The person in charge for formalism in the SEP is Alan Weir. I am sure he would appreciate an email about that. Keep us posted.

Formalists take rules for granted. That's Platonism.

That doesn't make sense automatically because formalism is a program for foundations, platonism is an ontological claim. And idk what post of MU it is. — Lionino

I was only relating what @Metaphysician Undercover said. Of course it doesn't make sense :-)

I'm not really too expert on formalism versus Platonism, I have a very lay-person understanding of those philosophical terms.

I don't see what this all has to do with your claim that a concept like a number, 5, could have a physical instantiation . Fingers are fingers, and are therefore physical instantiations of fingers, not of numbers, not matter how many of them you have. — Metaphysician Undercover

5 is an attribute of the fingers on your hand, would you grant me at least that?

I think of fingers as a physical instantiation of the concept of 5. But if you disagree, then we must be using the word differently. I'm ok with that. How about representation, in the same sense that the first cave man to kill five mastodons and make five marks in the ground to keep track.

Wittgenstein took up this issue in the Philosophical Investigations, showing why there is a lot more involved with learning a language than simple ostensive definition. Abstraction is very complex, and with complex concepts like number, an explanation of what it is about the thing which is being shown, which is being referred to with the word, is a requirement. — Metaphysician Undercover

As far as I know, Wittgy utterly failed to understand Cantor's diagonal argument; therefore his mathematical judgment is deficient in my opinion.

Perhaps abstraction is difficult to define or pin down with words. But we all know it when we see it, and with practice we become good at using it.

A person cannot simply look at the fingers on a hand and apprehend the concept 5. — Metaphysician Undercover

This is manifestly false. Not a matter of opinion or interpretation or language. Flat out false. On the contrary, it is exactly through the experience of looking at one's hand that one at first does apprehend the number 5; and only later, by analogy and induction, all the other natural numbers. Most of the others are far too big to have any such convenient physical representation. Our introduction to the natural numbers, our first concept of them, is by counting the things around us when we are babies. I"m no expert on child development, but there must come a time that a small human looks at his or her hand, and says, "Five." I just did it myself as an experiment. I looked at my hand and saw five. I believe you when you say you don't. You just lack the abstraction and math genes.

An explanation about quantity, or counting is required. The concept 5 is learned from the explanation, not from the ostensive hand, therefore the hand is not a physical instantiation of the number. — Metaphysician Undercover

Oh no. 5 is learned by bijection with the fingers, not with counting. Counting is a higher function. Bijection is more primitive or intuitive. If you've seen a mother cat missing a kitten from her litter, she is not going "One, two, three ..." She's comprehending the total number instinctively and knowing when she's one short.

It can be judged by anyone. The issue though, is that many, like yourself refuse to make such a judgement. — Metaphysician Undercover

There is a modern trend of misspelling judgment, and I can't let it go by. No middle 'e' in judgment.

You say that there is no truth or falsity to mathematical axioms, they are simply tools which cannot be judged for truth. Since mathematicians tend to think this way, they are not well suited for judging truth or falsity of their axioms. But I've shown how axioms can be judged for truth. If an axiom defines a word or symbol in a way which is inconsistent with the way that the symbol is used, then it is a false axiom. — Metaphysician Undercover

There's none in the theory. When you're thinking ABOUT the theory, you can have truth if you like. There is no "truth" in the axioms of group theory, but they are true about the symmetries of a triangle.

So for example, if a mathematical axiom defines "=" as meaning "the same as", — Metaphysician Undercover

There is no such axioms. You make stuff up then tilt your lance at strawmen.

yet in applied mathematics the mathematicians use "=" to mean "has the same value as", then the axiom makes a false definition. — Metaphysician Undercover

You are the only one making up these definitions so that you can disagree with them. Your characterizations of what mathematicians do is in your imagination.

This axiom will be misleading to any "pure mathematician" who uses it to produce a further conceptual structure with that axiom at the base, just like if anyone else working in speculative theories in other fields of science starts from a false premise. False propositions are fascinating, sometimes leading to theories which are extremely useful, because they are designed for the purpose at hand. — Metaphysician Undercover

You're off on your own thing here.

