Ombre determinanti #4
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Trovata la matematica di Pierce quando parla dei radicali in traccia2/articolo/TraduzionePassoChiavePeirce.docx . |
on PierceBeginning with his first paper on the "Logic of Relatives" (1870), Peirce extended the theory of relations that Augustus De Morgan had just recently awakened from its Cinderella slumbers. Much of the mathematics of relations now taken for granted was "borrowed" from Peirce, not always with all due credit; on that and on how the young Bertrand Russell, especially his Principles of Mathematics and Principia Mathematica, did not do Peirce justice, see Anellis (1995).[64] In 1918 the logician C. I. Lewis wrote, "The contributions of C.S. Peirce to symbolic logic are more numerous and varied than those of any other writer—at least in the nineteenth century."[91] Beginning in 1940, Alfred Tarski and his students rediscovered aspects of Peirce's larger vision of relational logic, developing the perspective of relation algebra. Relational logic gained applications. In mathematics, it influenced the abstract analysis of E. H. Moore and the lattice theory of Garrett Birkhoff. In computer science, the relational model for databases was developed with Peircean ideas in work of Edgar F. Codd, who was a doctoral student[92] of Arthur W. Burks, a Peirce scholar. In economics, relational logic was used by Frank P. Ramsey, John von Neumann, and Paul Samuelson to study preferences and utility and by Kenneth J. Arrow in Social Choice and Individual Values, following Arrow's association with Tarski at City College of New York. On Peirce and his contemporaries Ernst Schröder and Gottlob Frege, Hilary Putnam (1982)[86] documented that Frege's work on the logic of quantifiers had little influence on his contemporaries, although it was published four years before the work of Peirce and his student Oscar Howard Mitchell. Putnam found that mathematicians and logicians learned about the logic of quantifiers through the independent work of Peirce and Mitchell, particularly through Peirce's "On the Algebra of Logic: A Contribution to the Philosophy of Notation"[85] (1885), published in the premier American mathematical journal of the day, and cited by Peano and Schröder, among others, who ignored Frege. They also adopted and modified Peirce's notations, typographical variants of those now used. Peirce apparently was ignorant of Frege's work, despite their overlapping achievements in logic, philosophy of language, and the foundations of mathematics. Peirce's work on formal logic had admirers besides Ernst Schröder: |
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Qui il grafo delle relazioni tra le classi trovate. Da vedere su desktop, no mobile |
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Lavoro finito con estrapolazione relazioni tra materie. |
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Aggiunto il libro di Boole mathematical analysis of logic. (traccia2/math/BooleMathematicalAnalysisOfLogic.pdf). |
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Idea da controllare: Usare algebra di Pierce per derivare concetti su materie in bibliometria. Locgical sequence is a simpler conception than causal sequence, because every causal Una citazione è una sequenza causale, quindi una implicazione. Dai libri si trova relazione tra le classi Simile a concetto di functional di Hilbert vedi #2 (comment) |
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Sono riuscito a cancellare la versione doppia. Il giorno dom 27 set 2020 alle ore 00:42 luca giusti [email protected]
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Ciao Mauro, Luca Il giorno ven 7 ago 2020 alle ore 13:22 Mauro Bertani <
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Quel testo di pierce, io non lo ricordo Il dom 27 set 2020, 09:16 ellegi69 [email protected] ha scritto:
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Articoli sulle citazioni: Sono partito da una citazione trovata nell'elenco di Gnoli e seguendo le references sono arrivato ad articoli degli anni 50-70 che ipotizzano una struttura della letteratura scientifica che rimanda a lavoro di matematica su spazi multidimensionali metrici. Quello che stiamo facendo noi.
