Podcasts about peano

Play Episode Listen Later May 28, 2024 23:23

Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Understanding Gödel's completeness theorem, published by jessicata on May 28, 2024 on LessWrong. In this post I prove a variant of Gödel's completeness theorem. My intention has been to really understand the theorem, so that I am not simply shuffling symbols around, but am actually understanding why it is true. I hope it is helpful for at least some other people. For sources, I have myself relied mainly on Srivastava's presentation. I have relied a lot on intuitions about sequent calculus; while I present a sequent calculus in this post, this is not a complete introduction to sequent calculus. I recommend Logitext as an online proof tool for gaining more intuition about sequent proofs. I am familiar with sequent calculus mainly through type theory. First-order theories and models A first-order theory consists of: A countable set of functions, which each have an arity, a non-negative integer. A countable set of predicates, which also have non-negative integer arities. A countable set of axioms, which are sentences in the theory. Assume a countably infinite set of variables. A term consists of either a variable, or a function applied to a number of terms equal to its arity. An atomic sentence is a predicate applied to a number of terms equal to its arity. A sentence may be one of: an atomic sentence. a negated sentence, P. a conjunction of sentences, PQ. a universal, x,P, where x is a variable. Define disjunctions (PQ:=(PQ)), implications (PQ:=(PQ)), and existentials (x,P:=x,P) from these other terms in the usual manner. A first-order theory has a countable set of axioms, each of which are sentences. So far this is fairly standard; see Peano arithmetic for an example of a first-order theory. I am omitting equality from first-order theories, as in general equality can be replaced with an equality predicate and axioms. A term or sentence is said to be closed if it has no free variables (that is, variables which are not quantified over). A closed term or sentence can be interpreted without reference to variable assignments, similar to a variable-free expression in a programming language. Let a constant be a function of arity zero. I will make the non-standard assumption that first-order theories have a countably infinite set of constants which do not appear in any axiom. This will help in defining inference rules and proving completeness. Generally it is not a problem to add a countably infinite set of constants to a first-order theory; it does not strengthen the theory (except in that it aids in proving universals, as defined below). Before defining inference rules, I will define models. A model of a theory consists of a set (the domain of discourse), interpretations of the functions (as mapping finite lists of values in the domain to other values), and interpretations of predicates (as mapping finite lists of values in the domain to Booleans), which satisfies the axioms. Closed terms have straightforward interpretations in a model, as evaluating the expression (as if in a programming language). Closed sentences have straightforward truth values, e.g. the formula P is true in a model when P is false in the model. Judgments and sequent rules A judgment is of the form ΓΔ, where Γ and Δ are (possibly infinite) countable sets of closed sentences. The judgment is true in a model if at least one of Γ is false or at least one of Δ is true. As notation, if Γ is a set of sentences and P is a sentence, then Γ,P denotes Γ{P}. The inference rules are expressed as sequents. A sequent has one judgment on the bottom, and a finite set of judgments on top. Intuitively, it states that if all the judgments on top are provable, the rule yields a proof of the judgment on the bottom. Along the way, I will show that each rule is sound: if every judgment on the top is true in all models, then t...

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LW - Understanding Gödel's completeness theorem by jessicata

Play Episode Listen Later May 28, 2024 23:23

Link to original articleWelcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Understanding Gödel's completeness theorem, published by jessicata on May 28, 2024 on LessWrong. In this post I prove a variant of Gödel's completeness theorem. My intention has been to really understand the theorem, so that I am not simply shuffling symbols around, but am actually understanding why it is true. I hope it is helpful for at least some other people. For sources, I have myself relied mainly on Srivastava's presentation. I have relied a lot on intuitions about sequent calculus; while I present a sequent calculus in this post, this is not a complete introduction to sequent calculus. I recommend Logitext as an online proof tool for gaining more intuition about sequent proofs. I am familiar with sequent calculus mainly through type theory. First-order theories and models A first-order theory consists of: A countable set of functions, which each have an arity, a non-negative integer. A countable set of predicates, which also have non-negative integer arities. A countable set of axioms, which are sentences in the theory. Assume a countably infinite set of variables. A term consists of either a variable, or a function applied to a number of terms equal to its arity. An atomic sentence is a predicate applied to a number of terms equal to its arity. A sentence may be one of: an atomic sentence. a negated sentence, P. a conjunction of sentences, PQ. a universal, x,P, where x is a variable. Define disjunctions (PQ:=(PQ)), implications (PQ:=(PQ)), and existentials (x,P:=x,P) from these other terms in the usual manner. A first-order theory has a countable set of axioms, each of which are sentences. So far this is fairly standard; see Peano arithmetic for an example of a first-order theory. I am omitting equality from first-order theories, as in general equality can be replaced with an equality predicate and axioms. A term or sentence is said to be closed if it has no free variables (that is, variables which are not quantified over). A closed term or sentence can be interpreted without reference to variable assignments, similar to a variable-free expression in a programming language. Let a constant be a function of arity zero. I will make the non-standard assumption that first-order theories have a countably infinite set of constants which do not appear in any axiom. This will help in defining inference rules and proving completeness. Generally it is not a problem to add a countably infinite set of constants to a first-order theory; it does not strengthen the theory (except in that it aids in proving universals, as defined below). Before defining inference rules, I will define models. A model of a theory consists of a set (the domain of discourse), interpretations of the functions (as mapping finite lists of values in the domain to other values), and interpretations of predicates (as mapping finite lists of values in the domain to Booleans), which satisfies the axioms. Closed terms have straightforward interpretations in a model, as evaluating the expression (as if in a programming language). Closed sentences have straightforward truth values, e.g. the formula P is true in a model when P is false in the model. Judgments and sequent rules A judgment is of the form ΓΔ, where Γ and Δ are (possibly infinite) countable sets of closed sentences. The judgment is true in a model if at least one of Γ is false or at least one of Δ is true. As notation, if Γ is a set of sentences and P is a sentence, then Γ,P denotes Γ{P}. The inference rules are expressed as sequents. A sequent has one judgment on the bottom, and a finite set of judgments on top. Intuitively, it states that if all the judgments on top are provable, the rule yields a proof of the judgment on the bottom. Along the way, I will show that each rule is sound: if every judgment on the top is true in all models, then t...

speech define closed ea generally assume judgments intuitively pq theorem srivastava completeness rationalist lesswrong peano

Fondazione C.R.Tortona: "La Notte dei ricercatori-Intervista tra le stelle" al Liceo Peano.

Play Episode Listen Later Oct 4, 2023 5:30

Fondazione C.R.Tortona: Questa mattina in Buongiorno Pnr Valeria Ferrari intervista la professoressa Mutti del Liceo Peano per quanto riguarda "La Notte dei ricercatori-Intervista tra le stelle", che si terrà venerdì 6 Ottobre dalle 20.30 alle 23.30.

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AF - A Proof of Löb's Theorem using Computability Theory by Jessica Taylor

