computer architecture now known as the von Neumann architecture, which includes the basics of a modern electronic digital computer including:
A processing unit that contains an arithmetic logic unit and processor registers;
A control unit that contains an instruction register and a program counter
A memory unit that can store data and instructions;
External storage; and
Input and output mechanisms;
John von Neumann with the IAS machine, sometimes called the “von Neumann Machine”, stored in the the basement of Fuld Hall from 1942–1951 (Photo: Alan Richards)
The same year, in software engineering, von Neumann invented the so-called merge sort algorithm which divides arrays in half before sorting them recursively and then merging them. von Neumann himself wrote the first 23 page sorting program for the EDVAC computer in ink. In addition, in a pioneering 1953 paper entitled Probabilistic Logics and the Synthesis of Reliable Organisms from Unrealiable Components, von Neumann was first to introduce stochastic computing, though the idea was so groundbreaking that it could not be implemented for another decade or so (Petrovik & Siljak, 1962). Related, von Neumann created the field of cellular automata through his rigorous mathematical treatment of the structure of self-replication, which preceded the discovery of the structure of DNA by several years.
Although influential in his own right, throughout his life, von Neumann made sure to acknowledge that the central concept of the modern computer was indeed Turing’s 1936 paper On Computable Numbers, with an Application to the Entscheidungsproblem (Fraenkel, 1972)
”von Neumann firmly emphasised to me, and to others I am sure, that the fundamental conception is owing to Turing — insofar as not anticipated by Babbage, Lovelace and others.” — Stanley Fraenkel (1972)
Consultancies
“The only part of your thinking we’d like to bid for systematically is that which you spend shaving: we’d like you to pass on to us any ideas that come to you while so engaged.”
— Excerpt, Letter from the Head of the RAND Corporation to von Neumann (Poundstone, 1992)
Throughout his career in America, von Neumann held a number of consultancies for various private, public and defense contractors including the National Defense Research Council (NDRC), the Weapons Systems Evaluation Group (WSEG), the Central Intelligence Agency (CIA), the Lawrence Livermore National Laboratory (LLNL) and the RAND Corporation, in addition to being an advisor to the Armed Forces Specials Weapons Project, a member of the General Advisory Committee of the Atomic Energy Commission, of the Scientific Advisory Group of the United States Air Force and in 1955 a commissioner of the Atomic Energy Commission (AEC).
Personality
Despite his many appointments, responsibilities and copious research output, von Neumann lived a rather unusual lifestyle for a mathematician. As described by Vonnauman and Halmos:
“Parties and nightlife held a special appeal for von Neumann. While teaching in Germany, von Neumann had been a denizen of the Cabaret-era Berlin nightlife circuit.” — Vonneuman (1987)
The parties at the von Neumann’s house were frequent, and famous, and long. — Halmos (1958)
John von Neumann with his wife Klari Dan and their dog (Photo: Alan Richards)
His first wife, Klara, said that he could count everything except calories.
von Neumann also enjoyed Yiddish and dirty jokes, especially limericks (Halmos, 1958). He was a non-smoker, but at the IAS received complaints for regularly playing extremely loud German march music on the gramophone in his office, distracting those in neighboring offices, including Albert Einstein. In fact, von Neumann claimed to do some of his best work in noisy, chaotic environments such as in the living room of his house with the television blaring. Despite being a bad driver, he loved driving, often while reading books, leading to various arrests and accidents.
Von Neumann in the Florida Everglades in 1938 (Photo: Marina von Neumann Whitman)
As a Thinker
Stanislaw Ulam, one of von Neumann’s close friends, described von Neumann’s mastery of mathematics as follows:
“Most mathematicians know one method. For example, Norbert Wiener had mastered Fourier transforms. Some mathematicians have mastered two methods and might really impress someone who knows only one of them. John von Neumann had mastered three methods: 1) A facility for the symbolic manipulation of linear operators, 2) An intuitive feeling for the logical structure of any new mathematical theory; and 3) An intuitive feeling for the combinatorial superstructure of new theories.”
Biographer Sylvia Nasar describes von Neumann’s own “thinking machine” by the following, now well-known anecdote regarding the so-called “two trains puzzle”:
Two bicyclists start twenty miles apart and head toward each other, each going at a steady rate of 10 m.p.h. At the same time, a fly that travels at a steady 15 m.p.h. starts from the front wheel of the southbound bicycle and flies to the front wheel of the northbound one, then turns around and flies to the front wheel of the southbound one again, and continues in this manner till he is crushed between the two front wheels. Question: what total distance did the fly cover?
There are two ways to answer the problem. One is to calculate the distance the fly covers on each leg of its trips between the two bicycles and finally sum the infinite series so obtained. The quick way is to observe that the bicycles meet exactly an hour after they start so that the fly had just an hour for his travels; the answer must therefore be 15 miles. When the question was put to von Neumann, he solved it in an instant, and thereby disappointed the questioner: “Oh, you must have heard the trick before!”
“What trick,” asked von Neumann, “all I did was sum the infinite series.”
—Excerpt, A Beautiful Mind (Nasar, 1998)
As a Supervisor
In the paper Szeged in 1934 (Lorch, 1993) Edgar R. Lorch describes his experience of working as an assistant for von Neumann in the 1930s, including his duties:
Attending von Neumann’s lectures on operator theory, taking notes, completing unfinished proofs and circulating them to all American university libraries;
Assisting von Neumann in his role as the editor of the Annals of Mathematics by reading through every manuscript accepted to the publication, underlining greek letters in red and german letters in green, circling italics, writing notes to printers in the margins and going once per week to the printers in order to instruct them in the art of typesetting;
Translating von Neumann’s numerous 100-page papers into English;
“His fluid line of thought was difficult for those less gifted to follow. He was notorious for dashing out equations on a small portion of the available blackboard and erasing expressions before students could copy them.”
