Philosophical notes:

I think mathematics is nothing but the investigation of ideal constructs.

By a construct it is meant a set of objects fulfilling a set of statements written in a logical language.

The set of statements after which the construct is characterized is called the 'discourse of the construct'.

The statements in the discourse of the construct are produced recursively from a set of axioms by rules of inference, the axioms are statements that are taken to represent what is intuitively judged to be true characterizations of a set of primitive concepts.

A primitive (not definable) symbol is either composite or not.

A primitive symbol is said to be composite only if it refers to a concept that is most naively understood as being reducible to simpler concepts.

A primitive symbol is said to be atomic only if it refers to a concept that is most naively understood as not being reducible to simpler concepts.

A primitive symbol is dummy (sub-atomic) iff it is non referring.

In principle all empirical predicates are composite, also all novel fictional characters are composite.

A composite primitive that have no more than finitely many components is said to be a finite composite primitive.

If any discourse of a construct has every component of every composite primitive of it represented by a primitive of it, then this discourse is said to be: primitively transitive.

A construct is ideal iff its discourse is primitively transitive having no infinite composite primitives.

From the above a construct with a discourse having all its primitives as dummy or atomic is always ideal.

Primitives like equality, Part, membership, contact, Label, addition, Choice, negation are all atomic.

Primitives like multiplication, exponentiation, hyper-exponentiation, implication, binconditional etc.. are all composite.

Whether Conjunction or Disjunction is atomic is optional, I tend to hold conjunction as atomic and disjunction as composite. 

All known mathemtaical theories PRA, PA, Z, ZFC, Mereology, Mereotopology, are about ideal constructs.

Older mathematics is written in a semi-formal manner, like for example arithmetic, since the primitives they are based upon are so naive that one can work with in a consistent manner without the need for the full explicit formal logical machinery. However they end up speaking about the same construct or actually sub-constructs of those the known formal theories speak about.

Of course it is preferable to have a discourse that is based on naive primitives, preferably atomic, the naive characterization of which yields strong results, membership is a good example of such a primitive. Composite primitives used in the discourse of an ideal construct are also preferably reducible to 'hand-fully' finitely many primitives, like those of multiplication and exponentiation.

Addition might possibly be understood as a non referring primitive symbol, so it is of the most simple primitives, anyhow this might not really be the case since addition have strong similarity to concatenation of symbols. 

The main motive behind this definition is of having the complete story of a construction from the simplest primitives for it to be called ideal, neither empirical theories nor customary fictional stories known in literature speaks about ideal constructions in the sense outlined here, since non of the primitives in them are decomposed to their component primitives in them, since those are not really about outlining the whole constructive process, this leaves only mathematical theories speaking about ideal constructions, logic itself although presented using primitives in a constructive manner as outlined above, however it is not concerned primarily with construction and the ideal construct it leads to is just a singleton set, too trivial to be considered mathematical, or if so can be designated as very weakly mathematical.

I don't think it is possible for other concepts to overlap with mathematics so defined, i.e. there cannot exist some primitive concepts the discourse about which leads to an ideal construct that is NOT viewed as being mathematical. At most such constructs (if available) would be regarded as very weakly mathematical.

On the other hand it is very difficult to conceive of a mathematical theory that is not about an ideal construct. Accordingly mathematics is just being about ideal construction.

EMPTY SYMBOLS LINE OF THOUGHT

This is the traditional formalism view of mathematics being an empty game of symbols. Along the above explanation of mathematics one can easily say that mathematics is the collection of all consistent collections of statements written with empty symbols. Since those strings of symbols are non referring then one can in principle say that mathematics is about those symbols themselves and meaning given to primitives are just heuristic gestures to discover those empty symbolic games much as Newton's Apple or Philosopher's Stone had motivated research that lead to discoveries in physics and chemistry. An even more radical stance is to drop the consistency requirement and say that mathematics is nothing but string manipulation of empty symbols, even if those were inconsistent, on the basis of pragmatic considerations of discovering consistent subsets of the later, and by then would insistent on the concept of "non triviality", however non triviality is acceptable only in the view of a possible heuristic value in discovering theorems that would later be proved to follow from a consistent subset of those inconsistent non trivial theories, so still consistency would be needed for the final acceptance of those theorems.

