Date: Jun 15, 2013 2:59 PM
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,
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