—Excerpt, John von Neumann: As Seen by his Brother by N.A. Vonneuman (1987)
Later years
In 1955, Von Neumann was diagnosed with what was likely either bone, pancreatic or prostate cancer (accounts differ on which diagnosis was made first). He was 51 years old. Following two years of illness which at the end confined him to a wheelchair, he eventually died on the 8th of February 1957, at 53 years old. On his deathbed, he reportedly entertained his brother by reciting the first few lines of each page from Goethe’s Faust, word-for-word, by heart (Blair, 1957).
President Dwight D. Eisenhower (left) presenting John von Neumann (right) the Presidential Medal of Freedom in 1956
He is buried at Princeton Cemetery in Princeton, New Jersey alongside his lifelong friends Eugene Wigner and Kurt Gödel. Gödel wrote him a letter a year before his death, which has been made public. The letter is discussed in detail by Hartmanis (1989) in his working paper The Structural Complexity Column. An excerpt is included below:
Letter from Kurt Gödel to von Neumann, March 20th 1956
Dear Mr. von Neumann:
With the greatest sorrow I have learned of your illness. The news came to me as quite unexpected. Morgenstern already last summer told me of a bout of weakness you once had, but at that time he thought that this was not of any greater significance. As I hear, in the last months you have undergone a radical treatment and I am happy that this treatment was successful as desired, and that you are now doing better. I hope and wish for you that your condition will soon improve even more and that the newest medical discoveries, if possible, will lead to a complete recovery.
[…]
I would be very happy to hear something from you personally. Please let me know if there is something that I can do for you. With my best greetings and wishes, as well to your wife,
Sincerely yours,
Kurt Gödel
P.S. I heartily congratulate you on the award that the American government has given to you
Interview on Television
Remarkably, there exists a video interview with von Neumann on the NBC show America’s Youth Wants to Know in the early 1950s (below):
For anyone interested in learning more about the life and work of John von Neumann, I especially recommend his friend Stanislaw Ulam’s 1958 essay John von Neumann 1903–1957 in the Bulletin of the American Mathematical Society 64 (3) pp 1–49 and the book John von Neumann* by Norman Macrae (1992).
Sincerely,
Jørgen
References
Blair, C. 1957. Passing of a Great Mind. Life Magazine. Feb. 25th. 1957.
Fraenkel, S. 1972. Letter to to Brian Randell, 1972, quoted in Jack Copeland (2004) The Essential Turing, p. 22
Henderson, H. 2007. Mathematics: Powerful Patterns Into Nature and Society*. New York: Chelsea House
Leonard, R. 2010. Von Neumann, Morgenstern, and the Creation of Game Theory*. Cambridge University Press. pp. 42
Macrae, N. 1992. John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More*. Pantheon Press.
Nasar, S. 1998. A Beautiful Mind*. Simon & Schuster
Petrovic, R. & Siljak, D. 1962. “Multiplication by means of coincidence”. ACTES Proc. of 3rd Int. Analog Comp. Meeting.
Poundstone, W. 1992.Prisoner’s Dilemma: John von Neumann, Game Theory and the Puzzle of the Bomb*. Anchor Books.
Regis, E. 1992. Johnny Jiggles the Planet. The New York Times, November 8th, 1992, Section 7, Page 12.
Ulam, S. 1958. John von Neumann 1903–1957. The Bulletin of the American Mathematical Society64 (3) pp 1–49.
Van Hove, L. 1958. von Neumann’s Contributions to Quantum Theory. Bull. Amer. Math. Soc. 64 (3), pp. 95-99.
Our setting is the open unit disk, D and T the unit circle, in the complex plane, C. By H² is meant the standard Hardy space, the Hilbert space of holomorphic functions in D having square-summable Taylor coefficients at the origin. As usual, H² will be identified with its space of boundary functions, the subspace of L² (of normalized Lebesgue measure m on T) consisting of the functions whose Fourier coefficients with negative indices vanish.
A Toeplitz operator is the compression to H² of a multiplication operator on L². The operators of the paper’s title are compressions of multiplication operators to proper invariant subspaces of the backward shift operator on H². An effort has been exerted to make the paper reasonably self-contained.
Some preparation is needed prior to precise definitions.
We let P denote the
orthogonal projection on L²with range H². The operator P is given explicitly as a Cauchy integral:
We shall need to deal with certain unbounded Toeplitz operators.
The operator T_ϕ, the Toeplitz operator on H² with symbol ϕ, is defined by
We let S denote the unilateral shift operator on H². Its adjoint, the backward
shift, is given by
Clearly, a truncated Toeplitz operator does not have a unique symbol.
The compression of S to K_u will be denoted by S_u. Its adjoint, S*_u, is the
restriction of S* to K_u. The operators S_u and S*_u are the truncated Toeplitz operators with symbols z and z, respectively.
The next section contains most of the needed background on the spaces.
Background Materials;
The bulk of it can be found in standard sources;
The model space, as is well known, carries a natural conjugation, an antiunitary involution C, defined by
It preserves the model spaces.
An operator A on model space K_u is called C-symmetric if CAC = A*. S. R. Garcia and M. Putinar study the notion of C-symmetry in the abstract sense. They give many examples, including our truncated Toeplitz operators (at least those with bounded symbols). The following result is essentially theirs.
Let T(K_u) denote the space of all bounded truncated Toeplitz operators defined on the model space K_u.
We have the following results which are called defect operators or rank one operators in terms of reproducing kernel Hilbert spaces.
Condition for A_phi=0;
In this section will characterize the zero truncated Toeplitz operators whose related with its symbol.
The symbol of the truncated Toeplitz operator is not unique. The following corollary shows this…
Characterizations of Truncated Toeplitz Operators;
The bulk of the proof will be accomplished in two lemmas.
The following result is that the space of truncated Toeplitz opertaors is closed in the weak operator topology.
Characterization by Shift Invariance;
Given a bounded operator A on model space K_u, we let Q_A denote the associated quadratic form on K_u,
We shall say that A is shift invariant if
whenever f and Sf are in K_u.
If this happens then, by the polarization identity, we also have
whenever f,g, Sf,Sg are belong to the model space K_u.
Truncated Toeplitz operators can be characterize by using the following theorem.
The following theorem shows that J_w is an isometry between model spaces.
