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Premessa:
•Henri Poincaré (1854-19= 13)
•<= /span>
•Non elaborò una compiuta filosofia, ma si limitò a compiere alcune analisi, senza voler costruire una teoria glo= bale della scienza.
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Convenzione
•<= /span>Proposizione della quale non può essere data né dimostrazi= one logica né verifica sperimentale esauriente <= /div>
•<= /span>
•<= /span>E’ una premessa e uno strumento indispensabile di un discorso logico= -concettuale
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•Tale concetto fu usato per la prima volta da Poincaré per definire la natura degli assiomi e dei postulati delle geometrie, con particolare riferimento agli sviluppi delle geometrie non eucli= dee.
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Poincaré e le convenzioni…
= 226;Il concetto di convenzione viene introdotto da Poincarè in quanto egli nega:
•
•= la validità della teoria kantiana in riferimento = agli assiomi e ai postulati della geometria
•
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Kant
GIUDIZI SINTETICI A PRIORI
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•ESPERIENZ= A           = ;            &n= bsp;    +
•
•   GIUDIZI SINTETICI A PRIO= RI   =3D
•       
•        = ;  SCIENZA&#= 13;
•Critica della ragion pura
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•GIUDIZI : aggiungono un predicato ad un soggetto
•<= /span>
•SINTETICI= : il predicato dice qualcosa di nuovo rispetto ad esso
•<= /span>
•A PRIORI : non possono derivare all’esperienza
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•Poicaré si domanda qual è la natura = degli assiomi della geometr= ia:
•
•“Sono dei giudizi sintetici a priori, come diceva Kant? Allora = essi si imporrebbero a noi con una tale forza che non potremmo concepire la proposizione = contraria, né costruire su di essa posizion= e un edificio teorico. Non ci sar= ebbero geometrie non euclidee”
•        = ;            &n= bsp;             La scienza e l’ipotes= i
= •
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<= map name=3D"map_s3">
= 226;Escludendo la natura di verità a priori degli assiomi geometrici, “Dobbiamo per questo concludere che gli assiomi geometrici sono verità sperimentali? Ma non si speriment= a su delle rette o su delle circonferenze ideali; si sperimenta solo su oggetti materiali”.
•        = ;            &n= bsp;        
•        = ;            &n= bsp;            La scienza e l’ipotesi
•        = ;            &n= bsp;            = ;            &n= bsp;            = ;            &n= bsp;            = ;            &n= bsp;            = ;            &n= bsp;       
= •
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•“= ;Perciò gli assiomi non sono giudizi sintetici a priori, né fa= tti sperimentali: sono convenzioni.”
•
•        = ;            &n= bsp;            La scienza e l’ipotesi
•
•Le conclusioni di Poincarè
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…e quindi …
•non ha senso porsi il problema di quale <= span style=3D'position:absolute;top:35.0%;left:12.17%;width:79.4%;height:6.0%'>= geometria sia vera;
•
•la scelta di una geometria si basa sulla = comodità; <= /div>
•
•gli uomini hanno scelto la geometria euclidea, <= /span>perché permette di costruire una fisi= ca più s= emplice e più comoda.