Sorry, I don't understand what you mean by "meta-false". I am talking about "literally false". — Metaphysician Undercover

You said that the axioms are false.

But you misunderstand what that means. Take the powerset axiom. Suppose as you claim, it's false. Then we would be studying the class of set theories that lack powersets. It would be interesting math. It's actually done. The powerset axiom is one that is often assumed to be false, so that we can work out the set theory that doesn't depend on it.

But you think that by an axiom being false, it's committing some kind of metaphysical no-no or faux pas. That's what I mean by meta-false. You think that axioms aren't properly defined or conceptualized or something. NOT that they are literally propositions that are to be taken as false. That's an entirely other thing.

False to me, means not corresponding with reality. — Metaphysician Undercover

If 2 + 2 is 5, then I am the Pope.

That is a true statement that does not correspond with reality.

Ahab is captain of the Pequod. That is a true statement that does not correspond with reality, since both Ahab and the Pequod are fictional entities.

So you are wrong. But isn't the knight move a truth of chess that does not correspond to reality? Axioms in math are like that. Statements assumed true in a fictional context so as to work out the consequences.

For example, if someone says that in the use of mathematics, "=" indicates "the same as", but in reality, when mathematicians use equations, "=" means "has the same value as", then the person who said that "=" indicates "the same as" has spoken a falsity. Do you agree that this would be an instance of "literally false"? — Metaphysician Undercover

I can't sort it out. That paragraph broke my parser. An instance of literally false? I have no idea how to approach this question; nor would I be inclined to do so even if I did.

That doesn't help. Numbers form discrete units, and discrete units cannot model an idealized continuum. — Metaphysician Undercover

Not sure how the discrete/continuous thing got into this convo. Do you mean the set of real numbers doesn't contain all the real numbers? What's a discrete unit? Starting from the natural numbers we can logically construct the real numbers, even there you're wrong.

There is an inconsistency between these two, demonstrated by those philosophers who argue that no matter how many non-dimensional points you put together, you'll never get a line. The real numbers mark non-dimensional points, the continuum is a line. The two are incompatible. — Metaphysician Undercover

If you didn't get the number line in high school, there is not much I can say. The modern mathematical theory of the real numbers is logically unimpeachable. We can indeed start from the empty set and the axioms of ZF, and construct a model of a continuum; that is, an infinite totally-ordered Archimedean set. And we can show that all models of such a thing are isomorphic. So we can indeed construct the real numbers out of discrete units.

It's call the arithmetization of analysis. It's a thing in late 19th century math. Basically founding math, including calculus and continuous processes, on set theory.

https://en.wikipedia.org/wiki/Arithmetization_of_analysis

I was only relating what Metaphysician Undercover said. — fishfry

I am aware.

5 is an attribute of the fingers on your hand, would you grant me at least that? — fishfry

No, I'd say "it has five fingers" is an attribute of your hand. An easy way to think of attributes, is as what something has, a property. So ask yourself, do the fingers on your hand have 5. It doesn't make any sense to say that your fingers have the number 5 as an attribute. Number is a value, and values are proper to the subject, not the object. 5 is not an attribute in the way you propose it's a value.

I think of fingers as a physical instantiation of the concept of 5. But if you disagree, then we must be using the word differently. I'm ok with that. How about representation, in the same sense that the first cave man to kill five mastodons and make five marks in the ground to keep track. — fishfry

Using what word differently, instantiation, or 5? As I said before, I don't believe that numbers have any physical instantiations. Numbers are values and values do not have physical instantiations. So I don't understand what you're asking.

This is manifestly false. Not a matter of opinion or interpretation or language. Flat out false. — fishfry

OK, so we have a difference of opinion, and you are extremely convinced that you know the truth, and my opinion is false. This indicates to me that unless you can prove to me the truth of your opinion, then discussion is pointless. Maybe you can explain it to me. Imagine a person with no understanding of number, a young child just learning to speak for example. You believe that this person can stare at one's own fingers and abstract the concept 5, without any explanation. Please explain how this would be done.