The Nonlinear Library: Alignment Forum Weekly

Play Episode Listen Later Aug 16, 2023 5:28

Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: A Proof of Löb's Theorem using Computability Theory, published by Jessica Taylor on August 16, 2023 on The AI Alignment Forum. Löb's Theorem states that, if PA⊢□PA(P)P, then PA⊢P. To explain the symbols here: PA is Peano arithmetic, a first-order logic system that can state things about the natural numbers. PA⊢A means there is a proof of the statement A in Peano arithmetic. □PA(P) is a Peano arithmetic statement saying that P is provable in Peano arithmetic. I'm not going to discuss the significance of Löb's theorem, since it has been discussed elsewhere; rather, I will prove it in a way that I find simpler and more intuitive than other available proofs. Translating Löb's theorem to be more like Godel's second incompleteness theorem First, let's compare Löb's theorem to Godel's second incompleteness theorem. This theorem states that, if PA⊢¬□PA(⊥), then PA⊢⊥, where ⊥ is a PA statement that is trivially false (such as A∧¬A), and from which anything can be proven. A system is called inconsistent if it proves ⊥; this theorem can be re-stated as saying that if PA proves its own consistency, it is inconsistent. We can re-write Löb's theorem to look like Godel's second incompleteness theorem as: if PA+¬P⊢¬□PA+¬P(⊥), then PA+¬P⊢⊥. Here, PA+¬P is PA with an additional axiom that ¬P, and □PA+¬P expresses provability in this system. First I'll argue that this re-statement is equivalent to the original Löb's theorem statement. Observe that PA⊢P if and only if PA+¬P⊢⊥; to go from the first to the second, we derive a contradiction from P and ¬P, and to go from the second to the first, we use the law of excluded middle in PA to derive P∨¬P, and observe that, since a contradiction follows from ¬P in PA, PA can prove P. Since all this reasoning can be done in PA, we have that □PA(P) and □PA+¬P(⊥) are equivalent PA statements. We immediately have that the conclusion of the modified statement equals the conclusion of the original statement. Now we can rewrite the pre-condition of Löb's theorem from PA⊢□PA(P)P. to PA⊢□PA+¬P(⊥)P. This is then equivalent to PA+¬P⊢¬□PA+¬P(⊥). In the forward direction, we simply derive ⊥ from P and ¬P. In the backward direction, we use the law of excluded middle in PA to derive P∨¬P, observe the statement is trivial in the P branch, and in the ¬P branch, we derive ¬□PA+¬P(⊥), which is stronger than □PA+¬P(⊥)P. So we have validly re-stated Löb's theorem, and the new statement is basically a statement that Godel's second incompleteness theorem holds for PA+¬P. Proving Godel's second incompleteness theorem using computability theory The following proof of a general version of Godel's second incompleteness theorem is essentially the same as Sebastian Oberhoff's in "Incompleteness Ex Machina". Let L be some first-order system that is at least as strong as PA (for example, PA+¬P). Since L is at least as strong as PA, it can express statements about Turing machines. Let Halts(M) be the PA statement that Turing machine M (represented by a number) halts. If this statement is true, then PA (and therefore L) can prove it; PA can expand out M's execution trace until its halting step. However, we have no guarantee that if the statement is false, then L can prove it false. In fact, L can't simultaneously prove this for all non-halting machines M while being consistent, or we could solve the halting problem by searching for proofs of Halts(M) and ¬Halts(M) in parallel. That isn't enough for us, though; we're trying to show that L can't simultaneously be consistent and prove its own consistency, not that it isn't simultaneously complete and sound on halting statements. Let's consider a machine Z(A) that searches over all L-proofs of ¬Halts(''⌈A⌉(⌈A⌉)") (where ''⌈A⌉(⌈A⌉)" is an encoding of a Turing machine that runs A on its own source code), and halts only when finding su...

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AF - A Proof of Löb's Theorem using Computability Theory by Jessica Taylor

Play Episode Listen Later Aug 16, 2023 5:28

Link to original articleWelcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: A Proof of Löb's Theorem using Computability Theory, published by Jessica Taylor on August 16, 2023 on The AI Alignment Forum. Löb's Theorem states that, if PA⊢□PA(P)P, then PA⊢P. To explain the symbols here: PA is Peano arithmetic, a first-order logic system that can state things about the natural numbers. PA⊢A means there is a proof of the statement A in Peano arithmetic. □PA(P) is a Peano arithmetic statement saying that P is provable in Peano arithmetic. I'm not going to discuss the significance of Löb's theorem, since it has been discussed elsewhere; rather, I will prove it in a way that I find simpler and more intuitive than other available proofs. Translating Löb's theorem to be more like Godel's second incompleteness theorem First, let's compare Löb's theorem to Godel's second incompleteness theorem. This theorem states that, if PA⊢¬□PA(⊥), then PA⊢⊥, where ⊥ is a PA statement that is trivially false (such as A∧¬A), and from which anything can be proven. A system is called inconsistent if it proves ⊥; this theorem can be re-stated as saying that if PA proves its own consistency, it is inconsistent. We can re-write Löb's theorem to look like Godel's second incompleteness theorem as: if PA+¬P⊢¬□PA+¬P(⊥), then PA+¬P⊢⊥. Here, PA+¬P is PA with an additional axiom that ¬P, and □PA+¬P expresses provability in this system. First I'll argue that this re-statement is equivalent to the original Löb's theorem statement. Observe that PA⊢P if and only if PA+¬P⊢⊥; to go from the first to the second, we derive a contradiction from P and ¬P, and to go from the second to the first, we use the law of excluded middle in PA to derive P∨¬P, and observe that, since a contradiction follows from ¬P in PA, PA can prove P. Since all this reasoning can be done in PA, we have that □PA(P) and □PA+¬P(⊥) are equivalent PA statements. We immediately have that the conclusion of the modified statement equals the conclusion of the original statement. Now we can rewrite the pre-condition of Löb's theorem from PA⊢□PA(P)P. to PA⊢□PA+¬P(⊥)P. This is then equivalent to PA+¬P⊢¬□PA+¬P(⊥). In the forward direction, we simply derive ⊥ from P and ¬P. In the backward direction, we use the law of excluded middle in PA to derive P∨¬P, observe the statement is trivial in the P branch, and in the ¬P branch, we derive ¬□PA+¬P(⊥), which is stronger than □PA+¬P(⊥)P. So we have validly re-stated Löb's theorem, and the new statement is basically a statement that Godel's second incompleteness theorem holds for PA+¬P. Proving Godel's second incompleteness theorem using computability theory The following proof of a general version of Godel's second incompleteness theorem is essentially the same as Sebastian Oberhoff's in "Incompleteness Ex Machina". Let L be some first-order system that is at least as strong as PA (for example, PA+¬P). Since L is at least as strong as PA, it can express statements about Turing machines. Let Halts(M) be the PA statement that Turing machine M (represented by a number) halts. If this statement is true, then PA (and therefore L) can prove it; PA can expand out M's execution trace until its halting step. However, we have no guarantee that if the statement is false, then L can prove it false. In fact, L can't simultaneously prove this for all non-halting machines M while being consistent, or we could solve the halting problem by searching for proofs of Halts(M) and ¬Halts(M) in parallel. That isn't enough for us, though; we're trying to show that L can't simultaneously be consistent and prove its own consistency, not that it isn't simultaneously complete and sound on halting statements. Let's consider a machine Z(A) that searches over all L-proofs of ¬Halts(''⌈A⌉(⌈A⌉)") (where ''⌈A⌉(⌈A⌉)" is an encoding of a Turing machine that runs A on its own source code), and halts only when finding su...

theory speech proof ea observe pp turing halts theorem jessica taylor rationalist godel peano

Human iPSC-derived neural stem cells display a radial glia-like signature in vitro and favorable long-term safety in transplanted mice

PaperPlayer biorxiv neuroscience

Play Episode Listen Later Aug 4, 2023

Link to bioRxiv paper: http://biorxiv.org/cgi/content/short/2023.08.04.551937v1?rss=1 Authors: Luciani, M., Garsia, C., Beretta, S., Petiti, L., Peano, C., Merelli, I., Cifola, I., Miccio, A., Meneghini, V., Gritti, A. Abstract: Human induced pluripotent stem cell-derived neural stem/progenitor cells (hiPSC-NSCs) are a promising source for cell therapy approaches to treat neurodegenerative and demyelinating disorders. Despite ongoing efforts to characterize hiPSC-derived cells in vitro and in vivo, we lack comprehensive genome- and transcriptome-wide studies addressing hiPSC-NSC identity and safety, which are critical for establishing accepted criteria for prospective clinical applications. Here, we evaluated the transcriptional and epigenetic signatures of hiPSCs and differentiated hiPSC-NSC progeny, finding that the hiPSC-to-NSC transition results in a complete loss of pluripotency and the acquisition of a radial glia-associated transcriptional signature. Importantly, hiPSC-NSCs share with somatic human fetal NSCs (hfNSCs) the main transcriptional and epigenetic patterns associated with NSC-specific biology. In vivo, long-term observation (up to 10 months) of mice intracerebrally transplanted as neonates with hiPSC-NSCs showed robust engraftment and widespread distribution of human cells in the host brain parenchyma. Engrafted hiPSC-NSCs displayed multilineage potential and preferentially generated glial cells. No hyperproliferation, tumor formation, or expression of pluripotency markers was observed. Finally, we identified a novel role of the Sterol Regulatory Element Binding Transcription Factor 1 (SREBF1) in the regulation of astroglial commitment of hiPSC-NSCs. Overall, these comprehensive in vitro and in vivo analyses provide transcriptional and epigenetic reference datasets to define the maturation stage of NSCs derived from different hiPSC sources, and to clarify the safety profile of hiPSC-NSCs, supporting their continuing development as an alternative to somatic hfNSCs in treating neurodegenerative and demyelinating disorders. Copy rights belong to original authors. Visit the link for more info Podcast created by Paper Player, LLC

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Crayons, Touchdowns, and a Gallery of Monsters

