tag:blogger.com,1999:blog-3931579729864611467.post6786000848928797254..comments2023-10-31T05:37:16.659-07:00Comments on A Guy in the Pew: Is Mathematics Discovered or Invented?Chuck Blanchardhttp://www.blogger.com/profile/01417638725063186710noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-3931579729864611467.post-83021260617742845652008-04-29T16:19:00.000-07:002008-04-29T16:19:00.000-07:00For an introductory level work there's really none...For an introductory level work there's really none better than Stewart Shapiro's book "Thinking About Mathematics". It assumes very little in terms of mathematical and philosophical background. It is aimed at undergraduate students, so in some parts it might seem a bit slow, but the overview is still quite good. Shapiro covers the three major theories of the early 20th century (logicism, formalism, and intuitionism) before moving on to more contemporary ideas. One point to be warned about, however: Shapiro is an advocate of mathematical structuralism, so his presentation of that view as a solution to all the problems is not entirely unbiased.<BR/><BR/>For more advanced overviews, the entries in the Oxford Handbook of Philosophy of Mathematics and Logic (also edited by Shapiro, and perhaps the largest handbook I've ever seen) are generally quite good, and most of them cite good references.<BR/><BR/>For the specific topic of mathematical truth the starting point remains Paul Benacerraf's paper "Mathematical Truth". His papers "What Mathematical Truth Could Not Be" are also quite good, though in parts they expect some knowledge of mathematics and logic. A lot of the contemporary issues regarding the nature of mathematical truth focus on what kinds of philosophical conclusions can be drawn from metamathematical theorems.<BR/><BR/>Finally, there's still a large influence of the idea, due to W.V.O. Quine, amongst others, that the only kinds of mathematics we should even believe are true are the things used in the natural sciences. Quine's classic paper on this issue, and on ontology more generally, is "On What There Is".<BR/><BR/>Hopefully this is helpful/interesting. Not knowing the precise audience of this blog, I've tried to steer away from work that requires any substantial philosophical or mathematical background. Start with Shapiro's text, and see where you go from there.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3931579729864611467.post-27737304600382687482008-04-29T10:57:00.000-07:002008-04-29T10:57:00.000-07:00anonymous:This was very useful and interesting. T...anonymous:<BR/><BR/>This was very useful and interesting. Thanks! The references would be most welcome!Chuck Blanchardhttps://www.blogger.com/profile/01417638725063186710noreply@blogger.comtag:blogger.com,1999:blog-3931579729864611467.post-7462786226248547112008-04-29T08:33:00.000-07:002008-04-29T08:33:00.000-07:00I'd be delighted for some pointers to some introdu...I'd be delighted for some pointers to some introductory reading. Thanks!+Nicholashttps://www.blogger.com/profile/08438350082586972258noreply@blogger.comtag:blogger.com,1999:blog-3931579729864611467.post-63268023533577646262008-04-29T01:48:00.000-07:002008-04-29T01:48:00.000-07:00It's an interesting question, one that has occupie...It's an interesting question, one that has occupied both amateur and professional philosophers for a long time. I think it's probably fair to say that no one accepts a fully Platonistic view anymore (in the strict sense of accepting exactly what Plato said about mathematics), but there are lots of full-blooded realists around. For an interesting (and extremely influential) discussion of the debate, see Paul Benacerraf's 1973 paper "Mathematical Truth".<BR/><BR/>Lately, the major viewpoints are the following:<BR/>1) Structuralism, the view that mathematics is the science of structures. This does well at explaining the applicability of mathematics, for those of us who believe in a structured universe. Getting the details right has involved some pretty sophisticated logic, though.<BR/><BR/>2) Neo-logicisim, one of the current views most directly inspired by Plato. Holds that numbers, sets, etc really do exist, and that we know about them by certain 'abstraction principles', the classic example of which is Hume's Principle. This view is primarily associated with Crispin Wright, Bob Hale, and their students.<BR/><BR/>3) Nominalism, the view that mathematical entities don't exist at all, but are a useful (eg, for science) fiction. <BR/><BR/>Overall, the trend seems to be leaning towards the 'discovered' side of the debate, but nominalism is certainly a very active and real option. If you're interested in any one of these views in particular (or all of them) I'll happily point you to some good introductory reading.Anonymousnoreply@blogger.com