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:

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> > 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.

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> > 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.

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> > 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.

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> > 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).

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> > Now in first order logic quantification is allowed over Objects only,

> > predicate symbols are not quantified over.

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> > 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.

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> > 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.

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> > 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.

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> > 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.

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> > The above logical system clearly has the power to interpret second

> > order arithmetic!

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> > Actually along the same lines I think every piece of mathematics can

> > be said to be interpretable in a logical system.

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> > So logicism seems to be the case for the bulk of traditional

> > mathematics, and possibly for any mathematical endeavor?

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> > 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.

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> > 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.

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> > The representative set of a symbol goes generally along the following

> > lines.

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> > For monadic symbol $, it is the set of all x such that $(x) is True.

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> > For any n_adic symbol $, is the set of all <x1,...,xn> tuples such

> > that $(x1,..,xn) is True.

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> > A permutation over any set is a bijection from that set to that set.

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> > 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.

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> > 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).

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> > 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.

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> > Identity is a clearly invariant relation.

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> > 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.

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> > 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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