ConseggioLigure/seed-instruct-lij-eng huggingface dataset:
Traduxi sto testo da-o zeneise à l’ingleise:
Ancheu çerti filòsofi da matematica çercan de dâ conto de sta forma de reçerca e di seu produti coscì pe comme son, tanto che di atri sottolinian un ròllo ch’o va a-o de la da sola interpretaçion fin à l’analixi critica.
A traduçion in ingleise do testo a l’é:
Today, some philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that goes beyond simple interpretation to critical analysis.
Quæ a l’é a traduçion ingleise de sto testo?
A filosofia grega in sciâ matematica a l’ea fortemente influensâ da-o studio da geometria.
A traduçion in ingleise do testo a l’é:
Greek philosophy on mathematics was strongly influenced by their study of geometry.
Traduxi sto testo da-o zeneise à l’ingleise:
«Donca o 3, pe fâ un exempio, o representava unna moltitudine de unitæ, e donca o no l’ea "davei" un numero».
A traduçion in ingleise do testo a l’é:
Therefore, 3, for example, represented a certain multitude of units, and was thus not "truly" a number.
Traduxi sto testo da-o ligure à l’ingleise:
Ciù in là into tempo quelle primme idee di Greghi in scî numei en stæte imbösæ da-a scoverta de l’irraçionalitæ da reixe quaddrâ do numeo doî.
A traduçion in ingleise do testo a l’é:
These earlier Greek ideas of numbers were later upended by the discovery of the irrationality of the square root of two.
Traduxi in lengua ingleise:
Segondo a lezzendia, i discepoli de Pitagora ean mai tanto traumatizzæ da sta scoverta ch’avieivan ammassou Ippaso de mòddo ch’o no pòsse fâ conosce a seu idea eretica.
A traduçion in ingleise do testo a l’é:
According to legend, fellow Pythagoreans were so traumatized by this discovery that they murdered Hippasus to stop him from spreading his heretical idea.
Traduxi da-o ligure à l’ingleise:
O l’é un enigma profondo che da unna parte e veitæ matematiche pan avei unn’inevitabilitæ irrexistibile, da ma l’atro a fonte da seu «veridiçitæ» a l’arresta eluxiva.
A traduçion in ingleise do testo a l’é:
It is a profound puzzle that on the one hand mathematical truths seem to have a compelling inevitability, but on the other hand the source of their "truthfulness" remains elusive.
Traduxi da-o zeneise à l’ingleise:
Con l’anâ avanti do secolo, l’interesse do comenso o s’é allargou à unn’esploraçion averta di asciömi fondamentæ da matematica, dæto che o metodo de studio asciomatico o l’ea stæto dæto pe descondou fin da-i tempi d’Euclide, do 300 a.C. pöcassæ, comme base naturale da matematica.
A traduçion in ingleise do testo a l’é:
As the century unfolded, the initial focus of concern expanded to an open exploration of the fundamental axioms of mathematics, the axiomatic approach having been taken for granted since the time of Euclid around 300 BCE as the natural basis for mathematics.
Ti peu tradue sto testo in ingleise?
Inta matematica, coscì comme inta fixica, l’ea nasciuo de idee neuve e inaspëtæ, e da lì a no guæi saieiva arrivou di cangi scignificativi.
A traduçion in ingleise do testo a l’é:
In mathematics, as in physics, new and unexpected ideas had arisen and significant changes were coming.
Comm’à l’é a traduçion ingleise de sto testo?
No penso che e difficoltæ tra a filosofia e a matematica clascica seggian difficoltæ autentiche; penso ascì che e interpretaçioin filosòfiche da matematica che ne ven offerto de chì e de là seggian sbaliæ, e che a coscì dita «interpretaçion filosofica» a segge pròpio quello do quæ a matematica a no l’à de beseugno.
A traduçion in ingleise do testo a l’é:
I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that "philosophical interpretation" is just what mathematics doesn't need.
Traduxi da-o ligure à l’ingleise:
Tanti matematichi de profescion en stæti matematichi realisti; se veddeivan à liatri mæximi comme descrovitoî d’oggetti che se presentan in sce unna base naturale.
A traduçion in ingleise do testo a l’é:
Many working mathematicians have been mathematical realists; they see themselves as discoverers of naturally occurring objects.
