Date: Jun 15, 2013 7:31 AM Author: Zaljohar@gmail.com Subject: Why I think mathematics is really logic. 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),

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