On the contrary, it is exactly through the experience of looking at one's hand that one at first does apprehend the number 5; and only later, by analogy and induction, all the other natural numbers. — fishfry

Come on fishfry, say something reasonable. This is ridiculous. You are asserting that the number 5 is the first number that a person learns.

Oh no. 5 is learned by bijection with the fingers, not with counting. Counting is a higher function. Bijection is more primitive or intuitive. If you've seen a mother cat missing a kitten from her litter, she is not going "One, two, three ..." She's comprehending the total number instinctively and knowing when she's one short. — fishfry

Now it's time for me to say that I think you are wrong. I never learned bijection with my fingers, I learned how to count. We learned how to count to ten. Then we were given examples of the quantities which each name signified, but that was only after we learned how to count. Learning how to count was first because that's how we memorized the names, and their order. Once the names were memorized we could learn the quantity signified by the name. We did not learn bijection, that's a much more complex skill then simply memorizing the order of some words. All simple arithmetic was a matter of memorizing. Did you not use flash cards?

Cats don't do bijections, nor do young children learning about numbers. The mother cat knows each kitten intimately, and knows when one is missing because she misses it. She does not count them in any way.

There is a modern trend of misspelling judgment, and I can't let it go by. No middle 'e' in judgment. — fishfry

Sorry, the devil made me do it. For some reason, out of all the words that have multiple spellings British/American mainly, people on this forum complain about judgement/judgment. Why is this worthy of a correction? You didn't correct me when I spelled color colour.

If 2 + 2 is 5, then I am the Pope.

That is a true statement that does not correspond with reality. — fishfry

That's nonsense. There is nothing to relate "2+2=5" to you being the pope. So this conditional is clearly false, not true as you claim. If 2+2 is 5, how could that make you the Pope, there's no logical connection to support your claim of truth.

Statements assumed true in a fictional context so as to work out the consequences. — fishfry

Uh huh, fictional statements which are assumed to be true. That's contradiction. Do you mean a counterfactual? Obviously they are not assumed to be true. You and I seem to have a completely different idea as to what constitutes truth, so I think we'd better leave that alone.

An instance of literally false? — fishfry

"Literally false" was your terminology. Why pretend not to understand it?

"They're meta-false, as I understand you. They're not literally false. If the powerset axiom is false, you get set theory without powersets. You don't get some kind of philosophical contradiction. You are equivocating levels."

This discussion has degenerated. Let's evacuate.

Sorry, the devil made me do it. For some reason, out of all the words that have multiple spellings British/American mainly, people on this forum complain about judgement/judgment. Why is this worthy of a correction? You didn't correct me when I spelled color colour. — Metaphysician Undercover

Oh that's interesting. Is judgement with the extra 'e' a Britishism? I was not aware of that.

This discussion has degenerated. Let's evacuate. — Metaphysician Undercover

Ok.

Is judgement with the extra 'e' a Britishism? — fishfry

I don't know, but there are lots of US/Brit differences, the common one being the "o/ou", which most are familiar with. I'm Canadian so I'm stuck in between, getting it from both sides. For us, the 'proper' way is the Brit way, which my spellcheck hates. I have the keyboard option for Canadian English, but it seems to default to US. There are some interesting nuances, such as the practice/practise difference. We would use "practise" as a verb, an activity, but if a professional like a doctor, or lawyer, sets up a practice, we have the other form as a noun. It's not a very useful distinction, and difficult to figure out when you're writing, so screw it! What's the point in such formalities?

Funny, I never realised 'judgment' was even a possible spelling, even though I have obviously seen it tens of times; my brain simply filled the 'e' in. 'Judgment' looks awful to me, like it would have to be pronounced /jud-gh-ment/.

Back to the question of formalism... How does a formalist typically account for the ontology of rules? What kind of existence do rules have? Consider the rule of how to spell "judgement" for example, how does that rule exist?