The adjoint of the Crofoot transform is given by
The Crofoot transform of a truncated Toeplitz operator is a
truncated Toeplitz operator on another model space.
The following lemmas can be used to prove the above theorem.
Last time I talked about how the Yoneda Lemma allows us to think about non-traditional spaces. Today we’ll look at a practical application of this to music theory.
Here’s the motivational problem:
Can we construct a geometric space that has chords as its points and also encodes useful music theory in it somehow?
I know that’s vague, but I’ll become more precise as we go.
Pitch Class Sets
What I’m going to describe here is usually encountered in a class on post-tonal music theory. The book Introduction to Post-Tonal Theory by Joseph N. Straus is excellent and where I originally learned it.
The term “post-tonal” can be misleading here because, for the most part, it is an extremely useful mathematical way of thinking about music theory that doesn’t particularly have to do with atonal music or 12-tone serialism.
The Western 12-tone scale is essentially formed by taking an octave and dividing it up into 12 parts.
Since an octave (or 12 semitones up or down) gives the same note we can mathematically think of things more clearly by just labeling a C with 0, a C# with 1, a D with 2 and so on up to labeling a B with 11.
When we back to C we “wrap around” and call it 0 again. Mathematically, this is just modulo 12 arithmetic.
A great way to visualize this is to draw a 12-sided figure with all the side lengths the same (a regular dodecagon).
Now if we take a C major chord: 0, 4, 7, then transposing it to a major chord 3 semitones up just amounts to adding every number by 3 to get 3, 7, 10.
In fact, given any set of notes, we have the operation of transposition by n:
where mod 12 means we add by wrapping around and consider 12=0, 13=1, 14=2, etc (because they’re the same notes!).
We can also do something called an inversion. The term in music theory means inverting the intervals.
Mathematically, this means negating every single number and then figuring out what this number is mod 12. So the inversion of the C major chord: [0, 4, 7] is [0, -4, -7]=[0, 8, 5] or if we really are considering “chords” then the order doesn’t matter so it is [0,5,8].
But this is just an f minor chord!
We call this operation I for “inversion.” It can be visualized as a reflection of the dodecagon as follows:
Doing Tₙ for all choices of n to [0,4,7] gives you all 12 majors chords, and if you do both I and Tₙ then you’ll get all 12 minor chords, too.
The operations of transpositions and inversions generates something called a group. In fact, visualizing with a regular 12-gon immediately tells us that the TI group is what mathematicians call D_12, the Dihedral group of symmetries of the dodecagon. It has 24 elements.
We call an unordered collection of numbers between 0 and 12 a pitch set, and we get that D_12 acts on the set of pitch sets. Let’s examine this in more depth before moving on.
Group Actions on Pitch Sets
We just proved that the orbit of [0,4,7] under this action consists of the collection of all major and minor triads (three-note chords).
This should already be an indication we’re on the right track. Major and minor triads are the foundation of music in the West.
Note that none of the triads are sent to themselves. In other words, given a non-trivial symmetry/combination of transpositions and inversions, we will always get a distinct new triad.
Mathematicians might say this in a fancy way: the set of major and minor triads is a torsor under the TI-action.
It turns out this is a “generic” phenomenon. If you choose a pitch set at random, you are likely (the probability is greater than 50%) to have chosen one that has this property of getting 24 distinct new chords by translating and inverting.
We could say that it has the property of having no TI-symmetry. It’s worth thinking about this for a moment. Take [0,6] and invert it to [0, -6]. This is the same as [0,6] mod 12. Inverting does not give us a new pitch set.
So the reason [0,4,7] never had this happen was that it didn’t have some sort of symmetry like that.
We’ll call a k-chord (read: an unordered chord with k notes in it) TI-symmetric if there is some choice of non-trivial transposition and inversion such that the chord is sent to itself.
Now, even though these are rarer, it turns out that for any choice of k, there is always a k-chord with this property. These exist for rather silly reasons. For example, [0,1,2, … , k] is always an example of such a chord (exercise: why?).
For less trivial examples you could take the whole-tone scale [0,2,4,6,8,10]. If you translate by 2, then you certainly get the same thing back again.
Inversion also fixes this 6-chord.
This tells us that up to inversion and transposition there are only 2 distinct whole tone scales (if you want overkill then the subgroup generated by T_2 has 12 elements, so the Orbit-Stabilizer Theorem tells us this fact).
Here is an interesting question from pure music theory, and to my knowledge, it’s still open (although I suspect it’s fairly easy to answer).
None of this was specific to dividing up an octave into 12 notes. Suppose you invent a tonal system with n notes instead. Then you’d have an action of D_n on the k-chords.
Is there a simple closed-form formula for the number of k-chords that are TI-symmetric? More importantly, for a given n, which k gives the most number of k-chords with TI-symmetry.
I should point out that if you rule out the “silly examples” of TI-symmetry given by a strictly chromatic scale, then there is actually utility in figuring this out. TI-symmetry has played a great role in the history of composition.
For example, the augmented triad, the French augmented sixth chord, the diminished seventh, the famous chord from Stravinsky’s Petrushka, the hexatonic scale, the whole tone scale, and the octatonic scale are all examples with TI-symmetry.
So, I think this is more than just a novelty problem.
I’ll sketch the idea now of forming our space classifying pitch class sets.
I won’t rely on any details from my Yoneda Lemma article. You can just trust that there is some notion of a “generalized space” used to classify objects that retains a lot of information on how these objects are related.
Pitch Class Sets
Recall that a pitch set (or chord) is just converting notes to numbers: 0 is C, 1 is C#, 2 is D, etc. A given collection of pitches can be expressed in a more useful notation when there isn’t a key we’re working in.
For example, a C major chord is (047).
You may be confused about why I’ve switched to (047) from [0,4,7]. I want a different notation for the whole “class set” (sometimes called an equivalence class) of chords you get by performing translations and inversions to [0,4,7].
A pitch class setis then saying that there are collections of chords we want to consider to be the same.
There are a few music theoretic reasons for this. For one, our choice of 0 is completely arbitrary. We could have made 0 correspond to A, and we should get the same music theory. This amounts to identifying all pitch sets that are the same after translation.
