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Posts Tagged ‘Gödel

“If a ‘religion’ is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one”*…

 

Godel-Tower_2880x1620_Lede

 

In 1931, the Austrian logician Kurt Gödel pulled off arguably one of the most stunning intellectual achievements in history.

Mathematicians of the era sought a solid foundation for mathematics: a set of basic mathematical facts, or axioms, that was both consistent — never leading to contradictions — and complete, serving as the building blocks of all mathematical truths.

But Gödel’s shocking incompleteness theorems, published when he was just 25, crushed that dream. He proved that any set of axioms you could posit as a possible foundation for math will inevitably be incomplete; there will always be true facts about numbers that cannot be proved by those axioms. He also showed that no candidate set of axioms can ever prove its own consistency.

His incompleteness theorems meant there can be no mathematical theory of everything, no unification of what’s provable and what’s true. What mathematicians can prove depends on their starting assumptions, not on any fundamental ground truth from which all answers spring.

In the 89 years since Gödel’s discovery, mathematicians have stumbled upon just the kinds of unanswerable questions his theorems foretold. For example, Gödel himself helped establish that the continuum hypothesis, which concerns the sizes of infinity, is undecidable, as is the halting problem, which asks whether a computer program fed with a random input will run forever or eventually halt. Undecidable questions have even arisen in physics, suggesting that Gödelian incompleteness afflicts not just math, but — in some ill-understood way — reality…

A (relatively) simple explanation of the incompleteness theorem– which destroyed the search for a mathematical theory of everything: “How Gödel’s Proof Works.”

* John D. Barrow, The Artful Universe

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As we noodle on the unknowable, we might spare a thought for Vilfredo Federico Damaso Pareto; he died on this date in 1923.  An engineer, sociologist, economist, political scientist, and philosopher, he made several important contributions to economics, sociology, and mathematics.

He introduced the concept of Pareto efficiency and helped develop the field of microeconomics.  He was also the first to discover that income follows a Pareto distribution, which is a power law probability distribution.  The Pareto principle,  named after him, generalized on his observations on wealth distribution to suggest that, in most systems/settings, 80% of the effects come from 20% of the causes– the “80-20 rule.” He was also responsible for popularizing the use of the term “elite” in social analysis.

As Benoit Mandelbrot and Richard L. Hudson observed, “His legacy as an economist was profound. Partly because of him, the field evolved from a branch of moral philosophy as practised by Adam Smith into a data intensive field of scientific research and mathematical equations.”

The future leader of Italian fascism Benito Mussolini, in 1904, when he was a young student, attended some of Pareto’s lectures at the University of Lausanne.  It has been argued that Mussolini’s move away from socialism towards a form of “elitism” may be attributed to Pareto’s ideas.

Mandelbrot summarized Pareto’s notions as follows:

At the bottom of the Wealth curve, he wrote, Men and Women starve and children die young. In the broad middle of the curve all is turmoil and motion: people rising and falling, climbing by talent or luck and falling by alcoholism, tuberculosis and other kinds of unfitness. At the very top sit the elite of the elite, who control wealth and power for a time – until they are unseated through revolution or upheaval by a new aristocratic class. There is no progress in human history. Democracy is a fraud. Human nature is primitive, emotional, unyielding. The smarter, abler, stronger, and shrewder take the lion’s share. The weak starve, lest society become degenerate: One can, Pareto wrote, ‘compare the social body to the human body, which will promptly perish if prevented from eliminating toxins.’ Inflammatory stuff – and it burned Pareto’s reputation… [source]

220px-Vilfredo_Pareto_1870s2 source

 

 

“Neoliberalization has meant, in short, the financialization of everything”*…

 

Succession

 

Investing and deal-making occupy an outsized role in popular depictions of “business” like HBO’s Succession and Showtime’s Billions. They also occupy an outsized share of our elite: Over the last five years, the nation’s top business schools have sent nearly thirty percent of their graduating classes into finance.

But the buying and selling of companies, the mergers and divestments, the hedging and leveraging, are not themselves valuable activity. They invent, create, build, and provide nothing. Their claim to value is purely derivative—by improving the allocation of capital and configuration of assets, they are supposed to make everyone operating in the real economy more productive. The practitioners are rewarded richly for their effort.

