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. 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, I think 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.