Traduxi in ingleise:
Çerti princippi (p.e. che pe ògni doî oggetti che segge unna serie d’oggetti ch’a consciste pròpio de quelli doî oggetti determinæ) se peuan interpretâ comme direttamente vëi, ma in sciâ base de sti prinçippi a continua propòsta d’ipòtexi a porriæ revelâse indeçiddibile.
A traduçion in ingleise do testo a l’é:
Certain principles (e.g., for any two objects, there is a collection of objects consisting of precisely those two objects) could be directly seen to be true, but the continuum hypothesis conjecture might prove undecidable just on the basis of such principles.
Ti peu tradue sto testo in ingleise?
Tanto a barma de Platon comme o platonismo en stæte de connescioin sensæ e no superfiçiale, da-o momento che e idee de Platon ne vegnivan, e ean influensæ, da-i pitagòrichi da Greçia antiga, ch’ean ben ben conosciui, e che creddeivan che o mondo o fïse – letteralmente – generou da-i numei.
A traduçion in ingleise do testo a l’é:
Both Plato's cave and Platonism have meaningful, not just superficial connections, because Plato's ideas were preceded and probably influenced by the hugely popular Pythagoreans of ancient Greece, who believed that the world was, quite literally, generated by numbers.
Traduxi da-o ligure à l’ingleise:
St’idea a mostra de someggianse con tante cöse stæte dite da Husserl in sciâ matematica, e sostëgnan l’idea de Kant che a matematica a segge scintetica a priori.)
A traduçion in ingleise do testo a l’é:
This view bears resemblances to many things Husserl said about mathematics, and supports Kant's idea that mathematics is synthetic a priori.)
Traduxi da-o zeneise à l’ingleise:
O platonismo integrale o l’é unna variante moderna do platonismo, in reaçion a-o fæto che se peu demostrâ l’existensa de varri insemmi d’entitæ matematica à segonda di asciömi e de regole d’inferensa deuviæ (comme a leze do mezo escluso e quello da çernia).
A traduçion in ingleise do testo a l’é:
Full-blooded Platonism is a modern variation of Platonism, which is in reaction to the fact that different sets of mathematical entities can be proven to exist depending on the axioms and inference rules employed (for instance, the law of the excluded middle, and the axiom of choice).
Traduxi da-o ligure à l’ingleise:
O realismo insemmistico (dito ascì platonismo insemmistico), poxiçion defeisa da-a Penelope Maddy, a l’é l’opinion pe-a quæ a teoria di insemmi a l’à da fâ con un unico universo d’insemmi.
A traduçion in ingleise do testo a l’é:
Set-theoretic realism (also set-theoretic Platonism) a position defended by Penelope Maddy, is the view that set theory is about a single universe of sets.
Traduxi sto testo in ingleise:
An attribuio o paradòsso a-a «çircolaritæ viçiosa» e pe affrontâlo an construto quella ch’an ciammou teoria di tipi rammificæ.
A traduçion in ingleise do testo a l’é:
They attributed the paradox to "vicious circularity" and built up what they called ramified type theory to deal with it.
Quæ a l’é a traduçion ingleise de sto testo?
Fiña o Russel o diva che st’assiöma o no l’ea basou in sciâ lògica.
A traduçion in ingleise do testo a l’é:
Even Russell said that this axiom did not really belong to logic.
Traduxi in lengua ingleise:
Frege o l’à avuo de beseugno da quinta regola basica pe poei dâ unna definiçion espliçita di numei, ma tutte e propietæ di numei peuan vegnî fæte derivâ da-o prinçipio de Hume.
A traduçion in ingleise do testo a l’é:
Frege required Basic Law V to be able to give an explicit definition of the numbers, but all the properties of numbers can be derived from Hume's principle.
Quæ a l’é a traduçion ingleise de sto testo?
Ma o permette a-o matematico de profescion d’anâ avanti co-o seu travaggio e de lasciâ a trattaçion de sti problemi a-o filòsofo ò a-o sciensiou.
A traduçion in ingleise do testo a l’é:
But it does allow the working mathematician to continue in his or her work and leave such problems to the philosopher or scientist.
