Kurt Gödel’s open world

#1
C C Offline
https://3quarksdaily.com/3quarksdaily/20...world.html

EXCERPTS: . . . The reason Kurt Gödel drew gasps of awe from colleagues as brilliant as Einstein and John von Neumann was because he revealed a seismic fissure in the foundations of that most perfect, rational and crystal-clear of all creations – mathematics. Of all the fields of human inquiry, mathematics is considered the most exact.

Unlike politics or economics, or even the more quantifiable disciplines of chemistry and physics, every question in mathematics has a definite yes or no answer. The answer to a question such as whether there is an infinitude of prime numbers leaves absolutely no room for ambiguity or error – it’s a simple yes or no (yes in this case). Not surprisingly, mathematicians around the beginning of the 20th century started thinking that every mathematical question that can be posed should have a definite yes or no answer. In addition, no mathematical question should have both answers. The first requirement was called completeness, the second one was called consistency.

The overarching goal of mathematics was to prove completeness and consistency starting from a fundamental, minimal set of axioms, much like Euclid had built up the grand structure of plane geometry starting with a handful of axioms in his marvelous ‘Elements’. Mathematicians had good reasons to be optimistic....

[...] The intellectual godfather of the mathematicians was David Hilbert, perhaps the leading mathematician of the first few decades of the twentieth century. ... Hilbert set out 23 open problems in mathematics ... The second among these problems was to prove the consistency of arithmetic using the kind of axiomatic approach developed by Bertand Russell and Alfred North Whitehead. Hilbert was confident that within a few decades at best, every question in mathematics would have a definite answer that could be built up from the axioms...

[...] When Hilbert gave his talk, Kurt Gödel was still six years away from being born. Thirty years later he would drive a wrecking ball into Hilbert’s dream ... Kurt became a staunch Platonist whose belief that mathematical objects existed in a world of their own without any human intervention only became deeper during his life. For Gödel, numbers, sets and mathematical axioms were as real as planets, bacteria and rocks, simply waiting to be discovered and existing independent of human effort. A large part of this existence stemmed from the sheer beauty of mathematical structures that Gödel and his colleagues were uncovering: how could such beautiful objects exist only under the pre-condition of discovery by ordinary human minds?

By 1930 the Platonist Gödel was ready to drop his bombshell in the world of mathematics and logic. [...] As often happens with great scientific discoveries, few people understood the significance of what had just happened. The one exception was John von Neumann, a child prodigy and polymath who was known for jumping ten steps ahead of people’s arguments and extending them in ways that their creators could not imagine...

[...] So what had Gödel done? ... in a nutshell, what Gödel had found using an ingenious bit of self-referential mapping between numbers and mathematical statements was that any consistent mathematical system that could support the basic axioms of arithmetic as described in Russell and Whitehead’s work would always contain statements that were unprovable. This ingenious scheme included a way of encoding mathematical statements as numbers, allowing numbers to “talk about themselves”.

What was worse and even more fascinating was that the axiomatic system of arithmetic would contain statements that were true, but whose truth could not be proven using the axioms of the system [...] Thus, the system would always contain ‘truths’ that are undecidable within the framework of the system. And lest one thought that you could then just expand the system and prove those truths within that new system, Gödel infuriatingly showed that the new system would contain its own unprovable truths, and ad infinitum. This is called the First Incompleteness Theorem.

The Second Incompleteness showed that such a system cannot prove its own consistency, leading to another paradox and in effect saying that any formal system that is interesting enough to prove its own consistency can do so only if it’s inconsistent. This was an even more damning conclusion. Far from getting rid of the paradoxes that Russell and Whitehead believed would be clarified if only one understood the axioms and the deductions from them well enough, Gödel showed that such paradoxes are as foundational a feature of mathematical systems as anything else. As far as Hilbert was concerned, he had uncovered a rotten foundation underlying mathematics that doomed Hilbert’s program forever.

[...] As sometimes happens when a great mind declares a truth in a scientific discipline with such finality, reactions can range from disbelief and denial to acceptance. [...] Gödel’s work had a seismic impact on that of many other thinkers. ... Most notably, Gödel’s ingenious scheme of having numbers represent both themselves as well as instructions to specify operations on themselves is, without him ever knowing it, the basis of digital computing.

Thus by the time he was 24 years old, Gödel had established himself as a logician of the first rank and immortalized his name in history. In the next few years his friends and colleagues spread his gospel around the world, most notably in the United States...

[...] Political events were also clearly not evolving favorably by the time Gödel first made his way to America. Austrian anti-Semitism had already had a long history ... After Jewish professors were all dismissed throughout Germany and Austria, Hilbert was asked by the new Nazi minister of education what mathematics was like at the University of Göttingen where he taught. “There is no mathematics anymore at Göttingen”, Hilbert retorted.

Gödel, as involved as he was with the search for mathematical truth, was not finely attuned to what was happening to politics in the country. [...] But even Gödel could not ignore what was happening to his colleagues at the university, and after some unpleasant episodes including one in which he was bullied on the streets by Nazi thugs and Adele [his wife] fended off their taunts with her umbrella, the couple decided to emigrate to America for good.

[...] Kurt and Adele arrived in Princeton, a place puckishly described by recent resident Albert Einstein as “a quaint, ceremonial village, full of demigods on stilts”. Gödel had never known Einstein before coming to America, and yet it was Einstein who, along with the Austrian economist Oskar Morgenstern, provided him with the friendship of his Viennese colleagues which he so missed. Einstein and Gödel made for an unlikely pair: the former gregarious, generous, earthy and shabbily dressed, always eyeing the world through a sense of humor; the latter often withdrawn, hyper-logical, critical and unable to lighten up. And yet these exterior differences hid a deep and genuine friendship that went beyond their common background in German culture.