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AlgebraizationsAn alternate approach to the semantics of first-order logic proceeds via abstract algebra. This approach generalizes the Lindenbaum–Tarski algebras of propositional logic. There are three ways of eliminating quantified variables from first-order logic that do not involve replacing quantifiers with other variable binding term operators: Cylindric algebra, by Alfred Tarski and colleagues; Tarski and Givant (1987) showed that the fragment of first-order logic that has no atomic sentence lying in the scope of more than three quantifiers has the same expressive power as relation algebra.[18]:32–33 This fragment is of great interest because it suffices for Peano arithmetic and most axiomatic set theory, including the canonical ZFC. They also prove that first-order logic with a primitive ordered pair is equivalent to a relation algebra with two ordered pair projection functions.[19]:803 |
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Frege Frege draws an important distinction between concepts on the basis of their level. Frege tells us that a first-level concept is a one-place function that correlates objects with truth-values (147). First level concepts have the value of true or false depending on whether the object falls under the concept. So, the concept F has the value the True with the argument the object named by 'Jamie' if and only if Jamie falls under the concept F (or is in the extension of F). ref. https://en.wikipedia.org/wiki/Function_and_Concept Aggiunto file /Traccia2/math/fregeFunctionAndConcept |
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Ho guardato su gmail ma non ho ritrovato il testo su Peirce che ti avevo Il giorno dom 27 set 2020 alle ore 08:34 luca giusti [email protected]
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Ok, forse era quello. Il giorno dom 27 set 2020 alle ore 11:12 Mauro Bertani <
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Aggiunto due libri [Peirce Benjamin, "Linear Associative Algebra", American Journal of Mathematics , 1881, Vol. 4, No. 1 (1881), pp. 97-229 link] Probabile articolo che Pierce cita come fondamentale del 1883 in traccia_2/Capitolo_con_passo_chiave_di_Peirce.docx |
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Se parli invece della matematica di Pierce è nella cartella traccia2 Il dom 27 set 2020, 11:07 Mauro Bertani [email protected] ha scritto:
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Nell'articolo nella parte su metodo pustulazionale, nel sottoparagrafo di Peano, l'assioma 4, contraddetto da Russell, risolto, crea la teoria dei tipi. È la classe membro di se stessa? Qui una trattazione più completa. Vedere anche di Frege Basic law 5 |
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Esempio di studio di argomenti trattati da una persona. Unito a classificazione a faccette e messi tutto su una retta si mettono vicino gli autori simili. qui l'esempio delle query. |
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La logica dei level e i reeb graph #3 (comment) usata nei round elimina il problema della teoria dei tipi di russell perchè stringhe di soggetto create da differenti strutture nei round si trasformano sempre in un punto di uno spazio vettoriale R^3. E' una cosa molto simile a quello che aveva fatto Pierce di unire la logica all'algebra vettoriale per il calcolo delle soluzioni di frasi logiche con variabili. Ma qui si tiene conto anche dei manifold e delle logiche di high level. Non fornisce soluzioni a frasi logiche con variabili ma permette di studiare le logiche di high level come se fossero uno spazio vettoriale R^3 eliminando il paradosso dei tipi e disponendo i tipi su level in funzione della loro complessità. Non ho più una matrice di relazioni ma uno spazio di punti. E quindi si può usare la topologia. Neighborhood. |
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Type theory Frege -concept and function- Ernst Zermelo in his (1908) A new proof of the possibility of a well-ordering (published at the same time he published "the first axiomatic set theory")[18] laid claim to prior discovery of the antinomy in Cantor's naive set theory. He states: "And yet, even the elementary form that Russell9 gave to the set-theoretic antinomies could have persuaded them [J. König, Jourdain, F. Bernstein] that the solution of these difficulties is not to be sought in the surrender of well-ordering but only in a suitable restriction of the notion of set" In 1923, Ludwig Wittgenstein proposed to "dispose" of Russell's paradox as follows: Thus, simple TT (type theory) and ZFC could now be regarded as systems that 'talk' essentially about the same intended objects. The main difference is that TT relies on a strong higher-order logic, while Zermelo employed second-order logic, and ZFC can also be given a first-order formulation. The first-order 'description' of the cumulative hierarchy is much weaker, as is shown by the existence of denumerable models (Skolem paradox), but it enjoys some important advantages." In mathematics, logic, and computer science, a type system is a formal system in which every term has a "type" which defines its meaning and the operations that may be performed on it. In a system of type theory, a term is opposed to a type. For example, 4, 2+2, and 2*2 are all separate terms with the type {nat} for natural numbers. Traditionally, the term is followed by a colon and its type, such as 2:{nat} - this means that the number 2 is of type {nat} . Beyond this opposition and syntax, only few can be said about types in this generality, but often, they are interpreted as some kind of collection (not necessarily sets) of the values that the term might evaluate to. It is usual to denote terms by e and types by \tau . How terms and types are shaped depends on the particular type system and is made precise by some syntax and additional restrictions of well-formedness. |
