Costruttivismo matematico

Da Wikipedia, l'enciclopedia libera.
(Reindirizzamento da Costruttivismo (matematica))
Vai alla navigazione Vai alla ricerca

Nella filosofia della matematica, attorno all’espressione costruttivismo si raccolgono una varietà di prospettive e programmi di ricerca che, sebbene raccolgano eredità storiche e muovano da considerazioni tra loro assai diverse[1] e non sempre compatibili[2][3], convergono tutte intorno all’obbiettivo di proporre una nozione di esistenza più esplicita[4] e distinta[5] da quella invece asseribile - all’interno del modello di volta in volta messo a punto per meglio catturare le proprietà dell'insieme, sistema o struttura oggetto di studio - a partire dalla premessa che ciascuna affermazione possegga un valore di verità determinato[6] (principio di bivalenza) e facente spesso leva sulla coerenza[7] del modello (attraverso l'invocazione del principio del terzo escluso o il ricorso alla dimostrazione per assurdo).[8]

A tale obbiettivo è strettamente connesso[9] un secondo aspetto comune, per quanto esso stesso causa di contesa[10][11][12]: la centralità[13] dell'elaborazione e adozione di una pratica matematica algoritmica[14]. Ciò comporta, da una parte, la riformulazione di molte definizioni[15] secondo criteri volti a conferire loro un contenuto quanto più possibile positivo[16], concreto[17] e numerico[18]; dall’altra, nell’intraprendere una dimostrazione d’un qualsiasi teorema, l’impiego di modalità argomentative e di operazioni capaci di condurre il ragionamento ad una conclusione sì di carattere generale, ma allo stesso tempo in linea di principio calcolabile[19] dalle informazioni costitutive delle premesse[20]. Da questo origina lo sviluppo della logica intuizionista[21][22] e di altre formalizzazioni, come pure l'interesse per i principi di onniscienza[23].

Altrettanto importante e condivisa è, però, la priorità attribuita al significato e al contenuto numerico rispetto alla verità[24], nonché alla matematica nei confronti della logica[25][26].

Quandanche motivati da posizioni antagoniste, vi sono alcuni approcci costruttivisti – ad esempio, quello progredito dalle riflessioni di Errett Bishop[27][28] – che sotto un profilo strettamente logico e matematico producono risultati immediatamente validi anche dal punto di vista della matematica classica.[29][30] Altri, come il lavoro di Markov e proseguito da Kushner oppure l'intuizionismo promosso da Brouwer e portato avanti da Heyting, no[31][32].

Note[modifica | modifica wikitesto]

