What is Mathematics?

I will here present my own personal account on what mathematics is.

I think that in the broader sense mathematics is the "exercise of formal

thought", hence it is a sort of mental sport, like Chess, but on symbolic

level, i.e. a symbolic game.

Here formal theory is taken to mean any set of strings of symbols

governed by string manipulation rules, in such a manner that

we can construct further strings of symbols from prior ones using

these string manipulation rules, and such that the whole symbolic

streaming is not affected by what those symbols stand for.

For example take ordinary arithmetic that we were taught at primary school,

one can see that we infer string of symbols from prior ones without us

really knowing what for example the symbol 1 or even symbol +

stands for, we have rules about addition and multiplications and equality

and those would produce strings of symbols from prior ones without any

need for us to care about what those symbols exactly stand for. Such

a symbolic inferential system is what I label as FORMAL. The merit of

those systems is that they provide a descriptive account of rule following

that can be applicable to any empirical discipline, i.e. they are

empirically neutral, I can substitute anything instead of 1,

and the system reach into the same results. So to give a specific example lets take 2+2=4, this is a string of symbols dictated by the axioms of arithmetic, we reached that by strict rule following string manipulation rules 4 is the name given to the term symbol 1+1+1+1, and 2 is the name given to the term symbol 1+1, and by equality rule any string can be replaced by its name without causing additional change to the prior to replacement status, so 1+1+1+1=4 leads to 2+2=4. No particular meaning of 1 is required nor of +, they can be taken to be blind symbols (non referring symbols), the flow of inference follows the rules of the game.

A formal system can be taken to mean a system in which all of its symbols

are non referring to any particular object, actually we can even consider

all non logical symbols of it to have no reference at all, and still the whole

system will work, since the inferences in the system has nothing to do with

the reference of its non logical symbols.

However such an extreme non referring formal symbol would be hard to verify if

it is consistent, and actually one of the most important aspects in examining these

symbolic games is their syntactical consistency, for if they are inconsistent then they would prove everything and won't be worth considering other than knowing that it must be avoided.

The simplest criterion for determining significance (non triviality)

of those games is to pose the syntactical criterion of non existence

of a trivial proof of inconsistency of the theory, this doesn't prove

the consistency of the system, but at least tell us that it is possibly

consistent, or even if inconsistent still some non trivial fragments

of it might be consistent. However if we say that mathematics is just those

non trivial symbolic games, then this would constitutes what is meant by

mathematics in the more general sense. It would include all fun symbolic

games, and definitely not all such games would be of scientific (factual)

value. So this is not how mathematics is usually tackled.

The customary mathematical experience is about formal systems

in the above sense that have some apriori justification of being

consistent. This is what I call as justified mathematics.

This will focus our examination of symbolic games to those

that are highly likely to be consistent and likely being of use in the scientific process.

An apriori justified matter is usually something that reflects a basic rule of

our thought whether analytic or synthetic. Analytic rules which are the ones

that can be known from just knowing the meaning of the words, pose no real problem.

However the apriori synthetic ones are somewhat vague,

I would consider those as basic elements of our thought

about basic observations or intuitions about basic observations around us. And I think they are

widely acceptable that requires no empirical validation,

since they are rules of our thought by which we can discover

things, so it in some sense comes prior to empirically

validated (a posteriori) material.

Now if a formal system can be a part of a discourse about

apriori justified concepts that didn't require the consistency

of that formal system in the first place, then this part-hood

will be considered as providing a justification for consistency

of this formal system. By then this formal system would be enrolled

in the mathematical systems, also the apriori justification of it

would be part of mathematics, even the contentful axiomatic system

having that particular justifying semantic is also part of mathematics.

Example of apriori synthetic justified piece of thought is that about

"succession", this came from observations around us, we see that things

can follow things in succession, so the imagery grasp of having a starting

object then having a successor to it, then a successor to that successor

and so on.. is a basic figment of our perception about the world; Now

the ordinary +1 interpreted as denoting the successor, i.e. x+1 means

just Successor or x, this is an analytic fix and is not a cause

of any problem because it didn't add anything other than

extra-wording spelling succession, now same applies to

multiplication which is just abbreviatory over addition.

Now examination of basic thought about succession as presented

to our sensorium reveals that any finite iterative successor of the

first object is reachable by a predicate that holds of

the first object and that transits from every object to the

next, this is trivially true, and is acceptable as a basic rule

of thought about finite successions from a starting object,

there is no qualms about it really, and thus is justified to

be held as non contradictory (doesn't give rise to inconsistent

formalization) when added to the basic axioms characterizing succession,

addition and multiplication. The whole system is pretty much justified

apriori, through a mix of apriori synthetic and analytic,

and so arithmetic which is a part of this apriori investigation

about succession, would be a formal system whose consistency is

justified apriori.

