They say all mathematicians are Platonists. The interesting thing is that this thing that is not about anything can be so surprising.
--Barry On Apr 16, 2013, at 3:53 PM, "Nicholas Thompson" <[email protected]> wrote: > Owen, > > One of the reasons that mathematical language can be so precise is that it > isn’t ABOUT anything, right? The minute one adds semantics …. the minute > one applies mathematics to anything … all the problems of ordinary language > begin to manifest themselves, don’t they? > > Nick > > From: Friam [mailto:[email protected]] On Behalf Of Owen Densmore > Sent: Tuesday, April 16, 2013 3:50 PM > To: The Friday Morning Applied Complexity Coffee Group > Subject: Re: [FRIAM] Isomorphism between computation and philosophy > > One has to be careful with nearly all the "impossibility" theorems: Arrow's > voting, the speed of light, Godel, Heisenberg, decidability, NoFreeLunch, ... > and so on. > > To tell the truth, Godel .. it seems to me .. says to the practicing > mathematician that the axioms have to be very carefully chosen. Its sorta > like linear algebra: a system can be over constrained .. thus contain > impossibilities, or under constrained thus have multiple solutions. > > But all I'm hoping for is any attempt to make the words Nick and others have > be as precise as a computer language. If this is the case, then we can use > the lovely computation hierarchy from FSA, to CFL to Turing/Church. But > then, most mathematicians know none of this structure either. Sigh. > > I wish philosophy had the same constraints where bugs could be found. On the > other hand, ambiguity can be a huge plus, as any spoken language shows. > > -- Owen > > > On Tue, Apr 16, 2013 at 3:39 PM, Barry MacKichan > <[email protected]> wrote: > Curious. Isn't the proof of Godel's theorem a special case of this? > > As I understand it, the proof is this: > > Consider the statement: This theorem is not provable. If it is false, the > theorem is provable. Since 'provable' implies true, this is a contradiction. > Therefore the theorem is true, which means it is true and not provable. > > The genius in Godel's method is that he created an isomorphism between the > domain of the previous paragraph, and arithmetic, and the isomorphism > preserves truth and provability. Thus the above theorem corresponds to a > statement in arithmetic that is true and not provable. What is this > statement, you might ask. Well, evidently it is far to complex to compute or > write down (although it would be interesting to see if more powerful > computers or quantum computers would change this.) > > Anyway, that true but non-provable theorem shows that number theory (aka > arithmetic) is incomplete -- that's the definition of incomplete in this > context. > > --Barry > > > On Apr 16, 2013, at 10:25 AM, Owen Densmore <[email protected]> wrote: > > On Sat, Apr 13, 2013 at 2:05 PM, Nicholas Thompson > <[email protected]> wrote: > Can anybody translate this for a non programmer person? > > > Nick's question brings up a project I'd love to see: an attempt at an > isomorphism between computation and philosophy. (An isomorphism is a 1 to 1, > onto mapping from one to another, or a bijection.) > > For example, in computer science, "decidability" is a very concrete idea. > Yet when I hear philosophical terms, and dutifully look them up in the > stanford dictionary of philosophy, I find myself suspicious of circularity. > > Decidability is interesting because it proves not all computations can > successfully expressed as "programs". It does this by using two infinities > of different cardinality (countable vs continuum). > > Does philosophy deal in constructs that nicely map onto computing, possibly > programming languages? > > I'm not specifically concerned with decidability, only use that as an example > because it shows the struggle in computer science for modeling computation > itself, from Finite Automata, Context Free Languages, and to Turing Machines > (or equivalently lambda calculus). > > I don't dislike philosophy, mainly thanks to conversations with Nick. And I > do know that axiomatic approaches to philosophy have been popular. > > So is there a possible isomorphism? > > -- Owen > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com > > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com
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