Do you disagree with the point that inference rules may themselves be a mathematical object?

A→B being defined (convention) exactly by what it gives in a truth table according to each value of A and B, and A&B, etc. — Lionino

The symbol '->' may be a primitive or defined from primitive symbols.

The truth or falsehood, in a model M, of a sentence of the form 'P -> Q' is determined by the definition of 'S is true in model M'.

Meanwhile, 'P -> Q' is a formula (if 'P' and 'Q' are formulas) or it stands for a set of formulas (if 'P' and 'Q' are meta-variables ranging over formulas). It is not something that is "defined". Rather, it is shown to be a formula from the defintion of 'is a formula'.

If any rules at all, the idea that every rule we may come up with is a platonic object is silly, especially when so many rules are absolutely dependant on us being around. If you are talking about rules of logic and mathematics, then wonder why it is only such rules that get a special status. — Lionino

Yes, I think this is the issue, why would some rules get special status, and if they do, how could we know which ones deserve that special status. For example if we say some rules are objective, and other rules are subjective, what would distinguish the two?

? Those are set up by convention. — Lionino

So it appears like you want to start with the basic premise that rules are fundamentally arbitrary. Why should we agree to some rules and not to others then? Why would we want to start with something like "truth tables" as the primary rule?

It seems to me, that rather than jump right into the process of deciding which rules to accept, and which not to accept, we ought to first determine precisely what a rule is, When we have a complete understanding of what a rule is, then we will be much better prepared for making such a choice, by having some understanding of what the consequences of that choice might be. So rather than start from a truth table, as the basis for which rules to accept, we should start with the definition of a rule, as the basis for which rules to accept.

For example if we say some rules are objective, and other rules are subjective, what would distinguish the two? — Metaphysician Undercover

The two key words you used. Social rules are (inter-)subjective because, as soon as we die, they are not carried out, the "rules" of physics are carried out independently of an observer.

Why should we agree to some rules and not to others then? — Metaphysician Undercover

For 2000 years at very least, people thought that the LNC was fundamental. Then came dialethism.

we ought to first determine precisely what a rule is — Metaphysician Undercover

we should start with the definition of a rule — Metaphysician Undercover

Dictionary.

Do you disagree with the point that inference rules may themselves be a mathematical object? — TonesInDeepFreeze

I haven't thought about it deeply, so no. The matter of mathematicalabstract objects naturally goes back to Plato. If numbers and sets and so on are mathematical objects, rules are, in some way, the relationships betwen those numbers. I am not sure and can't imagine how the relationship between universals has been tackled by platonists, if at all, so I can't give a strong judgement on the matter.

Inference rules may be rigorously defined as relations on the power set of the finite set of formulas cross the set of formulas. So, if sets are mathematical objects, then, as rules themselves are sets, rules also are mathematical objects.

Let S be the set of formulas. Let T be the set of finite subsets of S. Let PT be the power set of T. Let x be the Cartesian product. Then:

An inference rule is a subset of PT x S.

Every rule is a set of ordered pairs, such that for each pair <G P>, G is a finite set of formulas (the premises) and P is a formula (the conclusion).

For example, with that definition, the rule of modus ponens is:

{<G P> | P is a formula, and there is a formula Q such that G = {P -> Q, P}}

All the rules of natural deduction can be written in that manner.

And then 'proof' may be defined as a sequence of formulas such that latter entries are conclusions from previous premises per the rules. So, proofs also are mathematical objects.

In general, languages, syntaxes, axiom sets, inference rules, systems, theories, and interpretations are also formalizable as mathematical objects.

If that is the case, MU's argument simply dissolves and rules are subject to the same debate of nominalismXplatonism as numbers.

I have no comment about the other poster in this context.

But I am glad that I made my quite relevant point that rules also may be regarded as mathematical objects.

The two key words you used. Social rules are (inter-)subjective because, as soon as we die, they are not carried out, the "rules" of physics are carried out independently of an observer. — Lionino

It appears like you are confusing descriptive rules with prescriptive rules. This is why we need a good definition of what a rule is. The laws of physics describe the way things behave. Social laws prescribe the way we ought to behave. The latter requires an agent who understands, and intentionally conforms one's activity to follow the law, the former is an inductive conclusion derived from observations of how things behave.

Some philosophers mix the two, so that a social rule is just a descriptive principle of how people behave in general. I think this is done to avoid the fact that people choose to follow rules. But this is problematic, because many people step outside the bounds of social rules, so it would be faulty induction.

If that is the case, I think MU's argument simply dissolves and rules are subject to the same debate of nominalismXplatonism as numbers. — Lionino

Tones is arguing that rules are Platonic objects just like numbers are. If that is the case, then formalism does not escape Platonism, it is a deeper form of Platonism, just like I said.

