Mathematics is the study of consequences of pre-held assumptions.
We may divide truths into two kinds: receptive and consequential!
With the first kind I mean truth that is external to our makings and procedures, that impose itself on us, that our main stance from it is that of "receiving it", like for example that the earth orbits the sun, that the sun is bigger than earth, like water is composed of hydrogen and oxygen, etc.. those are truths that impinge itself on us and we have no choice but to accept them as such, and to discover them we need to use apparatus related to detecting them, primarily being our five senses, devices, etc..We can call such truths as reality kind of truths.
The other kind of truth is what I label as "consequential", here this kind of truth has a human role in delivering it, for example we begin with an arbitrary set of rules, and then examine where these rules lead us to? A simple example is to have a rule:
R is transitive (i.e.; if x R y R z then x R z),
and another rule stating that
there are objects x,y,z,u such that x R y R z R u R x,
then from these two rules we'll have the following 12 consequences:
x R z, x R u, x R x, y R u , y R x, y R y, z R x, z R y, z R z, u R y, u R z, u R u.
The fellowship of those consequences from the two input rules is a kind of TRUTH. This is a consequential fact. In the sense that we are obligated to hold that those outcomes are the results of applications of the input rules. The main difference between the consequential and the receptive kind, is that with the consequential kind we have control over the conditions from which we harvested that truth, while with the receptive kind we don't control the medium from which that truth sprang. Now consequential truth might lead to outcomes that might oppose receptive truths (reality truths), for example: if we input the rule that all Males have XX chromosomes, the rule that Napoleon is a male, then the outcome is Napoleon has XX chromosomes. Here this outcome is against reality, but as a consequence of the aforementioned rules, the consequence itself is TRUE, i.e. it is TRUE that if we hold the first inputs to be true then it follows that the outcome is true.
Emmanuel Kant once spoke about what he termed as ANALYTIC truths, like in the sentence "All bald men are bald". which is a kind of reductive-affirmation consequential rule. Here the consequential rule is about "are", it affirms the acquisition of any predicate before it, so the general rule is: ALL objects that possess predicates, phi, pi, etc... ARE in possession of any amount of those predicates. Another way is to break it downs to rules related to conjunction and implication: that is [A & B -> A] , [ A & B -> B]. This is just a rule following scenario, so what he called as "analytic" fact, is actually nothing but a kind of consequential truth.
Once Hume had said that mathematics is absolute because it deals with those kinds of analytic truths, so 2+2=4, comes from the roles assigned to the symbols 2, +, =. The complete story is if we hold the rules of the game of arithmetic, then we'll have 2+2=4 being an outcome (a consequence) of this game. This is absolute, because the rules are put in absolute manner. And it is a kind of consequential TRUTH.
Here I'm stating a similar stance that is:
Mathematics is the study of consequential truth.
So mathematics is about studying rule following games. I call them games because the choice of the primary rules is IMMATERIAL, we can even call them ARBITRARY, the most important is to harvest consequential truths in those games. The reality of the games, i.e. the stance of its primary rules and consequential outcomes from reality, is not relevant to mathematics itself, it is however relevant to its application, but not to mathematics per se.
As such mathematics need not essentially have an application. However, that doesn't mean that it ought not to. On the contrary SOME parts of mathematics do have many applications, and those parts have their primary rules related in some sense to reality, that's why it had applications, and thus were significant in increasing our knowledge about reality, some parts of reality are rule driven, so examining rule driven scenarios can be related to reality. Applied mathematics to reality is the part of mathematics whose primary rules can be said to have some connection to reality and so can be labeled as quasi-empirical or the alike.
Of course it would be understandable that only beneficial rule following games would survive, and be studied more. However, that is not truly the primary job of the mathematician, the mathematician is concerned with consequential truths in rule following games, she\he has nothing to do with justification of the primary rules of those games from the perspective of match-ability with reality.
So what I'm advocating here is a line in philosophy of mathematics that can be termed as: "Consequential-ism". So mathematicians negotiate themselves with scenarios fruitful in consequential thought and study the consequential outcomes of those scenarios.
So mathematics per se is not necessarily empirical or quasi-empirical, although some parts of mathematics can be so.
