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Re: Why I think mathematics is really logic.
Posted:
Jun 15, 2013 2:59 PM


On Jun 15, 6:04 pm, CharlieBoo <shymath...@gmail.com> wrote: > On Jun 15, 7:31 am, Zuhair <zaljo...@gmail.com> wrote: > > > > > > > > > > > What is logic and what is mathematics is indeed a very tricky > > question, many people would naively consider mathematics as nothing > > but machinery for generating symbols out of symbols, they see > > mathematics as strings of symbols generating strings of symbols. So > > mathematics just provide the necessary strings of symbols that other > > disciplines would use by "attaching" meaning belonging to those > > disciplines to those symbols. Under this perspective mathematics can > > be said to be prior to any kind of fairly complex knowledge that > > necessitate generating many string of symbols from prior ones. So > > mathematics in this sense would be prior to the known logical systems > > as well like propositional logic, first order, second order, > > infinitary logic, etc. > > > However mathematics is not commonly perceived to be so general, more > > commonly it is thought to be about some particular content mostly > > about general spatialtemporal relations, thought about 'structure' > > seem to be at the core of it. In this sense mathematics would be > > posterior to logic, since the later covers more general grounds. But > > however day after day I'm more of the opinion that particular > > mathematics is wholly interpretable in mere logic! and that the > > particular content it is thought to negotiate is really dispensable > > with at least in principle. > > > A Logical system mends itself with general inferences, so logical > > connectives take arguments that range over all values that can > > substitute those arguments. To me any naive extension of a logical > > system is a logical system, of course this extension must not include > > complex measures, and they must be of the kind that makes one feel as > > being "natural" extensions. I'll speak about one below so that one > > gets the sense of what I mean by "natural" here. > > > Lets take first order logic "FOL", I'll accept this as a pure piece of > > logic (the recursive mechanism and the use of natural indexing, > > function symbols, etc.. although mathematical tools per se, but > > they'll be considered here a legitimate logical tools and thus just > > part of logic). > > > Now in first order logic quantification is allowed over Objects only, > > predicate symbols are not quantified over. > > > However this kind of logic can be naturally extended into a logic that > > allows quantification over predicate symbols that take only Object > > symbols as arguments, i.e. predicates that hold of objects only, those > > can be called predicates of the first kind, now predicates that take > > predicates of the first kind as arguments are named as predicates of > > the second kind. Now as with the case of first order logic, we can > > impose the restriction that predicates of the second kind cannot be > > quantified over so only constant symbols denoting particular > > predicates of the second kind can be used in a formula (in FOL only > > constant predicate symbols of the first kind are allowed in formulas > > and they are not quantified over). Now this extension is consistent, > > and it is a natural extension of FOL, it employs no concept other than > > a general copy of what underlies first order logic but to a higher > > realm, so the motivation and the tools used are all logical, so the > > resulting system is to me a "pure" piece of logic also. Along the > > 'same' lines one can extend that system further to one that also > > quantifies over predicates of the second kind but leave those of the > > third kind non quantified. Now this can be further extended using the > > natural indexing commonly used in logical systems to cover all > > predicates of any nkind. We use 'sorted' formulas those will use > > indexed predicate symbols as P1, P2, P3,... each Pi is taken to range > > only over predicates of the i_th sort and of course only takes Pi1 > > predicates as arguments. A well sorted formula would be acceptable > > only if every atomic formula of it is of the general form Pi+1(Qi),
More precisely of the general form Pi+1 (Qi,...,Ti)
> > Objects are indexed with 0 (or otherwise left unindexed). This > > system is purely logical since FOL is logical system and it is a > > fragment of it and since every one step higher extension of a logical > > fragment of it is logical also, so by naive understanding about > > induction logicality would sweep into the whole system. > > > Now we also desire to Extend that system! But to do that along similar > > lines it becomes "necessary" to use an indexing beyond the naturals. > > > Now we'll use the index #, and P# would denote a predicate that range > > over predicates of the # sort. Now each P# predicate can take any Pi > > where i is a natural index as an argument, so P#(P1), P#(P2), ... are > > all atomic formulas, and also along the same lines P#+1(P#), P#+2(P# > > +1), etc.. are all atomic formulas. Now this is also very natural > > since # is above any natural index and all can be seen to be > > 'immediately' lower than it, i.e. the distance that 0 has from # is > > not really different from that any n has from #, so a predicate of the > > #sort can take any predicate of the nsort (where n is a natural) as > > an argument, this is just a naive extension of the previous system. So > > this system for the same reasons outlined above would also inherit the > > logicality of the prior system, and it *is* a pure piece LOGIC. > > > The use of index # came out of logical necessity to extend the system > > along the same lines, so it is a necessity that sprang out of strictly > > employing logical lines of extension, since it is a necessity raised > > within pure logical context, and then it is logical. > > > The above logical system clearly has the power to interpret second > > order arithmetic! > > > Actually along the same lines I think every piece of mathematics can > > be said to be interpretable in a logical system. > > > So logicism seems to be the case for the bulk of traditional > > mathematics, and possibly for any mathematical endeavor? > > > I don't think the above result comes into conflict with the criterion > > of permutation invariance to decide logical notions, but I'm not sure. > > > I'd like here to also present this permutation invariance (as how I > > understand it) of a logical notion. > > > Now a symbol $ is said to be logical iff the representative set of it > > is invariant under all permutations of the domain of discourse over > > which arguments of $ range. > > > The representative set of a symbol goes generally along the following > > lines. > > > For monadic symbol $, it is the set of all x such that $(x) is True. > > > For any n_adic symbol $, is the set of all <x1,...,xn> tuples such > > that $(x1,..,xn) is True. > > > A permutation over any set is a bijection from that set to that set. > > > Now if D is the domain of discourse over which all arguments of $ > > range, and if f is a bijection from D to D, then $ would be called as > > permutation invariant iff for each permutation f on D the set D* of > > all <f(x1),...,f(xn)> tuples where <x1,...,xn> is an element of the > > representative set of $, is the same set D. > > > Now this works for all logical connectives, identity, first order > > quantifiers, any n quantifier, even infinite quantifiers, also it > > works for second order quantifiers over predicates that can have > > extensions (non purely intentional predicates). > > > Now the domain of discourse for any logical connective is the set of > > all "propositions" which are statements that are can either be True or > > False. > > Permutations can be carried over all "atomic" sentences of the > > propositional discourse since that will enact permutations on non > > atomic sentences as well. And those are easily seen to be invariant. > > > Identity is a clearly invariant relation. > > > The universal quantifier is a symbol linking an object to a > > proposition, so its representative set would be the set of all > > <x,phi(x)> of course phi is fixed, but x would be any object, phi is > > of course true for every x. Now this is invariant under all > > replacements of x (notice that phi is Constant formula raning over ONE > > value, so it is only replaced by itself!) > > > This criterion (as displayed here) is a nice criterion to separate > > logical from nonlogical constants. However it doesn't speak a lot > > about what metalogical symbols are allowed in formulas (like whether > > the symbol # above is allowed or not), although it allows the meta > > logical symbol omega in infinitary logic which it grants as logic. > > > Anyhow I really prefer naive natural extensions of logical systems as > > a generating stream of logical systems, yet the above criterion might > > be needed to settle the final demarcation of logic from other > > disciplines. > > > Whether this is of importance or not, is something that the future > > would determine. Arguable reasoning about naively extending logical > > systems in the genre displayed above seem to be a "limited" kind of > > reasoning, so if just from that reasoning all mathematics can be > > derived then it is worthwhile promoting logicism. However still for > > interpreting mathematics in those systems one needs to negotiate > > thoughts about 'structures' i.e. some contentfull thoughts that > > mathematics is seem commonly to be about, since the logical motivation > > is too general for negotiating those particular thoughts, anyhow > > seeing that there is a logical mainframe within which those thoughts > > can be carried out is without any doubt helpful in guiding > > contemplations about those thoughts themselves. So I think logicism is > > important in providing logical guidance and of course as being an > > arbiter for mathematical thought. > > > Zuhair > > #1 Don't try to define math in terms of math. Besides being circular > reasoning, you are simply taking a subset of math, isolating and > formalizing it. Define math in informal nonmathematical terms. > > # 2. Know what level of abstraction you are at. Don't define science, > ... > > read more »



