Posts Tagged ‘Hilbert’
“Your life has a limit, but knowledge has none”*…
Kurt Gödel’s Incompleteness Theorems put a cap on what we can “know” from within the logical and perceptual systems in which we live. While Gödel’s focus was trained on mathematics, his insight has seen much broader application (e.g., here).
Alexander V. Gheorghiu argues that it’s not so simple– and that what we claim as “true” is not a ceiling we are forever failing to reach but something that changes as we climb toward it…
On the evening of 8 September 1930, the German mathematician David Hilbert stepped up to a microphone in Königsberg and addressed his countrymen by radio. He was 68, recently retired, and at the height of his fame. The talk was a defense of pure reason, delivered in the city of Immanuel Kant, and it ended with two phrases that would later be carved on his tombstone: Wir müssen wissen. Wir werden wissen. We must know. We shall know.
Are certain questions closed to the human mind? Are there truths about the world that we can never know? Hilbert thought that, at least within mathematics, this could not be the case. For him, mathematics was a game played with symbols: one could fix the right axioms as rules of the game and every mathematical problem could be solved. He was wrong.
The day before his Königsberg address, at a small academic meeting in the same city, a 24-year-old Austrian named Kurt Gödel had quietly announced that no such system for arithmetic could exist. Whatever reasonable set of rules you fix, something true must lie beyond them and thus beyond our knowledge. We can’t know. We won’t know. Gödel’s results, known as the Incompleteness Theorems, are landmark achievements of 20th-century logic. Recent work in philosophy and mathematics has begun to illuminate the gap they opened, and with it something deeper about the foundations of knowledge itself. The story begins in the same city, but two centuries earlier…
[Gheorghiu unpacks the arguments of Kant, Hilbert, and Gödel, then traces the work done since. He concludes…]
… In 2005, Tor Sandqvist, a doctoral student at Uppsala University, submitted a thesis that took up [Michael Dummett‘s]’s challenge: to justify the ordinary practices of mathematicians without any appeal to truth. Where Dummett had given the conditions a solution must satisfy, Sandqvist gave a solution. He produced a precise mathematical justification of deduction as mathematicians actually use it, grounded entirely in practices of assertion. The boxer’s fists were untied.
It was rediscovered a decade later in the Department of Computer Science at University College London, where a group of logicians recognised what Sandqvist had actually built, and developed it into a full framework on a par with the orthodox truth-conditional view. The framework now stands equal to the truth-based account, analogous to the situation after Gödel’s PhD thesis. What about mathematics? The UCL team have recently shown that the Dummett-Sandqvist framework gives a way to understand what Gödel’s Incompleteness Theorems are about, and it is not the limits of knowledge.
If meaning comes from practices of assertion, then our concepts are settled only as far as our practices have settled them. A system of definitions is a kind of constitution that determines something. The US Constitution, for example, is meant to enshrine a form of democracy. But what a constitution defines answers to how it is amended.
Principia Mathematica is a constitution for arithmetic. To read it as a constitution is to assume it is consistent; otherwise, it would be nonsense. So, when we amend it by asserting that ‘Principia Mathematica is consistent,’ we are not changing the subject: the new axiom says aloud only what the practice already did. Gödel’s theorems apply afresh to the enlarged constitution: the amended system has its own unprovable truth, its own consistency it cannot certify. No amendment is the last; each faithful extension opens room for the next. To grasp a concept like arithmetic is not to survey a finished structure; it is to hold a principle for going on: a rule for laying the next stretch of line, not a map of where the line ends. Dummett called this openness indefinite extensibility.
Faithful is the word doing the work, because not every amendment is. Gödel claimed the US Constitution could deliver a dictatorship. That does not make dictatorship a form of democracy. Whatever loophole he had in mind, it would have contradicted the practices the Constitution already enshrined. An amendment can betray what a constitution defines. But it can also unfold it, drawing out what the founding commitments demanded all along. Only the second kind is what Dummett has in mind.