To pull this structure out of TIDF"s Platonic cesspool, and give it a nominalist foundation, you need to address the problems which I stated above. How do we get beyond arbitrariness? What makes some rules more acceptable than others. This commonly leads to pragmaticism

As these references to pragmatic theories (in the plural) would suggest, over the years a number of different approaches have been classified as “pragmatic”. This points to a degree of ambiguity that has been present since the earliest formulations of the pragmatic theory of truth: for example, the difference between Peirce’s (1878 [1986: 273]) claim that truth is “the opinion which is fated to be ultimately agreed to by all who investigate” and James’ (1907 [1975: 106]) claim that truth “is only the expedient in the way of our thinking”. Since then the situation has arguably gotten worse, not better. The often-significant differences between various pragmatic theories of truth can make it difficult to determine their shared commitments (if any), while also making it difficult to critique these theories overall. Issues with one version may not apply to other versions, which means that pragmatic theories of truth may well present more of a moving target than do other theories of truth. While few today would equate truth with expediency or utility (as James often seems to do) there remains the question of what the pragmatic theory of truth stands for and how it is related to other theories. Still, pragmatic theories of truth continue to be put forward and defended, often as serious alternatives to more widely accepted theories of truth. — https://plato.stanford.edu/entries/truth-pragmatic/

It appears like you are confusing descriptive rules with prescriptive rules. — Metaphysician Undercover

No, I am giving examples of subjective rules and objective rules because those are the keywords you used, not the new two keywords. A subjective rule may be descriptive or prescriptive, an objective rule also may be either — otherwise prescriptive grammar wouldn't exist.

How do we get beyond arbitrariness? — Metaphysician Undercover

Application, just like 2000 years ago. During ancient times, mathematics was an empirical endeavor. Many mathematicians of today in fact take pride in their research being useless — meaning having no application.

Tones is arguing that rules are Platonic objects just like numbers are. If that is the case, then formalism does not escape Platonism, it is a deeper form of Platonism, just like I said. — Metaphysician Undercover

He said they are mathematical objects, not platonic objects.

The lying crank wrote, "Tones is arguing that rules are Platonic objects just like numbers are."

That's yet another of the crank's lies about me. The crank needs to stop lying about me.

But I am glad that I made my quite relevant point that rules also may be regarded as mathematical objects. — TonesInDeepFreeze

Yes, it was a good explanation.

I should add that the above does not opine that those things are platonic things. Moreover, there is not a particular sense in which I am saying they are things. Moreover, I'm not opining that saying "things" or "objects" requires anything more than an "operational" sense: we use 'thing' or 'object' in order to talk about mathematics, as those notions are inherent in communication; it would be extraordinarily unwieldy to talk about, say, numbers without speaking, at least, as if they are things of some sort. But, it is not inappropriate to discuss the ways such things as rules are or are not mathematical things of some kind. — TonesInDeepFreeze

I explicitly said that I do not claim platonism. And I explicitly said that I am not advocating any particular sense of the notion of object. And I even said that we may discuss ways in which rules are or are not mathematical things of some kind. And I said that even if we don't commit to mathematics as talking about objects, communication about mathematics would be extraordinarily difficult if we did not at least talk as if we are talking about objects.

I wrote it explicitly. Yet the liar crank flat out lies that I said the opposite. The crank has no shame.

He said they are mathematical objects, not platonic objects. — Lionino

More exactly, I said they may be regarded as objects, and that we may discuss in what sense they are or are not objects. But the crank runs all over that like a loose lawn mower. The crank lies at will.

Thank you.

Application, just like 2000 years ago. — Lionino

So truth for you is pragmatic then?

He said they are mathematical objects, not platonic objects. — Lionino

I don't see how one could ever distinguish between these two. When an idea is said to be an "object" this is Platonism, by definition. Platonism is the ontology which holds that abstractions are objects.

When an idea is said to be an "object" this is Platonism, by definition. — Metaphysician Undercover

Nominalists agree that if mathematical objects exist, they would be platonic objects. But nominalists deny that mathematical objects are real, some think they are useful fictions. There are other kinds of realism besides platonic, including psychological and physicalist.

Note: it is platonism with lower case, we are not talking about Plato when the discussion is modern mathematics.

So truth for you is pragmatic then? — Metaphysician Undercover

That is a deep topic in itself and, though related, distinct from the metaphysics of mathematics.

The crank says, "When an idea is said to be an "object" this is Platonism, by definition. Platonism is the ontology which holds that abstractions are objects."

