Myth and mathematics: A conceptualistic philosophy of mathematics I

infinite structures is necessary or is a plausible source of explanation. Perhaps an example will be helpful. An architect can describe quite precisely a skyscraper so tall it would crumble under its own weight if one at tempted to built it. In fact he can describe quite precisely an infinitely tall skyscraper, by using a clause analogous to (x)(Ey)Sxy. He can then reason and prove theorems about the skyscraper ( 'There are infinitely many washrooms'). All materials which go into the description seem quite finite, and it is far more plausible to look to that description, than to look to infinite abstract skyscrapers, or to other infinite structures, for an explanation of the resulting theorem. Even if there were some such structures external to us whatever that means we would still need to describe the particular structure under consideration. That is, it is highly implausible that we have any way of directly attaching ourselves to any such entity, so we must pick out the one we wish to discuss, distinguishing it from all of the many slightly different mathematical structures by some appropriate description. Thus we are appealing to our powers of description or our powers of meaning. We might as well stop with an appeal to those powers and not postulate unhelpful entities. Of course our powers of meaning are not arbitrary. There are constraints governing, so to speak, what we can do. Only certain concepts will give the sharpness wanted in mathematics, so our choices of concepts are not completely arbitrary. And once we have picked the initial concepts, we have no control over the consequences. There is some sort of objectivity and great definiteness both in the original limitations of choice, and in the uncontrollable consequences. I am not sure how much explanation can be given of this kind of objectivity, and we shall make no attempt to give an explanation here. The present goals are more limited. We wish to reorient the philosophy of mathematics by advocating a kind of finitistic and mentalistic conceptualism, while avoiding misleading pictures and unhelpful theories. Perhaps it might appear that our finitistic view amounts, in some roundabout way, to nothing but a formalistic position. That, however, is not the case, since there is no reason to suppose that our concepts and their consequences can be exhaustively described by a single forM Y T H A N D M A T H E M A T I C S 193 malism. The world may be finite, but with those finite materials we can describe infinite computations, similar to the one implicit in the consistency sentence (x)(Tx --+ vrx), which transcend the bounds of formal (mechanical) decidability. The few concepts used in mathematics have a quite substantial existence unlike most of the "objects" of mathematics since the concepts can be individually presented and grasped. General problems concerning meaning, the nature of concepts and other abstract objects, applications of mathematics, quasi-empirical sources of knowledge, and further issues will be considered in Part II. The present conclusion is that mathematics is based, not on reference to objects, but on a small finite number of special concepts. Those concepts are quasi-mechanical and they force certain conclusions in a quasi-mechanical way. That is the primary source of.mathematical knowledge. Perhaps these reflections help resolve the paradox that mathematical knowledge can be so clear and definite, while it is so difficult to say what one is talking about. A C K N O W L E D G M E N T S The original impetus for this work came primarily from Charles Chihara's book Ontology and the Vicious-Circle Principle and from Paul Bernays's paper 'On Platonism in Mathematics'. I am indebted to Professor Hao Wang for much sound advice and many stimulating conversations on the philosophy of mathematics over a number of years. Thanks are also due to Professor Glen Kessler whose criticisms and questions caused a number of improvements; to Mr. Jonathan Lear for his valuable criticisms and participation in tutorials on topics related to the present work; to Mr. Phil Ehrlich for stimulating interest in, and providing references on, the development of geometry; to Professor Charles Parsons for his generous contribution of time, and for many helpful suggestions; and to a number of other students and colleagues for helpful comments and criticism. Part II has profited especially from Philip Kitcher's criticisms of an earlier draft. Stephan Korner's 1962 book, The Philosophy of Mathematics, Harper and Brothers, New York, has points of contact with the present work, especially in Chapter VIII; there are also major differences, and I shall not be able to discuss the agreements and disagreements here. The influence of Frege is manifest, but philosophically we are at right angles, 1 9 4 L E S L I E T H A R P s i n c e t h e p r e s e n t p o s i t i o n is c l o s e r to a n o b j e c t f r e e i n t u i t i o n i s m . ( B e s ide s b e i n g v i r t u a l l y o b j e c t f r e e , t h e r e is a f u r t h e r m a j o r d i f f e r e n c e w i t h i n t u i t i o n i s m : w e d o n o t ins i s t t h a t t h e p r o p o s i t i o n s o f al l m a t h e m a t i c a l t h e o r i e s c o n t a i n a k i n d o f p r o v a b i l i t y o p e r a t o r , a l t h o u g h w e a d m i t s u c h t h e o r i e s . * ) I n a d d i t i o n , a g e n e r a l d e b t to t h e w o r k o f P r o f e s s o r S a u l K r i p k e is h e r e b y a c k n o w l e d g e d ; I d o n o t k n o w , h o w e v e r , t o w h a t e x t e n t h e w o u l d a g r e e w i t h t h e p r e s e n t v i ews .