It's showing that this pattern applies to fingers and to toy cars and lollies and so on - divorcing the pattern from the things being counted. Only after this pattern is understood does the child begin to ask about bigger numbers, and eventually to realise there is no biggest number.

So we get "One counts as a number" and "every number has a subsequent number" and discover that the pattern does not end, and then learn to talk of the whole as being unbounded and that infinite counts as being unbounded... iterating the "...counts as..." to invoke more language games.

Math as we know it piggy-backed the development of money. Money, first invented in Lydia, was the first abstract object, typifying value, but not specifying the value of what.

So abstraction isn't a philosophical folly. It's the result of an astounding innovation.

the first abstract object — frank

I might have said property - this counts as being mine. Basic idea is right.

I might have said property - this counts as being mine. Basic idea is right. — Banno

That existed before money. They bartered. The problem was that corruption in bartering was rampant. They would put the good dates on the top of the caravan, and it was just mud-balls below that. It was so bad that it inhibited trade.

Money set trade free from corruption because it was these little pieces of gold which were stamped to assure a specific weight and purity.

Next came banking, which was mainly invented by the Italians. Now we have virtual money, which allows economies to grow past their present means. The human world as we know it today is a result of money and banking.

I think the tricky bit is that philosophers hear "1 finger and 2 fingers make 3 fingers because 1 + 2 = 3," or even "1 finger and 2 fingers must make 3 fingers, because ..." and this sounds to them like the natural world obeying the "laws of mathematics" or some such. As if the fingers might try to add up some other way, but they would always fail, because there's a law.

But it's actually more like this: if I'm already committed to saying 1 and 2 make 3, then I'm also committed to saying 1 finger and 2 fingers make 3 fingers; if I didn't, I'd be inconsistent. Similarly, I can't say it works with fingers but not with train cars.

Children do have to learn, through trial and error, how much they're supposed to generalize. (Calling cows "doggies" and all that. And learning the difference between count nouns and mass nouns.) And of course what counts as success or failure is determined not by nature alone but also by the adults that mediate a child's understanding of nature.

What's difficult for us, in talking about mathematics, or about language, or about concepts, is that we want to pass over the generation upon generation of practice and refinement, to recreate the primordial scene in which someone, however far back, came up with a way of doing this sort of thing that worked, and we want to identify the features of the environment that enabled it to work, very much as if we expect there would only be one way. Some aspects of our thinking we find relatively easy to change, but some are so deeply embedded that we cannot quite imagine an alternative, so we think this way uniquely fits how the world is.

But it's not just a question of whether other ways of thinking were adequate to "our" needs, but recognizing that there was already adaptive behavior and already learning before there was any conception at all, and even our first conceptual steps were built on that.

Math as we know it piggy-backed the development of money. — frank

Are you saying there could have been a period when people had money, but didn't have amounts of money?

I agree with the spirit of your history lesson, that abstraction was a practical, observable, behavioral thing, but I don't understand the idea that money is the basis of math.

Do you mean the premiss that space can be infinitely divided, not merely conceptually, but also physically? — Ludwig V

No, I've repeated this numerous times now, "space" is purely conceptual. it doesn't make sense to talk about dividing space physically. Physically there is substance, and that's what is divided. And representing that substance as "space" which is infinitely divisible is what I called the false premise which produces Zeno's paradoxes.

But a physical limit to the process of division doesn't undermine the conceptual description. — Ludwig V

It means that the conceptual description is false. And, this falsity, because it is a falsity, produces the absurd conclusions which Zeno demonstrates.

We've already left Meta behind, since he has claimed numbers are not ordered... — Banno

As usual, a completely false and utterly ridiculous representation. I said it doesn't make sense to use "next" in a way which is not either spatial or temporal. If we switch the term to "order" rather than "next", this allows all types of hierarchy such as good/bad, big/small, etc.. But the principle of the hierarchy, and the order of things within the category still needs to be defined. There is no such thing as simply "order" in the general sense. And to have a next implies a direction, which implies either a temporal or spatial ordering.

Therefore we cannot avoid expressing the order itself in spatial or temporal terms. If the scale is big and small for example, then for there to be an order one of the two extremes must be prior to the other, and this turns out to be a temporal order. If there was a supposed order which was infinite in all ways it could not be an order, because infinite possibility is disorder.