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La geometria euclidea
Il metodo assiomatico=
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•In un sistema deduttivo la dimostrazione di un teorema consiste nel far vedere = che esso è una conseguenza logica necessaria di alcune proposizioni pre= cedenti, queste a loro volta devono essere dimostrate e così via
R= 26;Devono quindi esistere un certo numero di proposizioni che si accettano come ver= e, dette postulati<= span style=3D'font-family:Arial;mso-ascii-font-family:Arial;mso-hansi-font-fami= ly: Arial;font-size:64%'> o assiomi. Essi devono essere:
•<= /span> = COERENTI - = due teoremi da essi dedotti non possono essere in  contraddizione tra loro;
• = COMPLETI - da essi si deve poter dedurre ogni teorema del sistema;
• = INDIPENDENTI – nessuno di essi deve essere una conseguenza logica degli = altri;
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•La questione della completezza e della coerenza è stata oggetto di molte controversie filosofiche:
R= 26;Kant  considera gli enti matematici = come oggetti sostanziali in un = regno di “pura intuizione”, = indipendenti dalle definizioni e = dagli atti individuali della mente = umana: non esiste il problema = della coerenza
R= 26;I “formalisti” si interessano soltanto del procedimento logico formale del = ragionamento sulla base dei = postulati
R= 26;Ciò accorda alla matematica tutta la libertà che le è necessaria per la <= /span>teoria e le applicazioni, ma impone al formalista di dimostrare che i suoi assiomi (arbitrari) non sono in = contraddizione
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R= 26;La via assiomatica= , completamente indipendente dalle considerazioni filosofiche, è la via naturale per accostarsi a un soggetto matematico in modo da districare l’intrecciarsi delle relazioni tra i vari fatti e rilevare l’impalcatura = logica della costruzione.
R= 26;A volte il concentrarsi sulla struttura formale piutt= osto che sul significato int= uitivo dei concetti rende più facili le generalizzazioni  e le applicazioni che in <= /span>un procedimento più intuitivo potrebbero essere trascurate.
<= ![if !vml]>3Dthamedium
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3DEuklid2
R= 26;Fin dai tempi di Eraclito la geometria ha rappresenta= to il prototipo di scienza ded= uttiva.
R= 26;L’opera di Euclide rappresenta una sintesi orga= nica delle conoscenze mat= ematiche dei suoi tempi ed è ispirata a fini didattici. <= /div>
R= 26;Gli ELEMENTI si compongono di 13 libri, nei quali si trova espost= a = sistematicamente tutta la geometria elementare. =
R= 26;Ogni libro inizia con un gruppo di PROPOSIZIONI; esse sono seguite da altre pro= posizioni che sono invece veri e propri problemi e teoremi.
•I principi fondamentali esposti negli “ELEMENTI” si distinguono in tre categorie:
•
1.= TERMINI o definizioni;
2.= POSTULATI di natura geometrica;
3.= NOZIONI COMUNI, postulati anch’essi, ma= di portata più generale;
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•Ai termini del primo libro seguono nel testo originale i 5 postulati:
•
1.Si ammette di poter condurre da qualsiasi punto= ad ogni altro punto = una linea retta;
2.
= 2.e che ogni retta terminata si possa prolungare continuamente per = dritto;
3.
3.e che con ogni centro e con ogni distanza si po= ssa descrivere un = circolo;
4.
4.e che tutti gli angoli retti siano uguali fra di loro;
5.
= 5.= e che se una retta, incontrandone altre due, forma angoli interni da una stessa parte minori di d= ue angoli retti, le due rette prolungate continuamente si incontrano = dalla parte in cui sono gli angoli minori di due retti.
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•V postulato di Euclide:
•
•= Se una retta, incontrandone altre due, forma angoli interni da una stessa parte minori di due angol= i retti, le due rette prolungate continuamente si= incontrano dalla parte in cui sono gli angoli <= /i>minori di due retti. =
•
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Dimostrazione sperimentale
•<= /span>Questo postulato a differenza degli altri non ha carattere “finito”
•<= /span>La massima lunghezza di qualunque oggetto o raggio luminoso è FINITA
= 226;Non si può verificare sperimentalmente
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•IL POS= TULATO E’ QUINDI INDIPE= NDENTE O NO DAGLI ALTRI POSTULATI?
•
= 226;Se esso fosse una conseguenza logica necessaria de= gli altri, sarebbe possibile eliminarlo come assioma e darne invece una dimostrazione servendosi degli altri postulati euclidei= .
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Proclo (IV sec a.C.)
•<= /span>Nuova definizione di retta parallela: la retta parallela a una retta data è il luogo dei punti aventi dalla retta una distanza fissata
•<= /span>IL PROBLEMA E’ SOLO SPOSTATO…i due postulati sono identi= ci!
= •
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