Interplace

Play Episode Listen Later Jul 14, 2023 9:53

Hello Interactors,My last post on fractals led me to refamiliarized myself with the man who coined the term, Benoit Mandelbrot, and his influential work on the fractal-like wonders of nature. I didn't realize he was following in the footsteps of 19th century mathematicians critical of the absolutist purity of Euclidean geometry – themes I recently explored here and here. My journey led me to a memory of a plane landing on a plane and the complexities that surface on the surface.Please don't be shy. Leave a comment or a like. Or just hit reply with a smiley face and a hello!Now let's go…I have a childhood memory, fueled by a crayon drawing, of watching a plane land at the Des Moines airport. My dad was returning home after a business trip. Over time, this memory transformed into a riddle most likely inspired by high school calculus. The riddle posed a question: as the distance between the plane and the runway progressively decreases, when does it equal zero? My pondering was rooted in the observation that, at a microscopic level, the rubber of the tire and the rough surface of the concrete never truly merge into zero. The presence of black streaks on the tarmac from rubber left behind served as evidence. According to classical physics, at an atomic level, the distance between a landing plane and the runway approaches zero but never truly reaches it.This is because the outermost electron clouds of the atoms in both the tires and the runway surface repel each other due to electromagnetic forces, creating a minute gap between them, measured in angstroms (10 to the power of -10 meters). However, from a practical standpoint, classical mechanics tells us that at a macroscopic level, the plane does make contact with the runway and eventually comes to a stop. Classical mechanics focuses on the behavior of objects on a larger scale, which outweighs the effects observed at the microscopic level. The mechanics of "touchdown" do not rely on atomic physics to achieve zero distance for the safe arrival of our loved ones.In my childhood crayon drawings, I depicted the runway as a straight line and the plane's wheels as a circle. Yet, this representation itself is a macroscopic interpretation of reality. If we were to examine my marks with a magnifying glass, we would see fragmented wax resting on the textured paper's peaks and valleys rather than perfectly straight lines or round circles. Similarly, we would find fragments of rubber deposited on the peaks and valleys of the concrete runway.In the realm of high school calculus, the line representing the runway and the circle representing the wheel would be precisely drawn on rigid gridded paper using a plastic flowchart template, akin to the tools my dad used to pseudocode his COBOL programs he no doubt was debugging with his colleagues in Toronto.Mathematically, I would have described the landing as the height of the plane decreasing as a function of time, incorporating concepts like velocity and acceleration. This interplay between decreasing height and time signifies the plane's motion until it decelerates and reaches a minimum altitude, indicating touchdown. I would have positioned the circle of my plastic template precisely on the flat line, accompanied by an equation describing the moment of touchdown.However, in 1982, two years before I was in calculus and the year I was learning geometry, mathematician Benoit B. Mandelbrot published "The Fractal Geometry of Nature," a highly influential book. Mandelbrot's work highlighted the importance of mathematics that deviated from the traditional Euclidean curves and shapes. Introduced by ‘modern' mathematicians like Georg Cantor and Giuseppe Peano a century earlier, the days of regarding mathematics as absolutely pure and unquestioning were being questioned.Mandelbrot offers why we were set on this smooth, well-worn trajectory of Euclidian mathematical purity,“The fact that mathematics, viewed by its own creators as ‘absolutely pure,' should respond so well to the needs of science is striking and surprising but follows a well-worn pattern. That pattern was first set when Johannes Kepler concluded that, to model the path of Mars around the Sun, one must resort to an intellectual plaything of the Greeks–the ellipse. Soon after, Galileo concluded that, to model the fall of bodies toward the Earth, one needs a different curve–a parabola. And he proclaimed that ‘the greatest book [of nature]...is written in mathematical language and the characters are triangles, circles and other geometric figures…without which one wanders in vain through a dark labyrinth.' In the pithy words of Scottish biologist D'Arcy Thompson: ‘God always geometrizes.'”Of the work of Cantor's set theory and Peano's space-filling curves, the theoretical physicist and mathematician Freeman J Dyson wrote,“These new structures were regarded by contemporary mathematicians as ‘pathological.' They were described as a ‘gallery of monsters,' kin to the cubist painting and atonal music that were upsetting established standards of taste in the arts at about the same time. The mathematicians who created the monsters regarded them as important in showing that the world of pure mathematics contains a richness of possibilities going far beyond the simple structures that they saw in nature.”Mandelbrot's research delved into the exploration of fractals, which he described as broken shapes, distinct from the smooth Euclidean curves. These fractals opened new possibilities, allowing for the modeling of complex phenomena found in nature. Mandelbrot's fractal geometry was brought to life through computer-generated images of landscapes and clouds, reflecting the generative algorithms found in nature. These images showcased the jagged, impure, and fractured lines that emerged, challenging the simplicity of Euclidean shapes.Mandelbrot emphasized that drawing a line between just two points on a square Euclidean plane oversimplifies reality. Instead, he considered the fracturing that occurs when lines connect every point in a square or a cube. In fact, the term "fractal" itself derives from the Latin adjective "fractus," meaning "broken." Mandelbrot highlighted the relevance of fractals lying between the shapes of Euclid, akin to fractions lying between integers.Mandelbrot offers that “When mathematicians concluded about a century ago that the seemingly simple and innocuous notion of ‘curve' hides profound difficulties, they thought they were engaging in unreasonable and unrealistic hairsplitting. They had not determined to look out at the real world to analyze it, but to look in at an ideal in the mind. The theory of fractals shows that they had misled themselves.”Mandelbrot's work demonstrated that the seemingly simple crayon drawing of my dad's plane landing concealed profound difficulties. My self-imposed brain teaser was was not an exercise in unreasonable hair-splitting, but rather an analysis of the real world. Fractals, I now know, provide a mathematical framework to quantify irregularities found in natural structures and allow for the analysis and modeling of complex systems exhibiting patterns at different scales.Mandelbrot's groundbreaking ideas expanded on Cantor and Peano to illuminate the vast possibilities and richness of mathematics beyond the limitations of traditional Euclidean structures. These concepts empower us to better understand the complexities of the natural world and prevent us from being misled by overly idealized notions. Thanks to their work, we are better equipped to explore and comprehend the intricate beauty of the natural world. Even the jagged wax deposits of the line depicting a runway in my childhood drawing. This is a public episode. If you would like to discuss this with other subscribers or get access to bonus episodes, visit interplace.io

La Notte del Liceo Classico, al Peano e in tutta Italia

Play Episode Listen Later May 5, 2023 13:36

Anche quest'anno il liceo tortonese aderisce all'evento che unisce tanti licei sul territorio nazionale. Appuntamento a questa sera per assistere a varie rappresentazioni e divulgazioni. Nello spazio condotto da Brocks, l'evento viene illustrato dalla docente Manuela Bonadeo.

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Sabato sotto i portici le Bancarelle Scientifiche con tutte le scuole cittadine

Il podcast di Piergiorgio Odifreddi: Lezioni e Conferenze.

Play Episode Listen Later Oct 7, 2022 20:20

Torna l'iniziativa con gli esperimenti scientifici illustrati al pubblico dagli studenti di tutte le scuole di ogni ordine e grado. Nello spazio condotto da Stefano Brocks (lun-ven 16-18), l'approfondimento con insegnanti e studenti del liceo Peano, capofila dell'iniziativa.

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"Morsi" di Marco Peano e "Il ritorno di Sira" di Maria Duenas

Il cacciatore di libri

Play Episode Listen Later Aug 24, 2022

"Morsi" di Marco Peano (Bompiani). Siamo nel 1996 in Piemonte, a Lanzo Torinese. La protagonista è Sonia, una bambina timida e solitaria che a volte viene lasciata dai genitori a casa della nonna. Una casa di cui Sonia ha paura. Inizialmente sembra un romanzo di formazione, ma improvvisamente in questa storia c'è una svolta, una scena chiave (di cui non possiamo anticipare nulla) che porta il romanzo sul piano del perturbarte e del fantastico, senza però perdere la credibilità. Un romanzo sul passaggio dall'infanzia all'adolescenza e poi all'età adulta, in cui crescere è visto come qualcosa di spaventoso. Dopo dodici anni la scrittrice spagnola Maria Duenas arriva in libreria con "Il ritorno di Sira" (Mondadori - traduz. Eleonora Mogavero e Giuliana Carraro), sequel di "La notte ha cambiato rumore" che aveva venduto in tutto il mondo oltre 5 milioni di copie e aveva ispirato la serie tv ""Il tempo del coraggio e dell'amore". Torna, dunque, la protagonista Sira che da sarta a Madrid si era trasformata in spia dei Servizi segreti britannici. In questo secondo volume siamo nel 1945, Sira sposato il collega Marcus e si sposterà in quattro luoghi: Gerusalemme (durante il mandato britannico), Londra, Madrid e Tangeri. Ancora una volta collaborerà con i servizi segreti britannici, lavorerà per la BBC e seguirà il viaggio di Eva Peron in Spagna.

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Odifreddi al Circolo dei Lettori: storie di animali, dagli elefanti alle lumache di mare.

