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When one for example define a formal theory like ZFC, or even a formal theory of arithmetic, now such theories has infinitely many strings of symbols, that serve as its axioms, theorems etc.. Now proofs can be understood as collections of those strings that are closed under inference rules, and even those rules are also expressed by stings of symbols. Now the whole machinery is assumed to go on infinitely.

My question is from where we bring the necessary ink to write down all of these symbols?

If we are defining a formal system then we must be assuming that it can be written in some realm. Otherwise it makes no sense to speak about writing down something that cannot be written.

I'm of course not speaking about pieces of mathematics that can be written by an actual computer, I'm speaking about theories of theoretic mathematics that has an infinite output. The physical world doesn't have an infinite supply of material. So are we presupposing platonism here? That is, a hypothetical world where all the symbols and sentences of a formal theory or actually of a formal langauge can be written. A world that can provide such unlimited supply of ink?

If we don't want to make that pre-supposition, then how we can account for speaking about such theories and carrying inferences in them, if there is no world in which their symbols are guaranteed to be written? It appears to me that without that pre-supposition we don't really have a formal theory.

If that is pre-supposed then a formal theory must start with that hypothetical stipulation of such a world. Like for example in saying:

IF there exists a world in which we can write down all sentences of the language of arithmetic and any set of those sentences (theories), then in that world we write the following .. and we state our axioms, inference rules, provability criteria, etc...

This explicit statement would make mathematics clearly based on the Platonism thesis, since without it there is no guarantee for any of its inferences to be carried out.

Zuhair
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  • I don’t see why formalism isn’t valid here. – Q the Platypus Aug 03 '20 at 13:39
  • We can speak the English language even if we don't assume that there is "some realm (??)" where all English sentences are written out. So I really don't understand your point. – Maximilian Janisch Aug 03 '20 at 13:51
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    @QthePlatypus formalism that I know of is the statement that mathematics is string manipulation rules of empty symbols. My question is: empty symbols written where? Why we think this is possible in the first place? Our physical world cannot guarantee writing all strings belonging to a formal theory like PA, ZFC, and the alike actually of any formal system with infinite potential. As far as I know formalism doesn't address that. – Zuhair Aug 03 '20 at 13:53
  • @MaximilianJanisch, the matter is different with formal theories, those speak of the totality of their sentences, etc.. and can be themselves treated as objects (infinite) within other theories, you don't have this with English. Anyhow I don't know how English or any human natural language is defined, I guess they can still have formal definition with an infinite ouput, if that is so then defining English after all possible outputs of it, would be a definition that pre-supppose a kind of platonism because this cannot be made in real life. – Zuhair Aug 03 '20 at 14:01
  • @Zuhair Let me put it differently: From Wikipedia: "A central idea of formalism 'is that mathematics is not a body of propositions representing an abstract sector of reality, but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess.'" What I want to say is that formalists don't care "where your ink is is." Just like they don't care "where your chess game is" (or that there is no way to fit all possible chess games in the universe). There are just certain rules to follow. – Maximilian Janisch Aug 03 '20 at 14:05
  • @MaximilianJanisch, yes but the FORMALISM itself won't make sense without such a pre-supposition. If you want to encompass this pre-supposition within the rule following game, so be it, but by then we'll know where we are starting from. Its a formulism that starts with a platonic pre-supposition. Without that the whole formalism makes no sense, since its not guaranteed to exist in the first place. There is no meaning in writing a detailed peace treatise for two enemies which do keep fighting and non of them ever desire to have peace in the first place. – Zuhair Aug 03 '20 at 14:14
  • @Zuhair “It is formalism that starts with a platonic pre-supposition.“ That is YOUR assumption, no? “Since it is not guaranteed to exist in the first place.“ That makes no sense to me. We can play chess without assuming Platonism. We can do math without assuming Platonism. You have used the word „exist“ here already with an underlying platonic flavor. “ There is no meaning in writing a detailed peace treatise [...]“ Of course there is no point in doing math with somebody that disagrees with your rules. Would you play chess against a Pigeon? Would you do math with a Pigeon? – Maximilian Janisch Aug 03 '20 at 14:19
  • @MaximilianJanisch, I don't know how actual playing with chess compares with playing with infinite collections of sentences like formal theories. There is a difference between playing chess and defining chess. If you define chess by the actual games, and what can actually be played, then this is a definition of chess that doesn't presuppose platonism, Yes. But if you define chess as all of what can be played, i.e. all possible games. Then this would be a theoretic definition, and it would pressupoe a realm in which all those games can take place (which is not the physical world of us). – Zuhair Aug 03 '20 at 14:28
  • @Zuhair No the chess rules are „the queen moves like this and the rook like this, etc.“ Formalists don’t care „where“ all the chess games that these rules apply to are. You can do something very similar for writing down math symbols (like „$a\implies b$ can always be replaced by $\neg a\lor b$“, etc.) If everyone agree then it works pretty well in my opinion (many chess games are being played and many interesting results reached in mathematics.) – Maximilian Janisch Aug 03 '20 at 14:32
  • @MaximilianJanisch, you can play some math without assuming platonism Yes. But when you speak about totalities of all sentences provable in some system, and that totality is infinite, then here you are assuming plantonism, at least as far as writing those system is concerned, because if there is no realm in which those totalities of symbols exist, then you are speaking about something that doesn't exist at all, which doesn't make sense. I'm not speaking about all kinds of mathematics, I'm speaking about formal theories with infinite output. – Zuhair Aug 03 '20 at 14:35
  • @Zuhair We seem to have different goals. I am just looking for a set of conventions under which it is possible to do math such that no one disagrees. This is entirely possible without Platonism. Your goal appears to be a different one so I am sorry but it seems I can’t help you here. – Maximilian Janisch Aug 03 '20 at 14:37
  • @MaximilianJanisch, I'm not speaking about the collection of rules in a theory, those can be finite of course, and those are writable, and there is no need to assume a platonic world in which they need to be written. I'm speaking about the totalities of all theorems of an axiomatic system, what we call as a formal theories. I'm saying if we are speaking about such totalities, and if those were infinite, then for our discourse about them to make sense, then we must be pre-supposing some kind of platonism, at least as far as writing them is concerned. – Zuhair Aug 03 '20 at 14:47
  • @MaximilianJanisch, yes mathematics is agreeable, the analytic consequences (theorems) of a set of premisses is agree-able. Yet when it comes into questions about where all of those sentences themselves exist? Its here where differences start. To me formal systems with extensive meta-theories must supply an account justifying their meta-theories? Thanks for your correspondenc. – Zuhair Aug 03 '20 at 14:57
  • @Zuhair Thank you too . I think there is no way that we will agree on the question at hand because in the end I don't care "where the sentences are" (and I am not sure if that is even a sensible question.) – Maximilian Janisch Aug 03 '20 at 17:12
  • (+1) Sensible or not, this is a question that has been bothering me for decades. – Calum Gilhooley Aug 03 '20 at 22:09
  • Questions like where the theory is written? from where the primitives of rule following games come (how we came in touch with them)? Where the analytic stream of rule followship itself is? what is its nature? In which world semantics of those systems are granted etc.. are all non mathematical questions I agree. But those questions are among the subject matter of philosophy of mathematics. – Zuhair Aug 11 '20 at 10:42

