Dude. According to that survey, a small minority think the foundation of math is set theory. There aren't a lot of experts in phil of math, but they would all roll their eyes at that. :grin:

So math is just language games, right? — frank

You could say that. The point though is that if a numeral refers to a number which is an object, and that object is said to be an idea in someone's mind, then it would be a different object in each mind. We are all distinct individuals with different bodies, different minds, and different ideas. It could not be the case that the idea in my mind (if we call it an object) is the same object as the idea in your mind even if we each refer to our ideas with the same word. We might use the same name "1", and even be trained to describe it with the same words, but it's still not the same idea.

You might consider the beetle in the box analogy. We use the same word, "beetle", and we might even describe it in the same way, but we still have distinct objects. The only way to assume that the numeral refers to the same object for distinct individuals, is to assume that the object is independent. That's Platonism. For whatever reasons, I do not know, @Banno insistently denies the obvious, to say that a numeral refers to a number, which is an object, is classic platonism. No one ought to be surprised by this. Western ideology is firmly based in idealism.

Here's me thinking you were following along...

• Benacerraf’s identification problem
• Gödel’s incompleteness theorems shows that set theory cannot fully capture all mathematical truths
• Category theory, topology, homotopy type theory don’t naturally live in the set-theoretic universe

There aren't a lot of experts in phil of math — frank

Compare and contrast... https://survey2020.philpeople.org/survey/results/5030?aos=47 ...note change in AOS.

We could go in to a discussion about whether the view expressed here is structuralism or constructivism, if you like. But none is a majority opinion, even amongst those who study in the area.

So you've just completely changed your mind here. You were quoting ZFC as if it were scripture a few pages back. Then you were a deflationary nominalist. Now.. I have no idea what you are. I think you might be constructing your view as you go along.

Perhaps you misunderstood...? Yep, we've moved on to address some of the shortages in structuralism, that it is overly formal, doesn’t explain how humans come to understand or manipulate numbers, what it might be to get a mathematical answer right and how we can still have objects in a structural framework...

You could say that. The point though is that if a numeral refers to a number which is an object, and that object is said to be an idea in someone's mind, then it would be a different object in each mind. — Metaphysician Undercover

An object in your mind is called a mental object. An object in your hand is a physical object. An abstract object is something that isn't physical, but it's not simply mental either.

The only way to assume that the numeral refers to the same object for distinct individuals, is to assume that the object is independent. — Metaphysician Undercover

That is correct.

Perhaps you misunderstood...? — Banno

Could be.

An object in your mind is called a mental object. An object in your hand is a physical object. An abstract object is something that isn't physical, but it's not simply mental either. — frank

:meh: What of quantification?

What of quantification? — Banno

What about it?

:grin: Enough with your suggestions; say something.

We've shown how quantification can be handled without invoking abstract objects at all — it’s rule-based, normatively grounded, and socially coordinated.

We've shown how quantification can be handled without invoking abstract objects at all — it’s rule-based, normatively grounded, and socially coordinated. — Banno

Ok. I don't object to that. I doubt you can do the same thing for ZFC, though. So are you now suspicious that ZFC might be bullshit?

An abstract object is something that isn't physical, but it's not simply mental either. — frank

This is platonism. The abstract object is independent from minds, but accessed by them.

What of quantification? — Banno

Quantification doesn't require platonism. The proposition that a numeral represents a thing which is a number is platonism. But we can quantify without that premise. For example, we can do a bijection between the numerals and the things to be quantified. The presumption of "numbers" is superfluous in this case.

This is platonism. The abstract object is independent from minds, but accessed by them. — Metaphysician Undercover

Yes. I know. You pretty much came up with Frege's argument all by yourself. That's pretty cool.

Ok. I don't object to that. I doubt you can do the same thing for ZFC, though. So are you now suspicious that ZFC might be bullshit? — frank

:roll:

Is that a no?

A no to what? Set your account out. Say something. Do the work.

A no to what? Set your account out. Say something. Do the work. — Banno

I've just been observing the different stances people are taking. The only book on phil of math I've read is Mary Tiles' book. After reading it, I realized the ways that set theory is conceptually objectionable, which might not be surprising since Cantor was a mystic, and his mathematical views were directly related to mysticism. Tiles doesn't campaign against set theory by any means, but she does leave the reader with the thought that we may one day rethink the whole thing. It may be that Aristotle was right after all.

That's pretty cool. — frank

Much appreciated, thank you.

I've just been observing the different stances people are taking. — frank

Yes, you read these threads as people and their interactions rathe than as about ideas.

You said this:

We've shown how quantification can be handled without invoking abstract objects at all — it’s rule-based, normatively grounded, and socially coordinated. — Banno

Now apply that strategy to the empty set. You'll find that you can't. Set theory is fundamentally platonic. Eject platonism, and you've ejected set theory.

You'll find that you can't. — frank

Why not? I have nothing in my pocket, therefore I have nothing. :meh:

Why not? I have nothing in my pocket, therefore I have nothing. — Banno

It doesn't sound like you know what a set is.

