```Date: Jun 15, 2013 2:59 PM
Author: Zaljohar@gmail.com
Subject: Re: Why I think mathematics is really logic.

On Jun 15, 6:04 pm, Charlie-Boo <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 spatial-temporal 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 n-kind. 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 Pi-1> > 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 un-indexed).  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 n-sort (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 non-logical constants. However it doesn't speak a lot> > about what meta-logical 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 non-mathematical terms.>> # 2. Know what level of abstraction you are at.  Don't define science,> ...>> read more »
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