  1. ^ Varieties of Constructive Mathematics, p. 2.
    «From a philosophical point of view, there is more to RUSS and INT than the mere adjunction of certain principles to BISH. In RUSS, every mathematical object is, ultimately, a natural number: constructions take place within a fixed formal system, functions are Godel numbers of the algorithms that compute them, and so on. On the other hand, INT is based on Brouwer's intuitionistic philosophy, including an analysis of the notion of an infinitely proceeding, or free choice, sequence.»
  2. ^ Varieties of Constructive Mathematics, p. 52.
    «Markov's Principle is false in many models of intuitionistic logic; indeed, according to Brouwer, it is incompatible with the intuitionistic notion of negation and the general idea of a choice sequence.»
  3. ^ Varieties of Constructive Mathematics, p. 2.
    «A [...] reason for paying particular attention to BISH is that every proof of a proposition P in BISH is a proof of P in RUSS and in INT. Indeed, the last two varieties may be regarded as extensions of BISH. In RUSS, the main principle adjoined to BISH is a form of Church's thesis that all sequences of natural numbers are recursive. In INT, two principles are added which ensure strong continuity properties of arbitrary real-valued functions on intervals of the line.»
  4. ^ Varieties of Constructive Mathematics, p. 1.
    «We engage in constructive mathematics from a desire to clarify the meaning of mathematical terminology and practice - in particular, the meaning of existence in a mathematical context.»
  5. ^ Schizophrenia in Contemporary Mathematics, p. 5.
    «Four principles stand out as basis: [...] (D) Meaningful distinctions deserve to be maintained.»
  6. ^ Varieties of Constructive Mathematics, p. 10.
    «The classical mathematician believes that every mathematical statement P is either true or false, whether or not he possesses a proof or disproof of P.»
  7. ^ Appendix B. Aspects Of Constructive Truth, p. 353.
    «[...] the constructivist attitude to questions of consistency. For the constructivist, consistency is not a hobgoblin. It has no independent value; it is merely a consequence of correct thought. The consistency of some particular formal system, even if it were constructively proved, would not be a very interesting result for the constructive mathematician.»
  8. ^ Elements of Intuitionism, p. 5.
    «Thus, while, to a platonist, a mathematical theory relates to some external realm of abstract objects, to an intuitionist it relates to our own mental operations: mathematical objects themselves are mental constructions, that is, objects of thought not merely in the sense that they are thought about, but in the sense that, for them, esse est concipi. They exist only in virtue of our mathematical activity, which consists in mental operations, and have only those properties which they can be recognized by us as having.It is for this reason that the intuitionistic reconstruction of mathematics has to question even the [...] most celebrated principle [...] the law of excluded middle: since we cannot, save for the most elementary statements, guarantee that we can find either a proof or a disproof of a given statement, we have no right to assume, of each statement, that it is either true or false; nor, therefore, to offer as a proof of a theorem a demonstration that it is derivable from the assumption either of the truth or of the falsity of some as yet undecided proposition,»
  9. ^ Varieties of Constructive Mathematics, pp. v-vi.
    «In the classical interpretation, an object exists if its non-existence is contradictory. There is a clear distinction between this meaning of existence and the constructive, algorithmic one, under which an object exists only if we can construct it, at least in principle.»
  10. ^ Lectures on Constructive Mathematical Analysis, p. 17.
    «The intuitive concept of algorithm, which is entirely sufficient for recognizing this or that specific prescription as an algorithm, is however too vague to yield rigorous proofs of theorems characterizing the class of algorithms as a whole. The urge to make the notion of algorithm sufficiently sharp and accessible to study by mathematical means led to the development of precise concepts of an algorithm. [...] shall use as a precise concept of algorithm the notion of normal algorithm proposed by Markov [...]»
  11. ^ The Constructivization of Abstract Mathematical Analysis, p. 410.
    «There are the formalizers of constructivity, whose formal systems have little relevance to the constructivization of existing mathematics; the recursive-function theorists, who base constructivity on an ad hoc assumption which is more of an impediment than a tool [...]»
  12. ^ Elements of Intuitionism, p. 175.
    «[...] Markov's principle is definitely incorrect intuitionistically, since it is inconsistent with the theory of lawless sequences [...]»
  13. ^ Lectures on Constructive Mathematical Analysis, p. 17.
    «One of the central places in mathematics is occupied by algorithms»
  14. ^ Varieties of Constructive Mathematics, p. 1.
    «What do we mean by an algorithm? We may think of an algorithm as a specification of a step-by-step computation [...] which can be performed, at least in principle [...] in a finite period of time; moreover, the passage from one step to another should be deterministic.»
  15. ^ Constructive analysis, p. 64.
    «[...] a single concept of classical mathematics may split into two or more distinct concepts when looked at constructively.»
  16. ^ Constructive Analysis, p. 3.
    «The task of making analysis constructive is guided by three basic principles. First, to make every concept affirmative.»
  17. ^ Constructive Analysis, p. 2.
    «It appears then that there are certain mathematical statements that are merely evocative, that make assertions without empirical validity. There are also mathematical statements of immediate empirical validity, which say that certain performable operations will produce certain observable results [...]. Mathematics is a mixture of the real and the ideal, [...]. The realistic component of mathematics - the desire for pragmatic interpretation - supplies the control which determines the course of development and keeps mathematics from lapsing into meaningless formalism. The idealistic component permits simplifications, and opens possibilities which would otherwise be closed. The methods of proof and the objects of investigation have been idealized to form a game, but the actual conduct of the game is ultimately motivated by pragmatic considerations.»