We also want to identify sets that are the same after inversion. In the previous post on this topic, I showed that if we label the vertices of a dodecagon, this amounts to a reflection symmetry.
The reflections together with the translations generate the dihedral group, so we are secretly letting it act on the set of all tuples of numbers 0 to 11, where each number only appears once and without loss of generality we can assume they are in increasing order.
Thus a pitch class set is just an equivalence class of a chord under this group action. It is not the direction I want this post to go, but given such a class, there is always a unique representative that is usually called the “prime form” (basically the most “compact” representative starting with 0).
Check out Straus’s book for more information on that. It is the standard way to talk about post-tonal theory.
The Geometric Space
The set of all “chords” should have some sort of useful topology on it. For example, [0,1,2,3] should be related to [0,1,2,4], because they are the same chord except for one note.
I don’t think doing something obvious like defining a distance based on the coordinates works. If you try to construct the lattice of open sets by hand based on your intuition, a definition might become more obvious. In any case, the topology isn’t important here.
Call this space of chords X.
Now we have a space with a group action on it. One might want to merely form the quotient space X/G.
The quotient map X → X/G will be 24 to 1 at most points, but it will also forget which chords were fixed by elements of the group. Part of the “theory” in music theory is to remember that information.
This is why I propose making the quotient stack [X/G]. This is one of those fancy spaces I wrote about last time. It is the moduli space of pitch class sets. It seems like an overly complicated thing to do, but here’s what you gain.
You now have a “space” whose points are the pitch class sets. If that class contains 24 distinct chords, then the point is an “honest” point with no extra information.
The fiber of the quotient map contains the 24 chords, and you get to each of them by acting by the elements of (i.e. it is a torsor under the TI group action).
Now consider something like the pitch class set [0,2,4,6,8,10]. The fiber of the quotient map only contains elements: (02468T) and (13579E). The stack will tag these points with D_6, which is the subgroup of symmetries which sends this chord to itself.
Now that I’ve drawn this, I can see that many of you will be skeptical about its simplicity.
Think of it this way:
The bottom thing is the space I’m describing. Each point in the space is tagged with the prime form representative together with the subgroup of symmetries that preserve the class.
That’s pretty simple.
Yet it remembers all of the complicated music theory of the top thing! If the topology was defined well, then studying this space may even lead to insights on how symmetries of classes are related to each other.
Further Reading
As far as I can tell, I’m the only person to propose this exact construction, but it does seem to be a special case of the construction of Guerino Mazzola. He uses topos theory in his amazing book The Topos of Music: Geometric Logic of Concepts, Theory, and Performance.
Dmitri Tymoczko has a construction similar to this, but he works one key and chord at a time to make orbifolds. His book A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice would be a great starting place for someone interested in these spaces but not ready for topos theory.
A visual proof of the Gauss Bonnet Theorem for triangles on spheres!
Spherical geometry is a beautiful, and very visual, area of mathematics, with weird properties (such as that the angles of triangles don’t sum to 180 !!!).
Understanding a little bit of Spherical Geometry
The Gauss-Bonnet Theorem for triangles on spheres is a special, but rather beautiful, case of the more general Gauss-Bonnet Theorem.
The theorem can be understood visually
First, let’s visualise what a triangle on a sphere is. Below are two visualisations of the same triangle.
Source: Geogebra. See https://www.geogebra.org/m/sPx39Zfd
To draw a triangle on a sphere, we pick 3 points, and then we draw lines connecting them. But drawing a ‘line’ on a sphere isn’t an obvious concept. So instead we draw a curve between the two points. Visually, its the curve you’d expect to draw if you tried to make the curve as short, or as ‘direct’, as possible.
We then consider the angles on the triangle. But this triangle is on a sphere, not on a flat sheet of paper! How do we work out the angle?
There is an intuitive answer. As the curves connecting the endpoints of the triangle get closer together, they look more and more ‘straight’. In the picture below, we zoom in really close to illustrate this.
Visually, this should give you the sense that we can make sense of the idea of an angle for a triangle on a sphere.
If you’re interested in how to define the angles on spheres then a slightly more detailed approach is below. But the intuitive understanding is enough for the rest of the proof, so its not essential to do this section to understand the rest.
Below, we take 3 points on a sphere, B, C, D and create a triangle. We then create 3 planes, using the centre of sphere and (C,D), (D,B) or (C,B).
Made using Geogebra
The above picture is a bit scary, so the picture below shows how we create a plane using two of the points and the centre.
Then we define the angle between two sides using the planes: as planes are ‘flat’ objects, like sheets of paper in 3D space, we can define the angle between two planes.
Stating the Theorem
Finally we state the Theorem! But, one last thing. We measure angles in radians. That just means that everything is measure compared to ‘2*pi’. We say 2pi radians is the same as 360 degrees. So pi radians is the same as 180 degrees. And pi/2 radians is the same in 90 degrees. This is similar to how you can measure distance in terms of kilometres, but you can also measure distance in terms of miles.
However, if you feel much more comfortable using degrees, I will also give the formula using degrees. (I prefer radians because all the results become so much prettier)
We label the three angles in the triangle a, b, c. Then we get the following formula for the area of the triangle:
Formula using radians
Or, using degrees instead of radians
formula using degrees
A surprising consequence
This formula for the area of the triangle already has some amazing consequences. Using the formula when the angles are measured in degrees, if the sum of the angles was equal to 180 degrees, then the area of the triangle would be 0. This is obviously nonsensical. As the area is greater than zero, we can conclude that the sum of the angles of spherical triangle must be greater than 180 degrees, or (measured in radians), greater than pi radians.
Proof
Finally, we prove the theorem!
First we find the area of a much simpler object. This object we call a ‘(spherical) lune’.
source: Dexter’s Notes, Geometry IB
We take two planes, and look at the area of the sphere ‘captured’ between them. The area of an entire sphere is 4pi. So if the angle between them is, say, c radians, then the area captured between it is 4c. For instance, if the angle between our planes was a right angle, that is 90 degrees or pi/2 radians, then the area captured would be 2pi, or half the total area.