There is the point of view that even if mathematics is not about reality-matching rule following games, yet it is about some kind of platonic realm matching. In other words mathematicians are somehow perceiving a kind of reality that is not the concrete physical reality, and that mathematical truths can be seen as a kind of receptive truth about that realm. This is what is generally labeled as mathematical Platonism.
The main problem with this viewpoint is that it is too strong, for although a working mathematician when he brings about some new mathematical rules or define new objects, he's indeed working within such context, that it appears as if he\she is "discovering" those, but at the and of the day, no such claim would be granted or even posed. What is granted is the consequential facts of his work, and it is that what counts. While its stance from such realm or even from reality is not even posed. That even if one day we reached philosophically speaking into the conclusion that "TRUTH" itself must be defined or characteriszed as "matchability" with some realm, and so consequential truths themselves being truths must be realized (i.e. match a piece) in some realm, lets say here a platonic realm; that is platonism is necessary to account for consequntial truths themselves; even then, still this would not be a mathematical fact, it would be related to philosophy of mathematics, but not to mathematics per se.
This philosophical line doesn't assert any of the known prior logical schools in philosophy of mathematics, so it is not to be confused with logicism, nor with formalism (which is the nearest to it), nor with nominalism, or structuralism, nor with fictionalism.
Here I'm not asserting that the rule following games must be purely logical (has no extra-logical primitive concepts), or that mathematics is nothing but a string manipulation rules of meaningless (empty) symbols. I'm simply saying that the rule following game can be of ANY motivation even empirically motivated, but if the sole study of it was as a rule following device, and the study was just for the consequential load of it, then it is a mathematical study. It can be a meaningful game, where meaning itself is stipulated through rules, so it would be a meaning rule manipulating system, it can also be an empty string of symbols rule manipulating system. It doesn't matter how many meanings are there and their stance from reality or even from a platonic realm, or from apriori concepts, what counts is the consequential bearings of those games.
So mathematics is the study of consequential truths in rule following games.
Of course this study does include heuristics involved in bringing about such games, and bringing about useful definitions and proofs in those games, for although those heuristics themselves are not decided by the primary rules themselves, yet they are finally about consequential bearing within those scenarios. Those heuristics can virtually have imports from any realm of thought actually, logical, apriori, structural, platonic, pragmatic, empirical, formal, etc.. so there is no limit to it, and it constitutes the mathematical ingenuity no doubt. But mathematics cannot be defined after those heuristics, since such a thing is clearly un-definable. That ingenuity can be understood as the tool that leads us to bring about mathematical systems, rather than it itself being identified with mathematics. Mathematics is the consequential bearings that this ingenuity finds in what it creates of rule following games. So philosophy of mathematical practice is not identical with that of mathematics.
A very radical dichotomy of pure versus applied mathematics along that line of thought is to say that:
Pure mathematics: is the study of consequence in pure rule following games.
Applied mathematics: is the study of consequence in impure rule following games.
In pure math every aspect of the game that is not relevant to the rule following streaming in it is shed out! i.e. its not part of the game.
In applied math there are some aspects that are not related to rule following streaming, that the rule following game is enforced to be meant with.
So arithmetic as presented usually is pure math, but arithmetic when presented with numbers as counting sets, is applied arithmetic to sets. Geometry if restricted to be about a certain contemplation about space, then its applied math, i.e. if there is for example a particular reference to what a point is, like being a mereological atom in some mereological space, this would be geometry applied to mereological spaces. If a point is an n-tuple of reals in an n-diamentional space, then this is geometry applied to sets of n-tuples of reals. All of these are examples of applied geometry. But when geometry is not attached to these contents, i.e. presented with a point being primitive, also *distance* being primitive, etc.., then here we are having a pure formal system that sheds away any particular meaning to what "point", "distance", etc.. means. So in applied mathematics one can realize two parts in the system, one is the pure rule following part, and the other is a dispensible with part that is not necessary to the rule following machinary carried in the system, but can be seen as carried on these rule following machinary. While in pure systems it is only the rule following machinary that is depicted.
Of course we would have many questions about those rule following games, that are outside of rule followship and even outside what is and what can be carried on rules in those games. Questions about the very nature of those games themselves. For example where games with infinite syntax are written? where the games are realized? how we come to know about such games? what are rule following streams themselves? in which world they exist? how are they related to our consciousness? etc.. All these are clearly non-mathematical questions, they are questions about mathematics yes, but they are not mathematical in nature. Those questions belong to philosophy of mathematics.