Does this work, or are the efforts largely wasted? One might default to the assumption that an industry attracting so much talent and generating so much profit must be creating enormous value. But the elaborate financial engineering of the 2000s, which attempted an alchemy-like conversion of high-risk loans into rock-solid assets, and then placed highly leveraged bets against their performance, led to the collapse of some established Wall Street institutions, massive bailouts for others, and a global economic meltdown. Mergers and acquisitions, meanwhile, appear largely to be exercises in wheel-spinning: “M&A is a mug’s game,” explains Roger Martin in the Harvard Business Review, “in which typically 70%–90% of acquisitions are abysmal failures.”…

Hedge funds and venture capital funds appear to badly underperform simple public market indexes, while buyout funds have performed roughly at par over the past decade. Of course, some funds deliver outsized returns in a given timeframe; even a random distribution has a right tail. And there are managers whose strong and consistent track records suggest the creation of real value.

In other words, most fund managers are generating the results that one might expect from an elaborate game of chance—placing bets in the market with odds similar to a coin flip. With enough people playing, some will always find themselves on winning streaks and claim the Midas touch, at least until the coin’s next flip. Except under these rules of “heads I win, tails you lose,” they collect their fees regardless…

In the U.S., finance, insurance and real estate (FIRE) sector now accounts for 20 percent of GDP– compared with only 10 percent in 1947.  The thorough and thoughtful analysis– and critique–  of the frothier components of that sector excerpted above is noteworthy, beyond its quality, for it’s origin; it is an early product of a new conservative think tank, American Compass.

Read it in full: “Coin-Flip Capitalism: A Primer.”

Pair with “What Kind of Country Do We Want?“, a resonant essay from the amazing Marilynne Robinson.

(image above: source)

* “Neoliberalization has meant, in short, the financialization of everything. There was unquestionably a power shift away from production to the world of finance… Neoliberalization has not been very effective in revitalizing global capital accumulation, but it has succeeded remarkably well in restoring, or in some instances (as in Russia and China) creating, the power of an economic elite. The theoretical utopianism of neoliberal argument has, I conclude, primarily worked as a system of justification and legitimation for whatever needed to be done to achieve this goal.”  — David Harvey, A Brief History of Neoliberalism

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As we look beyond price to value, we might recall that it was on this date in 1936 that Alan Turing submitted his paper, “On Computable Numbers” for publication; its full title was “On Computable Numbers, with an Application to the Entscheidungsproblem.”  In answer to Hibert’s and Ackermann’s 1928 challenge, Turing demonstrated that some purely mathematical yes-no questions can never be answered by computation; more technically, that some decision problems are “undecidable” in the sense that there is no single algorithm that infallibly gives a correct “yes” or “no” answer to each instance of the problem.  In Turing’s own words: “…what I shall prove is quite different from the well-known results of Gödel … I shall now show that there is no general method which tells whether a given formula U is provable in K.”

Turing followed this proof with two others, both of which rely on the first. And all rely on his development of type-writer-like “computing machines” that obey a simple set of rules and his subsequent development of a “universal computing machine”– the “Turing Machine,” a key inspiration (to von Neumann and others) for the development of the digital computer.

220px-Alan_Turing_Aged_16 source

 

“Nothing happens until something moves”*…

 

Universe

 

What determines our fate? To the Stoic Greek philosophers, fate is the external product of divine will, ‘the thread of your destiny’. To transcendentalists such as Henry David Thoreau, it is an inward matter of self-determination, of ‘what a man thinks of himself’. To modern cosmologists, fate is something else entirely: a sweeping, impersonal physical process that can be boiled down into a single, momentous number known as the Hubble Constant.

The Hubble Constant can be defined simply as the rate at which the Universe is expanding, a measure of how quickly the space between galaxies is stretching apart. The slightest interpretation exposes a web of complexity encased within that seeming simplicity, however. Extrapolating the expansion process backward implies that all the galaxies we can observe originated together at some point in the past – emerging from a Big Bang – and that the Universe has a finite age. Extrapolating forward presents two starkly opposed futures, either an endless era of expansion and dissipation or an eventual turnabout that will wipe out the current order and begin the process anew.

That’s a lot of emotional and intellectual weight resting on one small number…

How scientists pinned a single number on all of existence: “Fate of the Universe.”

[Readers might remember that the Big Bang wasn’t always an accepted paradigm— and that on-going research continues to surface challenges.]

