Posts Tagged ‘logic’
“No law of nature, however general, has been established all at once; its recognition has always been preceded by many presentiments.”*…
Laws of nature are impossible to break, and nearly as difficult to define. Just what kind of necessity do they possess?
… The natural laws limit what can happen. They are stronger than the laws of any country because it is impossible to violate them. If it is a law of nature that, for example, no object can be accelerated from rest to beyond the speed of light, then it is not merely that such accelerations never occur. They cannot occur.
There are many things that never actually happen but could have happened in that their occurrence would violate no law of nature. For instance, to borrow an example from the philosopher Hans Reichenbach (1891-1953), perhaps in the entire history of the Universe there never was nor ever will be a gold cube larger than one mile on each side. Such a large gold cube is not impossible. It just turns out never to exist. It’s like a sequence of moves that is permitted by the rules of chess but never takes place in the entire history of chess-playing. By contrast, if it is a law of nature that energy is never created or destroyed, then it is impossible for the total energy in the Universe to change. The laws of nature govern the world like the rules of chess determine what is permitted and what is forbidden during a game of chess, in an analogy drawn by the biologist T H Huxley (1825-95).
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Laws of nature differ from one another in many respects. Some laws concern the general structure of spacetime, while others concern some specific inhabitant of spacetime (such as the law that gold doesn’t rust). Some laws relate causes to their effects (as Coulomb’s law relates electric charges to the electric forces they cause). But other laws (such as the law of energy conservation or the spacetime symmetry principles) do not specify the effects of any particular sort of cause. Some laws involve probabilities (such as the law specifying the half-life of some radioactive isotope). And some laws are currently undiscovered – though I can’t give you an example of one of those! (By ‘laws of nature’, I will mean the genuine laws of nature that science aims to discover, not whatever scientists currently believe to be laws of nature.)
What all of the various laws have in common, despite their diversity, is that it is necessary that everything obey them. It is impossible for them to be broken. An object must obey the laws of nature…
But although all these truisms about the laws of nature sound plausible and familiar, they are also imprecise and metaphorical. The natural laws obviously do not ‘govern’ the Universe in the way that the rules of chess govern a game of chess. Chess players know the rules and so deliberately conform to them, whereas inanimate objects do not know the laws of nature and have no intentions.
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Scientists discover laws of nature by acquiring evidence that some apparent regularity is not only never violated but also could never have been violated. For instance, when every ingenious effort to create a perpetual-motion machine turned out to fail, scientists concluded that such a machine was impossible – that energy conservation is a natural law, a rule of nature’s game rather than an accident. In drawing this conclusion, scientists adopted various counterfactual conditionals, such as that, even if they had tried a different scheme, they would have failed to create a perpetual-motion machine. That it is impossible to create such a machine (because energy conservation is a law of nature) explains why scientists failed every time they tried to create one.
Laws of nature are important scientific discoveries. Their counterfactual resilience enables them to tell us about what would have happened under a wide range of hypothetical circumstances. Their necessity means that they impose limits on what is possible. Laws of nature can explain why something failed to happen by revealing that it cannot happen – that it is impossible.
We began with several vague ideas that seem implicit in scientific reasoning: that the laws of nature are important to discover, that they help us to explain why things happen, and that they are impossible to break. Now we can look back and see that we have made these vague ideas more precise and rigorous. In doing so, we found that these ideas are not only vindicated, but also deeply interconnected. We now understand better what laws of nature are and why they are able to play the roles that science calls upon them to play.
“What is a Law of Nature?,” Marc Lange explains in @aeonmag.
* Dmitri Mendeleev (creator of the Periodic Table)
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As we study law, we might send inquisitive birthday greetings to Federico Cesi; he was born on this date in 1585. A scientist and naturalist, he is best remembered as the founder of the Accademia dei Lincei (Lincean Academy), often cited as the first modern scientific society. Cesi coined (or at least was first to publish/disseminate) the word “telescope” to denote the instrument used by Galileo– who was the sixth member of the Lincean Academy.
“We do not inherit the earth from our ancestors, we borrow it from our children”*…
… and the interest rate on that loan is rising.