  18. ^ Constructive Analysis, p. 3.
    «Our program is simple: to give numerical meaning to as much as possible of classical abstract analysis.»
  19. ^ Varieties of Constructive Mathematics, p. 1.
    «we say 'performed, at least in principle', for it is possible for an algorithm to require an amount of time greater than the age of the universe for its complete execution. We are not concerned here with questions of complexity or efficiency.»
  20. ^ Apartness and Uniformity, p. 2.
    «In particular, if we give an intuitionistic-logic-based proof of the existence of a mathematical object x with a certain property P (x), then we can extract from our proof an algorithm that enables us to construct (compute) an object ξ and then to prove that P (ξ) holds.»
  21. ^ Apartness and Uniformity, p. 2.
    «Why do we make this logical shift? We do so in order to make our presentation fully constructive/computational [...] intuitionistic logic (plus an appropriate set- or type-theoretic foundation [...] automatically excludes nonconstructive arguments.»
  22. ^ Schizophrenia in Contemporary Mathematics, p. 1.
    «[...] involving a re-interpretation of the usual connectives and quantifiers, which permits the expression of certain important distinctions of meaning which the classical terminology does not.»
  23. ^ The crisis in contemporary mathematics, p. 511.
    «[...] principle of omniscience [...] denote the statement that it is possible to make an infinite computation of the type [...] described [...]»
  24. ^ The crisis in contemporary mathematics, p. 508.
    «[...] what are historically regarded as problems about truth are actually problems about meaning. I believe that if we agree on the meaning of such statements, then we can settle the question of their truth relatively easily. There is only one basic criterion to justify the philosophy of mathematics, and that is, does it contribute to making mathematics more meaningful.»
  25. ^ Varieties of Constructive Mathematics, p. 11.
    «Constructive mathematics is not based on a prior notion of logic; rather, our interpretations of the logical connectives and quantifiers grow out of our mathematical intuition and experience.»
  26. ^ Schizophrenia in Contemporary Mathematics, p. 2.
    «The experts now routinely equate the panorama of mathematics with the productions of this or that formal system. Proofs are thought of as manipulations of strings of symbols. Mathematical philosophy consists of the creation, comparison, and investigation of formal systems.Consistency is the goal. In consequence meaning is debased, and even ceases to exist at a primary level. The debasement of meaning has yet another source, the wilful refusal of the contemporary mathematician to examine the content of certain of his terms,such as the phrase "there exists. " He refuses to distinguish among the different meanings that might be ascribed to this phrase. Moreover he is vague about what meaning it has for him. When pressed he is apt to take refuge in formalities, declaring that the meaning of the phrase and the statements of which it forms a part can only be understood in the context of the entire set of assumptions and techniques at his command. Thus he inverts the natural order, which would be to develop meaning first, and then to base his assumptions and techniques on the rock of meaning. Concern about this debasement of meaning is a principal force behind constructivism.»
  27. ^ Constructive Analysis, p. 3.
    «This development is carried through with an absolute minimum of philosophical prejudice concerning the nature of constructive mathematics. There are no dogmas to which we must conform.»
  28. ^ Techniques of constructive analysis, pp. x-xi.
    «We should make it clear that we are not advocating the exclusive use of intuitionistic logic in mathematics. That logic is, we believe, the natural and right one to use when dealing with the constructive content of mathematics. To abandon classical logic in those fields (such as the higher reaches of set theory) where constructivity is of little or no significance makes no sense whatsoever. Nevertheless, it is remarkable how much mathematics actually has what Bishop called “a deep underpinning of constructive truth”.»
  29. ^ Intuitionism As Generalization, p. 124.
    «the approach to mathematics based on intuitionistic logic [...] in its simplest form, is a generalization of classical mathematics that accomodates both classical and computational models.»
  30. ^ Apartness and Uniformity, p. viii.
    «One other important feature of Bishop’s constructive mathematics is that it is completely consistent with classical mathematics - mathematics using classical logic.»
  31. ^ Varieties of Constructive Mathematics, p. 2.
    «Every proposition P in BISH has an immediate interpretation in CLASS, and a proof of P in BISH is also a proof of P in CLASS. This is not true in our other two varieties: Russian constructivism (RUSS) and Brouwer's intuitionism (INT).»
  32. ^ Techniques of constructive analysis, p. 5.
    «to regard Brouwer’s mathematics as inconsistent with its classical counterpart is a serious oversimplification of the situation, since the two types of mathematics are in many respects incomparable.»

Bibliografia[modifica | modifica wikitesto]

Introduzioni[modifica | modifica wikitesto]

Analisi reale, complessa e funzionale[modifica | modifica wikitesto]

Algebra[modifica | modifica wikitesto]

Topologia[modifica | modifica wikitesto]

Teoria degli insiemi costruttiva[modifica | modifica wikitesto]

Teoria dei tipi intuizionista[modifica | modifica wikitesto]

  • Johan Georg Granström, Treatise on Intuitionistic Type Theory, Springer Dordrecht, 2011, ISBN 978-94-007-1736-7.

Fondamenti e storia[modifica | modifica wikitesto]

Logica intuizionista[modifica | modifica wikitesto]

Raccolte[modifica | modifica wikitesto]

  • Harold Mortimer Edwards, Essays in Constructive Mathematics, Springer, 2005, ISBN 0-387-21978-1.

Voci correlate[modifica | modifica wikitesto]

Altri progetti[modifica | modifica wikitesto]

Controllo di autoritàGND (DE) 4165105-4
  Portale Matematica: accedi alle voci di Wikipedia che trattano di matematica
Estratto da "https://it.wikipedia.org/w/index.php?title=Costruttivismo_matematico&oldid=138145993"