My notes on the Definition of Mathematics.
I'll use here some personal terminologies, I hope it won't be disturbing.
We may divide truths into two kinds: receptive and consequential.
With the first kind I mean truths that are 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 see were those rules lead us to. A simple example is to have a rule that: R is transitive (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 affirmation consequential rule. Here the consequential rule is about "are", it affirms the acquisition of a predicate before it, so the general rule is ALL objects that possess predicate phi ARE in possession of predicate phi. 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 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 meant 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 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 to 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 they he is "discovering" those, but at the and of the day, no such a 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.
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 much meaning 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.
What are the drawbacks of that personal line of thought of mine about philosophy of mathematics.
We may divide truths into two kinds: receptive and consequential.
With the first kind I mean truths that are 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 see were those rules lead us to. A simple example is to have a rule that: R is transitive (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 affirmation consequential rule. Here the consequential rule is about "are", it affirms the acquisition of a predicate before it, so the general rule is ALL objects that possess predicate phi ARE in possession of predicate phi. 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 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 meant 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 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 to 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 they he is "discovering" those, but at the and of the day, no such a 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.
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 much meaning 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.
What are the drawbacks of that personal line of thought of mine about philosophy of mathematics.
Comments (31)
I do not particularly appreciate the use of the term "truth" in the context of mathematics. The quite dominant Correspondence theory of truth defines "truth" as:
In epistemology, the correspondence theory of truth states that the truth or falsity of a statement is determined only by how it relates to the world and whether it accurately describes (i.e., corresponds with) that world. Correspondence theories claim that true beliefs and true statements correspond to the actual state of affairs. This type of theory attempts to posit a relationship between thoughts or statements on one hand, and things or facts on the other.
Mathematics never talks about the real, physical world, because claiming such a thing, is simply a constructivist heresy.
Mathematics is exclusively and only about theorems that are provable from the construction logic of the abstract, Platonic world, i.e. the theory, generated by a particular arbitrary set of axioms.
Since we do not have access to the axiomatic construction logic of the real, physical world, i.e. the theory of everything (ToE), mathematics cannot possibly ever be about the real, physical world. Ever.
Therefore, the use of the term "truth" in mathematics, is always heretical, and I do not wish to participate in the propagation of that kind of horrible heresies.
Quoting Zuhair
These consequences are merely provable from these rules. They are not "true" in any possible fashion. Provability (PR) and correspondence-theory truth (CT) have absolutely nothing to do with each other.
CT truth and logical truth (LT) have also nothing to do with each other. For example, in "var b = true" the variable b does not correspond to anything in the real, physical world, but we cannot deny that it is logically true, if only, because we defined it to be.
Provability (PR) and logical truth (LT) have also nothing to do with each other. For example, in first his incompleteness theorem, Gödel encodes a statement that is logically true but not provable.
A lot of wrong views and other errors are the result of confusing CT, LT, and PR.
Quoting Zuhair
Mathematics is fundamentally reductionist.
In mathematics, reductionism can be interpreted as the philosophy that all mathematics can (or ought to) be based on a common foundation, which for modern mathematics is usually axiomatic set theory.
Of course, I totally disagree with the idea that any choice of axioms would be better than any other choice. Therefore, I utterly reject the idea that set theory would be a better foundation than any other Turing-complete axiomatization. So, I must protest against this particular detail with the following formalist objection:
Formalism holds that mathematical statements may be thought of as statements about the consequences of certain string manipulation rules. According to formalism, mathematical truths are not about numbers and sets and triangles and the like—in fact, they are not "about" anything at all. Formalism is thus silent on the question of which axiom systems ought to be studied, as none is more meaningful than another from a formalistic point of view.
(Note: I also protest against the use of the term "mathematical truths" in the fragment above.)
How do you both deal with that people wrestle with axioms as much as theorems? The Hilbert program's legacy (and logicism) is on the one hand to find a unifying logical system for all mathematics (which failed, strictly speaking), but the more pernicious legacy is thinking of mathematical theories as given by their axioms. Which is true in a logical/model theoretic sense, but what's really interesting about axiomatisations is how they allow mathematical structures to flower from them.
For a mathematical analogy; imagine that you have an idea of a structure you want to capture the behaviour of, and you have the behaviour, taking a 'pre-image' of that behaviour (through provability) to the axioms which engender it is just as vital a part of mathematics. If ever you've tried to axiomatise a structure you'd see that there's a reciprocality between the structure's concept and its mathematical definition.
A weird aspect of math is that "behaviour" means, "respectful of particular invariants", such as e.g. :
a ? b = b ? a
commutativity or so. In (software) programming, it means being able to execute a particular function, e.g.
a->f(b)
where "a" is capable of looking up a function "f" and apply it to a given argument. I personally find programming with invariants to be much harder than with functions ... but ultimately probably also more powerful.
Quoting fdrake
With the term "structure", do you mean structure as in "algebraic structure", i.e. a set with a collection of operations? Such as a group (K,+) of set K and the addition operation?