The main objection is that consistency of formal theories cannot be proved in formal theories with empty primitives since the later would be stronger (except in very weak scenarios) and begs the question of consistency even more than the former. So there must be a 'meaningful' formal theory (i.e. with referring primitives) in which consistency is proved, and whose consistency itself is self evident! So meaning trips in again in an inescapable manner. 

Still the above argument doesn't completely refute this line of thought, still it can be claimed that proving consistency is not about consistency per se but rather about our knowledge of consistency, whether a collection of statements is consistent or not has nothing to do with whether we know it is consistent or not. Just because we need some 'meaningful' method to prove consistency has nothing to do with the status of consistency of the theory in question nor it has anything to do with the need of this theory for that meaning, the meaning used in the proof doesn't necessarily seep into the theory proved consistent, so the subject matter of mathematics remains neutral to those considerations. 

According to this line of thought our interest in mathematics is at best pragmatic,i.e. of it being a tool used by other disciplines to outline particular rule following themes peculiar to those disciplines. So mathematics provides a kind of non specific rule following scenarios, i.e. some of the sentences of mathematics might have the property that when we substitute the variables in those sentences by such and such objects in a simultaneous manner (all occurrences of a variable are substituted by the Same object per substitution) then the sentences would still retain the truth value assigned to them by the prior truth stipulations on the empty symbols before substitution, so in this way mathematics provides a kind of a mantle of rule following. So pragmatically speaking we don't even need full proofs of consistency, all what we need is to engage with any theory that is likely to be consistent, likelihood of consistency is just enough for practical reasons, and along the same line even inconsistent theories that are non trivial, i.e. in the sense that they are so stipulated that they make it easy to discover theorems in consistent subsets of them, even those are to be considered in mathematical investigation. So this string manipulation line of non referring symbols still retain its vitality. All meaning invoked in the investigation of mathematics is to be discarded after we reach to the string manipulation desired, so meaning is just an aid and it is not in any manner viewed as being 'essential' about mathematics. Accordingly since mathematics are all consistent theories extending logic that are written with dummy symbols then in principle it would be expected that most of possible mathematics would not have any application at all, so applicability of mathematics is not a point against this line of thinking, that the current mathematics has a lot of application is just because it deliberately cast away a huge bulk of non applicable mathematics, and concentrate its investigation on those that may have fruitful applications, even a harsher stance is to go say that the strings of symbols themselves are 'fictional', i.e. 'don't exist', so mathematics would turn to be a fictional mantle of rule following. This is a harsh pragmatic instrumental characterization of mathematics. Whether it is true or not is something that no one can ascertain. It has the advantage of freeing mathematics from metaphysical commitments. However it is so difficult to ascertain and distances mathematics from how it is motivated and practiced. So we need to resort to some clear criterion that is more in alliance with how mathematics is practiced; and how it is practiced does involve meaning.
 
However it is clear that mathematics is to be based on primitives that are so simple that are either empty or look as if they are empty symbols, i.e. minimally referring symbols, and such that inferences can be made about them semi-formally in an flawless manner without the need for the full explicit logical apparatus. Those must be naively understood, and motivates strong intuitions around them, and above all must be trivially true. 

If we are to seek a meaning for mathematics then the main motive for mathematics, without doubt, pivots around 'construction', but of course that starting from the simplest possible primitive concepts and then building upwardly definable predicates or composite primitives. 

This account discourage the empty symbols line of thought of being the entire explanation of mathematics, although it doesn't role it out, and encourages the ideal construction principle outlined above.

WHY DISCOURAGING THE EMPTY LINE OF THOUGHT?

