Posts Tagged ‘logic’
“If the doors of perception were cleansed everything would appear to man as it is, infinite”*…
For 50 years, mathematicians have believed that the total number of real numbers is unknowable. A new proof suggests otherwise…
Infinity comes in many sizes. In 1873, the German mathematician Georg Cantor shook math to the core when he discovered that the “real” numbers that fill the number line — most with never-ending digits, like 3.14159… — outnumber “natural” numbers like 1, 2 and 3, even though there are infinitely many of both.
Infinite sets of numbers mess with our intuition about size, so as a warmup, compare the natural numbers {1, 2, 3, …} with the odd numbers {1, 3, 5, …}. You might think the first set is bigger, since only half its elements appear in the second set. Cantor realized, though, that the elements of the two sets can be put in a one-to-one correspondence. You can pair off the first elements of each set (1 and 1), then pair off their second elements (2 and 3), then their third (3 and 5), and so on forever, covering all elements of both sets. In this sense, the two infinite sets have the same size, or what Cantor called “cardinality.” He designated their size with the cardinal number ℵ0 (“aleph-zero”).
But Cantor discovered that natural numbers can’t be put into one-to-one correspondence with the continuum of real numbers. For instance, try to pair 1 with 1.00000… and 2 with 1.00001…, and you’ll have skipped over infinitely many real numbers (like 1.000000001…). You can’t possibly count them all; their cardinality is greater than that of the natural numbers.
Sizes of infinity don’t stop there. Cantor discovered that any infinite set’s power set — the set of all subsets of its elements — has larger cardinality than it does. Every power set itself has a power set, so that cardinal numbers form an infinitely tall tower of infinities.
Standing at the foot of this forbidding edifice, Cantor focused on the first couple of floors. He managed to prove that the set formed from all the different ways of ordering natural numbers (from smallest to largest, for example, or with all odd numbers first) has cardinality ℵ1, one level up from the natural numbers. Moreover, each of these “order types” encodes a real number.
His continuum hypothesis asserts that this is exactly the size of the continuum — that there are precisely ℵ1 real numbers. In other words, the cardinality of the continuum immediately follow ℵ0, the cardinality of the natural numbers, with no sizes of infinity in between.
But to Cantor’s immense distress, he couldn’t prove it.
In 1900, the mathematician David Hilbert put the continuum hypothesis first on his famous list of 23 math problems to solve in the 20th century. Hilbert was enthralled by the nascent mathematics of infinity — “Cantor’s paradise,” as he called it — and the continuum hypothesis seemed like its lowest-hanging fruit.
To the contrary, shocking revelations last century turned Cantor’s question into a deep epistemological conundrum.
The trouble arose in 1931, when the Austrian-born logician Kurt Gödel discovered that any set of axioms that you might posit as a foundation for mathematics will inevitably be incomplete. There will always be questions that your list of ground rules can’t settle, true mathematical facts that they can’t prove. As Gödel suspected right away, the continuum hypothesis is such a case: a problem that’s independent of the standard axioms of mathematics.
These axioms, 10 in all, are known as ZFC (for “Zermelo-Fraenkel axioms with the axiom of choice”), and they undergird almost all of modern math. The axioms describe basic properties of collections of objects, or sets. Since virtually everything mathematical can be built out of sets (the empty set {} denotes 0, for instance; {{}} denotes 1; {{},{{}}} denotes 2, and so on), the rules of sets suffice for constructing proofs throughout math.
In 1940, Gödel showed that you can’t use the ZFC axioms to disprove the continuum hypothesis. Then in 1963, the American mathematician Paul Cohen showed the opposite —you can’t use them to prove it, either. Cohen’s proof, together with Gödel’s, means the continuum hypothesis is independent of the ZFC axioms; they can have it either way.
In addition to the continuum hypothesis, most other questions about infinite sets turn out to be independent of ZFC as well. This independence is sometimes interpreted to mean that these questions have no answer, but most set theorists see that as a profound misconception.
They believe the continuum has a precise size; we just need new tools of logic to figure out what that is. These tools will come in the form of new axioms. “The axioms do not settle these problems,” said Magidor, so “we must extend them to a richer axiom system.” It’s ZFC as a means to mathematical truth that’s lacking — not truth itself.
Ever since Cohen, set theorists have sought to shore up the foundations of infinite math by adding at least one new axiom to ZFC. This axiom should illuminate the structure of infinite sets, engender natural and beautiful theorems, avoid fatal contradictions, and, of course, settle Cantor’s question…
Two rival axioms emerged that do just that. For decades, they were suspected of being logically incompatible.
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In October 2018, David Asperó was on holiday in Italy, gazing out a car window as his girlfriend drove them to their bed-and-breakfast, when it came to him: the missing step of what’s now a landmark new proof about the sizes of infinity. “It was this flash experience,” he said.