Late Notes [Aug 15, 2020]: the definition of mathematics presented here highly overlaps with that of Logic, that it appears to suit being a definition of logic rather than of mathematics. The totality of all consequential truths in rule following games seems to be equal to LOGIC in its entirety, a sort of PAN-LOGIC where logic is in its full extensional form, i.e. the full exemplification of logic.
The consequential reap (i.e. consequential truths) of a rule following game seems to be just a piece of logic.
So the above definition of mathematics that we gave seems to serve as a definition of logic by extension. Which is the counterpart to a definition of logic by intension.
Logic by intention is what enables having a rule following game! That is, Logic is that without which no rule following game is possible. So logic is what's necessary for having rule following games in the first place.
This intentional definition of logic is exemplified fully in its extensional definition, that even we can present it as a set:
Logic = { R: \exists T (T is a rule following game /\ R is a consequential truth in T) }
In English: logic is the set of all consequential truths in rule following games.
All in all, the method depicted here seems to suit a definition of Logic rather than mathematics.
The above definition of logic is in reality a definition of "whole logic" which is logic in the most general sense, and as such there is no difference between logic and mathematics. But if we look close into the intentional definition of logic which is logic being what's necessary to have rule following games, then this particular intent can be captured by saying that logic is meant with some kind of essential rule following games; that is, games that have their premises being some rules that are seen as too general rules of consequential thought. So logic in that sense is the study of rule following games whose premisses (axioms, inference rules, substitution rules,..) are general rules of consequential thought. The idea is that those systems would be needed when one wants to coin a rule following game, so they are basic rule following systems, they are about rule followship itself and in the general sense of it. This would serve as a definition of "pure" logic, which is not the same as the logic in its entirety [by extension] definition that we gave above which is a definition of "whole logic". Now pure mathematical theories need not have its axioms being about general rules of consequential thought, they can be ANY set of axioms, as long as we are not loading it with information that is not necessary for the rule following streaming, then its pure mathematics. As such pure mathematics is something larger than pure logic. However, "whole mathematics" and "whole logic" are identical.
So we have pure logic being a subset of pure mathematics, as such pure mathematics other than pure logic can be said to be a case of applied logic since it is loading its systems with information that is not necessary for a rule following game to be as such, i.e. it has some of it axioms that are not general rules of consequential thought, so it is applied logic to whatever those additional (extra-logical) axioms are capturing, even though the game is pure (doesnt have information that is not necessary for the post-axiomatic consequential streaming), yet it is still applied logic to a pure mathematical game. Applied mathematics on the other hand is a proper superset of pure mathematics, and so of pure logic, so it would be applied logic too. That's why the entirety of mathematics and logic are the same.
Against consequentialsim
Now lets go in the opposite direction to see by contrast how matters would fair.
The above can be seen as in some sense triumphing syntactical over semantic aspects. However, here we'll examine the opposite, which is by the way the most natural way to look at matters, that is semantics comes first and syntax is nothing but a tool to describe those semantics. So here we'll say that pure mathematics is about some world of objects, much as how biology is about the world of living objects, botany about the plants world, geology about rocks world, etc.. Now we need to put that as: mathematics is about the ... world? Now apparently the most attractive answer to fill in the blank is "Platonic". And I'd think the minimal characterization of that world is as a world in which every rule following game can be written (i.e. it provides enough material for the syntax of any rule following game we can describe to be written in a full manner) and can be realized (i.e. provides models in which those rule following games are true of). Clearly this will not be our physical universe. The next thing is to consider this platonic world as disjoint of our physical world, i.e. it doesn't overlap with it.
Now if we adopt that, then from the perspective of consequentialism this is seen as an applied mathematics to the platonic realm.
Of course, the burden of proof would rest on the shoulders of the claimer of that position to account for existence of such an ideal world in the first place, and to account for claiming existence of such a world being necessary for defining mathematics in reference to it.