Traduxi in lengua ingleise:
Hilbert o voeiva demostrâ a coerensa di scistemi da matematica co-o partî da l’ipòtexi che l’«aritmetica finitäia» (saieiva à dî un sottoscistema de l’aritmetica abituale di numei intreghi poxitivi, stæto çernuo pe-o seu carattere filosoficamente no controverso) a fïse coerente.
A traduçion in ingleise do testo a l’é:
Hilbert aimed to show the consistency of mathematical systems from the assumption that the "finitary arithmetic" (a subsystem of the usual arithmetic of the positive integers, chosen to be philosophically uncontroversial) was consistent.
Traduxi sto testo in ingleise:
Coscì, pe demostrâ che tutti i scistemi asciòmichi da matematica seggian coerenti comme matëia de fæto, beseugna pe primma cösa suppoñe a coerensa de un scistema matematico ch’o segge ciù fòrte che o scistema za stæto demostrou comme coerente.
A traduçion in ingleise do testo a l’é:
Thus, in order to show that any axiomatic system of mathematics is in fact consistent, one needs to first assume the consistency of a system of mathematics that is in a sense stronger than the system to be proven consistent.
Traduxi da-o ligure à l’ingleise:
Atri formalisti, comme Rudolf Carnap, Alfred Tarski e Haskell Curry conscideran a matematica comme l’investigaçion do scistema di asciömi formali.
A traduçion in ingleise do testo a l’é:
Other formalists, such as Rudolf Carnap, Alfred Tarski, and Haskell Curry, considered mathematics to be the investigation of formal axiom systems.
Traduxi da-o zeneise à l’ingleise:
Ciù tanti l’é i zeughi che studiemmo, e megio a l’é.
A traduçion in ingleise do testo a l’é:
The more games we study, the better.
Traduxi sto testo in ingleise:
A critica prinçipâ do formalismo a l’é che e idee matematiche effettive ch’interessa i matematichi en ben ben lontañe da-i zeughi de manipolaçion de strinche stæti mensunæ chì de d’ato.
A traduçion in ingleise do testo a l’é:
The main critique of formalism is that the actual mathematical ideas that occupy mathematicians are far removed from the string manipulation games mentioned above.
Traduxi sto testo da-o zeneise à l’ingleise:
Brower, o fondatô do movimento, o sostegniva che i oggetti da matematica nascian da-e forme a priori da-e voliçioin ligæ a-a perçeçion da matëia empirica.
A traduçion in ingleise do testo a l’é:
Brouwer, the founder of the movement, held that mathematical objects arise from the a priori forms of the volitions that inform the perception of empirical objects.
Quæ a l’é a traduçion ingleise de sto testo?
L’asciöma da çernia o ven refuou ascì inta ciù parte de teorie intuiçionistiche di insemme, sciben ch’o vëgne açettou inte quarcheduña de seu verscioin.
A traduçion in ingleise do testo a l’é:
The axiom of choice is also rejected in most intuitionistic set theories, though in some versions it is accepted.
Traduxi in ingleise:
Segondo sta vixon, a matematica a l’é un exerçiçio de l’intuiçion de l’òmmo, e no un zeugo con di scimboli denottæ da un çerto scignificato.
A traduçion in ingleise do testo a l’é:
In this view, mathematics is an exercise of the human intuition, not a game played with meaningless symbols.
Traduxi sto testo in ingleise:
Da mæxima, tutti i atri numei intreghi en definii da-a seu poxiçion drento de unna struttua, a linia di numei.
A traduçion in ingleise do testo a l’é:
Likewise all the other whole numbers are defined by their places in a structure, the number line.
Traduxi in ingleise:
À tutte e mainee, a seu affermaçion de base a fa referensa solo a-o tipo de entitæ ligou à un oggetto matematico, e no a-o tipo d’existensa di oggetti ò de struttue matematiche (dita à un atro mòddo, no a-a seu ontologia).
A traduçion in ingleise do testo a l’é:
However, its central claim only relates to what kind of entity a mathematical object is, not to what kind of existence mathematical objects or structures have (not, in other words, to their ontology).
Traduxi sto testo da-o zeneise à l’ingleise:
Se pensa che e struttue aggian unn’existensa reale, ma astræta e inmateriale.