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Interpretation For example, in the language of rings, there are constant symbols 0 and 1, two binary function symbols + and ·, and no binary relation symbols. (Here the equality relation is taken as a logical constant.) Again, we might define a first-order language L, as consisting of individual symbols a, b, and c; predicate symbols F, G, H, I and J; variables x, y, z; no function letters; no sentential symbols. Given a signature σ, the corresponding formal language is known as the set of σ-formulas. Each σ-formula is built up out of atomic formulas by means of logical connectives; atomic formulas are built from terms using predicate symbols. The formal definition of the set of σ-formulas proceeds in the other direction: first, terms are assembled from the constant and function symbols together with the variables. Then, terms can be combined into an atomic formula using a predicate symbol (relation symbol) from the signature or the special predicate symbol "=" for equality (see the section "Interpreting equality" below). Finally, the formulas of the language are assembled from atomic formulas using the logical connectives and quantifiers. Interpretations of a first-order language
The information specified in the interpretation provides enough information to give a truth value to any atomic formula, after each of its free variables, if any, has been replaced by an element of the domain. The truth value of an arbitrary sentence is then defined inductively using the T-schema, which is a definition of first-order semantics developed by Alfred Tarski. The T-schema interprets the logical connectives using truth tables, as discussed above. Thus, for example, φ & ψ is satisfied if and only if both φ and ψ are satisfied. This leaves the issue of how to interpret formulas of the form ∀ x φ(x) and ∃ x φ(x). The domain of discourse forms the range for these quantifiers. The idea is that the sentence ∀ x φ(x) is true under an interpretation exactly when every substitution instance of φ(x), where x is replaced by some element of the domain, is satisfied. The formula ∃ x φ(x) is satisfied if there is at least one element d of the domain such that φ(d) is satisfied. An extensional definition of a concept or term formulates its meaning by specifying its extension, that is, every object that falls under the definition of the concept or term in question. For example, an extensional definition of the term "nation of the world" might be given by listing all of the nations of the world, or by giving some other means of recognizing the members of the corresponding class. An explicit listing of the extension, which is only possible for finite sets and only practical for relatively small sets, is a type of enumerative definition REF: interpretation |
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JOURNAL ARTICLE |
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La base delle stringhe di soggetto è la logica del primo ordine a più variabili. We have here a function whose value is always a truth-value. We called such functions of one argument concepts. we call such functions of two arguments relations. Frege, Gottlob. Function und Begriff: Vortrag, gehalten in der Sitzung vom 9. Januar 1891 der Jenaischen Gesellschaft für Medicin und Naturwissenschaft. Рипол Классик, 1891. |
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De Morgan The supreme law of syllogism of three terms, the law which govern all of possible case, and which every variety of expression must be brought before inference can be made, is this; any relation of X to Y compound with any relation of Y to Z, give a relation of X to Z (compound ratio of magnitude of Euclide) Let the premise X.LY, Z..MY . These premises are identical to X..lY and Y..M^-1Z of which all the inference is X..lM^-1Z When by the word syllogism we are agree to mean a composition of two relations into one, we open the field in such a manner that the invention of the middle term, and the component relations which give the compound relation of the conclusion, is seen to costitute the act of mind, which ia always occurring in the effort of the reasoning power. It is to algebra that we must look for the most habitual use of logical form. Not that onymatic relations are found in frequent occurrance: but so soon as the syllogism is consider under the aspect of combination of relations, it became clear that there is more of syllogism, and more of its variety, in algebra than in other subject whatever, though the matter of relations, pure quantity, is itself a small variety Ref: de Morgan, A. "On the Logic of Syllogism: IV; and on the Logic of Relations,“transactions of the cambridge Philosophical Society”, 10: 331-358; ora in A. de Morgan." On the Syllogism and Other Logical Writings (1860): 208-246. |
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Ok. Il lun 28 set 2020, 09:33 ellegi69 [email protected] ha scritto:
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From APUPA created by bertanimauro: bertanimauro/APUPA#5
Se le righe delle matrici sono i coefficienti dei polinomi e l'hyperdeterminant trasforma le righe in altri coefficienti, tali coefficienti sono delle ombre, ossia delle costanti che nascondono polinomi.
Come determinante di laplace.
Già notazione di Leibniz di chiamare i coefficenti della matrice a_ij è una forma di ombra. Poi gli a_ij nascondono le incognite e le sottoparti dei determinanti nascondono parti di polinomi e il determinante nasconde un sistema di polinomi
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