The essential difference between that approach and consequentialism, is that the PRE-AXIOMATIC justification of truth of the axioms of pure mathematical theories is part of mathematics. So, the axioms must match some platonic piece. Not only that, they might even have a completely non-rule following game approach to some conceptualization about parts of that platonic realm, and this would also count as mathematics. So mathematics is not formal or consequential anymore, the formal and consequential systems are just out-phrasing that informal concept which is there in the platonic world, and this can even open the possibility for some mathematics that cannot be captured in a rule following manner?
The problem is that how do I know then when a piece of thought is about that platonic world or not?
One possible answer, is that the pre-axiomatic informal aspect must not be complex enough to require complex informal arbiters on truths. To make an example, let’s take formal ethics, here we are also dealing with abstractions like good and evil, benefit and harm, gain and loss, etc... However, the discipline of ethics does contain a lot of argumentation that is pretty much complex and all runs at the informal level, and there are many arbiters on truths in that field that are informal, so such a discipline would not qualify as being about a part of the Platonic realm. So, we reserve the platonic realm to be about notions that do not rely on complex informal pre-axiomatic arbiters on their truth. So, the kind of informal pre-axiomatic truths mathematical systems are about are in some sense trivial. This suits well arithmetic, and geometry axiomatization, which are indeed very trivial and having no complex philosophical or otherwise informal arbiters on their pre-axiomatic truths. This stands in contrast to for example applied physics, which demands sensual verification as arbiters on truths in them, which can be pretty much of a complex process. So, to make matters clearer, we'd say that its not enough to separate (disjoint) the physical world form the plantonic world, we need also to separate any world that demands complex informal arbiters on truths in it, i.e. call such worlds as "complex informal worlds", from the platonic world.
So, mathematics is about the kinds of truths whose main bulk of investigation is through putting them in rule following machineries and examining them there, and even further axiomatization of them is only aided by results of prior consequential truths obtained from working in these rule following systems.
So, we realize our platonic world to be mainly about such kinds of concepts, that is those that are mainly investigated through formal reasoning about them.
So, the Platonic realm is a world in which all possible rule following games can be written, AND where each consistent rule following game can be realized in (be true of) a part of it, AND that does not overlap with any complex informal world.
So as seen here the Platonic world pre-requisite is just to account for existence and realization of rule following games in an unlimited manner. And the purity of that realm is related to be mostly about reasoning in rule following machineries, and that the informal aspects to be very minimal that doesn't demand informal complex expertise about them.
To be noted is that the complexity spoken about here is about having *informal* arbiters on pre-axiomatic truths, this means that FORMAL arbiters on pre-axiomatic truths can be complex and are acceptable. This happens for example when we want to extend a certain formal system, so we experiment before hand with possible axioms and see where each of those FORMALLY leads to, and if a plausible preference is there then we'll adopt that axiom, this is a kind of pre-axiomatic reasoning that is aided by complex formal workup, so its not a sole informal arbiter. This is allowed in mathematical systems.
This account conflicts with consequentialism in the sense that there is a subject matter for pure mathematics, and that it’s in reality an applied consequentialism to a certain world. And that it is about conceptualization carried in the platonic world, and not just mere consequential reap! Not only that, it might even be possible to have strictly informal pure mathematics?
The main problem with this view point is that the platonic world is mysterious! While rule following games are clear and can be agreed upon.
Another attack:
That consequentialism is not really the essence of mathematics! due to the following reasons:
1. A mathematical system doesn't necessarily need a particular formal (rule following game) to express a content, we can dismiss a particular formalization for another formal system as long as the content is expressed equivalently. So, the particular formal system is not necessary, the content is what matters.
2. If a formal system leads to results that doesn't meet the original informal concept it’s supposed to capture, then we'll dismiss of that system.
3. The axiomatics is not decided from the system, i.e. the deductive streaming within the system had no effect on selecting the axioms
4. The definitions and arguments used in a proof in a formal system many times are not traceable to the primary rules of the system, its the original informal content that help spark the useful definitions and ideas behind many arguments and not the axioms themselves.
5. We can have informal mathematics and so discard away with all formal presentation of a mathematical concept
[Note: I'm using here the world formal to mean a rule following game actually, although formal should be defined as a rule following game played with variables (named gaps that can be filled with any element of the universe of discourse), but i think as far as the present discourse, they'd turn to be equivalent in effect]
Now to debunk those arguments we'd say
1. As regards 1 above, even though we can have equivalent formulations, yet each formalization has its own merits that deserves studying, it might lead to different extensions, also under different assumptions those axiomatizations might not be equivalent, so unless the axiomatization is just done by simple renaming of parts of the axioms, each axiomatization has its formal merits and so it’s worth studying on its own right.