I was thinking some days ago that, though I'm not sure what the favored way to do this is, if pressed to define the natural numbers I would just construct them: 1 is a natural number, and if n is a natural number then so is n+1. I would define them in exactly the same way we set up mathematical induction. (Which is why I commented to Metaphysician Undercover that the natural numbers "being infinite" is not part of their definition, as I see it, but a dead easy theorem.) — Srap Tasmaner

You just show that it is limitless which is how "infinite" is defined, so there is no difference and you are not getting away from it being so, by definition.

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

The prerequisite platonist premise.

It's not platonic. — Banno

The usual denial. That "1 counts for a number" rather than signifying a quantitative value, is platonic. That's what platonism does, it makes values which are inherently subjective mental features, into countable independent objects. This is a faulty attempt to portray what is fundamentally subjective (of the subject) as something objective (of the object)

So we get "One counts as a number" and "every number has a subsequent number" and discover that the pattern does not end, and then learn to talk of the whole as being unbounded and that infinite counts as being unbounded... iterating the "...counts as..." to invoke more language games. — Banno

Your statement "every number has a subsequent number" is a stipulation. Therefore it is something produced by design, definition, it is not something that we "discover". So you continue in your misguided attempt to justify mathematical platonism.

No, I've repeated this numerous times now, "space" is purely conceptual — Metaphysician Undercover

Are you a cartoon character? Do you know SpongeBob?

Are you a cartoon character? Do you know SpongeBob — frank

People trapped in a perpetual vortex are those likened to Plato's famously quoted analogy of 'The Cave:.

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

...

It's not platonic. — Banno

The only way that "1" can refer to an object called "a number", instead of referring to distinct ideas in the minds of individual subjects is platonism. Platonism is the only way that "1" can refer to the same thing (a number, an object) for multiple people. Otherwise "1" refers, for you, to the idea you have in your head, for me, to the idea I have in my head, and so on. This is the way that values such as mathematical values are presumed to be objective rather than being subjective like many other values. It's known as platonism.

Finally, you ask whether we're talking about a generalization or a rule, which sounds quite a bit like asking me if mathematics is discovered or invented. It's an unavoidable issue, and I've suggested before where my intuitions lie, which of course involves answering "neither". — Srap Tasmaner

You are right of course. Like you, I am disinclined to back either option. But I prefer to treat each claim as a comparison or analogy and to note similarities and differences between the language-games. This may appear to be a cop-out, but I think it is more judicious than drawing up battle-lines. The same goes for intuitions, and you give a good example. There is, I think, a similar phenomenon wherever people acquire in-depth expertise; it's not something we are born with, but something that is born of long and intimate acquaintance with the relevant skills.

I'll only add that I think too often we think we can fruitfully approach this issue by staring really hard at the natural numbers or at triangles and circles to figure out what they really are and where they came from, when we would do better to look at the practice of mathematics to see what's going on there. — Srap Tasmaner

Wittgenstein is very good on this, as I'm sure you know. It is important. I'm fond of the adage that a rich diet of examples is very helpful. That is also part of Wittgenstein's practice.

Now what I would maintain is that the two are for all intents and purposes the same. That is, the ellipsis as it stands does not tell us how to continue on, and so falls to the sort of view expressed by Kripke; but we dissolve this by insisting that there is a correct way to carry on, given by the model theoretical account. — Banno

It seems clear to me that Wittgenstein would agree with you:-

201"] That there is a misunderstanding here is shown by the mere fact that in this chain of reasoning we place one interpretation behind another, as if each one contented us at least for a moment, until we thought of yet another lying behind it. For what we thereby show is that there is a way of grasping a rule which is not an interpretation, but which, from case to case of application, is exhibited in what we call “following the rule” and “going against it”.

You can't follow a rule or go against it until you start applying it. Kripke's mistake was to demand that everything is settled in advance. There's a lot of discussion of similar ideas in the Blue Book (see p.34, 36 etc.)

At the same time, youa'e right that we can introduce further rules that effectively stabilize new ways of speaking. We can take an earlier practice and add a counts as norm that extends it. In this sense, following a rule can include treating a construction as if it were something more, because we have adopted criteria that make that treatment correct within the extended game. — Sam26

Paper money is a good example.

Calling on procedure alone is insufficient. We need there to be stuff to perform the procedure on. — Banno

You are right that not all rules are of the same kind. In addition to procedural, there are constitutive rules.