Play Episode Listen Later Mar 20, 2022 37:48

Nell'Occidente cristiano-giudaico – anche su indicazione del Genesi dove si ordina all'uomo di sottomettere i pesci del mare, gli uccelli del cielo e via dicendo – si è guardato per millenni agli animali come fonte di cibo, forza lavoro o, nel migliore dei casi, compagnia. Ma i nodi di questa visione fondata sulla presunta superiorità umana rispetto alle altre specie stanno ormai venendo al pettine, con tutte le catastrofiche conseguenze che ha avuto e avrà sulla natura e sul pianeta.In realtà, gli animali hanno gli stessi nostri diritti di abitare la Terra e, se si indaga nella letteratura, nella filosofia e soprattutto nelle scienze, si scopre che spesso hanno aiutato l'uomo a progredire, lo hanno ispirato o indirizzato nelle scoperte. In questo libro Piergiorgio Odifreddi, con la sua straordinaria capacità di metterci sempre un nuovo tarlo razionale nel cervello, fa una sorprendente carrellata di storie di scienza che, oltre all'uomo, hanno avuto per protagonisti degli animali. Si passa così dai conigli che, con la loro proverbiale prolificità, hanno esemplificato i numeri di Fibonacci ai ragni il cui filo resistentissimo, notò il chimico-scrittore Primo Levi, si solidifica secondo un processo più efficace di quelli messi a punto dall'uomo: per trazione. Il curioso, coltissimo e originale percorso di Odifreddi si snoda poi tra le rane e le torpedini di Galvani (queste ultime già utilizzate, secondo Plinio, nell'antichità per fare degli elettroshock naturali) e i moscerini di Morgan, indispensabili per gli studi sull'ereditarietà. E che dire del cane di Pavlov che (come le oche di Lorenz) ebbe lo straordinario merito di spostare l'attenzione degli psicologi dall'introspezione all'osservazione dei comportamenti? Eccezionali insegnamenti ci sono giunti da api e formiche, scimpanzé e mucche (quella di Jenner, pioniere dei vaccini). E poi, perché mai il gatto di Peano riesce sempre a cadere in piedi?Insomma, siamo ancora convinti di poter fare a meno degli animali…? Forse no, visto che è stata una semplice lumaca di mare a darci un'avveniristica lezione sulle sinapsi (tema su cui è fioccato più di un premio Nobel per la Medicina)! ************************************************************************* Evento svoltosi il 7 Febbraio 2022 presso il Circolo dei lettori, via Bogino 9, Torino con Antonella Frontani a cura di Cento per Cento Lettori. --- Send in a voice message: https://podcasters.spotify.com/pod/show/vito-rodolfo-albano7/message

Marco Peano "Morsi"

il posto delle parole

Play Episode Listen Later Mar 7, 2022 17:06

Marco Peano"Morsi"Bompiani Editorehttps://www.bompiani.it/Tutto ha inizio con una ragazzina che gioca nella neve. Si chiama Sonia, sono le vacanze di Natale del 1996 – quelle della grande nevicata – e lei deve passarle suo malgrado a casa della nonna. Siamo a Lanzo Torinese, un paesino di mezza montagna dove ogni cosa sembra rimasta ferma a cinquant'anni prima. Compresa la casa cigolante e ingombra di mobili in cui vive nonna Ada, schiva, severa vecchia che nella zona ha fama di guaritrice (ma chissà, forse è altro), per la quale Sonia prova un affetto distante. La scuola ha chiuso prima del previsto a causa di quello che tutti chiamano “l'incidente”: la professoressa Cardone, acida insegnante di italiano, si è trincerata nella sua aula e durante una lezione – di fronte a una classe segregata e terrorizzata – ha fatto qualcosa di indicibile. Qualcosa che adesso, mentre Lanzo un po' alla volta si svuota per via delle feste e dell'incessante vento ghiacciato, sembra riguardare tutti gli abitanti. Toccherà a Sonia, insieme al suo amico Teo, ragazzino di famiglia contadina educato alla voracità, affrontare l'incubo in cui sono precipitati. Complici per forza, Sonia e Teo si avventurano nel biancore accecante della neve col distacco curioso di chi non ha pregiudizi e forse proprio per questo può sperare nella salvezza. Ma che cos'è la salvezza? Andar via, cambiare vita? O restare e tentare di resistere?Un romanzo lucido e terribile, divertito e tagliente, che si misura con i grandi temi – la paura, la crescita – e reinventa le regole del gioco. Una storia sulla fatica di cavarsela in un mondo a misura di adulti, quando gli adulti escono di scena e ti lasciano solo.Marco Peano è nato a Torino nel 1979, ed è editor di narrativa italiana per la casa editrice Einaudi. Ha pubblicato nel 2015 il suo primo romanzo, L'invenzione della madre (minimum fax), un successo di critica e di pubblico, Premio Volponi opera prima e Premio Libro dell'anno di Fahrenheit.IL POSTO DELLE PAROLEascoltare fa pensarehttps://ilpostodelleparole.it/

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Ospite l'autore di "Morsi", Marco Peano

Le Mattine pt 2 - Ora Daria

Play Episode Listen Later Feb 22, 2022 7:32

ospite morsi peano

"Il francese" di Massimo Carlotto e "Morsi" di Marco Peano

Il cacciatore di libri

Play Episode Listen Later Feb 5, 2022

Il noir come lente di ingrandimento delle contraddizioni della società. È quello che accade con i romanzi di Massimo Carlotto come "Il francese" (Mondadori) in cui lo scrittore mette a fuoco gli aspetti peggiori e il malcostume nella provincia veneta e si concentra sulla criminalità che gira intorno alla prostituzione. Il protagonista è Toni Zanchetta, soprannominato "il francese". Lui ama definirsi un macrò, per darsi una parvenza di eleganza. In realtà è un protettore come tutti gli altri, un manipolatore, un uomo ripugnante. Una delle ragazze della sua maison scompare e lui è il primo sospettato. Carlotto sceglie di raccontare questa storia entrando nella testa del macrò, seguendo i suoi pensieri e la sua parabola discendente, ma senza alcuna indulgenza. Per i criminali di questa storia non c'è redenzione. Nella seconda parte parliamo di "Morsi" di Marco Peano (Bompiani). Siamo nel 1996 in Piemonte, a Lanzo Torinese. La protagonista è Sonia, una bambina timida e solitaria che a volte viene lasciata dai genitori a casa della nonna. Una casa di cui Sonia ha paura. Inizialmente sembra un romanzo di formazione, ma improvvisamente in questa storia c'è una svolta, una scena chiave (di cui non possiamo anticipare nulla) che porta il romanzo sul piano del perturbarte e del fantastico, senza però perdere la credibilità. Un romanzo sul passaggio dall'infanzia all'adolescenza e poi all'età adulta, in cui crescere è visto come qualcosa di spaventoso.

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Rerun of P1: Peano Addition

Breaking Math Podcast

Play Episode Listen Later Jan 27, 2022 34:58 Very Popular

On this problem episode, join Sofía and guest Diane Baca to learn about what an early attempt to formalize the natural numbers has to say about whether or not m+n equals n+m. [Featuring: Sofía Baca; Diane Baca] --- This episode is sponsored by · Anchor: The easiest way to make a podcast. https://anchor.fm/app Support this podcast: https://anchor.fm/breakingmathpodcast/support

addition sof rerun baca peano

Matematica: che forza, che bellezza! | Piergiorgio Odifreddi

TEDx Talks Italia

Play Episode Listen Later Jan 17, 2022 18:04

"La matematica migliora il mondo?" "Un giorno Talete andò in gita alle piramidi e misurò la loro altezza sfruttando la loro ombra, e qualche proprietà dei triangoli simili. Da quel momento, la matematica non ha più smesso di essere usata per risolvere i problemi scientifici più svariati, teorici e applicati: non a caso, Galileo diceva che essa è il linguaggio della natura, la lingua in cui è scritto il grande libro dell'universo". Piergiorgio Odifreddi ha studiato matematica in Italia, negli Stati Uniti e in Unione Sovietica ed ha insegnato logica presso l'università di Torino e la Cornell University. Collabora a La Repubblica, l'Espresso e le Scienze. Ha vinto nel 1998 il premio Galileo dell'Unione Matematica Italiana. Nel 2002 il premio Peano della Mathesis e nel 2006 il premio Italgas per la divulgazione. Tra i suoi libri "Il Vangelo secondo la Scienza" (Einaudi 1999), "C'era una volta un paradosso" (Einaudi 2001), "Le menzogne di Ulisse" (Longanesi 2004), "Il matematico impertinente" (Longanesi 2005) e "Perché non possiamo essere cristiani (e meno che mai cattolici)" (Longanesi 2007). Da Mondadori ha pubblicato "Il matematico impertinente" (2007).