2. As regards 2, that is only true in applied mathematics as defined above. The discarded system if found consistent can be true of another application, nevertheless if its consistent it would deserve studying in its own right! So, it might be discarded from being an application of the concerned concept, but it’s not discarded from mathematics altogether, so it’s still mathematical!
3. Regarding 3, that is true, but again if we treat the axioms as arbitrarily chosen, then their choice is immaterial, and again the choice of those axioms is only important as far as application is concerned and not for the pure rule following games. As far as PURE rule following is concerned the axioms are just initiators of proofs, and as such they are part of the rule following game, and nothing more to them is attached. So, the objection also only applies to applied mathematics and it drops in pure mathematics (defined above).
4. Regarding 4, this only pertains to applied mathematics (defined above), still other formalizations within the system are of interest and not only those used in definitions and proofs, and still other argumentation can be used in proofs. So, it’s only a relative argument against the system standing for an application, but not for the system overall. Afterall it’s about provability in the system anyway, so all of those definitions and argumentation are dispensable with actually, it’s only necessary for us humans to capture what's going one with the consequential streaming, but after all its about that streaming itself and nothing else.
5. Regarding 5, the proponent of it must bring about an example of essentially informal mathematics, that is one that cannot be formalized? i.e. cannot be put in a rule following game and be studied as the consequential streaming inside that game. Not only that it should be shown not to lead to useful formalizations. If that can be shown, then the above consequentialism line of thought fails. If that stands correct then it would really be very difficult to characteriza what's mathematics is about by then, since the above demarcation of the platonic realm would be blurred, since there can be a complex informal mathematical system and so the platonic realm would overlap with some complex informal worlds, then we need to see which ones are permitted to overlap with and which ones are forbidden? A hard task! I think such a kind of conceptualization called as "essentially informal mathematics", I'd think it’s better be re-labelled as quasi-mathematics, since it’s difficult for us to really make a clear definitional demarcation of the world that mathematics would be about by then.
However, on the other hand if we restrict the informal to the pre-axiomatics, then that is there, but its trivial in its informal state, and its only unraveled by the formal theories it leads to, so although the very nature of that informal piece itself might elude full formalizability, but it’s only the formalizations stemming from it that matters, so such concepts can be called as pre-mathematical concepts. Mathematics would be about the parts of those concepts that can be captured formally.
Justification
How do we justify using mathematics?
The answer is that if we adopt consequentialism, then this is justified on its own right, i.e., it has an apriori justification, it stands on its own shoulders, since clearly rule fellowship patterns are easily granted to be true by just playing the games. They are games, so there is no justification involved! The proof justifies the fellowship of the output from the input. So, its apriori, analytic.
If we adopt the platonic line presented above, then here we'll need justification, and this must be a convincing one! Because we are going against what we naturally experience. The natural experience is that we can form many logically CONSISTENT stories but most of them are LIES. So, what makes us think it’s useful to conceptualize in a realm which grants realization of all consistent systems in parts of it? Such a free enterprise for consistent formalization, might be damaging rather than helping as far as factuality is concerned.
The only answer that I have in mind is NATURALISM. That thinking along such lines of Platonism have been shown to increase our understanding of our own world, it has been shown to greatly aid in applications. Although to complete this argument one must show that this contribution is not anecdotal, i.e. made by chance, and so must provide a kind of INDISPENSABILITY argument for such role. I think any mathematical system that would proceed (other than consequentialism) using some kind of alternative realm to that of Platonism, would be in some sense more complex. And I think that under PRAGMATIC reasoning, it’s that who hold such an alternative view who must bring it about and show that its more useful than the usual platonic approach. Until that it appears that, besides consequentialism, Platonist presentation is to be held.
From all of that it appears that consequentialism is the nearest to what mathematics is! However, its blind to some philosophical aspects concerning mathematics, those seem to be met by a platonic approach, which in reality supports consequentialism in general though might differ with it on what’s pure mathematics.