(However, note that if you use degrees the formula is a bit less elegant. We then have to normalise 90 degrees and adjust by a factor of pi, so the area, when using degrees is given by 4*pi*c/180, and when c=90 we get the area = 4*pi*90/180 = 2pi as before)
Ok, so we can find the surface area of a much simple shape. But what about our triangle?
Next we use a sneaky trick. We do the same as the above, but three times. When we shade this on the sphere below, we can see that a triangle is in the region which is shaded all three colours (which are merged to form a brownish colour)
source: Dexter’s Notes, Geometry IB.
What happens when we sum the areas of each part? The triangle appears in all three of the spherical lunes, so its area is counted three times. So, if we sum the area of the lune, we get the total surface area of the sphere plus four times the area of our triangle. That’s because our triangle’s area is ‘double counted’ on each side, and there are two sides. Moreover, we know the area of each lune! That’s because the angle of the lune determines its area, and the angle of each lune in this case is simply the angle between two sides of the triangle.
Thus, we can conclude:
And we are done!
If you have any thoughts or improvements, please leave them in the comments below! I’m also newly on twitter, asethan_the_mathmo.
What the philosophy of mathematics is useful for? Max Black, the author of The Nature of Mathematics (1933), thought the main task of the foundation of mathematics (and, consequently the main task of any philosophy of mathematics) would be to elucidate “and analyze the notion of integer or natural number”
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Before we begin, you should know that I’m not actually going to present a proof of the Riemann Hypothesis.
This article is about a fictional object known as the field with one element, sometimes denoted Fᵤₙ. You can probably guess why: F is for field and “un” is the French word for 1.
When I first heard about this in grad school, I thought it was a joke. The object is “Fun” and it doesn’t really exist. How could people actually take this seriously?
But a lot of great mathematicians have done quite a bit of work on it: Jacques Tits, Alain Connes, Yuri Manin, and more.
A proper formulation could have major consequences for multiple branches of math, including computational complexity theory, noncommutative geometry, Arakelov geometry, and algebraic number theory (of course others too).
For the purpose of this article, we’ll focus on why it could possibly shed some light on the Riemann Hypothesis. So let’s get started.
What are fields?
Fields are one of the fundamental objects of mathematics. They are just an abstraction of what we already can do with the real numbers.
A field is a set with two operations (addition and multiplication) in which there is an additive identity, 0, and a multiplicative identity, 1, and every element (except 0) has an inverse for both operations.
That’s a lot of words, but it’s much simpler than it looks if you just think about the real numbers, ℝ, as the model for this.
Every real number has an additive inverse. For example, the additive inverse of 3 is just -3 because 3+(-3)=0.
Every non-zero real number has a multiplicative inverse. For example, the multiplicative inverse of 3 is (1/3). This is because 3*(1/3)=1.
For the same reason, ℚ, the set of rational numbers is also a field. So is ℂ, the set of complex numbers (imaginary numbers). If you’ve never seen this, try to work out things like the multiplicative inverse of 1+i to convince yourself.
Not all common numbers systems are fields. The set of integers, ℤ, do not form a field because, although there are addition and multiplication operations, 3 has no multiplicative inverse (1/3 is not an integer).
Not all fields are infinite. In fact, the integers with “mod 3” arithmetic, denoted ℤ/3, is a field with three elements {0,1,2}.
Mod 3 arithmetic is where you add and multiply as usual and then “loop around” like a clock: 0,1,2, 3=0, 4=1, 5=2, 6=0, …. (negatives work too: -1=2, etc).
We can check:
1+2=0. This shows that 1 is the additive inverse of 2 (and vice-versa).
2*2=4=1. This shows that 2 is its own multiplicative inverse.
Those were the only two difficult numbers to check, and so ℤ/3 is indeed a field.
Fields are quite well-understood at this point. Notice something weird happens in the finite field case. If you add 1 to itself 3 times you get the additive identity: 1+1+1=0. But if you do this in the infinite examples I showed, this never happens.
If you add 1 to itself a finite number of times and get 0, the number of times you need to do this is called the characteristic of the field. We’ll denote that with the letter p.
The characteristic of ℤ/3 is 3.
If adding 1 to itself never gives you 0 no matter how many times you do it, the characteristic is 0.
Important terminology: I’ll keep referring to positive characteristic in this article. This is just the standard way of referring to characteristic not equal to 0. In other words, p>0.
Most first courses on abstract algebra will prove an amazing fact: In the finite case, the characteristic is always a prime number! Moreover, the number of elements in a finite field is always a prime power: pⁿ (and p is the characteristic of such a field).
Conversely, for any prime power, there is an explicit construction for the field with that many elements. So, there are fields with 2 elements, 3 elements, 4 elements, 5 elements, 7, 8, 9, etc. There are no fields with 6 elements or 10 elements.
For completeness sake, not all infinite fields have characteristic 0. It’s easy to get the wrong impression from this limited explanation.
This brings us to a sticking point: There is no field with one element! (Namely, 1 is not a prime power).
Before getting more into that, let’s take a detour to see why one might hope there is such a thing.
The Riemann Hypothesis over finite fields
This is where things get really cool.
It turns out that “doing geometry” over positive characteristic fields is often easier than in characteristic 0 (but also harder in other senses). I won’t get into why, since that’s an entire graduate course of subtlety.
So it is sometimes a great tool to take something you want to prove over ℂ and reduce to positive characteristic, prove it there, and then try to lift it back to characteristic 0 somehow (this was actually a major point of my thesis).
It’s a bit complicated to define what geometry means in positive characteristic, but we can rely on a fairly accurate analogy. Geometry over ℝ or ℂ just means studying the shapes formed by the zero sets of polynomials.
So, if you’re over ℝ, have 2 variables, and a polynomial p(x,y)=y-x², the geometry it makes is the parabola you get when setting that equal to zero:
y-x²=0
Or more familiar to people: y=x²
I’ve talked about some of the weirdness of doing this over ℂ in other articles (The Hodge Conjecture and Falting’s Theorem and the Mordell Conjecture).