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LW - Logical Pinpointing by Eliezer Yudkowsky from Highly Advanced Epistemology 101 for Beginners

Play Episode Listen Later Dec 25, 2021 15:44

Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is Highly Advanced Epistemology 101 for Beginners, Part 11: Logical Pinpointing, published by Eliezer Yudkowsky. Followup to: Causal Reference, Proofs, Implications and Models The fact that one apple added to one apple invariably gives two apples helps in the teaching of arithmetic, but has no bearing on the truth of the proposition that 1 + 1 = 2. -- James R. Newman, The World of Mathematics Previous meditation 1: If we can only meaningfully talk about parts of the universe that can be pinned down by chains of cause and effect, where do we find the fact that 2 + 2 = 4? Or did I just make a meaningless noise, there? Or if you claim that "2 + 2 = 4"isn't meaningful or true, then what alternate property does the sentence "2 + 2 = 4" have which makes it so much more useful than the sentence "2 + 2 = 3"? Previous meditation 2: It has been claimed that logic and mathematics is the study of which conclusions follow from which premises. But when we say that 2 + 2 = 4, are we really just assuming that? It seems like 2 + 2 = 4 was true well before anyone was around to assume it, that two apples equalled two apples before there was anyone to count them, and that we couldn't make it 5 just by assuming differently. Speaking conventional English, we'd say the sentence 2 + 2 = 4 is "true", and anyone who put down "false" instead on a math-test would be marked wrong by the schoolteacher (and not without justice). But what can make such a belief true, what is the belief about, what is the truth-condition of the belief which can make it true or alternatively false? The sentence '2 + 2 = 4' is true if and only if... what? In the previous post I asserted that the study of logic is the study of which conclusions follow from which premises; and that although this sort of inevitable implication is sometimes called "true", it could more specifically be called "valid", since checking for inevitability seems quite different from comparing a belief to our own universe. And you could claim, accordingly, that "2 + 2 = 4" is 'valid' because it is an inevitable implication of the axioms of Peano Arithmetic. And yet thinking about 2 + 2 = 4 doesn't really feel that way. Figuring out facts about the natural numbers doesn't feel like the operation of making up assumptions and then deducing conclusions from them. It feels like the numbers are just out there, and the only point of making up the axioms of Peano Arithmetic was to allow mathematicians to talk about them. The Peano axioms might have been convenient for deducing a set of theorems like 2 + 2 = 4, but really all of those theorems were true about numbers to begin with. Just like "The sky is blue" is true about the sky, regardless of whether it follows from any particular assumptions. So comparison-to-a-standard does seem to be at work, just as with physical truth... and yet this notion of 2 + 2 = 4 seems different from "stuff that makes stuff happen". Numbers don't occupy space or time, they don't arrive in any order of cause and effect, there are no events in numberland. Meditation: What are we talking about when we talk about numbers? We can't navigate to them by following causal connections - so how do we get there from here? "Well," says the mathematical logician, "that's indeed a very important and interesting question - where are the numbers - but first, I have a question for you. What are these 'numbers' that you're talking about? I don't believe I've heard that word before." Yes you have. "No, I haven't. I'm not a typical mathematical logician; I was just created five minutes ago for the purposes of this conversation. So I genuinely don't know what numbers are." But... you know, 0, 1, 2, 3... "I don't recognize that 0 thingy - what is it? I'm not asking you to give an exact definition, I'm just trying to figure out what the heck you're ta...

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LW - Second-Order Logic: The Controversy by Eliezer Yudkowsky from Highly Advanced Epistemology 101 for Beginners

Play Episode Listen Later Dec 25, 2021 25:23

Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is Highly Advanced Epistemology 101 for Beginners, Part 14: Second-Order Logic: The Controversy, published by Eliezer Yudkowsky. Followup to: Godel's Completeness and Incompleteness Theorems "So the question you asked me last time was, 'Why does anyone bother with first-order logic at all, if second-order logic is so much more powerful?'" Right. If first-order logic can't talk about finiteness, or distinguish the size of the integers from the size of the reals, why even bother? "The first thing to realize is that first-order theories can still have a lot of power. First-order arithmetic does narrow down the possible models by a lot, even if it doesn't narrow them down to a single model. You can prove things like the existence of an infinite number of primes, because every model of the first-order axioms has an infinite number of primes. First-order arithmetic is never going to prove anything that's wrong about the standard numbers. Anything that's true in all models of first-order arithmetic will also be true in the particular model we call the standard numbers." Even so, if first-order theory is strictly weaker, why bother? Unless second-order logic is just as incomplete relative to third-order logic, which is weaker than fourth-order logic, which is weaker than omega-order logic - "No, surprisingly enough - there's tricks for making second-order logic encode any proposition in third-order logic and so on. If there's a collection of third-order axioms that characterizes a model, there's a collection of second-order axioms that characterizes the same model. Once you make the jump to second-order logic, you're done - so far as anyone knows (so far as I know) there's nothing more powerful than second-order logic in terms of which models it can characterize." Then if there's one spoon which can eat anything, why not just use the spoon? "Well... this gets into complex issues. There are mathematicians who don't believe there is a spoon when it comes to second-order logic." Like there are mathematicians who don't believe in infinity? "Kind of. Look, suppose you couldn't use second-order logic - you belonged to a species that doesn't have second-order logic, or anything like it. Your species doesn't have any native mental intuition you could use to construct the notion of 'all properties'. And then suppose that, after somebody used first-order set theory to prove that first-order arithmetic had many possible models, you stood around shouting that you believed in only one model, what you called the standard model, but you couldn't explain what made this model different from any other model -" Well... a lot of times, even in math, we make statements that genuinely mean something, but take a while to figure out how to define. I think somebody who talked about 'the numbers' would mean something even before second-order logic was invented. "But here the hypothesis is that you belong to a species that can't invent second-order logic, or think in second-order logic, or anything like it." Then I suppose you want me to draw the conclusion that this hypothetical alien is just standing there shouting about standardness, but its words don't mean anything because they have no way to pin down one model as opposed to another one. And I expect this species is also magically forbidden from talking about all possible subsets of a set? "Yeah. They can't talk about the largest powerset, just like they can't talk about the smallest model of Peano arithmetic." Then you could arguably deny that shouting about the 'standard' numbers would mean anything, to the members of this particular species. You might as well shout about the 'fleem' numbers, I guess. "Right. Even if all the members of this species did have a built-in sense that there was a special model of first-order arithmetic that was fleemer t...

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LW - Godel's Completeness and Incompleteness Theorems by Eliezer Yudkowsky from Highly Advanced Epistemology 101 for Beginners

Play Episode Listen Later Dec 25, 2021 20:21

Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is Highly Advanced Epistemology 101 for Beginners, Part 13: Godel's Completeness and Incompleteness Theorems, published by Eliezer Yudkowsky. Followup to: Standard and Nonstandard Numbers So... last time you claimed that using first-order axioms to rule out the existence of nonstandard numbers - other chains of numbers besides the 'standard' numbers starting at 0 - was forever and truly impossible, even unto a superintelligence, no matter how clever the first-order logic used, even if you came up with an entirely different way of axiomatizing the numbers. "Right." How could you, in your finiteness, possibly know that? "Have you heard of Godel's Incompleteness Theorem?" Of course! Godel's Theorem says that for every consistent mathematical system, there are statements which are true within that system, which can't be proven within the system itself. Godel came up with a way to encode theorems and proofs as numbers, and wrote a purely numerical formula to detect whether a proof obeyed proper logical syntax. The basic trick was to use prime factorization to encode lists; for example, the ordered list could be uniquely encoded as: 23 37 51 74 And since prime factorizations are unique, and prime powers don't mix, you could inspect this single number, 210,039,480, and get the unique ordered list back out. From there, going to an encoding for logical formulas was easy; for example, you could use the 2 prefix for NOT and the 3 prefix for AND and get, for any formulas Φ and Ψ encoded by the numbers #Φ and #Ψ: ¬Φ = 22 3#Φ Φ ∧ Ψ = 23 3#Φ 5#Ψ It was then possible, by dint of crazy amounts of work, for Godel to come up with a gigantic formula of Peano Arithmetic [](p, c) meaning, 'P encodes a valid logical proof using first-order Peano axioms of C', from which directly followed the formula []c, meaning, 'There exists a number P such that P encodes a proof of C' or just 'C is provable in Peano arithmetic.' Godel then put in some further clever work to invent statements which referred to themselves, by having them contain sub-recipes that would reproduce the entire statement when manipulated by another formula. And then Godel's Statement encodes the statement, 'There does not exist any number P such that P encodes a proof of (this statement) in Peano arithmetic' or in simpler terms 'I am not provable in Peano arithmetic'. If we assume first-order arithmetic is consistent and sound, then no proof of this statement within first-order arithmetic exists, which means the statement is true but can't be proven within the system. That's Godel's Theorem. "Er... no." No? "No. I've heard rumors that Godel's Incompleteness Theorem is horribly misunderstood in your Everett branch. Have you heard of Godel's Completeness Theorem?" Is that a thing? "Yes! Godel's Completeness Theorem says that, for any collection of first-order statements, every semantic implication of those statements is syntactically provable within first-order logic. If something is a genuine implication of a collection of first-order statements - if it actually does follow, in the models pinned down by those statements - then you can prove it, within first-order logic, using only the syntactical rules of proof, from those axioms." I don't see how that could possibly be true at the same time as Godel's Incompleteness Theorem. The Completeness Theorem and Incompleteness Theorem seem to say diametrically opposite things. Godel's Statement is implied by the axioms of first-order arithmetic - that is, we can see it's true using our own mathematical reasoning - "Wrong." What? I mean, I understand we can't prove it within Peano arithmetic, but from outside the system we can see that - All right, explain. "Basically, you just committed the equivalent of saying, 'If all kittens are little, and some little things ar...