Defusing Platonism:
The above method treats mathematics as games, more specifically rule following streaming that can be played under any assumptions, you can play it with real objects, with fictional objects, with mental constructs, with platonic objects, etc..., also the primary rules can be apriori, quasi-empirical, pragmatic, constructive, intuitive, experimental, etc.. But here I'll discuss if it is necessary to assume Platonism to account for our stipulation of those games, and especially if we are meant with all possible games.
Clearly there are mathematical systems whose stipulation implies the existence of infinitely many objects, now with these kinds of stipulation, clearly those cannot be written in the physical world of us, so for such a stipulation to make sense, then it must mean for them to be written in some realm! Not only that, should we think that there must be some realms in which those written strings of symbols are ture of, then those realms would be ones that have infinitely many objects, and this is true of set theory and many theories, then those also cannot be our physical world. In nutshell we'll need a realm in which those theories can be written and realized.
But, is it the case that we must have a platonic realm for those kinds of theories?
What was shown above in this account is that in order for the stipulation of such games to make sense, then it begs at least a realm in which they can be written, and even preferably also be realized. But nothing was shown about this realm needing to fulfill the full theses of Platonism (see: https://iep.utm.edu/mathplat/)
So, the more appropriate word is realism. I'll present here an explication about why Platonism (unless stipulated inside the rules of the game) need not be assumed for realizing those games.
I'll take all the games to be played in a first order logical milieu.
Now we can take the objects over which the variables of first order logic range to be simply Mereological atoms. That is objects that has no proper parts, so they only have themselves as the sole part. This way we minimize the *internal* characteristics of those objects, so that we are only engaged with them as they are related to each other after the rules stipulated in the games, what we can label as their structural properties. There would remain some internal characteristic to those objects that is not a structural property, that is them being atoms, but that characteristic is justified for making them as much possible be structurally approached. So, when we say "forall x" or "exists x" actually we are referring to atoms. Now the point is that those atoms need not be thought of being necessarily non-spatio-temporal entities? On the contrary we may think of our universe of discourse as an aggregate (a totality) of ideal spatio-temporal atoms. Understandably those would not pertain to our physical space and time complex, yet it simulates it but on another realm, a more ideal realm in which writing and realization of all rule following games is granted. So, this ideal realm has a kind of space and time, and two atoms cannot exist in whole at two places in that ideal world at the same time of that ideal world. This suffice the purpose of writing those theories in that world, since symbols are spatio-temporal entities. So, as far as the purpose of writing those theories is concerned, there is no need to assume Platonism, though there is a need to assume a kind of conditional realism in order for the stipulation of those games to make sense.
Now what about realizing those games, i.e. them being seen as true of something, and that thing is part of that ideal world; in other words, semantics. Is there a need to assume Platonism? I think that the answer is to the negative!
The reason is because we can take minimal requirements to be what's needed for that realization! That is, we need our ideal realm to function as a realizer of our games by just assuring at least ONE example of capture of all the rules of our games per game! In more professional terms, we need the ideal realm to grant the existence of a model per theory, and in that sense those models need not adhere to all of the platonic theses. The reason as said above is that we can take any first order theory, take its domain to be an aggregate of atoms; so, each constant symbol of that theory is just a name referring to a particular atom in that domain. Now what may be more difficult to get are the predicate symbols, i.e. relation and function symbols. Now those also can be understood as aggregates of ordered pairs of atoms in that domain. Now the ordered pair between any two atoms is itself an atom, it is the DIRECTION from one atom to the other, this direction can be fathomed as a mereological atom because we don't want anything to it other than the style of its connective action on atoms. That said we can take the domain of any first order theory to be an aggregate of ideal spatio-temporal atoms, and directions between those, which are in turn also ideal spatio-temporal atoms! This way, a model which is a tuple of sets and relations and functions on those sets, can be easily imaged as each of those domains (in multi-sorted theories) being the aggregates of atoms (including directional atoms), while the functions and relations to be the aggregates of directional atoms belonging to these aggregates. Now here when we say aggregates, we mean mereological totalities, that is fusions, so those are ontologically innocent, they don't result in increase of the material in the domains, they are just parts of those domains, and most importantly they are SPATIO-TEMPORAL entities, that is, none of those aggregates can exists in whole at two places at the same time in that ideal realm, so they don't violate spatio-temporality. Now in all mathematical theories written in first order language, one can take its primitives to be either atoms in the domain of discourse, or aggregates of atoms in that domain, so all of their signature is interpretable in an ideal spatio-temporal realm. This shows that there is no need for the full parcel of Platonism, and in particular non-spatio-temporality.