Asperó, a mathematician at the University of East Anglia in the United Kingdom, contacted the collaborator with whom he’d long pursued the proof, Ralf Schindler of the University of Münster in Germany, and described his insight. “It was completely incomprehensible to me,” Schindler said. But eventually, the duo turned the phantasm into solid logic.
Their proof, which appeared in May in the Annals of Mathematics, unites two rival axioms that have been posited as competing foundations for infinite mathematics. Asperó and Schindler showed that one of these axioms implies the other, raising the likelihood that both axioms — and all they intimate about infinity — are true…
There are an infinite number of infinities. Which one corresponds to the real numbers? “How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer.”
[TotH to MK]
* William Blake
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As we contemplate counting, we might spare a thought for Georg Friedrich Bernhard Riemann; he died on this date in 1866. A mathematician who made contributions to analysis, number theory, and differential geometry, he is remembered (among other things) for his 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, regarded as one of the most influential papers in analytic number theory.
“He told me that in 1886 he had invented an original system of numbering”*…
The rational numbers are the most familiar numbers: 1, -5, ½, and every other value that can be written as a ratio of positive or negative whole numbers. But they can still be hard to work with.
The problem is they contain holes. If you zoom in on a sequence of rational numbers, you might approach a number that itself is not rational. This short-circuits a lot of basic mathematical tools, like most of calculus.
Mathematicians usually solve this problem by arranging the rationals in a line and filling the gaps with irrational numbers to create a complete number system that we call the real numbers.
But there are other ways of organizing the rationals and filling the gaps: the p-adic numbers. They are an infinite collection of alternative number systems, each associated with a unique prime number: the 2-adics, 3-adics, 5-adics and so on.
The p-adics can seem deeply alien. In the 3-adics, for instance, 82 is much closer to 1 than to 81. But the strangeness is largely superficial: At a structural level, the p-adics follow all the rules mathematicians want in a well-behaved number system…
“We’re all on Earth and we work with the reals, but if you went [anywhere] else, you’d work with the p-adics,” [University of Washington mathematician Bianca] Viray explained. “It’s the reals that are the outliers.”
The p-adics form an infinite collection of number systems based on prime numbers. They’re at the heart of modern number theory… which is itself at the heart of computer science, numerical analysis, and cryptography: “An Infinite Universe of Number Systems.”
* Jorge Luis Borges, Labyrinths
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As we dwell on digits, we might send carefully-calculated birthday greetings to Klaus Friedrich Roth; he was born on this date in 1925. After escaping with his family from Nazi Germany, he was educated at Cambridge, then taught mathematics first at University College London, then at Imperial College London. He made a number of important contribution to Number Theory, for which he won the De Morgan Medal and the Sylvester Medal, and election to Fellowship of the Royal Society. In 1958 he was awarded mathematics’ highest honor, the Fields Medal, for proving Roth’s theorem on the Diophantine approximation of algebraic numbers.
“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”*…
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]
“The door handle is the handshake of the building”*…
Door handle and rose (1833–47), manufactured by Copeland & Garrett, Stoke-on-Trent. Victoria and Albert Museum, London
We have all become suddenly more aware of the moments when we cannot avoid touching elements of public buildings. Architecture is the most physical, most imposing and most present of the arts – you cannot avoid it yet, strangely, we touch buildings at only a very few points – the handrail, perhaps a light switch and, almost unavoidably, the door handle. This modest piece of handheld architecture is our critical interface with the structure and the material of the building. Yet it is often reduced to the most generic, cheaply made piece of bent metal which is, in its way, a potent critique of the value we place on architecture and our acceptance of its reduction to a commodified envelope rather than an expression of culture and craft.
Despite their ubiquity and pivotal role in the haptic experience of architecture, door handles remain oddly under-documented. There are no serious histories and only patchy surveys of design, mostly sponsored by manufacturers. Yet in the development of the design of the door handle we have, in microcosm, the history of architecture, a survey of making and a measure of the development of design and how it relates to manufacture, technology and the body.
For as long as there have been doors there have been door handles…
An appreciation of the apparati of accessibility: “Points of contact – a short history of door handles.”
* The Eyes of the Skin: Architecture and the Senses
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As we get a grip, we might send thoughtfully-wagered birthday greetings to a man whose thought open a great many (metaphorical) doors, Blaise Pascal; he was born on this date in 1623. A French mathematician, physicist, theologian, and inventor (e.g.,the first digital calculator, the barometer, the hydraulic press, and the syringe), his commitment to empiricism (“experiments are the true teachers which one must follow in physics”) pitted him against his contemporary René “cogito, ergo sum” Descartes…
“Neoliberalization has meant, in short, the financialization of everything”*…
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.
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