On 26/05/2013 10:17 PM, zuhair wrote: > On May 26, 4:49 pm, Nam Nguyen <namducngu...@shaw.ca> wrote: >> On 26/05/2013 3:52 AM, Zuhair wrote: >> >>> On May 26, 11:03 am, Nam Nguyen <namducngu...@shaw.ca> wrote: >>>> On 26/05/2013 12:52 AM, Zuhair wrote: >> >>>>> Frege wanted to reduce mathematics to Logic by extending predicates by >>>>> objects in a general manner (i.e. every predicate has an object >>>>> extending it). >> >>>> [...] >> >>>>> Now the above process will recursively form typed formulas, and typed >>>>> predicates. >> >>>> Note your "process" and "recursively". >> >>>>> As if we are playing MUSIC with formulas. >> >>>>> Now we stipulate the extensional formation rule: >> >>>>> If Pi is a typed predicate symbol then ePi is a term. >> >>>>> The idea behind extensions is to code formulas into objects and thus >>>>> reduce the predicate hierarchy into an almost dichotomous one, that of >>>>> objects and predicates holding of objects, thus enabling Rule 6. >> >>>>> What makes matters enjoying is that the above is a purely logically >>>>> motivated theory, I don't see any clear mathematical concepts involved >>>>> here, we are simply forming formulas in a stepwise manner and even the >>>>> extensional motivation is to ease handling of those formulas. >>>>> A purely logical talk. >> >>>> Not so. "Recursive process" is a non-logical concept. >> >>>> Certainly far from being "a purely logical talk". >> >>> Recursion is applied in first order logic formation of formulas, >> >> Such application isn't purely logical. Finiteness might be a purely >> logical concept but recursion isn't: it requires a _non-logical_ >> concept (that of the natural numbers). >> >>> and all agrees that first order logic is about logic, >> >> That doesn't mean much and is an obscured way to differentiate between >> what is of "purely logical" to what isn't. >> > > Yes I do agree that this way is not a principled way of demarcating > logic. I generally tend to think that logic is necessary for analytic > reasoning, i.e. a group of rules that make possible to have an > analytic reasoning. Analytic reasoning refers to inferences made with > the least possible respect to content of statements in which they are > carried, thereby rendering them empirically free. However this is too > deep. Here what I was speaking about do not fall into that kind of > demarcation, so it is vague as you said. I start with something that > is fairly acceptable as being "LOGIC", I accept first order logic > (including recursive machinery forming it) as logic, and then I expand > it by concepts that are very similar to the kind of concepts that made > it, for example here in the above system you only see rules of > formation of formulas derived by concepts of constants, variables, > quantifying, definition, logical connectivity and equivalents, > restriction of predicates. All those are definitely logical concepts, > however what is added is 'extension' which is motivated here by > reduction of the object/predicate/predicate hierarchy, which is a > purely logical motivation, and also extensions by the axiom stated > would only be a copy of logic with identity, so they are so innocuous > as to be considered non logical. > That's why I'm content with that sort of definitional extensional > second order logic as being LOGIC. I can't say the same of Z, or ZF, > or the alike since axioms of those do utilize ideas about structures > present in mathematics, so they are mathematically motivated no doubt. > NF seems to be logically motivated but it use a lot of mathematics to > reach that, also acyclic comprehension uses graphs which is a > mathematical concept. But here the system is very very close to logic > that I virtually cannot say it is non logical. Seeing that second > order arithmetic is interpretable in it is a nice result, it does > impart some flavor of logicism to traditional mathematics, and > possibly motivates logicism for whole of mathematics. Mathematics > might after all be just a kind of Symbolic Logic as Russell said. > > Zuhair > >>> similarly here >>> although recursion is used yet still we are speaking about logic, >>> formation of formulas in the above manner is purely logically >>> motivated. >> >> "Purely logically motivated" isn't the same as "purely logical". >> > > A part from recursion, where is the mathematical concept that you > isolate with this system?
I don't remember what you'd mean by "this system", but my point would be the following.
In FOL as a framework of reasoning, any form of infinity (induction, recursion, infinity) should be considered as _non-logical_ .
The reason is quite simple: in the language L of FOL (i.e. there's no non-logical symbol), one can not express infinity: one can express "All", "There exists one" but one simply can't express infinity.
Hence _infinity must necessarily be a non-logical concept_ . Hence the concept such the "natural numbers" can not be part of logical reasoning as Godel and others after him have _wrongly believed_ .
Because if we do accept infinity as part of a logical reasoning, we may as well accept _infinite formulas_ and in such case it'd no longer be a human kind of reasoning.
In fact in such case we'd consider ourselves as God.
For one, I'd certainly not consider myself so.
-- ---------------------------------------------------- There is no remainder in the mathematics of infinity.