Now there are two significant aspects about mathematics, the first is

it being formal thus enabling it to describe themes of rule following

that can be inhabited by any sort of objects, and the other part

is that of its apriori justification which will make it

about a principal aspect of our thought, thus giving strong grounds

for justifying its consistency and also for probability of having

applications in empirical sciences.

Now being apriori justified wont' restrict the aposteriori applicability

of the justified formal system, so the justified system won't be so limited

as to being just applicable to narrow rule following scenarios about

its justifying grounds, on the contrary being apriori is

"aposteriori neutral" or "empirically neutral". So this

justification won't limit the empirical applicability of the justified

system as compared to aposteriori justified formal theories (see below).

The apriori justification would be important for:

1. Justifying the claim of consistency of an already defined formal system

2. Can lead to the production a formal system from the outset

3. Can be important in extending the formal system by adding new axioms.

4. Provide semantics for the formal system that might be important in applications of it.

Examples of 1 is interpreting ordinary arithmetic in Peano arithmetic or simply

in more trivial kinds of arithmetic + the omega rule.

Example of 2 is Geometry where the formal system was

produced form the outset by basic thought about

spatial entities as trivially presented to our sensorium, which

are justified as apriori synthetic. Then after Geometry

was defined, one can go and use all of the formal system

of it (its syntax) as a non referring symbolic game the

symbols of which can stand used to refer to any empirically

validated object.

Example of 3 is extending the ordinary Peano arithmetic by an omega rule.

Example of 4 is definitely the use of arithmetic in most

of scientific disciplines, this can sometimes benefit

from basic thought about succession as an explanation

for the naturals.

One important issue is about aposteriori formal systems

and non mathematical apriori formal systems. Now for the

former ones, the formal system is already required to be

justified apriori before using it to explain empirical material,

no scientist would produce a theory of physics lets say,

that uses a formal system that he doesn't have any

justification of it being non contradictory in the first place,

that's why aposteriori axiomatic systems are NOT included in

the mathematical justification record because they didn't make

that justification by themselves, those systems only uses

these formal systems by assigning particular meanings to

their symbols as related to the subject matter of their investigation,

and the whole axiomatic system is actually fully contentful

rather than being formal, but say even if we took the formal

system used (the syntactical part) and make its symbols non

referring then still the formal system was not justified in

the first place by the aposteriori account of truth.

Should that be the case then it would become a legitimate

part of mathematics, but this never occurred so far,

and even if it occurs, it possibly won't have much

applications outside that particular aposteriori field

in which they were born.

Similarly apirori formal systems like formal ethics, formal ontology and

formal epistemology etc... all of those are just applied formal,

i.e. they require the formal system they use to be apriori

secured consistent before using it, and so the apriori material related to epistemology, ontology and ethics itself didn't contribute to the

justification of consistency of the formal systems they use.

So they are not part of mathematics. However should they be

the only source of justification for the formal systems they use

(i.e. they didn't require beforehand that the formal system

to be secured consistent by some other apriori investigation)

then they should be a legitimate part of mathematics,

but non of those is present so far, and even if that happened

it would likely be too particular that it won't have much

use outside the field in which it thrived.

In nutshell mathematics is about

"Non trivial symbolic games that have some apriori justification of being consistent"

However what remains unresolved is the exact nature of

mathematical objects, those objects that directly give

rise to the apriori justified formal thought?

what are those objects, are they present in reality or

are transcendental as rules of our thought needed prior

to observation? or in the admixture between those two?

are they totally dispensable by concrete objects and

so they are nothing but names of states about concrete

objects? are they fictions? this is another issue.

But I tend to think some of them are abstract objects

that do have a casual relationship with concrete objects!

they actually frame them into what they are and we

come to know about them from the effect of this framing

of those concrete objects, natural numbers might be

just objects that enforce concrete material to be

distributed as multiplicities, and us being concrete

objects sensed what happens to the distribution of

that concrete material by the abstract objects,

and thus came to know about them in an indirect manner,

so there is a causal relationship from those objects onto

the concrete world which in turn is transmitted to our sensorium.

The argument for mathematical objects being abstract

is that for example number 1 can be present "in whole"

at two places at the same time, this makes it in some

sense spatially ubiquitous, however that doesn't

mean that it is totally alien to space-time, it is still

spatio-temporal! but not delimited in the way concrete objects

are (cannot present "in whole" at two places at the same time,

a property we may label as "spatial-delimitation").

I don't see a clear argument that necessitates that causality

requires spatial-delimitation, in order to be possible?