Saying that it is a "group" automatically attaches a set of invariants. If you add enough invariants to the structure, i.e. you may use up all your degrees of freedom, then indeed, at some point there will only be one candidate definition that fits the bill. It could, for example, leave only one K possible. You could obviously also over-specify and propose the structure of something that cannot possibly exist.
I think that you can treat K as an unknown, or even one of the operations (why not?), but that will probably lead to specifying a non-trivial higher-order logic problem.
Are you trying to program a structure by attaching invariants?
Yes, but subject to the scrutiny of the historical method.
Quoting tim wood
Well, polymorphism is obviously permissible as long as it does not lead to fundamentally ambiguous situations ...
Polymorphism is first and foremost a language thing. I am not sure if it exists in the real, physical world. Maybe it does somewhere that I am not aware of. Polymorphism is obviously all over the place in software, like for example in duck typing:
Duck typing in computer programming is an application of the duck test—"If it walks like a duck and it quacks like a duck, then it must be a duck"—to determine if an object can be used for a particular purpose. With normal typing, suitability is determined by an object's type. In duck typing, an object's suitability is determined by the presence of certain methods and properties, rather than the type of the object itself.
So, yes, the practice is allowed, also in language in general (such as natural language), but the ever-present danger is: ambiguity.
Quoting tim wood
It is not necessarily the real world, or its situations, that are ambiguous. However, language definitely is: Every bug is an ambiguity and every ambiguity will sooner or later lead to a bug. The losing war against bugs is in reality one against ambiguity.
Although terminology is of course important for understanding, but I think one better try negotiate the essence of what's presented.
"Consequential truth" sounds very much like Coherence theory of truth.
In epistemology, the coherence theory of truth regards truth as coherence within some specified set of sentences, propositions or beliefs. The model is contrasted with the correspondence theory of truth. A positive tenet is the idea that truth is a property of whole systems of propositions and can be ascribed to individual propositions only derivatively according to their coherence with the whole.
"Coherence" in mathematics is obviously axiomatic (=reductionist). Fundamentally, I object to this view because it turns "truth" into some kind of calculation. Any such calculation can be not just arbitrarily faulty, but also fundamentally misguided.
Perhaps the best-known objection to a coherence theory of truth is Bertrand Russell's. He maintained that since both a belief and its negation will, individually, cohere with at least one set of beliefs, this means that contradictory beliefs can be shown to be true according to coherence theory, and therefore that the theory cannot work.
As Bertrand Russell argued, naive approach to coherence would clearly lead to contradictions.
However, what most coherence theorists are concerned with is not all possible beliefs, but the set of beliefs that people actually hold. The main problem for a coherence theory of truth, then, is how to specify just this particular set, given that the truth of which beliefs are actually held can only be determined by means of coherence.
At the same time, a selective approach is not really possible, because it is automatically circular.
I think that the better term for "consequential truth" in mathematics is "provability". It captures much better the status of mathematical theorems. If a theorem necessarily follows from a set of rules, then the theorem is "provable from" these rules. Why use the controversial term "truth" instead of "provability"? Especially given the fact that the rules from which a theorem is proven, do not need to be "true" in any sense ...
Eh. Structure's a placeholder there. Generally they'll be algebraic flavour, a collection of objects and operations or relators between them.
Quoting alcontali
I mean something like the following construction. Let's say we've fixed a vocabulary of symbols and interpretations for all the things we're considering. Say you've got sets of axioms [math]A_i[/math] and sets of inference rules [math]D_i[/math] that allow you to derive consequences [math]C_i[/math] from the axiomatic systems. Let's imagine that we have a bunch of consequences we know we need to reproduced; like 1+1=2 or 'avoid Russell's paradox", or "greater temperature contrasts between medium and immersed object yield more rapid temperature exchange" or something like that.
They're not really all the same thing, 1+1=2 is a specified theorem and a relation of formal objects, 'avoid Russel's paradox' is place a limitation on the space of theorems to avoid a nasty one, and 'greater temperature contrasts...' isn't even directed toward a mathematical object, it's trying to fit a bunch of mathematical objects to the world to describe a part of it.
Regardless, let's say that our end result is the grand 'ole [math]C[/math], the thing we wanna do implemented in a mathematical system. Let's just gloss over that the objects in [math]C[/math] need not be mathematical in nature; nature can shout NO to any model of it and formalisms can be wrongheaded or irrelevant. This is mostly setting up a flowchart rather than a mathematical argument. Anyway
We find out what axiomatic systems give what conclusions, but notice that the conclusions that we desire are the motivating feature in this diagram. Allow [math]C[/math] after the red arrow to be interacted with other objects and goals, and you have a picture of mathematical progress.
Pure formalism just gives you the black arrows, it does not give the sense of mathematical progress through the articulation and codification of ideas, just the dynamics of symbols, as if the ideas motivating them were completely irrelevant. Another way of putting it: formalism is just what we invent to get to where we need to go.
There seems to be an entire area of research on axiomatic subsystems of second-order arithmetic (Z2). It is apparently about crippling Peano arithmetic and check what's left over. Presburger and Robinson are examples of this, but there are also other variations.