A traduçion in ingleise do testo a l’é:
Structures are held to have a real but abstract and immaterial existence.
Comm’à l’é a traduçion ingleise de sto testo?
Se pensa che e struttue existan fin à tanto che ghe segge quarche scistema concreto ch’o pòsse exemplificâle.
A traduçion in ingleise do testo a l’é:
Structures are held to exist inasmuch as some concrete system exemplifies them.
Traduxi sto testo da-o ligure à l’ingleise:
Coscì comme o nominalismo, o pensceo post rem o denega l’existensa d’oggetti matematichi astræti con de propietæ che van a-o de là da seu poxiçion inte unna struttua relaçionale.
A traduçion in ingleise do testo a l’é:
Like nominalism, the post rem approach denies the existence of abstract mathematical objects with properties other than their place in a relational structure.
Traduxi sto testo da-o zeneise à l’ingleise:
L’opinion a l’é che a matematica a no segge universale a ch’a no l’existe inte nisciun senso reale feua che into çervello de l’òmmo.
A traduçion in ingleise do testo a l’é:
It is held that mathematics is not universal and does not exist in any real sense, other than in human brains.
Traduxi sto testo in ingleise:
À tutte e mainee, a mente de l’òmmo a no l’à de preteise particolæ in sciâ realtæ ò di metodi pe costrula da-a matematica.
A traduçion in ingleise do testo a l’é:
However, the human mind has no special claim on reality or approaches to it built out of math.
Traduxi sto testo da-o zeneise à l’ingleise:
A trattaçion de sta prospettiva ciù façile da attrovâ, e ciù conosciua (into ben e into mâ), a l’é Where Mathematics Come From, de George Lakoff e Rafael E. Núñez.
A traduçion in ingleise do testo a l’é:
The most accessible, famous, and infamous treatment of this perspective is Where Mathematics Comes From, by George Lakoff and Rafael E. Núñez.
Traduxi in ingleise:
Franklin, James (2014), «An Aristotelian Realist Philosophy of Mathematics», Palgrave Macmillan, Basingstoke; Franklin, James (2021), «Mathematics as a science of non-abstract reality: Aristotelian realist philosophies of mathematics», Foundations of Science 25.
A traduçion in ingleise do testo a l’é:
Franklin, James (2014), "An Aristotelian Realist Philosophy of Mathematics", Palgrave Macmillan, Basingstoke; Franklin, James (2021), "Mathematics as a science of non-abstract reality: Aristotelian realist philosophies of mathematics," Foundations of Science 25.
Traduxi da-o zeneise à l’ingleise:
Anche l’aritmetica euclidea stæta sviluppâ da John Penn Mayberry into seu libbro The Foundations of Mathematics inte Theory of Sets a l’é compreisa inta tradiçion realista aristotelica.
A traduçion in ingleise do testo a l’é:
The Euclidean arithmetic developed by John Penn Mayberry in his book The Foundations of Mathematics in the Theory of Sets also falls into the Aristotelian realist tradition.
Traduxi sto testo in ingleise:
Edmund Husserl, into primmo volumme de Logical Investigations, ciammou «protolegòmeni da lògica pua», o l’à criticou à fondo o psicologismo e o l’à çercou de piggiâne e distanse.
A traduçion in ingleise do testo a l’é:
Edmund Husserl, in the first volume of his Logical Investigations, called "The Prolegomena of Pure Logic", criticized psychologism thoroughly and sought to distance himself from it.
Traduxi da-o ligure à l’ingleise:
A sostëgne l’existensa d’entitæ matematiche comme megio spiegaçion de l’esperiensa, desreixando à sto mòddo l’idea che a matematica a segge distinta da-e atre sciense.
A traduçion in ingleise do testo a l’é:
It argues for the existence of mathematical entities as the best explanation for experience, thus stripping mathematics of being distinct from the other sciences.
Traduxi da-o zeneise à l’ingleise:
Sta conçeçion a l’é nasciua da l’affermaçion, vegnua delongo ciù popolare a-a fin do secolo vinti, che no fïse poscibile demostrâ l’existensa de nisciun di fondamenti da matematica.
A traduçion in ingleise do testo a l’é:
This grew from the increasingly popular assertion in the late 20th century that no one foundation of mathematics could be ever proven to exist.