The constitution of our language is our current practice for public discourse. As it is extensible in this way, what we claim as ‘true’ is not a fixed structure with a ceiling we are forever failing to reach, but something that can change. Gödel’s verdict never applied to mathematics, but only to some edition of the constitution. Within the Sandqvist system, we can attend to the extension systematically. Hilbert’s words outlive his programme: Wir werden wissen. We shall know. Not because the answers are waiting at the end of a ladder, but because truth goes exactly as far as we have built the rungs…
Eminently worth reading in full: “The Ladder of Knowledge,” from @aeon.co.
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As we stand on the shoulders of giants, we might note that today is National Nut Day.
“Sometimes the only way to move forward is to revisit the things in your past that were holding you back”*…
Adobe Flash was the language of choice for a generation of game developers, helping kickstart an indie revolution on the still-young web of the 1990s and 2000s. But it withered on the proprietary and insecure vine, and both web browsers and Adobe have now canned it, threatening countless games and interactive presentations with the memory hole. The Internet Archive comes to the rescue, not only archiving the flash files but emulating the player itself, allowing history to live on.
“The Internet Archive has begun emulating Flash Animations, Games and Toys in a new collection,” wrote archivist Jason Scott on Twitter. “It’s at https://archive.org/details/softwarelibrary_flash and it’s going to be past 1,000 items in 24 hours. You can add your own and get them running, and the animations have never ran smoother or better.”…
From our friends at Boing Boing: “Internet Archive turns on Flash emulation, already has 1000 items to check out.”
* “Barry Allen” (The Flash)
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As we celebrate that what’s old is new again, we might recall that it was on this date in 1915 that Albert Einstein presented the Einstein Field Equation to the Prussian Academy of Sciences. Einstein soon after elaborated it into the set of 10 equations that account for gravitation in the curved spacetime that he described in his General Theory of Relativity; they are used to determine spacetime geometry.
(German mathematician David Hilbert reached the same conclusion, and actually published the equation before Einstein– though Hilbert, who was a correspondent of Einstein’s, never suggested that Einstein’s credit was inappropriate.)
“I am incapable of conceiving infinity, and yet I do not accept finity”*…
Suppose you’re working at a hotel with infinitely many rooms in it, numbered 1, 2, 3, 4, 5, … all the way up forever and ever. (This is known as a Hilbert Hotel.) One evening when every single room is occupied, a traveler arrives and requests to be accommodated too. You’re the manager. What do you do to help the traveler?
Simple. You just ask each occupant to one room forward. 1 goes to 2, and 2 goes to 3, and so on. Every previous occupant gets a new room. And the first room is now open for the traveler.
The procedure above is characterized by an infinite number of actions or tasks to be carried out in a finite amount of time. Procedures with this character are known as supertasks…
More on the ins and outs of infinities at “Introducing Supertasks.” (More fun musings on infinity here and here; and more on Hilbert’s Hotel here.)
* Simone de Beauvoir, La Vieillesse
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As we muse on many, we might spare a thought for Hermann Hankel; he died on this date in 1873. A mathematician who worked with Möbius, Riemann, Weierstrass, and Kronecker (among others), he made important contributions to the understanding of complex numbers and quaternions… and to work begun by Bernard Bolzano on infinite series.
“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.
“I may be going nowhere, but what a ride”*…
Nine salvaged bikes were reassembled into a carousel formation. The bike is modular and can be dismantled, transported and reassembled. It is normally left in public places where it can attract a variety of riders and spectators.
From artist Robert Wechsler, the Circular Bike.
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As we return to where we started, we might send carefully-calculated birthday greetings to Stephen Smale; he was born on this date in 1930. A winner of both the Fields Medal and the Wolf Prize, the highest honors in mathematics, he first gained recognition with a proof of the Poincaré conjecture for all dimensions greater than or equal to 5, published in 1961. He then moved to dynamic systems, developing an understanding of strange attractors which lead to chaos, and contributing to mathematical economics. His most recent work is in theoretical computer science.
In 1998, in the spirit of Hilbert’s famous list of problems produced in 1900, he created a list of 18 unanswered challenges– known as Smale’s problems– to be solved in the 21st century. (In fact, Smale’s list contains some of the original Hilbert problems, including the Riemann hypothesis and the second half of Hilbert’s sixteenth problem, both of which are still unsolved.)
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