When you do this over ℚ, you get an interesting interplay between the topology and the number of integer solutions (Fermat’s Last Theorem and so on).
We can just do the same thing over finite fields. The geometry of p(x,y)=y-x² over ℤ/3 comes from plugging in and checking the zero set. It’s harder to visualize as “geometry”, but actually easier to work with since it’s finite.
In fact, we can just check {(0,0), (1,1), (2,1)} are the only three points. I’m glossing over some important details (like doing this projectively and making sure it’s nonsingular), but this way of thinking is good enough for getting the gist of things.
Zeta Functions
Suppose we start with a finite field with p elements, say F, and a “curve,” C, over that field (the zero set of a polynomial for simplicity).
We can count the number of points C has, N(1).
Then we can look at the same equation over the field with p² elements and call this N(2) and so on.
So N is a function. N(k) is just the number of points of C over the field with pᵏ elements.
The next part will look complicated, but it will simplify greatly.
Consider the function G(t)=N(1)t+N(2)t²/2+N(3)t³/3+…
The localZeta function of C is defined as the exponential of that:
Z(C, t)=exp(G(t)).
This might look crazy, but we can work out one example very easily to see that the definition is constructed to make things cancel out and simplify.
If we start with the polynomial p(x)=x, then the only solution to x=0 over any field is just x=0.
There is only one point no matter how many fields we check. Therefore, N(k)=1 for all k.
Let’s plug that in:
G(t)=t+t²/2+t³/3+…=-ln(1–t) for well-defined t, by basic Calculus (check the Taylor series).
Thus, Z(C, t)=exp(-ln(1-t))=1/(1-t).
Wow! See, it’s not bad at all.
The Weil Conjectures are a bunch of conjectures about Z(X, t) for any X (not just curves or points but higher-dimensional spaces, too) proposed by André Weil in 1949. They have been one of the main driving forces in research in algebraic/arithmetic geometry ever since.
Dwork, Grothendieck, and Deligne ended up proving them over several decades and many modern alternate proofs have been found.
The key takeaway is that one of the Weil Conjectures is the “Riemann Hypothesis” for these zeta functions. The general one is extremely technical, but Weil himself proved the Riemann Hypothesis for curves over finite fields.
The proof is relatively easy with the appropriate geometric machinery (for example, it’s left as an exercise in Hartshorne’s book Algebraic Geometry).
The Field With One Element
Okay. We’re finally ready to talk about the field with one element.
Remember, it doesn’t exist.
But the idea is to construct something that would let us do a type of generalized geometry.
Think about ℤ. It has the property that “reducing mod p,” for any prime p gives us the field with p elements (this is basically how we defined the field with p elements earlier).
This fact can be restated geometrically. There is a geometric space, X=Spec(ℤ), so that reducing mod p gives us exactly 1 point over the field with p elements, Fₚ, for each prime number p.
We worked out the local zeta function for a single point already! It’s just: 1/(1-t).
But we’re reducing these mod p to get a “local” zeta function. When you bundle these together to get the “global” zeta function of X, the appropriate way of doing this is by multiplying and keeping track of the prime (t→p⁻ˢ), we get:
the Riemann zeta function (from Riemann Hypothesis fame) in product form!
This probably looks ad hoc and random, but there are better-streamlined definitions that make this more natural and motivated.
This is what we’ve been looking for.
What if there were something called Fᵤₙ that acted like a finite field in such a way that we could treat X=Spec(ℤ) as a curve over it and then use Weil’s techniques for curves over finite fields to prove the Riemann Hypothesis?
People have done some pretty amazing work in this direction. Maybe one day we’ll have a Fᵤₙ proof of the Riemann Hypothesis.
The year 2020 might be the year in which I’ve received the most math problems from friends and seen the most math tricks shared on social media. It turns out that sharing math tricks with one another is one of the things people do most when they’re under lockdown. That fact is tempting me to wish for longer lockdowns.
Recently, I was sent one such math trick from a friend. He sent me a picture which showed an old man who seemed to be teaching mathematics in front of a classroom. He had his back facing a green chalkboard and his left hand pointing towards it as he said the following words to the class:
Photo by Max Fischer from Pexels
“80 bags of cement minus your age plus 40 will give you the last two digits of your birth year. Am I wrong?”
If you are anything like one of my friends who is very particular about tiny details, you would scream, “Hey! You can’t subtract ‘age’ from ‘bags of cement’. That’s blasphemy!”
But let’s ignore those intense details for a while. The “bags of cement” was, in my opinion, put there for comedic effect. Once we get rid of that, we can realize that this is a pretty neat math trick.
There will be a section of readers who would try this and it will work, while it will fail for others(it worked for me, by the way). This article is an attempt to explain all the possible reasons why the math works or doesn’t work for any particular person. So with the preambles out of the way, let’s start the real math.
WARNING: The math is going to get fairly complex so readers are going to have to explore the math while reading.
Stripping the Math Trick naked
First of all, we need to ignore the mention of “bags of cement” since the math wouldn’t work otherwise. In pure mathematical form, the math teacher’s claim goes like this:
80 — Age + 40 = Last Two Digits of Birth Year
In this form, it’s easy to see what is going on because this is the same as:
Age + Last Two Digits of Birth Year = 120 — — — —(Equation 1)
This means that the math teacher’s so-called “math trick” can be reworded in this less impressive way:
“Your age plus the last two digits of your birth year will give you 120. Am I wrong?”
Obviously, this gives away the math that is working under the hood and that is why it was coined in an ambiguous manner. However, once we strip the math trick naked, we can truly separate the math from the trick.
But is it true that Age + Last Two Digits of Birth Year = 120?
Relationship Between Age and Birth year
Photo by cottonbro from Pexels
In order to find out whether that relationship between Age and Last Two Digits of Birth Year was true, I decided to start with what I already knew about two similar quantities: Age and Birth Year. We all know that in order to calculate our current age, we do the following calculation:
Age = Current Year — Birth Year
In other words, Age + Birth Year = Current Year. Given that the year I received the message from my friend was 2020, we then have the identity
Age + Birth Year = 2020.