But why should we resort to the meaningful construction manner instead of the empty symbols line of thought, if the later free us from metaphysical commitments then why prefer the former which is obviously not a metaphysical free approach. Whatever theory written guided by the meaningful approach it would use symbols at the end, and it can always be reasoned that those symbols are empty and the meaningful reference is artificial and thus can be eliminated from the picture and what is left is just an empty symbolic game, so the meaningful approach is reducible to the empty line approach, so why insist on the former and discourage of the later? The reason is related to how mathematics is practiced, if it was the case that the meaning attached to symbols is just needed to discover those theories and then we can do without it and proceed in an empty manner, then the empty line of thought would have been preferred, at then we'd regard the meaning as 'discard-able' since there would be no further use of it other than the discovery it already lead to, this would have been a situation very much similar to Newton's apple, which was an inspiration to ponder about the concept of gravity, however no body needs to watch apples fall in order to understand concepts of gravity that Newton had put forth, nor anyone needs it to further investigate theories about gravity, so Newton's apple is not needed anymore and it is not in any manner considered essential to the subject matter of gravity, so the same stance could occur with formal systems were meaning could be considered as having a role similar to Newton's apple. However what is against this line of view is that unlike Newton's apple the meaning given to primitives proving the consistency of some theory is actually indispensable for that proof, since what we are confronted with are actually theories having INFINITELY many statements, and there is no way to see all those statements, so at the end we need to prove the axioms of the theory being true characterizations of some meaningful primitives that are considered as trivially true, there is no other way, if we strip the proving theory out of its meaning then it is just a jungle of symbols and it is formally speaking much stronger than the proved theory and from the strict formal point of view it even begs the question of consistency more than the proved theory, so it needs the meaning attached to it. Now if we don't have a proof of consistency then matters are even murkier, we'll need some meaningful engagement that support the likelihood of consistency, and this cannot be done in a non meaningful manner. What even make matters more difficult for the empty line of thought is that those meaningful approaches that proves consistency or its likelihood are not just needed for those purposes, actually pragmatically speaking those formal theories would be very difficult to deal with and understand if we strip them completely out of meaning, so difficult that it is almost impossible to work with them as such, and even further understanding of those theories like with the view of having a possible application does depend on them having meaning attached to them, not only that even pondering about extending those theories requires meaning, so meaning do not have the same status of Newton's apple at all, it is indispensable even at pragmatic level. However one must be cautious in giving any necessity status for meaning other than what is really just needed to explain the symbols in a formal theory, all equivalent meanings are to be understood as being the same in essence, and that essence is what the theory is about. It is agreeable that sometimes matters might go wrong and we may give the wrong attributes only to discard them later for better ones, such endeavors does give the image of supporting the empty symbols line of thought, however I still maintain that there would always remain some meaning curtailed to the theory common to all interpretations given to that theory, so it is not the case that we'll reach into an total emptiness of meaning. The case is actually of minimal reference that wrongly imparts non reference. So it would be much more fruitful to investigate the meaning connected to those theories instead of shying away form it just to get a general status of metaphysical freedom, which really make no advancement in understanding nor in progressing knowledge or working in mathematics. That's the main reason for dismissal of the empty symbols line of thought, although it might remain suitable for some weak areas of mathematics, but certainly not to be advocated for explaining and working with the whole of mathematics.

DEFENDING THE EMPTY SYMBOLS LINE OF THOUGHT

Still the empty symbols line of thought is not completely squashed, still those who hold this line of thought can say that even if meaning is indispensable to issues raised above still this meaning is just indispensable to "our knowledge" of those issues and not to those issues per se. Still one can 'in principle' maintain that the consistency of those systems has nothing to do with the meaning of those symbols, it has everything to do with the ANALYTIC machinery of logic, in other words the SET of all consistent collections of statements of empty symbols (i.e. the set of all sets of statements of consistent theories extending logic), which is held to be the subject matter of mathematics according to the empty symbols line of thought, is something that exclusively reflect the analytic power of logic and nothing else, that meaning is evoked in due course is nothing but an intermediate step, that even if we cannot dispense with practically speaking yet still this indispensability has nothing to do the mere status of extending logic per se. This line of thought views mathematics as having both necessary and analytic genre, the rules of logic are necessary to enable analytic (rule following) machinery so they are not a kind of statements that depends on 'intuition', they are stronger! they depend on 'necessity' which is something prior to intuitions since whatever intuition brings about must not be in conflict with those rules in order for it to enact analytic reasoning, now the further processing in logic is of course clearly analytic and also intuition free, so an 'intuition free' edifice of mathematics is raised here by this line of thought. Intuitions might even be *necessary* to know of those systems but still it is not held to be necessary for having those systems, so it is just a guide to and not an essence of what's going on in mathematics. So mathematics is traced to necessity of analytic thought (premises of logic) and Analytic reasoning (derivation of theorems from premises), no intuition is necessary for mathematics per se, but it is necessary for knowledge of mathematics. This gives a strong status of evidence based genre to mathematics that is metaphysical free at the same time. It is agreeable that it is in a sense an advantage to think of mathematics having such a characteristic. 