That's not a credible definition of 'platonism'.

https://plato.stanford.edu/entries/platonism-mathematics/ :

"Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. [emphasis added]"

"Mathematical platonism can be defined as the conjunction of the following three theses:

Existence.
There are mathematical objects.
Abstractness.
Mathematical objects are abstract.
Independence.
Mathematical objects are independent of intelligent agents and their language, thought, and practices [emphasis added]"

https://plato.stanford.edu/entries/platonism/ :

"Platonism is the view that there exist such things as abstract objects — where an abstract object is an object that does not exist in space or time and which is therefore entirely non-physical and non-mental [emphasis added]."

The crank doesn't know jack about logic, mathematics or philosophy of mathematics. But still he's good for serially shooting his ignorant mouth off about them.

Note: it is platonism with lower case, we are not talking about Plato when the discussion is modern mathematics. — Lionino

OK, I was unaware of that convention. We are definitely not talking about Plato, but modern day platonism (my spell check does not like the lower case).

But nominalists deny that mathematical objects are real, some think they are useful fictions. — Lionino

If the so-called mathematical objects are fictions then they are not really objects, but fictions. Therefore we cannot correctly refer to the tools of mathematics as "objects". If we're nominalist we'd say that anyone who speaks about mathematical objects is speaking fiction; fiction meaning untruth, because abstractions are simply not objects for the nominalist. And to call them objects would be false by the principles of that ontology.

That is a deep topic in itself and, though related, distinct from the metaphysics of mathematics. — Lionino

The point though, is that if we reject platonism, then we need some other ontology to support the reality of rules. We cannot defer to "intersubjectivity", because that makes a rule something between subjects, therefore outside the subject, and this external existence of rules is just a disguised platonism. Therefore we need to properly locate "rules" as properties of subjects, internal to thinking minds. I have my rules and you have your rules. Then the reality of agreement, convention, needs to be accounted for, and pragmaticism is designed for this purpose.

The crank writes, "If the so-called mathematical objects are fictions then they are not really objects, but fictions."

'Sam Spade is a detective'.

'The Maltese falcon is a statuette'.

'Sam Spade' and 'The Maltese falcon' are nouns. We regard a noun as referring to something, whether it ('it' is a pronoun that refers to something) is a material piece, a collection of material pieces, such things as a sports team, an experience, event, series of events, physical experience, mental experience, mental state, idea, concept, number, set, vector space, word, sentence, literary piece, story, narrative, etc. - now existing, once existing, predicted to exist in the future, conceived to possibly exist now, conceived to possibly to have existed in the past, or possibly conceived to exist in the future, or hypothetically now existing, once existing, Or, fictional instances ('instance' refers to something) of any of those ('those' is a pronoun to refer to some things) hypothetically.

But to accommodate a position that fictional things are not really things, the fictionalist could say, "When I say 'thing' or 'object' regarding mathematics, 'fictional' is tacit. I may be taken as saying 'fictional thing' or 'fictional object'".

The Maltese falcon is fictional, but I can still refer to the Maltese falcon without having to say each time "the fictional Maltese falcon". And it makes perfect sense for someone to ask, "What kind of thing is the Maltese falcon?" If the crank says I must answer, "It is a fictional thing. It is fictional statuette" then okay. But in context, it might be understood that we're talking about fictional things.

The number 1 is not a concrete object, but I can still refer to the number 1 without having to say each time "the fictional number 1". And it makes perfect sense for someone to ask, "What kind of thing is the number 1?" If the crank says I must answer, "It is a fictional thing. It is a number, which is a fictional thing" then okay. But, as mentioned, the fictionalist may take it as understood that we're talking about fictional things.

For anyone who doesn't understand what the number 1 is, it makes perfect sense to ask: "What kind of thing is the number 1. The answer, for a platonist is: "It is a mathematical object". The answer, for a nominalist is: "It's not a thing".

Pick your choice. Or, you can maintain your position as someone who doesn't understand what the number 1 is, and keep asking "What kind of thing is the number 1?"

To say, "the number 1 is a fictional thing," provides nothing by way of explanation, because we are left with essentially the same question; "What kind of thing is a fictional thing?". That's just an evasion; an indecisive move of avoidance practised by a slippery sophist. The sophist then slides off and starts talking about the number 1 as if it is a platonic object, hypocritically insisting that it is not.

What is the crank's definition of 'a fiction'?

If the so-called mathematical objects are fictions then they are not really objects, but fictions. — Metaphysician Undercover

And to call them objects would be false by the principles of that ontology. — Metaphysician Undercover

The fictionalist position is that mathematical truths are fictional (that was already explained further by me above), and, since they are nominalists, that there is no such thing as abstract objects. That is the position, engaging in a debate about "object this object that" is pointless as you will choose to tailor the meaning of "object" to suit your ends.

then we need some other ontology to support the reality of rules — Metaphysician Undercover

Ok. Psychologism, I choose you!

