by R. Passov
As the clouds of WWII darkened Austria, Kurt Gödel, the greatest logician of modern times, at Einstein’s urging, brought his two magnificent proofs to Princeton. There he would remain for almost forty years, never mentoring a graduate student, rarely lecturing, adding only one substantial but incomplete proof to the cannon of math.
Mildly underwhelmed by the impact of his discoveries, at Princeton he would gradually set aside math in favor of philosophy. Little of this work was published in his lifetime. But it was enjoyed by Einstein who for the last decade of his life, walked alongside as Gödel discussed a field Einstein once likened to ‘writings in honey.’
Gödel was born in 1906, into a German speaking family living in Brunn (Brno) Moravia, then part of the Austria-Hungary Empire, soon to be annexed by Germany, now part of the Czeck Republic. His father before passing unexpectedly in 1929, had built a successful textile business that would secure his family’s finances.
An early childhood struggle with Rheumatic fever left Gödel forever suspicious of the state of his health. He took his primary education at the local Realgymnasium. Modeled after the enlightened German system, the gymnasium offered “mental gymnastics, developing both mind and body.” Following his older brother, in 1921 he entered the University of Vienna.
Easily establishing his gifts, he studied theoretical physics, enjoyed the life of a student and earned a reputation for sleeping late.
In 1928 Gödel traveled to Bologna to hear a lecture from David Hilbert, an Ordinarius (full professor) at the University of Göttingen which, since the turn of the century, had been the preeminent place of study for higher mathematics.
Hilbert was born 1862 near Königsberg, Prussia, to a successful county judge who “…walked the same path everyday” stressing “…the Prussian virtues of punctuation, thrift … and diligence…” After a late start at the Gymnasium he entered the local university where he enjoyed a course of study defined by “…no specified requirements, no minimum units, no roll call, no examinations until the taking of the degree.” Where others had gone astray, Hilbert proved remarkably disciplined.
While pursuing his Habilitation, a second dissertation and a perquisite for the position of Privatdozent which offered the opportunity to lecture students for tips, he journeyed to the University of Leipzig to meet Felix Klein, a prominent geometer, soon to lead the University of Göttingen. Klein saw something in Hilbert and began a skillful mentorship, sending Hilbert across Europe to meet the great thinkers of the day.
When Descartes, in 1637, showed that the algebraic relationship of a line maps any point on a plane he re-discovered the symbiosis between functions in algebra and abstractions of geometry. Two hundred years later Lobachevsky would relax Euclid’s requirement that parallel lines remain forever parallel.
Out of this came the vectors and matrices of algebraic geometry. Then a calculus of vectors and finally, a calculus of tensors designed to work in higher-dimensional space unanchored by coordinates. It was a complicated addition that proved necessary for the development of General Relativity. And it was hard. So hard in fact that to understand it, Einstein needed a tutor who had trained under Klein.
Klein had defined the new geometries by the properties of the space in question that remain invariant to a change in basis – a change in orientation. Hilbert’s first major work, his Basis Theorem, published in 1888, established that “…there is always a finite number of basis for an invariant system.”
Hilbert’s theorem extended and simplified a proof so computationally complex it had been left alone for twenty years. His insight was to avoid constructing the object of his proof. Instead he established it out of logical necessity. While the outcome was of major importance, some recognized the deeper accomplishment: A proof leading math back to its Euclidian foundations, back to an ordered science of deducing theorems from a minimal set of independent postulates.
In 1895 Klein succeeded in luring Hilbert to Göttingen. There Hilbert began his ascent on the kingdom of mathematics. The coronation came at the Second Congress of Higher Mathematics, convened in Paris to mark the turn of the century. Nearly the last speaker, Hilbert quietly read through a program, upon publication condensed into twenty-three questions, that would define the contours of math for much of the twentieth century.
The second question (later expanded into two) was a challenge. Hilbert, like his predecessor Boole, believed feverishly in the perfection of math. This perfection lay in two properties of mathematics that, according to Hilbert, stood math above the natural sciences: The properties of Consistency and of Completeness.