I like to classify statements according to their backing as:

(1) Backed:

1_A: Strongly backed:
1_A_i: Empirical
1_A_ii: Necessary
1_A_iii: Analytic

1_B: Weakly backed
1_B_i: Intuitive
1_B_ii:Pragmatic

(2) Un-backed:
2_A: Ad-hoc
2_B: fictional
2-C: Against evidence.

So from the above mathematics belong to the non empirical strongly backed category.

This separates mathematics per se from knowledge about it, the later does require intuition to guide us to and through the mathematical realm. So the sole nature of mathematics might be strongly backed yet our knowledge of it might be weakly backed in the strong realms of it (like of strong set theories and large cardinals) since it depends on intuitions and pragmatic considerations, even sometimes down to the Ad-hoc and fictional.

Although I began with discouraging the empty symbols line of thought, yet I think it is the true one characterizing the essence of mathematics, although practically speaking the meaningful line of thought is indispensable to knowing of mathematics.



LOGIC & MATHEMATICS:

Frege wanted to reduce mathematics to Logic by extending predicates by objects in a general manner (i.e. every predicate has an object extending it).

To speak about this idea in more formal terms, we'll add a monadic symbol "e" ,denoting 'extension of' to the logical language, and stipulate the following formation rule:

if P is a predicate symbol then eP is a term.

Then axiomatize:

eP=eQ iff (for all x. P(x) <-> Q(x)).

Now the 'membership' relation would be *defined* as:

x E y iff Exist P. P(x) & y=eP

This is a second order definition of course. 

The base logic is second order with identity (axioms included) and e added.

This trial failed. 

However there has been many trials to salvage it. The general frame of all those trials is to restrict predicates eligible to have extensions.

So the formation rule will be modified to:

If P is a\an ... predicate symbol then eP is a term.

Examples: Replace "..." in the above rule with the following:

(1) first order 

(2) stratified

(3) Acyclic

By first order it is meant a predicate definable after a first order formula.
By stratified it is meant that all symbols in the formula except the logical connectives, quantifiers, punctuation, = and e will be labeled with naturals in such a manner that variables linked by = receive the same label, also any y and G in the formula y=eG will receive the same label, while predicate symbols will be labeled one step higher than their arguments, so if x is labeled with i, then P in P(x) must be labeled with i+1. Of course all occurrence of the same symbol are labeled by the same label. 

Accordingly the formula defining E above is not stratified.

Acyclic means that an acyclic polygraph can be defined on the formula such that each symbol (other than =,e,quantifier and the connectives and punctuation) is represented by one node and an edge will be stretched between any two nodes for each atomic formula having the symbols they stand for as arguments of.

/

Now it is clear that all the above qualifications are strictly syntactical and that they are connected to the issue of consistency of second order logic itself, so they are mere logical issues, they are not motivated by any so to say 'mathematical' concept per se.

Now I'm not sure of the strength of those approaches, but if they prove to interpret PA, then we can say that a substantial bulk of ordinary mathematics would be in some sense logical, since it would be interpreted in a basically logical theory since making modifications to ensure consistency of a fragment of second order logic is itself a pure logical issue.

Another trial along the same general lines is the following, but I'm not sure of it yet though.

We resort to typing predicates according to how they are defined, the idea is to have a sort of recursive definitional typing. 

object symbols shall be denoted by lower case.
predicate symbols shall be denoted in upper case
typed predicates shall be denoted by indexed predicate symbols.
starred predicate symbols represent  Constant predicates 
non starred predicate symbols represent Variables ranging over Constant predicates 
of the same index.

for example: P1 is a variable symbol ranging over all Constant predicates indexed with 1, so it ranges over Q1*, P1*, R1*,...., so it can only be substituted by those.

While P1* range over ONE predicate only.

All first order logic formulas have all predicate symbols in them being constant predicate symbols. And here they receive the index 1. The only exception is = predicate which would be left untyped thus it range over all OBJECTS.

Formation rules of typed formulas:

Rule 1. All first order logic formulas if we index all predicates (except =) in them with 1 and star them then they are typed formulas.