We can on the other hand have objects that are spatio-temporally

delimited, i.e. BOTH cannot be present in whole at two times at the same place

and cannot be present in whole at two places at the same time, now

still those can effect casual relationships, now causality

is not thought to be decreased when objects become just spatially delimited

(i.e. can present in whole at two times at the same place but not in two distinct places at the same time), which are most of the usual objects, so

dropping the requirement of temporal delimitation didn't affect

the possibility of effecting casual relationships, on the contrary there is a sense that it would increase the ability to effect casual relationships, since it gives the impression of more presence in space-time than the fleeting rapidly transiting objects, and so there is more chance of collision with concrete objects and thus more possibility of effecting casual relationships; so why dropping

spatial delimitation is expected to make effecting casual relationships

impossible? I don't see a clear argument for that? It actually also appears to have more presence in space-time than the concrete objects, although of course not touchable nor can be heard, so in this sense concrete objects no doubt are much easier to know of, but still I don't see a clear argument supporting that mathematical objects are 'outside of space-time', since this is a too vague statement to make. If they are so then how come we came to know about them being spatio-temporally non delimited? why should they have any such relationship to space and time if they are external to it? An object that do not effect casual relationships would be imagined as having Ghosty nature, something that is like zero dense, or in other words some object that no concrete object (i.e. spatially delimited) can collide with, so every concrete object will pass through it, since it cannot collide with it since it is zero dense, such an object cannot be seen because photons for example cannot be reflected from it, nor can be touched nor can be heard nor can it be presented to any of our five senses, such a ghosty object would be of the nature that it cannot effect casual relationship with any concrete object, and it can in principle be thought of being able to permeate through matter, passing through walls and barriers like ghosts in Cartoons. I don't think that any mathematician when he says that mathematical objects are abstract I say I don't think he means that they are so ghosty, because simply such objects won't be discover-able. On the other hands I think it is imaginable to think of objects that have ghosty nature and yet still are spatio-temporally delimited! i.e. being ghosty doesn't entail being spatio-temporally non delimited. The main point that I'm raising here is that when one says that an object cannot effect casual relationships with any concrete object then this entails that it has a ghosty nature as described above, now I don't see how being not spatially delimited (i.e. presence at two places at the same time) implies having such a ghosty nature? I think it is possible to be BOTH "not spatially delimited" and "not zero dense", why not? I don't see any reason to forbid that. It is true that being not spatially delimited can give the impression of being ghosty since the usual spatial barring on concrete objects is breached, this barring strangulate concrete objects to exist in just one place at a given moment of time, so if this is breached, thus it would indeed give the impression of permeating through spatio-temporal concrete material, and thus having a ghosty nature, but this impression is not a proof that this must be the case, and it is in some sense mysterious, there is no clear 'concrete' barrier that was breached (by non-spatial-delimitation) in order for us to claim that a ghosty nature is necessary for such a breach, and after all there is no clear evidence that being ghosty would enable such non spatial delimitation in the first place? what has been breach is a certain spatial barring property that pertains to concrete objects, but breaching a barring property is different from permeating through concrete objects, so that impression is not grounded at all; actually those non-spatially delimited mathematical objects do not have full ghosty material, because we do acquire knowledge about mathematical entities, so if those were ghosty then no such knowledge would have been possible unless there are routs of income of knowledge other than our sensorium, which is unlikely.

One implication of what I said above is that basic mathematical objects might be physically sensed though not as solid as concrete manner, but yet still they might be physically sensed through concrete material reflected from them reaching to our sensorium thus conveying information about them, this might be possible? so if we imagine for example the universals as non-delimited spatio-temporal objects that are not zero dense, attached to the concrete objects that they predicate, then number can be considered a higher level universal attached to the lower level universals they would predicate (since number is a property predicative of predicates in terms of their fulfillment grade, so 2 can be imagined as an object attached (in whole) to all predicates having double fulfillment, and each predicate having double fulfillment can be imagined as an object attached (in whole) to each of the two objects it predicates. Similarly sets can be imagined along the same vain, as predicates of predicates of .... etc. Other definable parts of mathematics are just analytic conservative extensions over those basic sensed objects. This seems rather awkward and unbelievable. However the above line of reasoning presented here shows that this cannot be ruled out, still I think that an empirical way to test it is rather very difficult if possible at all.

In nustshell I don't see that being spatially non-delimited implies not effecting casual relationships, and even the opposite direction is not justifiable. That said, I still see that it is possible to have a casual interaction between concrete objects (at least spatially delimited objects) and abstract objects (i.e. spatio-tempral objects that are not spatio-temporally delimited) and of course this will reach our sensorium since the effect of that interaction on the concrete material will be transmittable to our sensorium, since our sensorium is concrete; so the possibility of indirect casual relationship between us and those spatio-temporal non delimited objects can still hold, and this might explain at least why it is not impossible to know of those spatio-temporal non delimited objects (i.e. the mathematical objects) which are called usually as abstract, should those be truly existing.

I'm not sure if this makes sense. But even if so, surely not