Math as we know it piggy-backed the development of money. Money, first invented in Lydia, was the first abstract object, typifying value, but not specifying the value of what. — frank

Yes, I think that may well be fair. But I can't help observing the ancient Egyptians had ordinary arithmetic, which, it would seem, was primarily aimed at the logistics of huge work forces - rations, supplies, etc. Ancient Sumer, China and Lombardy all contributed. There's plenty of people to share in the credit and the blame.

What's difficult for us, in talking about mathematics, or about language, or about concepts, is that we want to pass over the generation upon generation of practice and refinement, to recreate the primordial scene in which someone, however far back, came up with a way of doing this sort of thing that worked, and we want to identify the features of the environment that enabled it to work, very much as if we expect there would only be one way. — Srap Tasmaner

Yes. I do like bits of history as a way of understanding something about our present practices. But I wouldn't want to treat history as sacrosanct in some way. There's nothing wrong with inventing language games to bring out one point or another. Wittgenstein does it all the time, so it can't be wrong, can it?

The only way that "1" can refer to an object called "a number", instead of referring to distinct ideas in the minds of individual subjects is platonism. Platonism is the only way that "1" can refer to the same thing (a number, an object) for multiple people. Otherwise "1" refers, for you, to the idea you have in your head, for me, to the idea I have in my head, and so on. This is the way that values such as mathematical values are presumed to be objective rather than being subjective like many other values. It's known as platonism. — Metaphysician Undercover

The problem with Plato's ideas is that he tries to apply the model of 3D physical objects to abstract objects. Both exist and can be referred to, but they are not the same kind of objects. Your idea that the only kind of object that is not a 3D physical object is an idea in the mind. Numbers are not just ideas in the mind, but are rooted as objects in our shared practices.

Numbers are not just ideas in the mind, but are rooted as objects in our shared practices. — Ludwig V

Let's be clear, numerals are objects in our shared practices. Numbers if they are assumed to be objects are nothing other than platonic objects.

The question is, what do you think a numeral like "1" refers to. If you think it refers to an object, in the type of "number", or the "mathematical" type, that is a platonic object. If you think it refers to an idea of quantitative value, or order, in your mind, that is meaning, not an object. If you think it refers to an object of shared practise in your mind, there is no such thing. Numerals are objects of shared practice in your minds, not numbers.

numerals — Metaphysician Undercover

Before we even get to the question of what a numeral refers to, you face an issue of what makes any given numeral count as a 1 (or as a numeral, or as a symbol). If each individual 1 is a token of the type <1>, you have to say what sort of thing the type is. That's not going to work out. A natural move to avoid types as abstract objects is to claim that the various numerals 1 belong to an equivalence class, but that's not so much an explanation as a restatement of our starting point, that each numeral 1 counts as a numeral 1, and it gives you no help actually defining the equivalence class.

Meta's errors include only thinking of something being either in the world or in the mind. So money, property and number, amongst other things, cause him great difficulty because they rely on communal intent. We might be tempted to express this as "they exist between minds", but that's not quite it, either. Some - many - things owe their existence to public rules, practices and recognition, and these need both minds (plural) and the world. Meta is trapped, as notes, because if numbers are only in the world, he owes us a story about where they are; and if they are only in the mind, he owes us a story about how we manage to do things with them in the world.

Numbers are not like rocks, nor are they like sensations.

That's part of the reason that he can't make sense of logical precedence, restricting himself to temporal or spatial precedence. His metaphysical picture cannot represent logical priority at all, since it's neither purely mental or purely of the world. And along with that go other things that rely on public standards for correctness, such as normative dependence, and rule-dependence.

The following makes his error particularly clear:

The only way that "1" can refer to an object called "a number", instead of referring to distinct ideas in the minds of individual subjects is platonism. Platonism is the only way that "1" can refer to the same thing (a number, an object) for multiple people. Otherwise "1" refers, for you, to the idea you have in your head, for me, to the idea I have in my head, and so on. This is the way that values such as mathematical values are presumed to be objective rather than being subjective like many other values. It's known as platonism. — Metaphysician Undercover

Notice that this odd position is blandly asserted, not supported by any argument.

He relies on presuming that all reference must be object-reference, that object-reference must be either mental or Platonic, and that public sameness requires numerical identity of a referent. Meta relies on an unargued slide: “same object” → “same referent” → “same use” He treats these as equivalent, but they are not. What is required for reference to function is not that we talk about the same object but that we have a public criteria for correctness. It's learning that public criteria that so clearly portrays; learning to count is learning to participate in public activities involving fingers and toy cars and slices of pizza. Numerals get their identity from roles in activities, not from reference to entities.