Example: for all x. P1*(x) -> x=eP1*

Rule 2. Un-starring predicate symbols in a typed formula results in a typed formula.

Example: for all x. P1(x) -> x=eP1 

is a typed formula.

Rule 3. quantifying over variable predicates of a typed formula results in a typed formula

so "Exist P1. for all x. P1(x) -> x=eP1"  is a typed formula.

Rule 4. If a formula F is a definitional formula of predicate Q after a typed formula Gn (G has the highest index of a predicate in it being n), and if all of those highest indexed predicates were constant predicates and if Q received the same index n, then F is a typed formula. 

In general F is a definitional formula of predicate Q after typed formula Gn means F is a formula
of the form "For all x. Q(x) iff Gn(x)".

Rule 5. For the same conditions in Rule 4, if any of the highest indexed predicates in Gn is a variable
symbol, then Q must receive index n+1 in order for F to be a typed formula.

Examples:

For all x. Qi+1(x) iff ~ Exist Pi. Pi(ePi) & x=ePi

For all x. Qi(x)  iff  Pi*(x) & ~Gi*(x)

are typed formulas.

Rule 6: a typed predicate symbol (any predicate symbol in a typed formula) only range over predicates that hold of OBJECTS only.

Rule 7: if a formula is a typed formula, then ALL of its sub-formulas are typed!

Rule 8: if Pi,Qj are typed formulas, then Pi|Qj is a typed formula. 

where "|" is the Seffer stroke.

Rule 9: all propositional logic equivalents of any typed formula is a typed formula.

So for example: " for all x. ~ [Qi+1(x) xor (~Exist Pi. Pi(ePi) & x=ePi)]" is a typed formula.

Now the above process will recursively form typed formulas, and typed predicates.

As if we are playing MUSIC with formulas.

Now we stipulate the extensional formation rule:

If Pi is a typed predicate symbol then ePi is a term.

The idea behind extensions is to code formulas into objects and thus reduce the predicate hierarchy
into an almost dichotomous one, that of objects and predicates holding of objects, thus enabling 
Rule 6.

What make matters enjoying is that the above is a purely logically motivated theory, I don't see any clear mathematical concepts involved here, we are simply forming formulas in a stepwise manner and even the extensional motivation is to ease handling of those formulas. A purely logical talk.

The surprise is that Second order arithmetic is interpretable in the above LOGICAL system.

Thus highly motivating Logicism!

So mathematics is formed in the womb of logic.

Frege was not far after all.

For a nice presentation of this theory see: 

http://zaljohar.tripod.com/typing.pdf

Note (I know now that this fails, but the general plan persists) see:

http://zaljohar.tripod.com/extensionallogic.txt

If this line of thought manage to work, then the empty symbols line of thought would be highly motivated! the reason is that even though mathematics is more meaningfully motivated after constructions or the more general 'structures', however seeing that it can be the result of theories that are logically motivated (which is way weaker) does impart a sense of Analyticity to it, and a sense of dispensability of meaning attached to symbols. So mathematics would more fit being machinery for analytic inference than it being about particular kind of reasoning involved with constructions and structures. 

Anyhow I tend to think that it is possibly the case that weak mathematics is logical or quasi-logical (near logical) and this might include most of traditional mathematics, but however stronger mathematics like that of strong set theories like ZF and stronger theories or the alike, those most probably involve concepts that are higher than mere logic or near logic, mostly those concepts would be best understood as being about constructions or more generally structures. On the other hand according to how constructiveness is defined here even logic itself can be encompassed by that definition, so the definition that I gave to mathematics in the head of this account stands.



FINAL WORDS:
Care must be taken not to infer that 'constructiveness' is ideational in nature. On the contrary it might be objective and human independent. When we are investigating ideal constructs, this doesn't mean that we are forming them in our minds or anything like that, it might as well mean that we are 'discovering' them. The wide applicability of mathematics shows that there is some overlap between constructiveness in nature and ideal constructiveness. However the details of that overlap is to be left to deep philosophical speculative intrigue. Accordingly, unlike the known Constructivist's school of mathematics, infinite sets, power sets, Choice and law of excluded middle all comfortably find their way into acceptance among mathematics according to the Ideal Constructive line of thought presented here.


Zuhair

May 25 2013