There’s much discussion of what’s causing the sudden-feeling spike in prices that we’re experiencing: pandemic disruptions, nativist and protectionist policies, the over-taxing of over-optimized supply chains, and others. But Robinson Meyer argues that there’s another issue, an underlying cause, that’s not getting the attention it deserves… one that will likely be even harder to address…
Over the past year, U.S. consumer prices have risen 7 percent, their fastest rate in nearly four decades, frustrating households and tanking President Joe Biden’s approval rating. And no wonder. High inflation corrodes the basic machinery of the economy, unsettling consumers, troubling companies, and preventing everyone from making sturdy plans for the future…
For years, scientists and economists have warned that climate change could cause massive shortages of major commodities, such as wine, chocolate, and cereals. Financial regulators have cautioned against a “disorderly transition,” in which the world commits only haphazardly to leaving fossil fuels, so it does not invest enough in their zero-carbon replacements. In an economy as prosperous and powerful as America’s, those problems are likely to show up—at least at first—not as empty grocery shelves or bankrupt gas stations but as price increases.
That phenomenon, long hypothesized, may be starting to actually arrive. Over the past year, unprecedented weather disasters have caused the price of key commodities to spike, and a volatile oil-and-gas market has allowed Russia and Saudi Arabia to exert geopolitical force.
“This climate-change risk to the supply chain—it’s actually real. It is happening now,” Mohamed Kande, the U.S. and global advisory leader at the accounting firm PwC, told me.
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How to respond to these problems? The U.S. government has one tool to slow down the great chase of inflation: Leash up its dollars. By raising the rate at which the federal government lends money to banks, the Federal Reserve makes it more expensive for businesses or consumers to take out loans themselves. This brings demand in the economy more in line with supply. It is like the king in our thought experiment deciding to buy back some of his gold coins.
But wait—is it always appropriate to focus on dollars? What if the problem was caused by too few goods? Worse, what if the economy lost the ability to produce goods over time, throwing off the dollars-to-goods ratio? Then what was once an adequate number of dollars will, through no fault of its own, become too many...
… if the climate scars on supply continue to grow, does the Federal Reserve have the right tools to manage? Stinson Dean, the lumber trader, is doubtful. “Raising interest rates will blunt demand for housing—no doubt. But if you blunt demand enough to bring lumber prices down, you’re destroying the economy,” Dean told me. “For us to have lower lumber prices, we can only build a million homes a year. Do you really want to do that?
“Raising rates,” he said, “doesn’t grow more trees.” Nor does it grow more coffee, end a drought, or bring certainty to the energy transition. And if our new era of climate-driven inflation takes hold, America will need more than higher interest rates to bring balance to supply and demand.
A provocative look at the tangled roots of our inflation, suggesting that “The World Isn’t Ready for Climate-Change-Driven Inflation,” from @yayitsrob in @TheAtlantic. Eminently worth reading in full. Via @sentiers.
* Native American proverb
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As we dig deeper, we might send carefully calculated birthday greetings to Frank Plumpton Ramsey; he was born on this date in 1903. A philosopher, mathematician, and economist, he made major contributions to all three fields before his death (at the age of 26) on this date in 1930.
While he is probably best remembered as a mathematician and logician and as Wittgenstein’s friend and translator, he wrote three paper in economics: on subjective probability and utility (a response to Keynes, 1926), on optimal taxation (1927, described by Joseph E. Stiglitz as “a landmark in the economics of public finance”), and optimal economic growth (1928; hailed by Keynes as “”one of the most remarkable contributions to mathematical economics ever made”). The economist Paul Samuelson described them in 1970 as “three great legacies – legacies that were for the most part mere by-products of his major interest in the foundations of mathematics and knowledge.”
For more on Ramsey and his thought, see “One of the Great Intellects of His Time,” “The Man Who Thought Too Fast,” and Ramsey’s entry in the Stanford Encyclopedia of Philosophy.
“My work consists of two parts; that presented here plus all I have not written. It is this second part that is important.”*…
On the occasion of it centenary, Peter Salmon considers the history, context, and lasting significance of Wittgenstein‘s revolutionary first work…
One hundred years ago, a slim volume of philosophy was published by the then unknown Austrian philosopher Ludwig Wittgenstein. The book was as curious as its title, Tractatus Logico-Philosophicus. Running to only 75 pages, it was in the form of a series of propositions, the gnomic quality of the first as baffling to the newcomer today as it was then.