Of course I'm speaking about the particular aspect of need or non need of Platonism as far as writing and realising mathematical theories is concerned, and I was in some sense minimalistic. But if one desires a full kind of explanation, like a realisation that captures properties as they are, then here perhaps Platonism would trip in. Let me explain this, for example lets take arithmetic, here I'm simply saying that its a game played with symbols and we'd be content with realizing it in a world of atoms and aggregates of atoms as displayed above, so numbers are just atoms in that game, however one may want to speak about applied arithmetic in which those numbers stand for being differential indicies of quantities of elements in sets, now it might be good to know that those numbers being atoms then we'd avoid the Julius Caesar problem, and there is nothing into them other than their function in being such indices, but although we can realise the whole of what's in the arithmetic game by aggregates of atoms, yet still we want more than that, we want to navigate the property itself that numbers are serving as indices of, so for example we know that number 1 is a name that stands for an atom, that atom indicate the quantity of elements in singleton sets, so it an indicator of singleton-ness, but what is to be a *singleton* itself? Now when we say a set is singleton, or as written in first order logic "x is singleton", then clearly singleton is a predicate, and is a predicate of sets here, now the predicate singleton is itself a non-spatio-temporal entity!!! It is the kind of entities that presents in whole at two distinct places at the same time. Now if arithmetic is in application, then this aspect would pop on, and this aspect is to be captured because it is what's our theory is manipulating (in application). So, ignoring this aspect in an applied arithmetic to sets, would be a kind of partial explanation as far as semantics is concerned. So, although we can possibly defuse platonism in the narrow sense, yet we may not be able to escape it in the more broader sense.
It needs to be understood that I'm not saying that such an ideal realm does exist should the realized game in it proves to be consistent. I'm saying that for our games to make sense when they are stipulated in a manner as to imply them having infinitely many strings of symbols, or when they are speaking of infinitely many objects, then for us to make sense of our stipulation, then we need to say that those are games played in an ideal realm in which they are granted of being written in the first place, and also provides a model for each of those games. But this ideal world is only to be stated correctely in a conditional manner, one can say it is a hidden axiom, it is one of the rules of the game really. This would make the presentation of our games complete and sensible. But it doesn't mean that it exists, even if the games were consistent! It only means that its existence is pre-supposed in the game, and its part of the game. What matters really is the rule following results; that's the mathematics in our games.
I personally think that none of those infinitely written games and those that speak of infinitely many objects, is real or that there is a real world in which they are realized. The point is that those games would only enrich our finite game following vocabulary, and it is those game following scenarios on the finite level and in particular those which can be written in our world and realized in our world, that are real! The rest of the games have their reality only pre-supposed by their stipulation, and therefore are not granted to be true, unless they can be shown in some sense to be indispensable to our scientific discoveries, which I doubt it can be shown in full, even though they can be shown to be indispensable PRACTICAL tools for such a purpose, but I think that all of those can be dispensed with by their finite initial segments, which can be re-written in full without them, and those are the only real and TRUE games, and the only ones which have applications. The other ones are FALSE, but their initial finite writable segments are TRUE, and thats what counts!
But anyway, no one can say for sure, which of those troublesome rule following games (those that inherently imply having quantities or qualities of objects that our physical world cannot supply or afford to have) is real in some realm or not? It appears to be an impossible mission, so what we can answer is whether it CAN be real, i.e. there is a potential for it being real in some realm, and that mostly follows from its consistency I suppose. But the problem is that most formal systems are ones whose consistency cannot be proved! So actually what we have at hand is a possibility of being consistent, and thus of being true in some world. Now, whether that world in which some game (mathematical theory) is true possess a kind of interaction with our physical universe that its benificial to know about? This is something that we don't know, but if so, then such games would be real and worth knowing as far as the quest of factual investigation of our universe and reality in general is concerned!
Since we don't know, then we may give it the benefit of doubt! So, this practically justifies investigating them.