I don't think I can add anything to your replies. I would likely just confuse the issue.

Then do we have broad agreement?

Notice that this odd position is blandly asserted, not supported by any argument. — Banno

Frege had a pretty persuasive argument for it.

Frege does provide an argument, not just an assertion. His framework addresses the public, objective character of numbers, which Meta simply assumes must be Platonically instantiated. Frege’s “objects” are still abstracta; Wittgenstein and Strawson show how this is overkill: we get public reference without treating numbers as objects at all. Learning to count (Srap) shows how participation in public rule-governed practices secures objectivity and coordination without invoking Platonism.

And then we have Benacerraf's identification problem. There are multiple equally valid set-theoretic constructions of the natural numbers. If numbers are “objects” in the Fregean or set-theoretic sense, which objects are they? There is no fact of the matter that uniquely picks one construction over another. In contrast, public reference and logical precedence do not require objecthood at all. Benacerraf's argument shows objecthood is doing no work, the Wittgenstinian account offers an alternative.

So that's deflationary nominalism. It's a minority view.

you’ve given something a name and a popularity rating.

Why should I care?

Why should I care? — Banno

Because you're procrastinating from whatever it is you're supposed to be doing right now?

There was a lot of strenuous protesting in this thread to the effect that infinity is a thing. Turns out you actually agree with Meta. Numbers aren't things. They're just elements of language games.

If each individual 1 is a token of the type <1>, you have to say what sort of thing the type is. That's not going to work out. — Srap Tasmaner

I don't understand you. In each instance where 1 is taken to be a token, the type is a symbol. And the type of symbol is mathematical. And the type of mathematical symbol is a numeral. How is there a problem with this?

if they are only in the mind, he owes us a story about how we manage to do things with them in the world. — Banno

I have no problem with that story. we are human beings with minds and free will, and we figure things out and do things. Don't you think that's the case?

Notice that this odd position is blandly asserted, not supported by any argument. — Banno

It appears like I didn't make the argument clear enough for you, when I stated it earlier. So, here it is.

If a numeral refers to an object, which is within a human mind, it is a different object for me as it is for you, due to the nature of subjectivity. My thoughts are not the same as your thoughts, so we'd have distinct objects being referred to because we have distinct minds. Therefore, since a numeral is supposed to refer "an object", not a bunch of different objects, and also to the same object for you, as it refers to for Srap, that object must be independent from both of you. The referenced "number", as "an object", must be an independent object This is known as a platonic object. Hence, assuming that a numeral refers to an object called a number, is platonism.

To state it simply, without assuming that the object referred to is an independent, platonic object, it is impossible that the numeral refers to the same object for distinct people, because we each have distinct minds with distinct thoughts. Then the numeral would refer to a bunch of different objects in different minds, instead of "an object", the specified "number". Therefore the assumption that a numeral refers to an object called a number is platonism.

He relies on presuming that all reference must be object-reference, — Banno

That's another one of your very absurd misrepresentations. I explicitly stated, in the passage you quoted, that the symbol might refer to an idea in a mind. Never did I imply that I believe all reference must be object-reference.

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".

nah. I’m at the shops, picking up a few groceries. You are just something to look at while I wait for the check out chick.

Wow! Being productive. :up:

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 — Metaphysician Undercover

So math is just language games, right?

There was a lot of strenuous protesting in this thread to the effect that infinity is a thing. Turns out you actually agree with Meta. Numbers aren't things. They're just elements of language games. — frank

Elements in a language game can be things - because we quantify over them... all these numbers are even, all those numbers are prime.

So math is just language games, right? — frank

Drop "just" and you might be getting there.

Wow! Being productive. :up: — frank

Despite my obvious addiction, I am still functional. But it's going to be 36℃ today, 42℃ tomorrow, so productivity occurs inside or in the early morning.

Elements in a language game can be things - because we quantify over them... all these numbers are even, all those numbers are prime. — Banno

You're trying to have your cake and eat it too.

Despite my obvious addiction, I am still functional. But it's going to be 36℃ today, 42℃ tomorrow, so productivity occurs inside or in the early morning. — Banno

-15C here. :confused:

You're trying to have your cake and eat it too. — frank

That's permitted, under the rules...

That's permitted, under the rules... — Banno

You have the minority view, so you must be following the minority rules. :razz:

You have the minority view — frank

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