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LW - Standard and Nonstandard Numbers by Eliezer Yudkowsky from Highly Advanced Epistemology 101 for Beginners

Play Episode Listen Later Dec 25, 2021 25:30

Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is Highly Advanced Epistemology 101 for Beginners, Part 12: Standard and Nonstandard Numbers, published by Eliezer Yudkowsky. Followup to: Logical Pinpointing "Oh! Hello. Back again?" Yes, I've got another question. Earlier you said that you had to use second-order logic to define the numbers. But I'm pretty sure I've heard about something called 'first-order Peano arithmetic' which is also supposed to define the natural numbers. Going by the name, I doubt it has any 'second-order' axioms. Honestly, I'm not sure I understand this second-order business at all. "Well, let's start by examining the following model:" "This model has three properties that we would expect to be true of the standard numbers - 'Every number has a successor', 'If two numbers have the same successor they are the same number', and '0 is the only number which is not the successor of any number'. All three of these statements are true in this model, so in that sense it's quite numberlike -" And yet this model clearly is not the numbers we are looking for, because it's got all these mysterious extra numbers like C and -2. That C thing even loops around, which I certainly wouldn't expect any number to do. And then there's that infinite-in-both-directions chain which isn't corrected to anything else. "Right, so, the difference between first-order logic and second-order logic is this: In first-order logic, we can get rid of the ABC - make a statement which rules out any model that has a loop of numbers like that. But we can't get rid of the infinite chain underneath it. In second-order logic we can get rid of the extra chain." I would ask you to explain why that was true, but at this point I don't even know what second-order logic is. "Bear with me. First, consider that the following formula detects 2-ness:" x + 2 = x 2 In other words, that's a formula which is true when x is equal to 2, and false everywhere else, so it singles out 2? "Exactly. And this is a formula which detects odd numbers:" ∃y: x=(2y)+1 Um... okay. That formula says, 'There exists a y, such that x equals 2 times y plus one.' And that's true when x is 1, because 0 is a number, and 1=(20)+1. And it's true when x is 9, because there exists a number 4 such that 9=(24)+1... right. The formula is true at all odd numbers, and only odd numbers. "Indeed. Now suppose we had some way to detect the existence of that ABC-loop in the model - a formula which was true at the ABC-loop and false everywhere else. Then I could adapt the negation of this statement to say 'No objects like this are allowed to exist', and add that as an axiom alongside 'Every number has a successor' and so on. Then I'd have narrowed down the possible set of models to get rid of models that have an extra ABC-loop in them." Um... can I rule out the ABC-loop by saying ¬∃x:(x=A)? "Er, only if you've told me what A is in the first place, and in a logic which has ruled out all models with loops in them, you shouldn't be able to point to a specific object that doesn't exist -" Right. Okay... so the idea is to rule out loops of successors... hm. In the numbers 0, 1, 2, 3..., the number 0 isn't the successor of any number. If I just took a group of numbers starting at 1, like {1, 2, 3, ...}, then 1 wouldn't be the successor of any number inside that group. But in A, B, C, the number A is the successor of C, which is the successor of B, which is the successor of A. So how about if I say: 'There's no group of numbers G such that for any number x in G, x is the successor of some other number y in G.' "Ah! Very clever. But it so happens that you just used second-order logic, because you talked about groups or collections of entities, whereas first-order logic only talks about individual entities. Like, suppose we had a logic talking about kittens and whether they're innocent. Here's a mod...

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LW - Second-Order Logic: The Controversy by Eliezer Yudkowsky from Highly Advanced Epistemology 101 for Beginners

Play Episode Listen Later Dec 25, 2021 25:23

Link to original articleWelcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is Highly Advanced Epistemology 101 for Beginners, Part 14: Second-Order Logic: The Controversy, published by Eliezer Yudkowsky. Followup to: Godel's Completeness and Incompleteness Theorems "So the question you asked me last time was, 'Why does anyone bother with first-order logic at all, if second-order logic is so much more powerful?'" Right. If first-order logic can't talk about finiteness, or distinguish the size of the integers from the size of the reals, why even bother? "The first thing to realize is that first-order theories can still have a lot of power. First-order arithmetic does narrow down the possible models by a lot, even if it doesn't narrow them down to a single model. You can prove things like the existence of an infinite number of primes, because every model of the first-order axioms has an infinite number of primes. First-order arithmetic is never going to prove anything that's wrong about the standard numbers. Anything that's true in all models of first-order arithmetic will also be true in the particular model we call the standard numbers." Even so, if first-order theory is strictly weaker, why bother? Unless second-order logic is just as incomplete relative to third-order logic, which is weaker than fourth-order logic, which is weaker than omega-order logic - "No, surprisingly enough - there's tricks for making second-order logic encode any proposition in third-order logic and so on. If there's a collection of third-order axioms that characterizes a model, there's a collection of second-order axioms that characterizes the same model. Once you make the jump to second-order logic, you're done - so far as anyone knows (so far as I know) there's nothing more powerful than second-order logic in terms of which models it can characterize." Then if there's one spoon which can eat anything, why not just use the spoon? "Well... this gets into complex issues. There are mathematicians who don't believe there is a spoon when it comes to second-order logic." Like there are mathematicians who don't believe in infinity? "Kind of. Look, suppose you couldn't use second-order logic - you belonged to a species that doesn't have second-order logic, or anything like it. Your species doesn't have any native mental intuition you could use to construct the notion of 'all properties'. And then suppose that, after somebody used first-order set theory to prove that first-order arithmetic had many possible models, you stood around shouting that you believed in only one model, what you called the standard model, but you couldn't explain what made this model different from any other model -" Well... a lot of times, even in math, we make statements that genuinely mean something, but take a while to figure out how to define. I think somebody who talked about 'the numbers' would mean something even before second-order logic was invented. "But here the hypothesis is that you belong to a species that can't invent second-order logic, or think in second-order logic, or anything like it." Then I suppose you want me to draw the conclusion that this hypothetical alien is just standing there shouting about standardness, but its words don't mean anything because they have no way to pin down one model as opposed to another one. And I expect this species is also magically forbidden from talking about all possible subsets of a set? "Yeah. They can't talk about the largest powerset, just like they can't talk about the smallest model of Peano arithmetic." Then you could arguably deny that shouting about the 'standard' numbers would mean anything, to the members of this particular species. You might as well shout about the 'fleem' numbers, I guess. "Right. Even if all the members of this species did have a built-in sense that there was a special model of first-order arithmetic that was fleemer t...

speech controversy beginners logic ea epistemology completeness eliezer yudkowsky rationalist godel peano

LW - Logical Pinpointing by Eliezer Yudkowsky from Highly Advanced Epistemology 101 for Beginners