1. The world is all that is the case.
1.1 The world is a totality of facts not of things.
1.11 The world is determined by the facts, and by their being all the facts.
1.12 For the totality of facts determines what is the case, and also whatever is not the case.
1.13 The facts in logical space are the world.And so on, through six propositions, 526 numbered statements, equally emphatic and enigmatic, until the seventh and final proposition, which stands alone at the end of the text: “Whereof we cannot speak, thereof we must remain silent.”
The book’s influence was to be dramatic and far-reaching. Wittgenstein believed he had found a “solution” to how language and the world relate, that they shared a logical form. This also set a limit as to what questions could be meaningfully asked. Any question which could not be verified was, in philosophical terms, nonsense.
Written in the First World War trenches, Tractatus is, in many ways, a work of mysticism…
Ludwig Wittgenstein’s Tractatus is as brilliant and baffling today as it was on its publication a century ago: “The logical mystic,” from @petesalmon in @NewHumanist.
* Ludwig Wittgenstein
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As we wrestle with reason and reality, we might recall that it was on this date in 1930 that Dashiell Hammett‘s The Maltese Falcon— likely a favorite of Wittgenstein’s— was published. In 1990 the novel ranked 10th in Top 100 Crime Novels of All Time list by the Crime Writer’s Association. Five years later, in a similar list by Mystery Writers of America, the novel was ranked third.
“Your job as a scientist is to figure out how you’re fooling yourself”*…
And like scientists, so all of us…
Science has shown that we tend to make all sorts of mental mistakes, called “cognitive biases”, that can affect both our thinking and actions. These biases can lead to us extrapolating information from the wrong sources, seeking to confirm existing beliefs, or failing to remember events the way they actually happened!
To be sure, this is all part of being human—but such cognitive biases can also have a profound effect on our endeavors, investments, and life in general.
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Humans have a tendency to think in particular ways that can lead to systematic deviations from making rational judgments.
These tendencies usually arise from:
• Information processing shortcuts
• The limited processing ability of the brain
• Emotional and moral motivations
• Distortions in storing and retrieving memories
• Social influence
Cognitive biases have been studied for decades by academics in the fields of cognitive science, social psychology, and behavioral economics, but they are especially relevant in today’s information-packed world. They influence the way we think and act, and such irrational mental shortcuts can lead to all kinds of problems in entrepreneurship, investing, or management.
Here are five examples of how these types of biases can affect people in the business world:
1. Familiarity Bias: An investor puts her money in “what she knows”, rather than seeking the obvious benefits from portfolio diversification. Just because a certain type of industry or security is familiar doesn’t make it the logical selection.
2. Self-Attribution Bias: An entrepreneur overly attributes his company’s success to himself, rather than other factors (team, luck, industry trends). When things go bad, he blames these external factors for derailing his progress.
3. Anchoring Bias: An employee in a salary negotiation is too dependent on the first number mentioned in the negotiations, rather than rationally examining a range of options.
4. Survivorship Bias: Entrepreneurship looks easy, because there are so many successful entrepreneurs out there. However, this is a cognitive bias: the successful entrepreneurs are the ones still around, while the millions who failed went and did other things.
5. Gambler’s Fallacy: A venture capitalist sees a portfolio company rise and rise in value after its IPO, far behind what he initially thought possible. Instead of holding on to a winner and rationally evaluating the possibility that appreciation could still continue, he dumps the stock to lock in the existing gains.
An aid to thinking about thinking: “Every Single Cognitive Bias in One Infographic.” From DesignHacks.co via Visual Capitalist.
And for a fascinating look of cognitive bias’ equally dangerous cousin, innumeracy, see here.
* Saul Perlmutter, astrophysicist, Nobel laureate
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As we cogitate, we might recall that it was on this date in 1859 that “The Carrington Event” began. Lasting two days, it was the largest solar storm on record: a large solar flare (a coronal mass ejection, or CME) that affected many of the (relatively few) electronics and telegraph lines on Earth.
A solar storm of this magnitude occurring today would cause widespread electrical disruptions, blackouts, and damage due to extended outages of the electrical grid. The solar storm of 2012 was of similar magnitude, but it passed Earth’s orbit without striking the planet, missing by nine days. See here for more detail on what such a storm might entail.
“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.
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