Then the reality of agreement, convention, needs to be accounted for, and pragmaticism is designed for this purpose. — Metaphysician Undercover

Ok, am I supposed to disagree? Formalism is still not disguised platonism, much less nominalism.

It is no mystery that we may build whatever system possible by tweaking the axioms and logical language we use for it. So why is it a surprise that there is some perceived "arbitrariness" to mathematics and logic when put under this light? Mod-60 arithmetic has been useless for us for most of history, and yet now our society relies fully on clocks. Putting mathematics in a heavenly pedestal is something I used to do.

The fictionalist position is that mathematical truths are fictional (that was already explained further by me above), and, since they are nominalists, that there is no such thing as abstract objects. That is the position, engaging in a debate about "object this object that" is pointless as you will choose to tailor the meaning of "object" to suit your ends. — Lionino

I think we've gotten beyond this talk of "objects". We've moved on to "rules", because rules are what formalism takes for granted. Platonism takes mathematical objects for granted, formalism takes rules for granted. These seem very similar to me, though you seem to have a desire to drive a wedge between platonism and formalism. That is why I said we need to look into the ontology of rules. If rules are supposed to exist in the same way that platonic objects are supposed to exist, then there is no real difference between the two.

Your reference to fictionalism inclines me to think that you want to portray rules as fictions. But this does not suit the nature of rules. Rules are guidelines for behaviour, principles of what one ought to do. Rules fall into a completely different category from fact and fiction and cannot be classed as either.

Ok, am I supposed to disagree? Formalism is still not disguised platonism, much less nominalism. — Lionino

We still haven't determined that yet, because we haven't determined how formalists account for the existence of rules. I think formalism generally takes rules for granted. Then the rules simply "are", just like platonic objects simply "are", and formalism is a form of platonism.

I think we've gotten beyond this talk of "objects". We've moved on to "rules", because rules are what formalism takes for granted. — Metaphysician Undercover

It has already been discussed that rules may be considered objects.

formalism takes rules for granted — Metaphysician Undercover

"Taking something for granted" means nothing in this context.

These seem very similar to me, though you seem to have a desire to drive a wedge between platonism and formalism. — Metaphysician Undercover

That is just projection from you. I am correctly stating these two are distinct views, one may be a formalist without being a platonist and vice-versa.

If rules are supposed to exist in the same way that platonic objects are supposed to exist, then there is no real difference between the two. — Metaphysician Undercover

That is just going back to square one but talking about rules instead of numbers, see my first line in this post.

inclines me to think — Metaphysician Undercover

Idc.

Then the rules simply "are", just like platonic objects simply "are", and formalism is a form of platonism. — Metaphysician Undercover

If you had a single reference to support this nonsensical claim, you would have given it already. You talk about "rules" without specifying what kind of rule you are talking about. If it is a mathematical rule, see the first line in this post and then this post.

We have made a full lap on this conversation, going over to something that has been discussed already. I ask that anyone, if forgetting, go back to the beginning of this page to remember what arguments were given. I am not going to reply anymore to anyone who is simply repeating what has already been said.

Welcome to another episode of 'A Day In The Life Of Muddlefizzle Undergarment Internet Crank':

Huston Lover: I saw 'The Maltese Falcon' again last night. What a great film.

Muddlefizzle Undergarment: Never heard of it.

Huston Lover: It's terrific. Humphrey Bogart plays this detective Sam Spade whose partner was murdered.

Muddlefizzle Undergarment: What do you mean "he plays"?

Huston Lover: What do you mean?

Muddlefizzle Undergarment: You said, "he plays". What is that?

Huston Lover: Humphrey Bogart was an actor. He plays the character Sam Spade.

Muddlefizzle Undergarment: A character? What's that?

Huston Lover: A character. A fictional person.

Muddlefizzle Undergarment: Fictional people don't exist.

Huston Lover: Right. They don't exist like you and I. But they exist in the stories. Like the statuette, the Maltese Falcon in the story. It doesn't exist like the Stanley Cup exists, but it is a fictional object

Muddlefizzle Undergarment: So you're a Platonist!

Huston Lover: I don't even know what that is. I was just trying to tell you about the movie.

Muddlefizzle Undergarment: As I've told the Platonists at The Philosophy Forum, there are no fictional objects; there are only actual objects. It is nonsense to talk about fictional objects. Like it is nonsense when mathematicians try to make you believe that numbers and sets are objects. That breaks the law of identity! So please don't try to make me believe that there are fictional people and fictional objects.