Play Episode Listen Later Dec 25, 2021 15:44

Link to original articleWelcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is Highly Advanced Epistemology 101 for Beginners, Part 11: Logical Pinpointing, published by Eliezer Yudkowsky. Followup to: Causal Reference, Proofs, Implications and Models The fact that one apple added to one apple invariably gives two apples helps in the teaching of arithmetic, but has no bearing on the truth of the proposition that 1 + 1 = 2. -- James R. Newman, The World of Mathematics Previous meditation 1: If we can only meaningfully talk about parts of the universe that can be pinned down by chains of cause and effect, where do we find the fact that 2 + 2 = 4? Or did I just make a meaningless noise, there? Or if you claim that "2 + 2 = 4"isn't meaningful or true, then what alternate property does the sentence "2 + 2 = 4" have which makes it so much more useful than the sentence "2 + 2 = 3"? Previous meditation 2: It has been claimed that logic and mathematics is the study of which conclusions follow from which premises. But when we say that 2 + 2 = 4, are we really just assuming that? It seems like 2 + 2 = 4 was true well before anyone was around to assume it, that two apples equalled two apples before there was anyone to count them, and that we couldn't make it 5 just by assuming differently. Speaking conventional English, we'd say the sentence 2 + 2 = 4 is "true", and anyone who put down "false" instead on a math-test would be marked wrong by the schoolteacher (and not without justice). But what can make such a belief true, what is the belief about, what is the truth-condition of the belief which can make it true or alternatively false? The sentence '2 + 2 = 4' is true if and only if... what? In the previous post I asserted that the study of logic is the study of which conclusions follow from which premises; and that although this sort of inevitable implication is sometimes called "true", it could more specifically be called "valid", since checking for inevitability seems quite different from comparing a belief to our own universe. And you could claim, accordingly, that "2 + 2 = 4" is 'valid' because it is an inevitable implication of the axioms of Peano Arithmetic. And yet thinking about 2 + 2 = 4 doesn't really feel that way. Figuring out facts about the natural numbers doesn't feel like the operation of making up assumptions and then deducing conclusions from them. It feels like the numbers are just out there, and the only point of making up the axioms of Peano Arithmetic was to allow mathematicians to talk about them. The Peano axioms might have been convenient for deducing a set of theorems like 2 + 2 = 4, but really all of those theorems were true about numbers to begin with. Just like "The sky is blue" is true about the sky, regardless of whether it follows from any particular assumptions. So comparison-to-a-standard does seem to be at work, just as with physical truth... and yet this notion of 2 + 2 = 4 seems different from "stuff that makes stuff happen". Numbers don't occupy space or time, they don't arrive in any order of cause and effect, there are no events in numberland. Meditation: What are we talking about when we talk about numbers? We can't navigate to them by following causal connections - so how do we get there from here? "Well," says the mathematical logician, "that's indeed a very important and interesting question - where are the numbers - but first, I have a question for you. What are these 'numbers' that you're talking about? I don't believe I've heard that word before." Yes you have. "No, I haven't. I'm not a typical mathematical logician; I was just created five minutes ago for the purposes of this conversation. So I genuinely don't know what numbers are." But... you know, 0, 1, 2, 3... "I don't recognize that 0 thingy - what is it? I'm not asking you to give an exact definition, I'm just trying to figure out what the heck you're ta...

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LW - Standard and Nonstandard Numbers by Eliezer Yudkowsky from Highly Advanced Epistemology 101 for Beginners

Play Episode Listen Later Dec 25, 2021 25:30

Link to original articleWelcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is Highly Advanced Epistemology 101 for Beginners, Part 12: Standard and Nonstandard Numbers, published by Eliezer Yudkowsky. Followup to: Logical Pinpointing "Oh! Hello. Back again?" Yes, I've got another question. Earlier you said that you had to use second-order logic to define the numbers. But I'm pretty sure I've heard about something called 'first-order Peano arithmetic' which is also supposed to define the natural numbers. Going by the name, I doubt it has any 'second-order' axioms. Honestly, I'm not sure I understand this second-order business at all. "Well, let's start by examining the following model:" "This model has three properties that we would expect to be true of the standard numbers - 'Every number has a successor', 'If two numbers have the same successor they are the same number', and '0 is the only number which is not the successor of any number'. All three of these statements are true in this model, so in that sense it's quite numberlike -" And yet this model clearly is not the numbers we are looking for, because it's got all these mysterious extra numbers like C and -2. That C thing even loops around, which I certainly wouldn't expect any number to do. And then there's that infinite-in-both-directions chain which isn't corrected to anything else. "Right, so, the difference between first-order logic and second-order logic is this: In first-order logic, we can get rid of the ABC - make a statement which rules out any model that has a loop of numbers like that. But we can't get rid of the infinite chain underneath it. In second-order logic we can get rid of the extra chain." I would ask you to explain why that was true, but at this point I don't even know what second-order logic is. "Bear with me. First, consider that the following formula detects 2-ness:" x + 2 = x 2 In other words, that's a formula which is true when x is equal to 2, and false everywhere else, so it singles out 2? "Exactly. And this is a formula which detects odd numbers:" ∃y: x=(2y)+1 Um... okay. That formula says, 'There exists a y, such that x equals 2 times y plus one.' And that's true when x is 1, because 0 is a number, and 1=(20)+1. And it's true when x is 9, because there exists a number 4 such that 9=(24)+1... right. The formula is true at all odd numbers, and only odd numbers. "Indeed. Now suppose we had some way to detect the existence of that ABC-loop in the model - a formula which was true at the ABC-loop and false everywhere else. Then I could adapt the negation of this statement to say 'No objects like this are allowed to exist', and add that as an axiom alongside 'Every number has a successor' and so on. Then I'd have narrowed down the possible set of models to get rid of models that have an extra ABC-loop in them." Um... can I rule out the ABC-loop by saying ¬∃x:(x=A)? "Er, only if you've told me what A is in the first place, and in a logic which has ruled out all models with loops in them, you shouldn't be able to point to a specific object that doesn't exist -" Right. Okay... so the idea is to rule out loops of successors... hm. In the numbers 0, 1, 2, 3..., the number 0 isn't the successor of any number. If I just took a group of numbers starting at 1, like {1, 2, 3, ...}, then 1 wouldn't be the successor of any number inside that group. But in A, B, C, the number A is the successor of C, which is the successor of B, which is the successor of A. So how about if I say: 'There's no group of numbers G such that for any number x in G, x is the successor of some other number y in G.' "Ah! Very clever. But it so happens that you just used second-order logic, because you talked about groups or collections of entities, whereas first-order logic only talks about individual entities. Like, suppose we had a logic talking about kittens and whether they're innocent. Here's a mod...

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LW - Godel's Completeness and Incompleteness Theorems by Eliezer Yudkowsky from Highly Advanced Epistemology 101 for Beginners

Play Episode Listen Later Dec 25, 2021 20:21

Link to original articleWelcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is Highly Advanced Epistemology 101 for Beginners, Part 13: Godel's Completeness and Incompleteness Theorems, published by Eliezer Yudkowsky. Followup to: Standard and Nonstandard Numbers So... last time you claimed that using first-order axioms to rule out the existence of nonstandard numbers - other chains of numbers besides the 'standard' numbers starting at 0 - was forever and truly impossible, even unto a superintelligence, no matter how clever the first-order logic used, even if you came up with an entirely different way of axiomatizing the numbers. "Right." How could you, in your finiteness, possibly know that? "Have you heard of Godel's Incompleteness Theorem?" Of course! Godel's Theorem says that for every consistent mathematical system, there are statements which are true within that system, which can't be proven within the system itself. Godel came up with a way to encode theorems and proofs as numbers, and wrote a purely numerical formula to detect whether a proof obeyed proper logical syntax. The basic trick was to use prime factorization to encode lists; for example, the ordered list could be uniquely encoded as: 23 37 51 74 And since prime factorizations are unique, and prime powers don't mix, you could inspect this single number, 210,039,480, and get the unique ordered list back out. From there, going to an encoding for logical formulas was easy; for example, you could use the 2 prefix for NOT and the 3 prefix for AND and get, for any formulas Φ and Ψ encoded by the numbers #Φ and #Ψ: ¬Φ = 22 3#Φ Φ ∧ Ψ = 23 3#Φ 5#Ψ It was then possible, by dint of crazy amounts of work, for Godel to come up with a gigantic formula of Peano Arithmetic [](p, c) meaning, 'P encodes a valid logical proof using first-order Peano axioms of C', from which directly followed the formula []c, meaning, 'There exists a number P such that P encodes a proof of C' or just 'C is provable in Peano arithmetic.' Godel then put in some further clever work to invent statements which referred to themselves, by having them contain sub-recipes that would reproduce the entire statement when manipulated by another formula. And then Godel's Statement encodes the statement, 'There does not exist any number P such that P encodes a proof of (this statement) in Peano arithmetic' or in simpler terms 'I am not provable in Peano arithmetic'. If we assume first-order arithmetic is consistent and sound, then no proof of this statement within first-order arithmetic exists, which means the statement is true but can't be proven within the system. That's Godel's Theorem. "Er... no." No? "No. I've heard rumors that Godel's Incompleteness Theorem is horribly misunderstood in your Everett branch. Have you heard of Godel's Completeness Theorem?" Is that a thing? "Yes! Godel's Completeness Theorem says that, for any collection of first-order statements, every semantic implication of those statements is syntactically provable within first-order logic. If something is a genuine implication of a collection of first-order statements - if it actually does follow, in the models pinned down by those statements - then you can prove it, within first-order logic, using only the syntactical rules of proof, from those axioms." I don't see how that could possibly be true at the same time as Godel's Incompleteness Theorem. The Completeness Theorem and Incompleteness Theorem seem to say diametrically opposite things. Godel's Statement is implied by the axioms of first-order arithmetic - that is, we can see it's true using our own mathematical reasoning - "Wrong." What? I mean, I understand we can't prove it within Peano arithmetic, but from outside the system we can see that - All right, explain. "Basically, you just committed the equivalent of saying, 'If all kittens are little, and some little things ar...