Let’s explore the above identity some more and see if we find anything interesting. Let Age = X₄X₃X₂X₁, Birth Year = Y₄Y₃Y₂Y₁, and Current Year = Z₄Z₃Z₂Z₁, where X₁, X₂, X₃, and X₄ each represent a digit of the person’s Age. Y₁, Y₂, Y₃, Y₄, Z₁, Z₂, Z₃, and Z₄ each represent digits in a similar manner.
Although most people reading this don’t have four-digit ages, I used this approach because the math still works. However, because of how complex the math is going to get if we crossover from A.D. to B.C., what we’re doing here doesn’t cover people who are older than 2020 years at the time of writing this. So I’m sure almost nobody reading this has been left out.
In that case, I need you to convince yourself that if X₄X₃X₂X₁+ Y₄Y₃Y₂Y₁ = 2020, then the following is also true:
X₄X₃X₂X₁ + Y₂Y₁ = Z₄Z₃20 — — — — (Equation 2)
This is one of the keys to proving or disproving Equation 1.
Now, let’s try and find the kinds of digits that satisfy Equation 2. To do so, we need to go through three steps of addition:
First of all, in the ones place, X₁ + Y₁ must result in either 0 or 10. If X₁ + Y₁ = 10 then 1 gets carried over into the tens place. Otherwise, nothing gets carried over.
Next, in the tens place, X₂ + Y₂ must result in either 2 or 12(whether or not there is a carryover of 1 from the ones place). If X₂ + Y₂ =12 then 1 gets carried over into the hundreds place. Otherwise, nothing gets carried over.
Finally, there is no digit in the hundreds or thousands place of Last Two Digits of Birth Year so we have X₄X₃ = Z₄Z₃ if there was no carryover from the hundreds place or X₄X₃ + 1 = Z₄Z₃ if there was a carryover of 1 from the hundreds place.
This analysis therefore implies that Equation 1(the math teacher’s claim) is not always true. In fact, his claim is only true if one of the following two things happens:
X₂ + Y₂ = 12 and X₄ = 0 = X₃
X₂ + Y₂ = 2 but X₄ = 0 and X₃ = 1
Furthermore, with lots of exploration using the three steps of addition, you can discover that
X₄X₃X₂X₁ + Y₂Y₁ = 20+100n — — — —(Equation 3a)
for any non-negative integer n. In other words,
Age + Last Two Digits of Birth Year = 20+100n — — — —(Equation 3b)
So it is easy to see that the math teacher’s claim(Equation 1) is the case when n = 1. However, that isn’t the only possible case. The sum of Age and Last Two Digits of Birth Year can be any number in the sequence: 20, 120, 220, 320,…. Let’s call these possible numbers “Magic Numbers”.
However, this begs another question. For any particular person or group of people, is it possible to determine what the corresponding Magic Number is?
Finding the Magic number
Near the beginning of the article, I already established that the math trick essentially degenerates into
Age + Last Two Digits of Birth Year = 120 — — — — (Equation 1)
but we’ve realized that we need to change 120 to other Magic Numbers depending on the situation. It’s important to note that all the Magic Numbershave their last two digits together always being 20.
Going back to the three steps of addition, it means that the main thing that matters is if there was a carryover of 1 from step 2(the tens place) to step 3. Whether or not there is a carryover of 1 determines what digit(s) should come before 20 in the digits of the Magic Number.
So what age groups or birth year groups will cause a carryover of 1 from step 2 to step 3?
Image by Karolina Grabowska from Pixabay
First, we need to consider that there might be carryovers from step 1. In step one, X₁ + Y₁ must result in either 0 or 10.
If X₁ + Y₁ = 0 then the only possible (X₁,Y₁) pair is (0,0)(no carryover to the next step).
If X₁ + Y₁ = 10 then the set of all possible (X₁,Y₁) pairs is {(1,9),(2,8),(3,7),(4,6),(5,5),(6,4),(7,3),(8,2),(9,1)}(carryover of 1 to the next step).
In step 2, X₂+Y₂ must result in either 2 or 12. After some exploration, we can discover that
If X₂ + Y₂ = 2 then the set of all possible (X₂,Y₂) pairs is {(0,1),(1,0)} if there was a carryover from the previous step or {(0,2),(1,1),(2,0)}otherwise.
If X₂ + Y₂ = 12 then the set of all possible (X₂,Y₂) pairs is {(2,9),(3,8),(4,7),(5,6),(6,5),(7,4),(8,3),(9,2)} if there was a carryover from the previous step or {(3,9),(4,8),(5,7),(6,6),(7,5),(8,4),(9,3)} otherwise.
This is all the information we need to answer one of our earlier questions: what age groups or birth year groups will cause a carryover of 1 from step 2 to step 3?
There is only a carryover from step 2 to step 3 when X₂ + Y₂ = 12. Based on the possible pairs for (X₂,Y₂) and (X₁,Y₁), this happens only when Last Two Digits of Birth Year is not in the range [0,20]. These are people with birth years in the ranges [1921,1999], [1821,1899], [1721, 1799],…,[21,99]. (Remember that all these years are in A.D., not B.C.)
On the other hand, there is no carryover if Last Two Digits of Birth Year is in the range [0,20]. These are people with birth years in the ranges [2000,2020], [1900,1920],[1800,1820],…,[0,20].
So if we modify the math trick, we need to first find out whether or not the last two digits(or the entirety) of the person’s birth year falls in a particular range. Then depending on whether or not there is a carryover of 1, we can decide what the Magic Number is. That will give us
Age + Last Two Digits of Birth Year = Magic Number
But we have to change this into a form that looked like the original math trick:
P — Age + Q = Last Two Digits of Birth Year
such that P + Q = Magic Number
With all that complex math out of the way, let’s modify the math trick.
The Modified Math trick
Our new math trick is only going to work under the following conditions:
Assumption 1: The subject(in other words, the person whose age you’re using) should use the age they would have attained if their birthday in 2020 had arrived already.
Assumption 2: The subject shouldn’t be more than 2020 years old.