:wink:

Then do we have broad agreement? — Banno

I was referring to the previous two posts. Beyond that, there's much that I agree with, but I still have puzzles (questions), which is not quite the same as disagreement. Partly, they centre on the questions about what it is for a mathematical object, such as a number, to exist. Partly, they centre on what the timeless present means in this context.

And all this by way of showing that some rules are not procedural at all; they are constitutive norms. — Banno

I agree with that. I don't have a problem about the timeless present in the case of constitutive norms. But in relation to procedures, I do. For the obvious reason, that a procedure takes place in time.

But we need another step - "1 counts as a number" - to get the procedure moving. — Banno

Of course. You may care to know that, as I understand it, the reason the Pythagoreans did not count 1 as a number was, at least partly, because they saw it as the source of all the other numbers. But don't we also need 0, as the starting-point?

I have a procedure for producing one natural number from another, but more to the point is that the natural numbers just are what you get when you do that. — Srap Tasmaner

That's reassuring! But I'm not quite clear what it means to "produce" a number. It's not as if we say to ourselves "I need another number here" and so instigate the procedure. Does your procedure create the numbers it produces from scratch or does it just produce another copy of the number????

Numerals get their identity from roles in activities, not from reference to entities. — Banno

You are not wrong. But now we are getting into trouble with the difference between numerals and numbers. I have a feeling, however, that we may need numbers in order to identify correspondences between numeral systems and perhaps even number systems with different bases.
I'm also getting puzzled about "to be is to be the value of a variable", or, more expansively, the idea that existence is defined within language games and the rejection of single (absolute?) criterion of existence across language games. I think that approach has a great deal to be said for it.

What I said, is that if a numeral is taken to refer to an object, a thing called a number, that object must be a platonic object. This is supported by the argument above. However, I do not believe that a numeral refers to an object called a number. I believe that it refers to an idea called a value. I believe that values are not objects, yet they are referred to. Therefore, in no way do I believe that all reference is "object-reference". — Metaphysician Undercover

I think many people believe that if something is referred to, it counts as an object.
It is true that we equate numbers with values, in the mathematical sense. That's to do with the uses that we put numbers to. So you are right to foreground what we do with numbers - or numerals if you prefer. But I think you slip up when you say that the numeral refers to an idea. That just resuscitates that argument you gave about numbers as ideas. The assignation of value in this context is public and shared, so it cannot be about ideas in our individual minds.

The only way to assume that the numeral refers to the same object for distinct individuals, is to assume that the object is independent. That's Platonism. — Metaphysician Undercover

I'm getting the impression that your objection is simply to the concept of an abstract object, which you call platonism. Would that be fair?

An object in your mind is called a mental object. An object in your hand is a physical object. An abstract object is something that isn't physical, but it's not simply mental either. — frank

Yes. Though there are lots of different kinds of physical object, not all of which can be held in your hand. Shadows, reflections, clouds, lightning, colours, sounds, surfaces, centres of gravity and on and on. Similarly with mental objects. Abstract objects also come in lots of different kinds.

For example, we can do a bijection between the numerals and the things to be quantified. The presumption of "numbers" is superfluous in this case. — Metaphysician Undercover

In the Roman number system "V" counts as five. The Chinese system has 五 (wǔ) for the same number. The ancient greeks used the letters of their alphabet as numerals, so five was the letter epsilon. If you just talk about numerals, you lose the equivalences across different systems.

I think many people believe that if something is referred to, it counts as an object. — Ludwig V

I think that would be an odd use of language, if every word referred to an object. Definitely not suitable for a rigorous logic. For example, we distinguish noun and verb, object from subject, subject from predicate. To disregard these distinctions would incapacitate logical procedures.

So you are right to foreground what we do with numbers - or numerals if you prefer. But I think you slip up when you say that the numeral refers to an idea. That just resuscitates that argument you gave about numbers as ideas. The assignation of value in this context is public and shared, so it cannot be about ideas in our individual minds. — Ludwig V

I actually don't mind when people refer to numbers as objects, that's the way I learned in school. But when people do this they need to respect the ontological consequences.

When you count something publicly, you share your assignment of value. Other people can observe and correct you if they think you make a mistake. This clearly is about ideas in our minds.

I'm getting the impression that your objection is simply to the concept of an abstract object, which you call platonism. Would that be fair? — Ludwig V

Banno was denying that the principles he asserted were platonist, and so I was trying to get him to acknowledge that they are. My objection was to the hypocrisy of publicly rejecting platonism then employing platonist principles.

In the Roman number system "V" counts as five. The Chinese system has 五 (wǔ) for the same number. The ancient greeks used the letters of their alphabet as numerals, so five was the letter epsilon. If you just talk about numerals, you lose the equivalences across different systems. — Ludwig V

That's exactly the reality of translation. In most cases there is no true equivalence "across different systems". The different language games come into being and evolve under different social contexts. The assumption of platonism, produces the idea of eternal unchanging objects which words refer to, and disables us from being able to understand the reality of the nonequivalent aspects.

