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. — Ludwig V

Again...

Well, what I was pointing to is the difference between a numeral and a number is in the use to which it is put; one counts with numbers, not numerals. "Numerals get their identity from roles in activities, not from reference to entities" is intended to point this out. The difference between numerals and numbers is not ontological, it is grammatical.

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

Herein lies much confusion, that can be sorted by looking at quantification.

Again, it hadn't occurred to me that this would be problematic. It's quite legitimate to move from "7 > 5" to "Something is greater than five", or "There is a thing greater than five". That doesn't commit us to bumping in to fives and sevens along with chairs and tables. Quantification tells us what a grammar ranges over, not what exists as a spatiotemporal object. The confusion here is between differing language games; to think that "object" only means tables and chairs and not 7 or fully incorporated companies.

Being an object is a role in a language game, not an ontological status.

Failing to recognise this is what sits behind the confusion of calling things "platonism" hereabouts.

if being is reduced to value, that's idealism, not necessarily platonist though, but most cases yes. — Metaphysician Undercover

Who said anything about reducing being to value?

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

Hierarchy, yes. Order not necessarily. Alphabetical order doesn't imply value.

What we were discussing was the act of assigning value, counting. — Metaphysician Undercover

Oh, dear. How can one assign a value without assigning it to something? In any case, counting chickens, for example, answers the question "How many" and assigns a value to the brood, if you like. But it doesn't assign any particular value to any of the chickens.

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? — Metaphysician Undercover

When I say that the President is bold, I am talking about the President, not the idea of the President. When I say that the President has executive power, I'm talking about the idea of the President. The idea of something is a different entity (if it is an entity at all) from the something that it is an idea of.

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. — Srap Tasmaner

OK. In that case, you carry out the procedure. What bothers me is the idea that a formula like S(n)=n+1 is not a set of instructions about how to do something, but actually does it. So someone might say that formula generates the infinity of numbers. That's not at all the same thing.

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. — Srap Tasmaner

I don't have a problem with that. Something like regularizing, tidying up, making explicit - even get a whole new perspective on something entirely familiar. I can see a point to that.

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

Yes. One would need a demonstration of the written instructions as well. It's the gesture of adding one to the total, letting one sheep through the gate, and one more, let through the next one and so on.

I've been thinking a little, as we've gone along, about the most famous "primitive" counting systems, — Srap Tasmaner

Yes. I do like half-way houses. They can be very instructive.

we might ask whether people using one counting system are doing something psychologically different from people using another, — Srap Tasmaner

It would depend on the details.

Being an object is a role in a language game, not an ontological status. — Banno

So the same thing will work for "abstract" and "platonism.". They're parts of a language game. You can't reject them without special pleading.

Godel said we perceive abstract objects. He would know.

Where a function will have exactly one result for each input, a procedure need not. — Banno

Thanks for that distinction. I wasn't aware.

I hadn't considered that someone would suppose that logical procedures are somehow temporal. I find that idea quite odd. — Banno

I'm glad you agree with me. I had noticed that people often speak as if the procedure (or function) somehow executed itself. Obviously a procedure or function only achieves the result if someone follows the instructions. In that case, talk of a function yielding a result is short-hand, omitting the proviso "when someone follows the instructions. Would that be right? The problem is the idea that the rule executes itself in advance of our following it.

do we want natural numbers or counting numbers? — Banno

OK. It depends on what you are doing. I was thinking of the point of origin on a graph, but that's not quite the same as counting numbers.

The difference between numerals and numbers is not ontological, it is grammatical. — Banno

So the numeral is the number in the way that lump of wood is the king in chess? Yes, that's much neater.
Ockham would be pleased.

The confusion here is between differing language games; to think that "object" only means tables and chairs and not 7 or fully incorporated companies. — Banno

Oh dear. I obviously made my point very badly. I was trying to get at the point that there are different kinds of object, that's all.

OK. It depends on what you are doing. I was thinking of the point of origin on a graph, but that's not quite the same as counting numbers. — Ludwig V

I think this is important - see how what we are up to changes what number system we are using?

letting one sheep through the gate, and one more, let through the next one and so on. — Ludwig V

I had to double-check but I never posted this! A couple times I wrote a post which contained exactly this point. (This post is what was left.) It would have gone something like this:

You can count sheep in a field just by looking but there are a number of challenges. A better way is to force them through a chute into another field or paddock or something, and then counting them as they come through is easy. It's interesting that you needn't care what order they come through in; you have your helper — the dog — start a number of fleeing-toward-the-chute processes that run concurrently, and you count them as they terminate. It doesn't even matter that they interfere with each other.

