On Dec 15, 4:43 am, rusi <rustompm...@gmail.com> wrote: > On Dec 14, 10:15 pm, Eelco <hoogendoorn.ee...@gmail.com> wrote: > > > 'Kindof' off-topic, but what the hell :). > > <deja-vu> > We keep having these debates -- so I wonder how off-topic it is... > And so do famous > CSists:http://research.microsoft.com/en-us/um/people/gurevich/opera/123.pdf > </deja-vu>
Well, you are right, there are some deep links here. My view of what is wrong with mainstream mathematics is its strange interpretation of the semantics of classical logic. (And I dont think any other schools get it quite right either; I think finitists may avoid the mistakes of others, but are rightfully accussed of being needlessly restrictive, for instance) This is best illustrated by means of the principle of explosion. It rests on assuming a contradiction, and then assigning rather peculiar semantics to them. What is typically left unstated are the semantics of symbol lookup, but apparently it is implicitly understood one can pick whatever value upon encountering a contradicting symbol. There is no well defined rule for the lookup of a twice-defined symbol. Of course the sane thing to do, to a mind grown up around computer languages, upon encountering a twice defined symbol, is not to continue to generate deductions from both branches, but to throw an exception and interrupt the specific line of reasoning that depends on this contradicting symbol right then and there. Conceptually, we can see something is wrong with these undefined semantics right away. A logical system that allows you to draw conclusions as to where the pope shits from assertions about natural numbers could not more obviously be broken. If you dont have this broken way of dealing with contradictions, one does not have to do one of many silly and arbitrary things to make infinity work, such as making a choice between one-to-one correspondence and subset-relations for determining the cardinality of a set; one can simply admit the concept of infinity, while useful, is not consistent, keep the contradiction well handled instead of having it explode in your face (or explode into the field of transfinite analysis; a consequece of 'dealing' with these issues by rejecting the intuitively obviously true relation between subset relations and cardinality), and continue reasoning with the branches of your argument that you are interested in. In other words, what logic needs is a better exception-handling system, which completes the circle with programming languages quite nicely. :) -- http://mail.python.org/mailman/listinfo/python-list