I'm coming to think that mathematics is nothing but the study of formal tautologies, i.e. the statements of the form that if such and such ... are assumed true then it follows that such and such... are true. The language of the formal theories can have extra-logical primitives, and it doesn't matter what kind of motivation its axioms are grounded in, it can be recreational, like chess problems, it can be a posteriori serving to capture empirically validated truths, or it can apriori synthetic meeting some strong intuition or philosophical argument, or it can be apriori analytic meaning, or even an apriori analytic syntactical consequence. What matters is the absolute consequence of pre-held assumptions in whatever language as far as that language is rigorous enough for those consequences to be ABSOLUTELY TRUE.
In this sense mathematics is ANALYTIC, and it is nothing but the arena of absolutely true analytic reasoning.
So mathematics is the logic of matters, whether those matters are logical or extra-logical. So in spirit LOGICISM is the correct philosophy of mathematics. This need not be confused with logicism in the sense of grounding mathematics in analytic meaning, no, here it is another kind of logicism that is more akin to the "if-then-ism". The most important matter is that the subject matter of mathematics is absolute pieces of analytic facts.
However, this definition of mathematics is subject to "trivialism" since most games so played might be trivially true because they may begin with inconsistent assumptions. So a justification against trivialism is demanded.
It is true that this justification is demanded, but this is not the job of mathematics itself, those justifications might belong to the subject matter of the discipline from which those axioms arose, and so it belongs to APPLIED mathematics, rather than to mathematics proper. However, it might be said that the axioms of pure mathematical systems need to have their justification being grounded in mathematical thought. And the latter is not an absolute analytic consequential fact, it is more of a deep penetrative speculation related to our reasoning about the world, which indeed can be apriori synthetic, thus removing the analytic edifice as a basis for mathematics. So grounding of the truth of arithmetic, Geometry, for example, needs to be part of mathematics proper and not applicative of another discipline like the philosophy of mathematics or linguistics or whatever. Here with mathematics thought of in this sense, it seems impossible to ground it in the logicism account given above, an attempt to ground it in ANALYTIC MEANING might explain arithmetic, but seem to mostly fail for Geometry. The problem is that Analytic meaning is itself dubious, it is not like analytic syntactical rule following substitutive games, which constitutes the above-mentioned account of logicism. For example is identity analytic, is the omega-rule about the naturals analytic of their meaning? would the stipulation that the set of all points on a plane equidistant from a common point, is the same as the closed figure in that plane having constant curvature at each of its points, be analytic? etc...
As far as I'm concerned here, the justification for those purely mathematical systems seems to be a matter of ART to the mathematician, the actual mathematics for him is in CREATING them, or exploring their consequences, though he is aware of them harboring some truth beyond his creativity. All of this talk about justification might assist in having a better insight into the truth of those axioms, and in how to find new true axiomatic mathematical systems but still all of that remains at the underground pre-mathematical creativity, it more appropriately speaking belongs to mathematical "investigation", or to hypothetical mathematics, but not to the FACTUAL mathematical matters, with which mathematics is identified, much as hypothetical physics is not Physics in the factual sense. What matters is the final arbiter of the truth of those subjects, in physics and empirical sciences, it is empirical testing, while in mathematics it is LOGICAL CONSISTENCY if that is assured, then there is no matter of triviality. However, it might be said that there is another aspect that is demanded from those systems to be included in factual mathematics, that in addition to them being consistent they must also be TRUE since what's the use of investigating a consistent False system like for example PA+PA is not consistent. However investigating models of those systems had been of mathematical interest, and actually, they may even find application in some fields of knowledge, so this argument is false. Every consistent formal system is important, actually, even inconsistent ones if their proof of consistency is NOT trivial are important, for they might have heuristic value in discovering true and consistent fragments of them.
So mathematics is the study of absolutely true analytic reasoning.
Internal and External meaning
There is some sense in which logic is ought to be about grasping Analytic truths, i.e. matters that are true in all possible worlds.
I think there are two kinds of such analytic statements
1. Analytic by syntax
2. Analytic by meaning
In the first kind, I prefer to put it in an implicational form, i.e. of the form "if Q, then P" to mean if Q is true, then P is true, where Q and P might be a schema of sentences or individual sentences, where Q|- P, i.e. P is a theorem of Q.
The second kind has no general syntactical form, but it is supposed
to capture a meaning that is external to it that
is true in all possible worlds, like identity theory for example.
The second kind is not easy to tackle. So I'll speak
of the first kind.
The first kind is about 'derivation', so it is not about
grasping some external meaning or about being faithful
to some principle or concept, it is rather about enacting
a specified role which is basically syntactical in nature,
although this can have some built-in semantics of truth in it, still that semantics is based on derivation rather than on capturing some external meaning.
To illustrate that we begin with the simplest form of role enaction, which is the logical connectives, let's take the 'implication' connective "->". Now, this connective connects two statement in such a manner that the resulting statement's truth is dependent on the truths of the individual statements it connects, this role of connecting and its effect on truth is not completely external to the connective symbol itself! To illustrate that we'd say that
the role of -> is to connect a true P on the left to a true Q on the right, or to connect a false P on the left to any sentence Q on the right, to result in a sentence (WHICH CONTAINS ->) that is true, or otherwise the sentence (WHICH CONTAINS ->) would be false.
So the role of -> on the truth flow within the sentence P->Q
is CHARACTERIZED by a phrase that uses -> in it also, so it is not definable in an "eliminable" manner.