* Albert Einstein

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As we center ourselves, we might spare a thought for Kurt Friedrich Gödel; he died on this date in 1978.  A  logician, mathematician, and philosopher, he is considered (along with Aristotle, Alfred Tarski— whose birthday this also is– and Gottlob Frege) to be one of the most important logicians in history.  Gödel had an immense impact upon scientific and philosophical thinking in the 20th century.  He is, perhaps, best remembered for his Incompleteness Theorems, which led to (among other important results) Alan Turing’s insights into computational theory.

Kurt Gödel’s achievement in modern logic is singular and monumental – indeed it is more than a monument, it is a landmark which will remain visible far in space and time. … The subject of logic has certainly completely changed its nature and possibilities with Gödel’s achievement.                  — John von Neumann

kurt_gödel source

 

“The laws of nature are but the mathematical thoughts of God”*…

 

euclid

2,300 years ago, Euclid of Alexandria sat with a reed pen–a humble, sliced stalk of grass–and wrote down the foundational laws that we’ve come to call geometry. Now his beautiful work is available for the first time as an interactive website.

Euclid’s Elements was first published in 300 B.C. as a compilation of the foundational geometrical proofs established by the ancient Greek. It became the world’s oldest, continuously used mathematical textbook. Then in 1847, mathematician Oliver Byrne rereleased the text with a new, watershed use of graphics. While Euclid’s version had basic sketches, Byrne reimagined the proofs in a modernist, graphic language based upon the three primary colors to keep it all straight. Byrne’s use of color made his book expensive to reproduce and therefore scarce, but Byrne’s edition has been recognized as an important piece of data visualization history all the same…

Explore elemental beauty at “A masterpiece of ancient data viz, reinvented as a gorgeous website.”

* Euclid, Elements

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As we appreciate the angles, we might spare a thought for Kurt Friedrich Gödel; he died on this date in 1978.  A  logician, mathematician, and philosopher, he is considered (along with Aristotle, Alfred Tarski— whose birthday this also is– and Gottlob Frege) to be one of the most important logicians in history.  Gödel had an immense impact upon scientific and philosophical thinking in the 20th century.  He is, perhaps, best remembered for his Incompleteness Theorems, which led to (among other important results) Alan Turing’s insights into computational theory.

Kurt Gödel’s achievement in modern logic is singular and monumental – indeed it is more than a monument, it is a landmark which will remain visible far in space and time. … The subject of logic has certainly completely changed its nature and possibilities with Gödel’s achievement.                  — John von Neumann

kurt_gödel source

 

Written by LW

January 14, 2019 at 1:01 am

“Mystery has its own mysteries”*…

 

Finally, an answer to a question that puzzled Cantor and Hilbert (proprietor of The Infinite Hotel) and challenged Cohen and Gödel…

In a breakthrough that disproves decades of conventional wisdom [and confounds common sense], two mathematicians have shown that two different variants of infinity are actually the same size. The advance touches on one of the most famous and intractable problems in mathematics: whether there exist infinities between the infinite size of the natural numbers and the larger infinite size of the real numbers…

Connecting the sizes of infinities and the complexity of mathematical theories:                        “Mathematicians Measure Infinities and Find They’re Equal.”

* “Mystery has its own mysteries, and there are gods above gods. We have ours, they have theirs. That is what’s known as infinity.”  – Jean Cocteau

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As we go big, we might spare a thought for Paul Erdős; he died on this date in 1996.  One of the most prolific mathematicians of the 20th century (he published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed), he is remembered both for his “social practice” of mathematics (he engaged more than 500 collaborators) and for his eccentric lifestyle (he spent his waking hours virtually entirely on math; he would typically show up at a colleague’s doorstep and announce “my brain is open”, staying long enough to collaborate on a few papers before moving on a few days later).

Erdős’s prolific output with co-authors prompted the creation of the Erdős number, the number of steps in the shortest path between a mathematician and Erdős in terms of co-authorships.  Low numbers are a badge of pride– and a usual marker of accomplishment: As of 2016, all Fields Medalists have a finite Erdős number, with values that range between 2 and 6, and a median of 3.  Physics Nobelists Einstein and Sheldon Glashow have an Erdős number of 2.   Baseball Hall of Famer Hank Aaron can be considered to have an Erdős number of 1 because they both autographed the same baseball (for number theorist Carl Pomerance).  Natalie Portman’s undergraduate collaboration with a Harvard professor earned her an Erdős number of 5; Danica McKellar (“Winnie Cooper” in The Wonder Years) has an Erdős number of 4, for a mathematics paper coauthored while an undergraduate at UCLA.

 source

 

Written by LW

September 20, 2017 at 1:01 am

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