Completeness would ensure that, in a finite number of steps, the process of math leaves no truth safe from discovery. Consistency, in turn, would ensure that once a truth is discovered by the process of mathematics it would remain forever true, never subject to contradiction. Proving these two conditions Hilbert believed, would quite the many who sought other philosophies which they hoped would contain the mess of discovery that unsettled the sciences at the turn of the century.
So secure was Hilbert in his beliefs that he turned rhetorical, proclaiming “There are no ignoramuses!”
Hilbert could be progressive. For example, when Emmy Noether, whom Einstein described as “…the most significant creative mathematical genius thus far produced since the higher education of women …” sought an opportunity to lecture at Gottingen, Hilbert graciously yet tersely offered that if higher mathematics is indifferent to gender then so should be the University of Gottingen’s math department.
But if you were not a believer Hilbert could bear down with an unforgiving intensity. Such was the case with Luitzen Egbertus Jan Brouwer, otherwise known as Bertus.
Non-Euclidian geometries forced the abandonment of Kant’s assertion that the Euclidian concept of space was an emergent property of the mind. The void gave Brouwer the opportunity to assert that inherent to human nature is not Euclidian space but instead the properties of the natural numbers. Higher mathematics, according to Brouwer, arises from a native capacity to “…add one object to another, one after another.” As such, mathematics comes before logic and so Hilbert’s dream of containing math in a finite number of steps is not realizable.
Tall, handsome, urban Brouwer had made substantial contributions to topology. Yet his other ideas made him an easy target. “Intellect” according to Brouwer, is the “downfall of man.” Religion “stupefies” the masses. And, while his wife toiled long hours as a pharmacist to support his writing, Brouwer argued that women entering the work force was sure to “ruin the value of work.”
While Einstein asked, “What is this frog and mouse battle among the mathematicians?” Hilbert saw the existential nature of the fight. Brouwer’s nativist argument, if accepted, would deny Hilbert his throne. After one of his better students momentarily switched sides, Hilbert went to extremes in expressing his antipathy toward the ideas and to the man himself.
Brouwer had enjoyed a longstanding position on the editorial board of Mathematische Annalen, the most prestigious of mathematical journals. Hilbert, having inherited the Managing Editor title from Klein, in an effort to oust Brouwer succeeded in causing the whole editorial board, including Einstein, to resign.
The sixth of Hilbert’s twenty-three questions, often overlooked, called for the “Mathematical treatment of the axioms of physics.” Hilbert had convinced himself that Physics would be best served confined to an axiomatized version of itself.
In 1905 Hilbert and his close colleague, H. Minkowski, witnessed Einstein publish the first of his two theories of relativity. Minkowski, having lectured to Einstein while at the University of Zurich, was surprised at such an accomplishment from one of his “lessor students.” He reformatted Einstein’s solution, introducing “world lines in a four-dimensional vector space.” Though Einstein remarked that Minkowski’s “…chalk is cheaper than grey matter…” the solution survives as the favored mathematical development of Special Relativity.
Hilbert and Minkowski would join forces in the hunt for the missing half of Einstein’s theory – a theory of gravity. After four promising years, likely the stronger of the two at understanding physics, Minkowski is cut down by a ruptured appendix. Hilbert, intent on seeing the dream realized, soldiers on. He hires a series of tutors, including Eugene Weiner who would go on to win a Nobel Prize in physics.
After a decade of effort, in November of 1915, believing he is close to an “axiomatized” theory of gravity, Hilbert invited Einstein to Göttingen for a series of lectures. As Einstein described a problem in physics, Hilbert saw a challenge in mathematics.
Several great thinkers circled as Einstein struggled to master the properties of a geometry that curved in the presence of matter. At first it seemed that Hilbert, by publishing a solution to Einstein’s Field Equations on November 15th, ten days ahead of Einstein, had won the race for the theory of General Relativity.
Ready to show off his “axiomatic solution to [the] great problem,” Hilbert again invited Einstein to Göttingen for a talk he planned to give on the 18th. Einstein declined the invitation and shortly afterward, in a letter to a friend, made his feelings toward Hilbert known:
The theory is beautiful beyond comparison. However, only one colleague has really understood it, and he is seeking to “partake” in it (Abraham’s expression) in a clever way. In my personal experience I have hardly come to know the wretchedness of mankind better than as a result of this theory and everything connected to it. But it does not bother me.