It is not like when I read the word "Horse" where it will direct me to a meaning that does not use the word "Horse" at all, actually, I can express this meaning by saying that: "it is an animal that is a mammal that ....." without the word "Horse" occurring in the definitional phrase at all, when I read the word "Horse", my mind is completely transfered to the image about horses, and I forget completely about the string of symbols "Horse", this is 'meaning by denotation' and it is an 'external referential' form of meaning.
We cannot define a logical connective without using it in the definition. So we can only characterize it.
Now it is known that all logical connectives can be defined in terms of the Sheffer stroke. But the Sheffer stroke itself cannot be definable without using the Sheffer stroke.
The basic two connectives of logic are negation and disjunction, and those are not definable without using themselves in the definition, so they can be characterized only.
However, those may be called the simplest derivability roles that can be, and there is a rule that specifies how they affect truths of the sentences (in which they occur) from the
truths of the statements they connect. That specification serves as to provide a "meaning" to those connectives, although that meaning is not external to them, I mean that meaning involves them, it doesn't eliminate them, it only specifies a functional role to them, similar thing can be said to specifications about what is a "variable" symbol, a "function" symbol and a "predicate" symbol, etc.... all of those are syntactical substitutional meanings.
So in some sense, it looks like "impredicative" form of definition that those connectives are given, although impredicative is not the proper word to use here.
Now the aim of the syntactical analytic is to build up a machinery of the deriving of strings of symbols from strings of symbols, and so on,... so there is a game of rules stipulated on strings of symbols, whereby those are derived step-wisely in a controlled mechanistic manner.
Now a sentence would be called Purely syntactically analytic if all of its symbols do not have external meaning, i.e. all of them only specify the roles they take in enacting the inference stated by the sentence, so they have no meaning outside of the sentence, so we can say that all of its meaning is contained (enclosed) within it. And I think that LOGIC is the study of the PURELY SYNTACTICALLY ANALYTIC statements.
However, we can have syntactic analytic statements that are not so "meaning contained", i.e. that are mostly fathomed
as referring to some external meaning, although the inference made in them is enacted in a syntactical manner.
for example 2+3=5, here the primitive binary function symbol "+" is usually fathomed as being a function on numbers that
assign its arguments (which are 2 and 3 here) to the number of elements in a disjoint union of the sets they number, i.e. it sends 3 and 2 to 5 which is the number of elements of a set that is the disjoint union of a set having 2 as the number of its elements and a set having 3 as the number of its elements. And actually one can think
of + as a set of ordered triplets (x,y,z) where z is the number of elements in a disjoint union of sets a,b having x as the number of elements of a and y as the number of elements of b.
So + can be reasoned as an object that does not involve the symbol + at all, so it can be assigned an "external" meaning, and actually, it is usually reasoned about and known through this kind of external assignment.
However the arithmetical inferences made using the + function are purely syntactic, they have nothing to do with this external assignment or any other external equivalent assignment, inferences about + are syntactically guided by axioms of PA, and the sentence
"if PA is true, and given the usual definitions of 2,3 and 5, then 2+3=5"
This sentence is a syntactical analytic sentence, and the crux of the reasoning in it is completely formal (i.e. syntactical), and whatever meaning assigned to + that is external to it then this meaning is not really needed for justifying the absolute truth of the above sentence.
Now I view mathematics as being about analytic sentences,
especially about the syntactical analytic since this is absolutely verifiable, however, should there be sentences
that are analytic by meaning, then those would also be
mathematical. Now Logic is about the purely analytic sentences, which are sentences using symbols that are, as said above, reasoned as having rules specifying their roles in the inference made in the sentence, so they have only internal meaning concerned solely with the derivation made by the sentence, so they are sentences in which their meaning is confined and enclosed within them. While other parts of mathematics (i.e. those that are outside logic) are those which care about analytic sentence containing primitives that can be assigned by (and usually discovered and fathomed by) an external meaning.
I'm really skeptical about the 'analytic by meaning' statements, even if those are verified as such, it appears that they are ought to be included among metaphysics rather than mathematics proper. However, I'm not so sure of that. It might be possible that syntactical expressions of those
'analytic by meaning' sentences might prove to be considered as "logic", and so would be among mathematics also. However, I doubt this really, because it is very hard to ascertain that such sentences can hold true in every possible world without them being involved in some metaphysical consideration.
So in nutshell mathematics is at least about the syntactical analytic whether it is purely so or not; and at the most, it is about every analytic sentence even those that are justified as 'analytic by meaning'.
However, this method must call for what constitutes "interest" in a mathematical statement, of course there are some trivial points, that their antecedents must be thought of being highly TRUE of this world, or at least of PART of this world, or at least proved indispensable to supplying heuristic means of aiding in discovering Part of this world;
definitely, the least is that they must be non-trivial, i.e. there is no present proof of their inconsistency, so if they are consistent or at least highly thought of being so, then they might provide a hypothetical description of PART of our world and thus proves heuristic in discovering this world in which we live. Otherwise one might coin mathematics for recreational purposes, but this is another matter.
Of course, it is clear that when we are at the grasp of sentences that are true in all possible worlds then they are true of our world, so if they are nontrivial, then definitely they may assist in discovering our world, this may in part explain why mathematics had succeeded in discovering our world.
However, we need to coin some criterion other than those externally oriented applicative interests. I think it must be related to the richness of formal provability a system can have, and this calls for maximality of formal provability!
The system which covers more and more syntactical analytic statements is what is aimed at so that we won't miss important pieces of knowledge that might aid us in discovering our world!!! I think one can reach at some structural criterion about the complexity of theoretic consequences of a certain formal system that makes it the point of interest in mathematics. However, this is something that is far remote from my knowledge.
Zuhair