I don't know, but there are lots of US/Brit differences, the common one being the "o/ou", which most are familiar with. I'm Canadian so I'm stuck in between, getting it from both sides. For us, the 'proper' way is the Brit way, which my spellcheck hates. I have the keyboard option for Canadian English, but it seems to default to US. There are some interesting nuances, such as the practice/practise difference. We would use "practise" as a verb, an activity, but if a professional like a doctor, or lawyer, sets up a practice, we have the other form as a noun. It's not a very useful distinction, and difficult to figure out when you're writing, so screw it! What's the point in such formalities? — Metaphysician Undercover

You managed to entirely misunderstand my answer.

Since judgement is apparently a Britishism, I accept that and have no trouble with it; no more than I have with colour. I speak perfect English. I left my spanner on the bonnet of the lorry!

So I did NOT realize that judgement with an e is a Britishism. I just thought it was the same common misspellings that's on my personal list of stuff that annoys me.

And now that I know it's a Britishism, the next time I'm about to chide someone for misspelling judgment, I'll first ask them if they're British; and if they are, I'll belay that chide.

Is that any more clear?

How does a formalist typically account for the ontology of rules? What kind of existence do rules have? Consider the rule of how to spell "judgement" for example, how does that rule exist? — Metaphysician Undercover

I explored this question somewhat in my Grundlagenkrise thread, specially in my chat with Banno, but there was no interest in the topic died after 3 days — folks prefer to go around circles about ethics instead and keep it shallow. The ontology of rules are ultimately derived from logic, be it first-order or second-order — and logical terms can be taken as primitives defined from their truth tables — and the usage of undefined terms, such as "line", "+", or, in the case of ZF, membership ∈.

It's call the arithmetization of analysis. It's a thing in late 19th century math. Basically founding math, including calculus and continuous processes, on set theory.

https://en.wikipedia.org/wiki/Arithmetization_of_analysis — fishfry

5 views per day. The title doesn't resonate with many apparently (including me). Nevertheless, an important movement.

I explored this question somewhat in my Grundlagenkrise thread, specially in my chat with Banno, but there was no interest in the topic died after 3 days — folks prefer to go around circles about ethics instead and keep it shallow. The ontology of rules are ultimately derived from logic, be it first-order or second-order — and logical terms can be taken as primitives defined from their truth tables — and the usage of undefined terms, such as "line", "+", or, in the case of ZF, membership ∈. — Lionino

If logic is following rules, as formalists seem to think, then to say that rules are derived from logic is circular. That's the issue with formalism to avoid the vicious circle, rules must exist as Platonic Forms. So formalism really cannot avoid Platonism, because the only ontologically coherent formalism is Platonism.

5 views per day. The title doesn't resonate with many apparently (including me). Nevertheless, an important movement. — jgill

Virtually every professional mathematician lives in the world created by this movement. Nobody notices because it's like fish not noticing water. It's just the familiar founding of analysis in set theory.

If logic is following rules, as formalists seem to think, then to say that rules are derived from logic is circular. — Metaphysician Undercover

Logic is not "following rules". Your argument is failed as it relies on a nonsensical definition.

In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules. — Wikipedia: Formulism (Philosophy of Mathematics)

However you frame it, rules are an essential aspect of formalism. So the ontology of rules needs to be addressed if we want to determine whether formalism can actually avoid Platonism, or whether it is as I say, just a deeper form of Platonism.

What "ontology of rules"?

Virtually every professional mathematician lives in the world created by this movement. Nobody notices because it's like fish not noticing water. — fishfry

Thus, were set theory removed mathematicians would perish. I think not. But mathematics would not be nearly as robust as it is today. My humble opinion.

Back in the late 1960s my advisor remarked on the separation of the nitty gritty at ground level and the efforts to fly high and look down on mathematics, an abstract perspective to see how the various parts fitted together and document how parts from one branch were like parts form another. He gave me a choice and I felt far more comfortable working in the lowlands, (particularly after learning a bit about algebraic topology). I came into the profession exploring convergence and divergence of analytic continued fractions and related material. Pretty much an extension of the efforts during the 1700s and 1800s to solidify those properties of series. Grubby stuff, but I still enjoy grovelling in it. :cool:

Thus, were set theory removed mathematicians would perish. — jgill

I said no such thing. If the freeway didn't exist, I'd take the old dirt road that people used before they built the freeway. I'd still get where I'm going. Set theory is the modern general framework for most math. That doesn't mean math couldn't get along without it. But if you want to do math these days, you have to use the language of set theory simply because everyone else does.

