By Penelope Maddy
Mathematicians are likely to give some thought to themselves as scientists investigating the good points of actual mathematical issues, and the wildly profitable program of arithmetic within the actual sciences reinforces this photo of arithmetic as an aim research. For philosophers, even if, this realism approximately arithmetic increases severe questions: What are mathematical issues? the place are they? How can we learn about them? delivering a scrupulously reasonable remedy of either mathematical and philosophical matters, Penelope Maddy right here delineates and defends a unique model of mathematical realism. She solutions the conventional questions and poses a tough new one, refocusing philosophical awareness at the urgent foundational problems with modern arithmetic.
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Extra resources for Realism in Mathematics (Clarendon Paperbacks)
Sample text
58 The thir d an d fina l anti-realis t schoo l o f though t I wan t t o consider her e is logicism, or really , the version of logicism advance d by the logica l positivists. ^ 9 Thi s versio n o f logicis m i s outrigh t Platonistic: arithmeti c i s th e scienc e o f somethin g objectiv e (be cause logi c i s objective), that something objective consists o f object s (numbers), and ou r logica l knowledg e i s a priori. I f this project ha d succeeded, th e epistemologica l problem s o f Platonis m woul d hav e been reduce d t o thos e o f logic , presumabl y a gain .
Wha t make s '2 + 2 = 4 ' tru e i s the natur e of the abstract entities 2 an d 4 an d the operation plus ; fo r me to kno w tha t ' 2 + 2 = 4' , thos e entitie s mus t pla y a n appropriat e causal rol e i n th e generation of my belief. But how ca n entities that don't even inhabit the physica l univers e take par t i n an y causa l interactio n whatso ever? Surely to be abstract i s also to b e causally inert. Thus, if Platonism is true, we can have no mathematica l knowledge. Assuming that we do have such knowledge, Platonism must be false.
I wil l leave th e thre e grea t schools a t thi s point. I don't clai m t o have refute d eithe r formalis m or conventionalism , though I hop e the profound difficultie s the y face hav e been drawn clearl y enough . Intuitionism I reject o n th e ground s give n above ; I assume tha t th e job o f th e philosophe r o f mathematic s i s t o describ e an d explai n mathematics, not t o reform it. Let m e retur n no w t o Platonism , th e vie w tha t mathematic s i s an objective science . Platonis m naturall y conflict s wit h eac h o f th e particular form s o f anti-realis m touched o n here—wit h intuition ism o n th e objectivit y of mathematica l entities , with formalis m on the statu s o f infinitar y mathematics , wit h logicis m o n th e nee d fo r mathematical existenc e assumption s goin g beyon d thos e o f logic— but th e Platonist' s traditiona l an d pures t opponen t i s the nominal ist, wh o simpl y hold s tha t ther e ar e no mathematica l entities .