Filozofija matematike

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Filozofija matematike je grana filozofije koja proučava pretpostavke, osnove i implikacije matematike. Njen cilj je da se razumeju priroda i metode matematike, i spozna mesto matematike u životima ljudi. Logička i strukturalna priroda matematike sama po sebi čini ovu studiju širokom i jedinstvenom među njenim filozofskim pandanima.[1][2]

Platonizam ili matematički realizam

Platonizam ili realizam postulira da matematika postoji u svom vlastitom svetu, paralelnom s našim.[3] Lako je uočiti da se matematika pojavljuje u skoro svim naukama. Osnovna misao je da je matematika nešto što već postoji i što matematičari istražuju. Ovo se može uporediti s Platonovim svetom ideja u kojem je naš vlastiti svet samo senka očitog. Aksiom unutar realizma je analogan fizičkom svetu prirodnih zakona. Problem s ovakvim pristupom je da se mora objasniti u kojem svetu se matematika nalazi, i kako je ona povezna s našim fizičkim svetom.

Poznati platonisti ili realisti su Pitagora, Rodžer Penrouz i Kurt Gedel.[4]

Formalizam

Formalizam zastupa stav da matematika u osnovi govori o manipulacijama nizovima informacija, tj. koristi se različitim pravilima kod kojih se menjaju simboli prema temeljnim pretpostavkama.[5] Ove temeljne pretpostavke su aksiomi koji se manipulacijom u skladu s određenim pravilima pretvaraju u teoreme. Na taj način se matematika može uporediti sa igrom, npr. šahom, gde se figure pomiču u skladu sa strogo određenim pravilima. Formalizam ne postavlja iste zahteve kao platonizam: mogu se odbaciti aksiomi i pravila, jer nisu „prirodni zakoni”, i ne postoji „perfektna” aksiomska struktura. Unutar formalizma ne postoji čvrsta veza između nauke i matematike, već se smatra da je slučajnost da ove strukture liče jedna na drugu, i ne postoji platonski svet ideja „iza” fizičkog sveta.

Problemi koje formalizam teško objašnjiva su Gedelovi teoremi nepotpunosti. Neki od poznatih formalista bili su David Hilbert[6] i Haskel Kari.

Logicizam ili logistika

Logicizam ili logistika uči da je matematika isto što i logika i da se može izvesti iz nje. Takvo stajalište podržavali su Bertrand Rasel i Alfred Nort Vajthed u Principia Mathematica čiji je krajnji cilj bio ujedinjenje filozofske logike i matematike. Ovakve ideje su danas uglavnom odbačene.[7][8]

Spoznajne teorije

Spoznajne teorije vide matematiku kao unutarnju funkciju ljudske svesti, što je prirodan sled naše perceptivne sposobnosti. Može se npr. videti da mozak jako reaguje na geometrijske predmete stvorene ravnim linijama, dok bezoblični predmeti ne daju iste jake reakcije kao što je to bilo u prvom slučaju. Dakle u ovom slučaju spoznajne teorije vide matematiku kao bitno podređenu biologiji. Matematika bi stoga bila elektrohemijski fenomen u ljudskom mozgu.

Socijalni konstruktivizam

Socijalni konstruktivizam smatra da se matematika mora promatrati kao socijalni predmet, kao deo društva, i njena unutarnja logika treba da sledi isti obrazac kao i drugi naučni procesi.[9]

Reference

  1. ^ Benacerraf, Paul, and Putnam, Hilary (eds., 1983), Philosophy of Mathematics, Selected Readings, 1st edition, Prentice-Hall, Englewood Cliffs, NJ, 1964. 2nd edition, Cambridge University Press, Cambridge, UK, 1983.
  2. ^ Dummett, Michael (1991 a), Frege, Philosophy of Mathematics, Harvard University Press, Cambridge, MA.
  3. ^ "Platonism in the Philosophy of Mathematics", (Stanford Encyclopedia of Philosophy)
  4. ^ Platonism in Metaphysics (Stanford Encyclopedia of Philosophy)
  5. ^ Zach, Richard (2019), Zalta, Edward N., ур., „Hilbert’s Program”, The Stanford Encyclopedia of Philosophy (Summer 2019 изд.), Metaphysics Research Lab, Stanford University, Приступљено 25. 5. 2019 
  6. ^ Kleene, Stephen (1971). Introduction to Metamathematics. Amsterdam, Netherlands: North-Holland Publishing Company. 
  7. ^ Tegmark, Max (februar 2008). „The Mathematical Universe”. Foundations of Physics. 38 (2): 101—150. Bibcode:2008FoPh...38..101T. arXiv:0704.0646Слободан приступ. doi:10.1007/s10701-007-9186-9. 
  8. ^ Tegmark 1998, стр. 1.
  9. ^ Ernest, Paul (1998), Social Constructivism as a Philosophy of Mathematics, State University of New York Press, Albany, NY.

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