I think not. But mathematics would not be nearly as robust as it is today. My humble opinion. — jgill

Yes ok. But people are into alternatives these days. Category theory and homotopy type theory are two big alternative foundational frameworks. It's all a matter of historical development.

Back in the late 1960s my advisor remarked on the separation of the nitty gritty at ground level and the efforts to fly high and look down on mathematics, an abstract perspective to see how the various parts fitted together and document how parts from one branch were like parts form another. He gave me a choice and I felt far more comfortable working in the lowlands, (particularly after learning a bit about algebraic topology). I came into the profession exploring convergence and divergence of analytic continued fractions and related material. Pretty much an extension of the efforts during the 1700s and 1800s to solidify those properties of series. Grubby stuff, but I still enjoy grovelling in it. :cool: — jgill

Sounds like fun.

I don't opine as to what that other poster has in mind. But:

Rules themselves may be mathematical objects. Languages, axioms, rules, systems, theories, and proofs can be defined and named in set theory. Even informally, when, for example, we say "by the rule of modus ponens", the rule of modus ponens is a thing named by 'the rule of modus ponens'.

What "ontology of rules"? — Lionino

The ontological status of rules. If rules are real, then they have some form of existence. Ontology is the study of what there is, and the possibility of different forms of existence. An ontological study of rules will determine if there is such a thing as rules, and if so what type of existence they have.

Consider it is the same sort of issue as the ontological status of numbers, for comparison. We can ask, is there such a thing as rules, just like we can ask is there such a thing as numbers. If we answer yes, there is such a thing as rules, then we may proceed to ask questions like are they objective, and if the answer to this is yes, then we are in Platonism. If we answer no, rule are only subjective, then we are faced with a whole lot of problems as to how such a thing as a rule could actually exist.

For example:

https://ontology.buffalo.edu/smith/articles/Hart_Rawls_Searle.pdf

Consider it is the same sort of issue as the ontological status of numbers — Metaphysician Undercover

Not quite the same.

If we answer yes, there is such a thing as rules, then we may proceed to ask questions like are they objective, and if the answer to this is yes, then we are in Platonism — Metaphysician Undercover

It doesn't work like that for numbers. In any case, one has to ask what kind of rules you are talking about. 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. As I said before, if you want to break it down to rules of logic, what is wrong with them being defined as:

and logical terms can be taken as primitives defined from their truth tables — Lionino

? Those are set up by convention.

Rules themselves may be mathematical objects. Languages, axioms, rules, systems, theories, and proofs can be defined and named in set theory. Even informally, when, for example, we say "by the rule of modus ponens", the rule of modus ponens is a thing named by 'the rule of modus ponens'. — TonesInDeepFreeze

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.

For mathematical fictionalism (a kind of nominalising program by Hartry Field), there is an implicit fictional operator when we talk truthfully about mathematics. So "2+2=4" is not true, but "according to arithmetic, 2+2=4" is true. A bit like "Zeus is the most powerful god" implicitly contains "Within Greek mythology..." if we want to speak truthfully. It is a strong proposal compared to other nominalists, but obviously it has its share of objections and issues.

https://plato.stanford.edu/entries/nominalism-mathematics/#MatFic

My point is that there may be many views as to what mathematical objects are or are not, including realism, fictionalism, nominalism... But that, in that inquiry, not just things like sets, numbers, algebras may be considered, but also rules.

and logical terms can be taken as primitives defined from their truth tables
— Lionino
Those are set up by convention. — Lionino

It's not clear to me what you're claiming. Example?

It's not clear to me what you're claiming. Example? — TonesInDeepFreeze

I am asking him if there is anything wrong with taking logical operators, such as & → ~, to be defined from their truth tables, instead of being mysterious platonic objects that float around in another dimensions.
Like 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.

Yes, I think that is what Metaphysician Undercover is talking about.