ConseggioLigure/seed-instruct-eng-lij huggingface dataset:
Translate from English to Ligurian:
For Quine, as for Wittgenstein and Austin, meaning is not something that is associated with a single word or sentence, but is rather something that, if it can be attributed at all, can only be attributed to a whole language.
The Ligurian (Genoese) translation is:
Pe Quine, comme pe Wittgenstein e Austin, o scignificato o no l’é quarcösa ch’o s’assòçia à unna sola paròlla ò frase, ma o l’é ciuffito quacösa che, s’a peu vegnî attribuia, a peu vegnî attribuia solo che à un lenguaggio intrego.
Translate from English to Ligurian (Genoese dialect):
The specific instances of vagueness that most interest philosophers of language are those where the existence of "borderline cases" makes it seemingly impossible to say whether a predicate is true or false.
The Ligurian (Genoese) translation is:
I caxi speçifichi de vaghessa ch’interessan ciù tanto i filòsofi do lenguaggio en quelli donde l’existensa de «caxi limite» a rende impossibile, à l’apparensa, dî se un predicato o l’é veo ò fäso.
Translate from English to Ligurian (Genoese dialect):
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics.
The Ligurian (Genoese) translation is:
A filosofia matematica a l’é quella ramma da filosofia ch’a studia e ipòtexi, i fondamenti e e implicaçioin de scense matematiche.
Translate this sentence to Genoese:
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.
The Ligurian (Genoese) translation is:
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.
Can you translate this text to Ligurian?
Greek philosophy on mathematics was strongly influenced by their study of geometry.
The Ligurian (Genoese) translation is:
A filosofia grega in sciâ matematica a l’ea fortemente influensâ da-o studio da geometria.
Translate from English to Genoese:
Therefore, 3, for example, represented a certain multitude of units, and was thus not "truly" a number.
The Ligurian (Genoese) translation is:
«Donca o 3, pe fâ un exempio, o representava unna moltitudine de unitæ, e donca o no l’ea "davei" un numero».
Translate to Genoese:
These earlier Greek ideas of numbers were later upended by the discovery of the irrationality of the square root of two.
The Ligurian (Genoese) translation is:
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.
What’s the Genoese translation of this sentence?
According to legend, fellow Pythagoreans were so traumatized by this discovery that they murdered Hippasus to stop him from spreading his heretical idea.
The Ligurian (Genoese) translation is:
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.
Can you translate this text to Ligurian?
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.
The Ligurian (Genoese) translation is:
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.
Translate from English to Ligurian:
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.
The Ligurian (Genoese) translation is:
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.
Translate from English to Ligurian (Genoese dialect):
In mathematics, as in physics, new and unexpected ideas had arisen and significant changes were coming.
The Ligurian (Genoese) translation is:
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.
Translate to Ligurian:
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.
The Ligurian (Genoese) translation is:
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.
Translate from English to Ligurian:
Many working mathematicians have been mathematical realists; they see themselves as discoverers of naturally occurring objects.
The Ligurian (Genoese) translation is:
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.
Translate from English to Genoese:
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.
The Ligurian (Genoese) translation is:
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.
What’s the Genoese translation of this sentence?
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.
The Ligurian (Genoese) translation is:
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.
What’s the Ligurian translation of this sentence?
This view bears resemblances to many things Husserl said about mathematics, and supports Kant's idea that mathematics is synthetic a priori.)
The Ligurian (Genoese) translation is:
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.)
Translate to Genoese:
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).
The Ligurian (Genoese) translation is:
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).
Translate to Ligurian:
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.
The Ligurian (Genoese) translation is:
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.
Translate this sentence to Ligurian:
They attributed the paradox to "vicious circularity" and built up what they called ramified type theory to deal with it.
The Ligurian (Genoese) translation is:
An attribuio o paradòsso a-a «çircolaritæ viçiosa» e pe affrontâlo an construto quella ch’an ciammou teoria di tipi rammificæ.
Translate from English to Ligurian (Genoese dialect):
Even Russell said that this axiom did not really belong to logic.
The Ligurian (Genoese) translation is:
Fiña o Russel o diva che st’assiöma o no l’ea basou in sciâ lògica.
Translate to Ligurian (Genoese):
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.
The Ligurian (Genoese) translation is:
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.
What’s the Ligurian translation of this sentence?
But it does allow the working mathematician to continue in his or her work and leave such problems to the philosopher or scientist.
The Ligurian (Genoese) translation is:
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.
Translate to Ligurian:
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.
The Ligurian (Genoese) translation is:
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.
Translate to Ligurian (Genoese):
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.
The Ligurian (Genoese) translation is:
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.
What’s the Genoese translation of this sentence?
Other formalists, such as Rudolf Carnap, Alfred Tarski, and Haskell Curry, considered mathematics to be the investigation of formal axiom systems.
The Ligurian (Genoese) translation is:
Atri formalisti, comme Rudolf Carnap, Alfred Tarski e Haskell Curry conscideran a matematica comme l’investigaçion do scistema di asciömi formali.
Translate this sentence to Ligurian:
The more games we study, the better.