The new math trick follows this procedure:
Ask the subject if the last two digits(together) of their birth year is in the range [0,20]. If it is go to step 2a, but otherwise go to step 2b.
For the next step, assume that the person’s age has four digits. If it has less than four digits ask the subject to put as many zeros in front of the age as needed to turn it into four digits. The four-digit form of the age is what you will use as the actual age.
(a)Ask the subject what the first two digits of the new Age is. If you take those two digits and follow it up with 20, the result is the Magic Number. So if the first two digits of their Age is AB, then the Magic Number is AB20.
(b)Ask the person what the first two digits of the new Age is. Add 1 to that number. Take the result and follow it up with 20 and what you get is the Magic Number. So if the first two digits together is CD and AB = CD + 1, then the Magic Number is AB20.
Age + Last Two Digits of Birth Year = AB20. So split AB20 into any two summands of your choice and write a new equation like this: P — Age + Q = Last Two Digits of Birth Year such that P + Q = AB20
Translate this into a beautiful sentence like: “P bags of cement minus your age plus Q will give you the last two digits of your birth year. Am I wrong?”
Smile as the subject stares in awe of your intelligence.
As an example, suppose my subject was a 122-year-old woman which implies that her four-digit age is 0122. That would mean that she was born in 1898. The Last Two Digits of Birth Year is 98 and that isn’t in the range [0,20] so I move to step 2b. The Magic Number is 220(or 0220) since CD=01 and AB=01.
If I decide to split the Magic Number as 220 = 132 + 88 then P = 132 and Q = 88 so that I can confidently tell the woman “132 bags of cement minus your age plus 88 will give you the last two digits of your birth year. Am I wrong?”
Conclusion
Try the modified math trick with your friends and let me know how it goes. It is certainly not as simple, short and attractive as the original but it works every time.
Photo by Gabby K from Pexels
However, as I’ve stated before, this isn’t guaranteed to work if your friend’s birthday in 2020 hasn’t arrived yet(that’s the reason for Assumption 1) or if your friend has managed to be more than 2020 years old. Out of those two issues, the former is more common. That is why this math trick is best used at the end of the year 2020 when everyone’s birthday would have passed.
But a math problem that is only guaranteed to work on only one day on onlyone year isn’t fun, is it? There are ways to modify this so that it works for everyone at the end of 2021, 2022, or any year at all. It certainly gets more complex but it still remains fun. I’m unlikely to explore that in an article, however. To all readers, I have to say what all good math textbooks like to say:
Here, u is the control force to balance it, M is mass of cart, m is the mass of pendulum ball.
Real life example of state variables
The train has arrived. There is no change in the position, speed and the traffic on the track.
The dynamic equation of the inverted pendulum on cart.
This is how the dynamic equations look like in MATLAB code
Equilibrium points of an inverted pendulum
The linearized state-space expression
State-space model of the linearised system
To gain altitude, the tail ailerons bend downwards to increases the angle of attack and hence, the lift. But that downwards force on the ailerons forces the airplane to dip down initially. On right is the altitude graph depicting the initial dip.
The transfer function with poles and zeros with b=1
John von Neumann at age 7 (1910)
Left: John von Neumann at age 11 (1915) with his cousin Katalin Alcsuti. (Photo: Nicholas Vonneumann). Right: The Neumann brothers Miklós (1911–2011), Mihály (1907–1989) and János Lajos (1903–1957)
Two portraits of John von Neumann (1920s)
From von Neumann’s Fellowship application to the International Education Board (1926)
The Auditorium Maximum at the University of Göttingen, 1935
Left: John von Neumann in the 1920s. Right: von Neumann, J (1923). Zur Einführung der transfiniten Zahlen (“On the introduction of transfinite numbers”). Acta Litterarum ac Scientiarum Regiae Universitatis Hungaricae Francisco-Josephinae, sectio scientiarum mathematicarum, 1, pp. 199–208.
Left: von Neumann, J. (1928). Zur Theorie der Gesellschaftsspiele (“On the Theory of Games of Strategy”). Right: First edition copy of Theory of Games and Economic Behavior (1944) by John von Neumann and Oskar Morgenstern (Photo: Whitmore Rare Books).
First edition copy of Mathematische Grundlagen der Quantenmechanik (1932)
Left: von Neumann, J. (1932). Proof of the Quasi-ergodic Hypothesis. Proceedings of the National Academy of Sciences 18 (1) pp. 70–82. Right: von Neumann, J. (1932). Physical Applications of The Ergodic Hypothesis. Proceedings of the National Academy of Sciences 18(3) pp. 263–266.
Institute for Advanced Study in Princeton, New Jersey (Photo: Cliff Compton)
Photo of part of the faculty at the Institute for Advanced Study, including its most famous resident Albert Einstein. John von Neumann visible in the background. 
Julian Bigelow, Herman Goldstine, J. Robert Oppenheimer and John von Neumann in front of MANIAC, the Institute for Advanced Study computer.
Left: The photo from von Neumann’s Los Alamos ID badge. Right: John von Neumann talking with Richard Feynman and Stanislaw Ulam in Los Alamos (Photo: )
von Neumann speaking at the American Philosophical Society in 1957 (Photo: Alfred Eisenstaedt)
John von Neumann with the IAS machine, sometimes called the “von Neumann Machine”, stored in the the basement of Fuld Hall from 1942–1951 (Photo: Alan Richards)
John von Neumann with his wife Klari Dan and their dog (Photo: Alan Richards)
Von Neumann in the Florida Everglades in 1938 (Photo: Marina von Neumann Whitman)
President Dwight D. Eisenhower (left) presenting John von Neumann (right) the Presidential Medal of Freedom in 1956
Source: Geogebra. See https://www.geogebra.org/m/sPx39Zfd
Made using Geogebra
Formula using radians
formula using degrees
source: Dexter’s Notes, Geometry IB 
source: Dexter’s Notes, Geometry IB.
Photo by Max Fischer from Pexels
Photo by cottonbro from Pexels
Image by Karolina Grabowska from Pixabay
Photo by Gabby K from Pexels