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Platonism doesn't have to be assumed when writing down formal theories. I'm going to give formalist answers to this however other philosophies of mathematics have there own answers.

In formalism mathematics is considered to be a system of symbols that are manipulated according to rules. However a key aspect of such systems of symbols is that you can define a new symbol systems that is isomorphic to the previous one.

So the system of arithmetic using Hindu-Arabic numbers is equivalent to the system that uses the Peano axioms. However the Hindu-Arabic numbers are more compact.

Some of these isomorphism are so compact that a finite string manipulation rule in one system is equivalent to an infinite number of string manipulations in another. So while we say informally that there are infinite axioms created by the first order axiom schema of induction; formally what is said is that there is an axiom schema which has certain text manipulation behaviours.

Formally we never deal with an infinite strings, rather we deal with finite strings that are isomorphic to infinite string systems.

Q the Platypus
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  • as far as I know PA cannot be finitely aximatized. So the formal game of PA is (by stipulation) played with infinitely many symbols. Truly each time you are making a proof you are taking finitely many symbols, that's true, but the whole game is indeed infinite. When we are defining PA we define the whole of PA, and this totality is an infinite totality. That applies to most formal systems that I know of. You can still maintain your formalist position since all of what a formalist care about is the the post-axiomatic analtyic reap. He doens't care about other aspects of the game. – Zuhair Aug 11 '20 at 08:32
  • continuation, ... the formalist seem to be not addressing questions about the formal games he is playing that are related to aspects other than rule following properties, so he doesn't care about the truth of his axioms either, nor care where the symbols comes from, etc... he only care about the rule following machinary, that if we have such and such rules (even if chosen arbitrarily) we'll be having such and such output. And honsestly this is what's agreed upon by all, and this constitute what's absolute in mathematics, that is the rule followship. – Zuhair Aug 11 '20 at 08:36
  • continuation.. the other properties that a game that the formalist herself defines seems to be irrelevant to her consideration as long as those properties are irrelevant to the rule following stream of his theory. That's the usual defence I suppose. But here I'm addressing what the formalist is ignoring, those other aspects of the game, like where its played?? Here I'd think that most of formal theories can only be played fully in some ideal world, we usually refer to it (though this might be wrong) as platonic, but its anyway an ideal world, which grant semantics and syntax of a formal theory – Zuhair Aug 11 '20 at 08:45
  • continuation... for the prupose of granting the syntax of a theory, it appears we don't need the full platonic thesis, it can be an ideal world with concrete (ideal spatio-temporal) entities that serves to be the symbols of the formal systems. The full plantonic thesis is however (I think) would be required for realising those theories, i.e. providing semantics for them, but that's another story. – Zuhair Aug 11 '20 at 08:48
  • I think if a formal system (in whole) can be written in the real world by a computer for example, then of course we don't need to assume some ideal world in which its written. But if it goes way beyond the limits of our physical world, that it cannot be seen as capable of been written in it, then here we are pre-supposing an ideal real which grant such writing in it, even if we didn't mention it explicitly. – Zuhair Aug 11 '20 at 10:46
  • 1st order PA can't be cannot be finitely aximatized yes. (2nd order can but that has its own issues). However formalists don't work in the infinite axiomisation of PA; they work in the "A finite number of axioms + an axiom schema" which reduces it down to a finite set. – Q the Platypus Aug 12 '20 at 00:30
  • It would be like saying that you can't write $\mathbb{N}$ with a finite number of symbols. Where clearly I have done just that. – Q the Platypus Aug 12 '20 at 00:32
  • @Q the Platypus a schema is NOT a sentence of the language of the theory, it is a sentence of the meta-language, so this is something else. PA has infintely many primary rules (axioms + inference rules +....). But of course each time one works on a proof then she is taking a finite set of sentences, that's OK, but that has nothing to do with PA itself, PA itself is the TOTALITY of all theorems. And this is an infinite set of objects. There are not twos about such a thing. Clearly such a thing cannot be written in our physical world, although an initial segment of it can, but not it itself. – Zuhair Aug 12 '20 at 11:09