
Re: LOGIC & MATHEMATICS
Posted:
May 28, 2013 11:02 PM


On 27/05/2013 10:44 PM, Nam Nguyen wrote: > 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 nonlogical 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 _nonlogical_ >>> 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 _nonlogical_ . > > The reason is quite simple: in the language L of FOL (i.e. there's no > nonlogical 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 nonlogical concept_ . Hence the > concept such the "natural numbers" can not be part of logical reasoning > as Godel and others after him have _wrongly believed_ .
By "logical reasoning" I meant "pure logical reasoning"
> 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.
Again, "pure logical 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.
NYOGEN SENZAKI 