Huston Lover: Then how can I tell you who Sam Spade and Brigid O'Shaughnessy and Kasper Gutman are? I mean, they're not living people or dead people, so what else can I call them except 'fictional people'? What can I call the Maltese Falcon if not 'a fictional object'?

Muddlefizzle Undergarment: You may call them 'fictions'. Otherwise, I'll post that you're a Platonist.

Huston Lover: Would that be bad?

Muddlefizzle Undergarment: Very bad. Because then I can say that you're a slippery sophist dragging us all into the Platonic sewer. But good for me, because then I'd have the satisfaction of exposing you as a slippery sophist sewer bound Platonist.

Huston Lover: Okay, but if Sam Spade and the Maltese Falcon can only be called 'fictions' and not 'a fictional person' and 'a fictional object', then how do I refer to them so that you know what they are?

Muddlefizzle Undergarment: Okay, you can call him 'a fictional detective' and you can call it a 'fictional statuette'. As long as you never say they are a fictional person and fictional object. Because, just like at The Philosophy Forum, I've explained that there are no fictional objects or people.

Huston Lover: Thanks for that, Muddlefizzle. But isn't a detective a person, and a statuette an object? So a fictional detective is a fictional person and a fictional statuette is a fictional object.

Muddlefizzle Undergarment: No! Not unless you want to be dirty slippery sophist sewer seeking Platonist rat!

Huston Lover: Okay, Muddlefizzle. But you should check out the movie.

Muddlefizzle Undergarment: I will not be doing that. I only watch documentaries. They're about real things.

I'd like to see the crank try to write mathematics in English without referring to sets, numbers, etc. as if they are things of some kind. Specifically, that requires avoiding the word 'it' to refer to things. ... Confound it, I did it again, I used the word 'thing'! The English language has that nasty habit of making it virtually impossible not to use the word 'thing'. What's up with that? The language must have been invented by satanists...I mean...platonists.

It has already been discussed that rules may be considered objects. — Lionino

...

I am correctly stating these two are distinct views, one may be a formalist without being a platonist and vice-versa.[/quote]

The question then, is how do you think that rules can be taken as objects, in a way which is not platonist? I already explained how the concept of "intersubjective objects" is inherently platonist.

You talk about "rules" without specifying what kind of rule you are talking about. — Lionino

I am talking about prescriptive rules, as I said, ones which people follow when they are doing what they should do, such as when they are doing mathematics. All prescriptive rules can be classed together as the same type, and this distinguishes them from descriptive rules which are commonly understood as inductive principles. Any further division of type is not necessary at this time.

If it is a mathematical rule, see the first line in this post and then this post. — Lionino

I don't deny that rules are subject to the platonist/nominalist debate. In fact that's what I am claiming, and that's what we are debating. I am saying that if you state that rules are understood as objects (whether or not those objects are claimed to be fictional is irrelevant), then you have taken a platonist stance in this debate. Further, I am claiming that all common formalist positions on this matter are platonist as well. I am not saying that it is impossible to produce a nominalist ontology of rules, I am saying that formlism relies on platonist ontology. And I've supported my claim with explanations. I am waiting for you to produce evidence which is contrary to what I am claiming, but you do not seem capable of finding any.

And I have no idea what Tones has been talking about.

The language must have been invented by satanists...I mean...platonists. — TonesInDeepFreeze

Our language is very much platonist. And this is the reason why Plato's writings are still available to us after more than two thousand years. He has had great influence over our culture, and those human beings in authoritative positions have had great respect for platonism, so platonism is extensive through our institutions. Therefore our culture and language have extensive platonist features. If you had education in the history of philosophy, you'd know this. Furthermore, the Catholic church was very much platonist influenced, and the Church, for an extended period of time, controlled the use of language through the use of force. This is known as The Inquisition.

The cranks says, "I have no idea what Tones has been talking about"

I do like when occasionally the crank speaks truthfully.

And characteristic of him to not know what is being talked about even when what is being talked about is what he's been talking about.

For starters, the crank throws around the word 'platonism' but he doesn't even know what it is.

Then the crank says, "Our language is very much platonist."

Right, it was Plato, Platonists, and platonists who installed the words 'it', 'thing' and 'object' and the need for them.

And notice no response to:

I'd like to see the crank try to write mathematics in English without referring to sets, numbers, etc. as if they are things of some kind. Specifically, that requires avoiding the word 'it' to refer to things. — TonesInDeepFreeze

or

What is the crank's definition of 'a fiction'? — TonesInDeepFreeze