[...] But the real reason Einstein so admired Gödel was likely because he shared Gödel’s unshakeable belief in the purity of the mathematical constructs governing the universe. Einstein who was not formally religious nevertheless always harbored a deep belief that the laws of physics exist independently of human beings’ abilities to identify and tamper with them – that was one reason he was so uneasy with the then standard interpretation of quantum mechanics which seemed to say that there was no reality independent of observers.

Gödel outdid him and went one step further, believing that even numbers and mathematical theorems exist independently of the human mind. It was this almost spiritual and religious belief in the objective nature of mathematical reality that perhaps formed the most intimate bond between the era’s greatest theoretical physicist and its greatest logician.

It also helped that Gödel got interested in Einstein’s general theory of relativity, once playing with the equations and startling Einstein by concluding that the theory allowed for the existence of closed timelike curves – in other words [...] For Gödel’s Platonic mind, this kind of result based purely on mathematics and without any physical basis was exactly the kind of absolute mathematical truth he believed in.

[...] For all of Gödel’s scathing remarks and frequent silence about other mathematicians’ work, he was profusely generous toward Paul Cohen when Cohen sent him his proof of the independence of the Continuum Hypothesis, a problem that Gödel himself had tried and failed to solve for more than twenty years.

Gödel’s peculiar obsessions and pedantry made him a difficult colleague, and his promotion of to full professor was held up until 1953 because the faculty feared he would be challenging to deal with when it came to the obligatory administrative matters that full professors had to busy themselves with. Once again von Neumann came to his friend’s rescue, asking, “If Gödel cannot call himself Professor, how can the rest of us?” But even after Gödel got promoted his insecurities did not leave him, and he kept on feeling a mixture of self-pity and suspicions of conspiracy on the part of the institute to demote or fire him. He could nonetheless be a very loyal friend and colleague...

After Einstein’s death in 1955 and von Neumann’s excruciatingly painful death in 1957, Gödel began to increasingly rely on Adele and Oskar Morgenstern [...] But the spark of genius that had lit the mathematical world on fire seemed to have gone missing. In his last few years, Gödel became obsessed with not just believing that there was a conspiracy against him but also one against a hero of his, the 18th century mathematician and polymath Gottfried Wilhelm Leibniz. He became convinced that there was a plot to keep Leibniz’s work hidden from the world.

[...] There was little that anyone could do to help. In 1977 Morgenstern himself received a diagnosis of terminal cancer and became paralyzed. His tragic last notes and letters indicate the struggle he was facing as Gödel increasingly came to rely on him, phoning him two or three times every day to communicate his latest worries, even as he himself was facing his own mortality. The last straw was when Adele fell sick and had to spend several months in a hospital. After Morgenstern, she had been his last link to the sane world, and in spite of neighbors and colleagues trying to help out, he stopped eating, convinced that he was being poisoned through his food ... When Adele came back the end was already there, and Gödel entered the hospital for the last time. The cause of death was “malnutrition”, although most people believed that slow suicide was the more likely explanation.

How do we deal with the legacy of someone like Gödel? Philosophically, Gödel’s theorems had such a shattering impact on our thinking because, along with two other groundbreaking ideas of 20th century science – Heisenberg’s Uncertainty Principle and quantum indeterminacy – they revealed that human beings’ ability to divine knowledge of the universe had fundamental limitations. ... Nonetheless, mathematics continued to thrive within the boundaries of his theorems...

But that also says something about human minds and points to a debate still raging – whether the mind itself is some kind of Turing machine. The implication of Gödel’s proof is that if the mind is indeed a machine, it will be subject to the incompleteness theorems and there will always be truths beyond our grasp. If on the other hand, the mind is not a machine, it frees it up from being described through purely mechanistic means.

Both choices point to a human mind and a world it inhabits that are “decidedly opposed to materialistic philosophy”. Beyond this possible truth is another one that is purely psychological. We can either feel morose in the face of the fundamental limits to knowledge that Gödel revealed, or we can revel, as the historian George Dyson put it, to “celebrate his proof that even the most rigid numerical bureaucracy contains the tools by which higher truth will always be able to effect an escape.”Gödel offers us an invitation to an open world, a world without end.

But what about the paradoxes of the man himself, someone devoted to the highest reaches of rational thought in the most logical of all fields of inquiry, and still one who seemed to have had an almost mystical belief in the spiritual certainty of mathematics and often gave in to the worst impulses of irrationality?

I think a clue comes from Gödel’s obsession with Leibniz in his last few years. Leibniz was convinced that this is the best of all possible worlds, because that is the only thing a just God could have created. Like his fellow philosophers and mathematicians, Leibniz was religious and saw no contradictions between science and faith [...] A few years before his mother Marianne’s death in 1961, Kurt wrote to her in a letter his belief that a God probably exists: “For what kind of sense would there be in bringing forth a creature (man), who has such a broad range of possibilities of his own development and of relationships, and then not allow him to achieve 1/1000 of it?”

Like his fellow philosopher Leibniz, Kurt Gödel could perfectly reconcile the rational and the transcendental. In doing this, he proved himself to be much more at home in the 18th century than the 20th. Perhaps that vision of a reconciliation between rational thought and seemingly irrational human frailty and belief will be, even more than his seminal mathematical discoveries, his enduring legacy... (MORE - missing details)
Reply
#2
Magical Realist Offline
Quote:”Gödel offers us an invitation to an open world, a world without end.

I'll take that world, even if it IS more uncertain.
Reply




Users browsing this thread: 1 Guest(s)