Traduxi da-o zeneise à l’ingleise:
Un argomento matematico o peu trasmette de fäscitæ da-a concluxon a-e premisse coscì comme o peu trasmette de veitæ da-e premisse a-a concluxon.
A traduçion in ingleise do testo a l’é:
A mathematical argument can transmit falsity from the conclusion to the premises just as well as it can transmit truth from the premises to the conclusion.
Traduxi in lengua ingleise:
O l’à dæto unn’espoxiçion detaggiâ de sto fæto inte New Directions.
A traduçion in ingleise do testo a l’é:
He gave a detailed argument for this in New Directions.
Traduxi da-o ligure à l’ingleise:
Se a matematica a l’é tanto empirica comme e atre sciense, aloa sto fæto o vorrieiva dî che i seu resultati son tanto fallibili e contingenti comme quelle de atre.
A traduçion in ingleise do testo a l’é:
If mathematics is just as empirical as the other sciences, then this suggests that its results are just as fallible as theirs, and just as contingent.
Traduxi sto testo da-o ligure à l’ingleise:
Pe unna filosofia da matematica ch’a çerche de scompassâ çertidun tra i difetti de metodologie de Quine e Gödel, piggiando di aspeti da ciaschedun de sti doî, se remanda à Realism in Mathematics de Penelope Maddy.
A traduçion in ingleise do testo a l’é:
For a philosophy of mathematics that attempts to overcome some of the shortcomings of Quine and Gödel's approaches by taking aspects of each see Penelope Maddy's Realism in Mathematics.
Ti peu tradue sto testo in ingleise?
O l’à comensou co-o «betweennes» di assiömi de Hilbert pe caratterizâ o spaçio sensa coordinâlo, e dapeu o l’à azzonto de relaçioin extra tra i ponti pe fâ o travaggio che in preçedensa o l’ea fæto da-i campi vettoriali.
A traduçion in ingleise do testo a l’é:
He started with the "betweenness" of Hilbert's axioms to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by vector fields.
Traduxi sto testo da-o ligure à l’ingleise:
Inte sto senso, no gh’é de problemi metafixichi ò epistemiològichi speciali da matematica.
A traduçion in ingleise do testo a l’é:
By this account, there are no metaphysical or epistemological problems special to mathematics.
Quæ a l’é a traduçion ingleise de sto testo?
À tutti i mòddi, tanto che inte unna vixon empirista a valutaçion a l’é unna speçie de confronto co-a «realtæ», i construttivisti soçiali fan nottâ che a direçion da reçerca matematica a l’é dettâ da-e mòdde do gruppo soçiale ch’o ê manifesta ò da-i beseugni da soçietæ ch’o â finançia.
A traduçion in ingleise do testo a l’é:
However, while on an empiricist view the evaluation is some sort of comparison with "reality", social constructivists emphasize that the direction of mathematical research is dictated by the fashions of the social group performing it or by the needs of the society financing it.
Ti peu tradue sto testo in ingleise?
Ma i costruttivisti soçiali sostëgnan che a matematica a segge fondâ de fæto in sce unna gròssa inçertessa: da maniman che a prattica da matematica a se sviluppa, a scituaçion da matematica ciù antiga a vëgne missa in dubio e a vëgne corretta inta mesua inta quæ a communitæ scientifica a ô domanda e a ô dexidia.
A traduçion in ingleise do testo a l’é:
But social constructivists argue that mathematics is in fact grounded by much uncertainty: as mathematical practice evolves, the status of previous mathematics is cast into doubt, and is corrected to the degree it is required or desired by the current mathematical community.
Comm’à l’é a traduçion ingleise de sto testo?
A natua soçiale da matematica a l’é ben missa in evidensa da-e seu subcoltue.
A traduçion in ingleise do testo a l’é:
The social nature of mathematics is highlighted in its subcultures.
Traduxi sto testo da-o zeneise à l’ingleise:
Pe-i costruttivisti soçiæ o processo de «fâ matematica» a l’é a creaçion effettiva do scignificato, mentre i realisti soçiæ ghe veddan unna carensa ò da capaçitæ de l’òmmo d’astrae, ò donca do bias cognitivo de l’òmmo, ò ancon de l’intelligensa collettiva di studioxi da matematica comme un impedimento a-a comprenscion de un universo reale d’oggetti da matematica.