After giving the “… impression that Einstein’s merit was confined to asking the right questions, while [he] provided the answers …” Einstein’s arguments cause Hilbert to see that “… his theory was actually based on a much wider array of assumptions than his axiomatic presentation had indicated.”
Hilbert revised his proof, relinquishing claims to priority. In a letter dated the 20th of December, Einstein reverts to form:
There has been a certain ill-feeling between us, the cause of which I do not want to analyze. I have struggled against the feeling of bitterness attached to it, and this with complete success. I think of you again with unmarred friendliness and ask you to try to do the same with me. Objectively it is a shame when two real fellows who have extricated themselves somewhat from this shabby world do not afford each other mutual pleasure.
For over seventy-five years speculation remained as to whether Hilbert had beat Einstein to General Relativity. Though the debate is likely forever settled in Einstein’s favor, there was another matter, perhaps deeper even than the structure of General Relativity, that also settled in Einstein’s favor.
Hilbert had come to Bologna in 1928 with sixty-seven mathematicians in tow. He felt partially vindicated in his quest for the dominance of math as John Von Neumann, a mathematician gifted beyond measure, thought he had a proof of the consistency of a formal system. Gushing with enthusiasm, Hilbert expanded the quest to include Completeness.
Gödel was moved by the great man. In the following year he produced a dissertation that proved the Consistency of ‘first-order logic.’ He then attempted, in his Habilitation, to prove the same for axiomatized math.
Starting with the most succinct set of axioms that support all properties of the natural numbers, to his surprise he showed that such a system is not Complete: “…there will always exist a truth whose nature is apparent but whose essence can’t be proven within the system.”
Next, Gödel proved that within such a system (within a mathematics built through a formal process beginning with a set of independent axioms from which all theorems are deduced) what is provable is only that the system is inconsistent. Hilbert never fully recovered from his shock.
Gödel did allow one of his later discoveries to see light. He published his “limit solution” to Einstein’s field equations for General Relativity. His was the first to show the mathematical possibility of “Rotating Universes” within which “… closed timelike word lines permit … time travel.
Einstein knew his proof of General Relativity was secure when it mapped the perturbation in Mercury’s orbit around the sun, evidence of the curvature of space-time. Enough has been written about his views toward math to support a range of theories. Mine is that Einstein believed that his physics is an ongoing test of just one of the infinite truths contained within mathematics.
Hilbert wanted math axiomatized, but he wanted more than that. He wanted to organize physics so that discovery would match his vision of an orderly process in which science is built one deductive brick after another. Einstein, it seems, wanted physics left open to where curiosity leads.
The argument between Hilbert and Brouwer can be recast as an argument over the nature of infinities – over the seeming impossibility of the sum of an ever-decreasing increment contained within a finite measure. Hilbert accepted that so long as the next member in the series grows proportionately smaller, such a series has a finite limit. Thus, the solution to Zeno’s paradox.
Brouwer does not accept such a resolution. But he may have accepted Gödel’s solution which was to assert that it is neither Euclid’s space nor the natural numbers that is native to the mind but rather the concept of a “possible infinity” – one that is conceptual and yet unrealizable.
- “Einstein’s Theory of Relativity,” Max Born, Dover Publications, 1962
- “Gödel meets Einstein,” Pale Yourgran, Open Court, 1999
- “Herman Minkowski, Relativity and the Axiomatic Approach to Physics,” Leo Corry, from: “Minkowski Spacetime: A Hundred Years Later,” Fundamental Theories of Physics, 165; Springer, 2010
- “Hilbert,” Constance Reid, Springer-Verlag, 1970
- “Math In Minutes,” Paul Glendinning, Quercus, 2011
- “The Computer from Pascal to von Neumann,” Herman H. Goldstine, Princeton University Press, 1972
- “The Development of Mathematics,” E. T. Bell, Dover Publications, 1940
- “The Feynman Lectures on Physics,” Volume II, Chapter 42