That's exactly the reality of translation. In most cases there is no true equivalence "across different systems". — Metaphysician Undercover

However, in the case of symbols used in calculation, an equivalence can be established.

My objection was to the hypocrisy of publicly rejecting platonism then employing platonist principles. — Metaphysician Undercover

So you think that "to be is to be the value of a variable" is a platonist principle? I know you sometimes use words in ways I find hard to understand. This seems to be another case.

When you count something publicly, you share your assignment of value. — Metaphysician Undercover

Very true. Except that ordinal numbers don't assign a value; that assigns a place in an order. Assigning a value in mathematics just means what you do when you substitute a specific number (or word or sentence) to a place in a formula that is designated for such "values".

This clearly is about ideas in our minds. — Metaphysician Undercover

No, it isn't. It is about whatever I am assigning a value to.

we distinguish noun and verb, object from subject, subject from predicate. — Metaphysician Undercover

In the context of traditional grammar, an object can be almost any noun, limited only by the specific subject and verb that you are talking about.

I think that would be an odd use of language, if every word referred to an object. — Metaphysician Undercover

Not all words refer to anything. That's why there's such a fuss about dragons and the present king of France.

So you think that "to be is to be the value of a variable" is a platonist principle? — Ludwig V

if being is reduced to value, that's idealism, not necessarily platonist though, but most cases yes. That's classical Pythagorean idealism, the cosmos is made up of mathematical objects.

Except that ordinal numbers don't assign a value; that assigns a place in an order. — Ludwig V

A place in an order, or hierarchy is a value.

No, it isn't. It is about whatever I am assigning a value to. — Ludwig V

What we were discussing was the act of assigning value, counting. That was the subject. Now you are changing the subject to claim that we were not talking about this act, but that we were talking about the thing which you assign the value to. Clearly we were not, as whatever it is assumed to be was not even mentioned.

Not all words refer to anything. That's why there's such a fuss about dragons and the present king of France. — Ludwig V

Why do you allow that sometimes when words refer to ideas (two, three, for example), they refer to things, but sometimes when words refer to ideas (dragons, present king of France), they do not refer to things? Why not just maintain consistency and recognize that these are all cases of words referring to ideas?

But I'm not quite clear what it means to "produce" a number. It's not as if we say to ourselves "I need another number here" and so instigate the procedure. Does your procedure create the numbers it produces from scratch or does it just produce another copy of the number???? — Ludwig V

Eh. A procedure, as I'm using the term here, accepts some input and yields some output. You show me a natural number, and I can show you another.

What I was suggesting was that we can replace our pre-theoretical understanding of counting with this system, consisting of exactly two rules (that 1 is a natural number, and every natural number has a successor), and we will (a) lose nothing, and (b) gain considerably in convenience for doing things that build on counting.

I consider (a) and (b) more or less facts, but there's nothing wrong with examining them closely. Philosophy spends a lot of time doing exactly this sort of thing, but not only philosophy. Linguistics is an easy example quite nearby, where people want to describe a great mass of complex behavior in terms of a smallish set of rules that could account for it. A more or less universal scientific impulse.

So the "axiomatization" of counting here is open to criticism, and I believe it will withstand it.

But it doesn't necessarily tell you what counting actually is.

It might. In a sense, when you come up with a little set of rules like this, if it works pretty well, then what you definitely have is a model of the behavior you want to understand. Whether that model reflects the underlying mechanisms of the behavior, or only simulates the behavior itself, relying on different mechanisms, that's not always perfectly clear. (In one formulation of Chomsky's program, it was of the utmost importance that you have a finite system of rules that can, through recursion, generate an infinite number of sentences, because the system has to be instantiated in a human brain.)

I've been thinking a little, as we've gone along, about the most famous "primitive" counting systems, the "1, 2, 3, many" type. Is "many" a number there? Not exactly. Is it open-ended enough that it might even apply to endless or unbounded sequences? Maybe, maybe not. What I'm trying to say is it might not be quite the same thing as us saying "1, 2, 3, 4 or more" or "1, 2, 3, more than 3", because in our system of counting numbers there is definitely no upper bound.

I suppose I'm bringing that up because we might ask whether people using one counting system are doing something psychologically different from people using another, but we might also ask if there is some difference that philosophy ought to be interested in. The latter, I suppose, would be something about the system itself, and the thoughts that it enables or doesn't, and therefore what would be available as truth, given such a system.

I don't have a problem about the timeless present in the case of constitutive norms. But in relation to procedures, I do. For the obvious reason, that a procedure takes place in time. — Ludwig V

I was using "procedure" as a generalisation of "function". Where a function will have exactly one result for each input, a procedure need not. I hadn't considered that someone would suppose that logical procedures are somehow temporal. I find that idea quite odd.

But don't we also need 0, as the starting-point? — Ludwig V

Again, it hadn't occurred to me that this wasn't obvious... do we want natural numbers or counting numbers? It's not needed, as such, unless you have nothing in your pocket.

That is, which game are you playing?