A traduçion in ingleise do testo a l’é:
Social constructivists see the process of "doing mathematics" as actually creating the meaning, while social realists see a deficiency either of human capacity to abstractify, or of human's cognitive bias, or of mathematicians' collective intelligence as preventing the comprehension of a real universe of mathematical objects.
Ti peu tradue sto testo in ingleise?
À tutti i mòddi, i metodi sviluppæ da Frege e Tarski pe-o studio do lenguaggio matematico en stæti ben ben esteixe da l’allievo de Tarski Tuttavia, i metodi sviluppati da Frege e Tarski, Richard Montague, da di atri linguisti che travaggian inta semantica formale pe mostrâ che a distinçion tra o lenguaggio matematico e quella naturale a porrieiva no ëse coscì grande comm’a pâ.
A traduçion in ingleise do testo a l’é:
However, the methods developed by Frege and Tarski for the study of mathematical language have been extended greatly by Tarski's student Richard Montague and other linguists working in formal semantics to show that the distinction between mathematical language and natural language may not be as great as it seems.
Traduxi sto testo in ingleise:
L’affermaçion che «tutte» e entitæ postulæ inte teorie scientifiche, compreiso i numei, dovieivan ëse açettæ comme vee a l’é giustificâ da l’olismo de conferma.
A traduçion in ingleise do testo a l’é:
The assertion that "all" entities postulated in scientific theories, including numbers, should be accepted as real is justified by confirmation holism.
Traduxi da-o ligure à l’ingleise:
Field o l’à sviluppou a seu vixon into fiçionalismo.
A traduçion in ingleise do testo a l’é:
Field developed his views into fictionalism.
Ti peu tradue sto testo in ingleise?
L’argomento o se basa in sce l’idea che se pòsse dâ un bon resoconto naturalistico in scî proçesci de pensceo à livello de proçesci do cervello pe-o raxonamento matematico insemme à tutto o resto.
A traduçion in ingleise do testo a l’é:
The argument hinges on the idea that a satisfactory naturalistic account of thought processes in terms of brain processes can be given for mathematical reasoning along with everything else.
Traduxi da-o ligure à l’ingleise:
Unn’atra linia de defeisa a l’é de sostegnî che i oggetti astræti seggian relevanti pe-o raxonamento matematico à un mòddo ch’o no l’é ligou a-a causa e ch’o no l’é analogo a-a perçeçion.
A traduçion in ingleise do testo a l’é:
Another line of defense is to maintain that abstract objects are relevant to mathematical reasoning in a way that is non-causal, and not analogous to perception.
Traduxi in lengua ingleise:
Comme exempio, an portou doe preuve de l’irraçionalitæ de .
A traduçion in ingleise do testo a l’é:
By way of example, they provide two proofs of the irrationality of .
Ti peu tradue sto testo in ingleise?
Paul Erdős o l’ea ben conosciuo pe-a seu noçion de un ipotetico «Libbro» ch’o l’aviæ dovuo contegnî e preuve matematiche ciù belle e e elegante.
A traduçion in ingleise do testo a l’é:
Paul Erdős was well known for his notion of a hypothetical "Book" containing the most elegant or beautiful mathematical proofs.
Traduxi sto testo in ingleise:
O dualismo e o monismo en e doe scheue de pensceo prinçipæ in sciô problema mente-còrpo, ascì ben che l’é vegnuo à formâse de vixoin intermedie che no peuan ëse compreise do tutto inte uña de doe categorie.
A traduçion in ingleise do testo a l’é:
Dualism and monism are the two central schools of thought on the mind–body problem, although nuanced views have arisen that do not fit one or the other category neatly.
Comm’à l’é a traduçion ingleise de sto testo?
Hart, W.D. (1996) «Dualism», in Samuel Guttenplan (org) A Companion to the Philosophy of Mind, Blackwell, Oxford, 265-7.
A traduçion in ingleise do testo a l’é:
Hart, W.D. (1996) "Dualism", in Samuel Guttenplan (org) A Companion to the Philosophy of Mind, Blackwell, Oxford, 265-7.