The Ligurian (Genoese) translation is:
Ciù tanti l’é i zeughi che studiemmo, e megio a l’é.
Can you translate this text to Ligurian?
The main critique of formalism is that the actual mathematical ideas that occupy mathematicians are far removed from the string manipulation games mentioned above.
The Ligurian (Genoese) translation is:
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.
Translate from English to Genoese:
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.
The Ligurian (Genoese) translation is:
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.
Translate this sentence to Genoese:
The axiom of choice is also rejected in most intuitionistic set theories, though in some versions it is accepted.
The Ligurian (Genoese) translation is:
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.
Translate to Genoese:
In this view, mathematics is an exercise of the human intuition, not a game played with meaningless symbols.
The Ligurian (Genoese) translation is:
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.
Translate to Ligurian (Genoese):
Likewise all the other whole numbers are defined by their places in a structure, the number line.
The Ligurian (Genoese) translation is:
Da mæxima, tutti i atri numei intreghi en definii da-a seu poxiçion drento de unna struttua, a linia di numei.
What’s the Genoese translation of this sentence?
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).
The Ligurian (Genoese) translation is:
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).
Translate to Genoese:
Structures are held to have a real but abstract and immaterial existence.
The Ligurian (Genoese) translation is:
Se pensa che e struttue aggian unn’existensa reale, ma astræta e inmateriale.
Translate from English to Ligurian:
Structures are held to exist inasmuch as some concrete system exemplifies them.
The Ligurian (Genoese) translation is:
Se pensa che e struttue existan fin à tanto che ghe segge quarche scistema concreto ch’o pòsse exemplificâle.
Translate from English to Ligurian (Genoese dialect):
Like nominalism, the post rem approach denies the existence of abstract mathematical objects with properties other than their place in a relational structure.
The Ligurian (Genoese) translation is:
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.
Translate this sentence to Genoese:
It is held that mathematics is not universal and does not exist in any real sense, other than in human brains.
The Ligurian (Genoese) translation is:
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.
Translate from English to Ligurian:
However, the human mind has no special claim on reality or approaches to it built out of math.
The Ligurian (Genoese) translation is:
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.
Translate from English to Genoese:
The most accessible, famous, and infamous treatment of this perspective is Where Mathematics Comes From, by George Lakoff and Rafael E. Núñez.
The Ligurian (Genoese) translation is:
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.
Translate from English to Genoese:
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.
The Ligurian (Genoese) translation is:
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.
Translate to Genoese:
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.
The Ligurian (Genoese) translation is:
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.
Translate to Ligurian (Genoese):
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.
The Ligurian (Genoese) translation is:
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.
Translate from English to Ligurian (Genoese dialect):
It argues for the existence of mathematical entities as the best explanation for experience, thus stripping mathematics of being distinct from the other sciences.
The Ligurian (Genoese) translation is:
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.
Translate to Ligurian (Genoese):
This grew from the increasingly popular assertion in the late 20th century that no one foundation of mathematics could be ever proven to exist.
The Ligurian (Genoese) translation is:
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.
Translate from English to Genoese:
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.
The Ligurian (Genoese) translation is:
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.
Can you translate this text to Ligurian?
He gave a detailed argument for this in New Directions.
The Ligurian (Genoese) translation is:
O l’à dæto unn’espoxiçion detaggiâ de sto fæto inte New Directions.
What’s the Genoese translation of this sentence?
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.
The Ligurian (Genoese) translation is:
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.
Translate from English to Ligurian (Genoese dialect):
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.
The Ligurian (Genoese) translation is:
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.
What’s the Ligurian translation of this sentence?
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.
The Ligurian (Genoese) translation is:
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.
Translate to Ligurian (Genoese):
By this account, there are no metaphysical or epistemological problems special to mathematics.
The Ligurian (Genoese) translation is:
Inte sto senso, no gh’é de problemi metafixichi ò epistemiològichi speciali da matematica.
Translate from English to Genoese:
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.
The Ligurian (Genoese) translation is:
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.
Translate from English to Genoese:
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.
The Ligurian (Genoese) translation is:
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.
Translate from English to Genoese:
The social nature of mathematics is highlighted in its subcultures.
The Ligurian (Genoese) translation is:
A natua soçiale da matematica a l’é ben missa in evidensa da-e seu subcoltue.
Can you translate this text to Ligurian?
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.
The Ligurian (Genoese) translation is:
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.
Translate this sentence to Ligurian:
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.
The Ligurian (Genoese) translation is:
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.
Translate from English to Ligurian (Genoese dialect):
The assertion that "all" entities postulated in scientific theories, including numbers, should be accepted as real is justified by confirmation holism.
The Ligurian (Genoese) translation is:
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.
Translate to Ligurian:
Field developed his views into fictionalism.
The Ligurian (Genoese) translation is:
Field o l’à sviluppou a seu vixon into fiçionalismo.
Translate to Ligurian (Genoese):
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.
The Ligurian (Genoese) translation is:
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.
Translate to Genoese:
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.
The Ligurian (Genoese) translation is:
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.