
Re: LOGIC & MATHEMATICS
Posted:
May 28, 2013 10:29 PM


On 28/05/2013 6:06 AM, Zuhair wrote: > On May 28, 7:44 am, Nam Nguyen <namducngu...@shaw.ca> 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_ . >> >> 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. >> > > I see, you maintain the known prejudice that the infinite is non > logical? hmmm... anyhow this is just an unbacked statement.
I did; you just don't recognize it apparently: my "The reason is quite simple:" paragraph.
> I don't see any problem between infinity and logic,
Well, then, why don't you express infinity with purely logical symbols, for us all in the 2 fora to see? Seriously, that would be a great achievement!
> also I do maintain that > logic with _infinite formulas_ *is* indeed logic, anyhow.
Anyway, First Order _Logic_ does _not_ admit _infinite formulas_ !
  There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI 