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The proof-theoretic ordinal of Peano Arithmetic is Epsilon-0

Iowa Type Theory Commute

Play Episode Listen Later Dec 11, 2021 14:10

In this episode, I outline the argument for why the proof-theoretic ordinal (in the sense of Rathjen, as presented last episode) is epsilon-0. My explanation has something of a hole, in explaining how one would go about deriving induction for ordinals strictly less than epsilon-0 in Peano Arithmetic. To help paper over this hole a little, I discuss a really nice recent exposition of encoding ordinals in Agda.

proof epsilon arithmetic ordinal agda peano

Cristiana Peano "Due Punti Festival"

il posto delle parole

Play Episode Listen Later Oct 1, 2021 18:05

Cristiana PeanoCristiana Peano, Presidente della Commissione Terza Missione e Comunicazione del Dipartimento di Scienze Agrarie, Forestali e Alimentari"Due Punti Festival"https://festivalduepunti.unito.it/chi-siamo/Il Festival Due Punti è una strada immaginaria che collega spazi fisici e mentali e ne riduce la distanzaRoberta Gorra, responsabile del progettoScopri gli appuntamenti di sabato 2 e domenica 3 ottobre 2021Sabato 2 ottobrehttps://festivalduepunti.unito.it/eventi/2021-10-02/Domenica 3 ottobrehttps://festivalduepunti.unito.it/eventi/2021-10-03/Il Festival Due Punti è un progetto di public engagement finanziato dall'Università degli Studi di Torino, coordinato dal Dipartimento di Scienze Agrarie, Forestali e Alimentari, organizzato in collaborazione con il Dipartimento di Scienze Veterinarie e con il Dipartimento di Lingue e Letterature Straniere e Culture Moderne.cibo, ecosistemi e cultura"Ci relazioniamo costantemente con gli organismi che abitano e plasmano i nostri stessi ecosistemi. Lo facciamo ad esempio per produrre cibo, ma anche quando decidiamo come gestire una foresta o un allevamento, curare un giardino o un animale da compagnia.Tema centrale del Festival è quindi la riflessione sul nostro ruolo, come individui e comunità, all'interno degli ecosistemi, di cui siamo parte integrante.Anche le narrazioni orientano le nostre relazioni: scrittori, traduttori, critici e artisti condivideranno i loro sguardi sui paesaggi che li circondano e li abitano.Le attività in programma permetteranno un incontro inter e multidisciplinare, mostrando ai cittadini e ai ricercatori come i confini della ricerca possano sfumare in un incontro tra saperi."IL POSTO DELLE PAROLEascoltare fa pensarehttps://ilpostodelleparole.it/

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Studenti del Peano esprimono "L'Europa che sogniamo"

Play Episode Listen Later May 20, 2021 48:10

La partecipazione alla Festa dell'europa è stata occasione per la classe 2' Scienze Umane per elaborare il proprio manifesto dei valori che i giovani auspicano che l'Unione Europea esprima. Nella trasmissione condotta da Brocks ("Citylife Musica e Notizie", da lun a ven 16-18), l'insegnante Simona Merlino introduce tre studentesse a presentare i principi enunciati nel loro elaborato: intervengono Alice Verna, Isabella Solari, Laura La Piana.

nella festa notizie studenti l'europa brocks scienze umane peano

“Una questione di grammatica razionale” di Giuseppe Peano

Liber Liber

Play Episode Listen Later Feb 16, 2021 1:21

In questo articolo del 1911 il matematico Giuseppe Peano si chiede se le classificazioni delle parole usate dalle grammatiche tradizionali (sostantivo, aggettivo, pronome, verbo...) dipendono solo da proprietà delle parole (proprietà formali) oppure se dipendono dagli enti a cui le parole si riferiscono (proprietà reali).

giuseppe questione di grammatica razionale peano

“Una questione di grammatica razionale” di Giuseppe Peano

Pagina Tre

Play Episode Listen Later Feb 16, 2021 1:21

Giuseppe Peano presenta questo articolo al Quarto Congresso Internazionale di Filosofia di Bologna del 1911. Il matematico piemontese si chiede se le classificazioni delle parole usate dalle grammatiche tradizionali (sostantivo, aggettivo, pronome, verbo…) dipendono solo da proprietà delle parole di quella precisa lingua (proprietà formali) oppure se dipendono dagli enti a cui le parole si […]

bologna filosofia giuseppe questione di grammatica razionale peano

Math Mutation 266: A Number Is A Number

Math Mutation

Play Episode Listen Later Dec 25, 2020 4:11

We explore Bertrand Russell's set-theoretic definition of whole numbers. (Send feeback to erik@mathmutation.com)

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“La numerazione binaria applicata alla stenografia” di Giuseppe Peano

Liber Liber

Play Episode Listen Later Dec 21, 2020 1:38

In questo articolo il matematico piemontese propone un nuovo codice per comunicare lettere, numeri e altri caratteri utilizzando il sistema numerico binario. Peano progetta anche una "macchina stenografica" che applica questo codice per la scrittura veloce.

giuseppe applicata peano

“La numerazione binaria applicata alla stenografia” di Giuseppe Peano

Pagina Tre

Play Episode Listen Later Dec 21, 2020 1:38

Giuseppe Peano presentò questo breve testo alla seduta dell’Accademia delle Scienze di Torino del 13 novembre 1898. In questo articolo il matematico piemontese propone un nuovo codice per comunicare lettere, numeri e altri caratteri utilizzando il sistema numerico binario. Peano progetta anche una “macchina stenografica” che applica questo codice per la scrittura veloce. Sistemi di […]

torino giuseppe sistemi scienze accademia applicata peano

Una Virtual Open Night per il Liceo Peano di Tortona

Play Episode Listen Later Dec 9, 2020 17:05

In tempi di didattica a distanza, anche gli eventi di orientamento si svolgono in modalità virtuale: ma la scuola tortonese non rinuncia al suo evento culturale, quest'anno trasferito sui social. La dirigente scolastica Maria Teresa Marchesotti illustra i contenuti dell'evento on line venerdì 11 alle 19.

virtual liceo tortona peano

Math Mutation 265: Defining Numbers, Sort of

Math Mutation

Play Episode Listen Later Nov 30, 2020 5:29

Is there a fundamental set of axioms we can use to define whole numbers? (Send feeback to erik@mathmutation.com)

numbers defining math sort infinity mathematics mutation geometry peano

La didattica a distanza: qualche disagio ma anche opportunità

PaperPlayer biorxiv neuroscience

Play Episode Listen Later Nov 17, 2020 13:07

Come si organizzano studenti e docenti del liceo Peano per il periodo in cui non si possono effettuare lezioni in presenza? Lo descrive Manuela bonadeo, docente di Storia dell'Arte, nello spazio condotto da Brocks su Radio Pnr (Lun-ven 16-18).

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Transient Maternal IL-6 boosts glutamatergic synapses and disrupts hippocampal connectivity in the offspring

Play Episode Listen Later Nov 2, 2020

Link to bioRxiv paper: http://biorxiv.org/cgi/content/short/2020.11.02.364356v1?rss=1 Authors: Mirabella, F., Desiato, G., Mancinelli, S., Fossati, G., Rasile, M., Morini, R., Markicevic, M., Grimm, C., amegandjin, C., Termanini, A., Peano, C., Kunderfranco, P., DiCristo, G., Zerbi, V., Lodato, S., Menna, E., Matteoli, M., Pozzi, D. Abstract: Early prenatal inflammatory conditions are thought to represent a risk factor for different neurodevelopmental disorders, with long-term consequences on adult brain connectivity. Here we show that a transient IL-6 elevation, occurring at vulnerable stages of early neurodevelopment, directly impacts brain developmental trajectories through the aberrant enhancement of glutamatergic synapses and overall brain hyper-connectivity. The IL6-mediated boost of excitatory synapse density results from the neuronautonomous, genomic effect of the transcription factor STAT3 and causally involves the activation of RGS4 gene as a candidate downstream target. The STAT3/RGS4 pathway is also activated in neonatal brains as a consequence of maternal immune activation protocols mimicking a viral infection during pregnancy. By demonstrating that prenatal IL-6 elevations result in aberrant synaptic and brain connectivity through the molecular players identified, we provide a mechanistic framework for the association between prenatal inflammatory events and brain neurodevelopmental disorders. Copy rights belong to original authors. Visit the link for more info

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Primo giorno di scuola: l'organizzazione al liceo Peano

Play Episode Listen Later Sep 15, 2020 12:12

Lezione in presenza per le classi prime e quinte, mista tra presenza e remoto per le altre. Le modalità didattiche e gli spazi del liceo tortonese vengono gestiti in modo da rispettare le norme antiCovid senza trascurare la didattica. Nello spazio condotto da Brocks (lun-ven 17.19) interviene Manuela Bonadeo, docente di Storia dell'Arte, per illustrare la partenza delle lezioni

primo storia scuola nello lezione liceo brocks l'organizzazione peano

La tecnologia come opportunità didattica, ai tempi del virus