Quoting fdrake
I think that they do not care, for example, in Z2 research. They're just interested in what the effect is of the crippling on the resulting arithmetic. For example:
Skolem arithmetic is the first-order theory of the natural numbers with multiplication, named in honor of Thoralf Skolem. The signature of Skolem arithmetic contains only the multiplication operation and equality, omitting the addition operation entirely.
It is actually interesting stuff. I am surprised that this botched version of arithmetic still seems to work.
Empiricism is about correspondence and Mathematics is about consequence.
Can you list your complete set of axioms for arithmetic in a system where godels theorem doesn't apply. ( it cannot be recursively innumerable )
I would say that tautology and contradiction have nothing to do with CT truth but the truth value of the rest of propositions are linked with CT truth. Second of all, we can define probability of a proposition to mean assigning truth value ( T or F) . In that case, l dont think godels theorem would work. It will be great if we can create a discussion to understand the incompleteness theorem.
If rules are arbitrary, would you allow contradictions in it.
Will a system have logically interlinked rules such that they have to be consistent and if so, they won't be arbitrary strictly speaking. Will a different system be created by only removing certain rules or creating new ones. How would you create new rules ?
Agreed. It establishes science and mathematics as epistemic domains instead of subject matters. They are indeed not about anything. They are knowledge-justification methods.
As regards provability and logical truth, one needs to be careful, what Godel has proved is that some theories cannot prove all statements made in their language, so there would be statements that the theory neither denies nor proves. That's all, this is about completeness of theories. From the perspective of this account this would be phrased as: some rule following games can have statements written in their language that has no consequential truth from the rules of those games nor do their negation has, because they are neither consequences of those rules nor is their negations. Godel demonstrated that for some theory T there can be a sentence S written in the language of T such that S is equivalent to statement "S is not provable in T", so obviously if T is consistent, then T cannot prove S. Be aware that this doesn't entail that S is a consequential truth of T, no! For S can be false and T be inconsistent! When we say that S is a logical truth, this is actually mean that S is a consequential truth of the theory T + Con(T), where "Con(T)" is the statement "T is consistent". In other words S is provable in T+Con(N), that's why its said to be a logical truth, in reality it means that it is a consequential truth from the rule following game T+Con(N), notice that it is not a consequential truth of T itself. "Logical truth" is provability in some system. So it is a relative concept. In other words S is not a logical truth of T, the logical truths of T are the theorems of T only.
I don't know what you mean that all propositions other than tautologies and contradictions has Correspondence truth. Correspondence with what? with Reality? what's Reality? I think this is mistaken, those propositions has no innate truths in them, rather truth is assigned to them by the system by dictation (if they are axioms) or by being consequential truths following from the rules of the game.
As regards your second reply, my answer is YES, inconsistent rule following games are definitely mathematical as far as our study of them is about the consequential truth of them.
I didn't get your question about how to create new rules, you simply stipulate them or they are consequences of newly stipulated rules? what's the problem?
Right, so your receptive/consequential distinction falls approximately along the same line as Hume's matters of fact vs. relations of ideas, which is now often aliased as synthetic/analytic, although one can certainly split semantic hairs and introduce further distinctions, like Kant's a priori/a posteriori. (For that matter, Hume's own treatment of relations of ideas amounted to more than just rule-following.) In any event, this is a well-trodden path - I was just wondering whether you thought you were bringing some fresh perspective and not just your personal terminology.
The idea that all there is to mathematics is manipulation of symbols according to fixed rules is true only within a comparatively limited, 18th century conception of mathematics, when mathematics was the mathematics, and it sprang from the same source as logic and rationality itself. Since then we have significantly broadened our ideas of mathematics - and for the matter, of logic and rationality as well. This is what @fdrake is getting at: the more interesting questions in the philosophy of mathematics revolve around the rules themselves, rather than the uncontroversial fact that conclusions can be deduced by following those rules.
I hope that you won't mind my questions as l am confused by your term, " consequential truth ".
.
T+con(N) would rule out any statements like " this is not provable " and so on. Other than that, what will be a consequential truth of T itself and how does it differ from consequential truth of T+con(N) as a S that is a logical truth in T should also be a logical truth in T+con(N) .
I will address your other points to but l need to get my head clear on these terms
https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics)
https://plato.stanford.edu/entries/formalism-mathematics/
Quoting Zuhair
Hmmm. What does "mathematics is about rule following games" mean then?
Quoting Zuhair
What does that mean? How is meaning "part of those rules?" For example we can do basic arithmetic from the Peano axioms without assigning any meaning to the symbols, then we can balance our checkbooks or do number theory using those symbols. But it's still formalism. There's no meaning inherent to the symbols.
So what does it mean for meaning to be part of the rules?
Totally agreed.
There is even an interesting paragraph on this principle in Wikipedia:
A philosophical defeat in the quest for "truth" in the choice of axioms
(Note: I personally consider it to be a victory.)
For good measure, and in order to stamp out the constructivist heresies that keep flying around, I propose to dig up and resurrect Luitzen Brouwer's dead body and to ritually kill him again. The problem is that Satan never dies completely; and unfortunately, neither does his notorious accomplice, Luitzen Brouwer, the worst constructivist heresiarch in the history of mankind.
Exactly! Very nicely put!