Zeno insists that we count the sheep — that is, the rational numbers — as we find them, in their natural order. But Cantor showed that there is a way to force them through a chute so that you can count them one-at-a-time. It's interesting that it turns out you cannot do this with the real numbers. (And I'll note again that we might take from Zeno not what we're usually told to, but a clever illustration that the rationals in their natural order do not form a sequence, or as an illustration simply of the reason: Zeno shows us that there is no smallest rational number greater than 0, and so there's no "first step". That was worth learning.)

But it doesn't necessarily tell you what counting actually is.
— Srap Tasmaner
Yes. One would need a demonstration of the written instructions as well. — Ludwig V

I was thinking more of (a) how we individuate objects in our environment, (b) how we consider some of them countable and some not, and especially (c) the idea of associating one list with another. There's quite a little leap in (c), because you have to recognize that two collections have structures that can be treated "isomorphically". In our case, the word "collection" seems a bit out of place, but it's not, because we know what kind of structure the natural numbers have without collecting them all. The rational numbers with that same order (that is, "<") do not have the same structure as the natural numbers, but you can order them differently so that they do. That's (d), the cattle chute, re-ordering a collection (even an open-ended one) so that you can map it onto another, or vice versa. Between (c) and (d) it's hard to say which is the bigger leap in imagination. I lean toward (c). When did shepherds start using notched sticks or knotted strings to count cattle? How on earth did they come up with such an idea? Extraordinary.

~~

I had to double-check but I never posted this! A couple times I wrote a post which contained exactly this point. — Srap Tasmaner

I think it's just a coincidence. I used this example because it occurred to me at the time, not because I had read it before.

When did shepherds start using notched sticks or knotted strings to count cattle? How on earth did they come up with such an idea? — Srap Tasmaner

I imagine that there was a problem on the second day that someone took someone else's sheep out and came back with fewer. There has to be an agreed record of how many sheep went out.

Zeno insists that we count the sheep — that is, the rational numbers — as we find them, in their natural order. — Srap Tasmaner

You are making me very curious about the rationals, reals, etc. But I think I'll leave them for another occasion. Thank you for your help. .. and you for yours.

I think it's just a coincidence. I used this example because it occurred to me at the time, not because I had read it before. — Ludwig V

That's what I meant. I was very pleased you had the same thought.

But Cantor showed that there is a way to force them through a chute so that you can count them one-at-a-time. It's interesting that it turns out you cannot do this with the real numbers. — Srap Tasmaner

Interesting metaphor. Does that make the real numbers like a tube of sausage mince? :chin:

Cheers. Interesting chat.

the real numbers — Banno

That's a step in the right direction. You have to switch from a count noun to a mass noun. Water from a fire hose. But even that's not good enough, because with an election microscope you can count individual molecules of water. Maybe the real numbers are closer to something like an electromagnetic field, something where the idea of counting instead of measuring is not just impractical but unthinkable.

Maybe there's no joy there. Still, forcing the unwieldy mass of rational numbers to line up single file to be counted was a master stroke.

Still, forcing the unwieldy mass of rational numbers to line up single file to be counted was a master stroke. — Srap Tasmaner

It's just that the extension of the idea of the real numbers seems to be somehow bigger than the extension of the idea of the natural numbers. We could express that by saying it appears the set of natural numbers is a subset of the set of reals.

Neither set is countable, but that sense that one is bigger than the other was expressed in terms of cardinality.

We could express that by saying it appears the set of natural numbers is a subset of the set of reals. — frank

The natural numbers are also a proper subset of the rationals, but they're the same size.

Here's another way to look at the difference: the Cartesian product of the natural numbers and the natural numbers is different set, certainly, which you can think of as ordered pairs or as the rational numbers with duplicates, but it's not any bigger and you could still lay them out on a line and you can count them. The Cartesian product of the real numbers and the real numbers is a plane: you go up a whole dimension.

Neither set is countable — frank

The natural numbers are countable.

The natural numbers are also a proper subset of the rationals, but they're the same size. — Srap Tasmaner

As is, there is a bijection between them.

The natural numbers are countable. — Banno

You couldn't finish counting them.

And yet they are countable. Look it up.

The natural numbers are also a proper subset of the rationals, but they're the same size. — Srap Tasmaner

You mean they have the same cardinality. Neither one really has a size.

And yet they are countable. Look it up. — Banno

Denumerable, yes. Let's not mistake that for countable in the common sense of the term. I think that's where some of the confusion in this thread is coming from.

Oh, frank. Ok.

Denumerable — frank

Which some authors prefer, but it means what other authors mean by "countable". So long as we know what we mean, "The natural numbers are violet" would do just fine.

Which some authors prefer, but it means what other authors mean by "countable". So long as we know what we mean, "The natural numbers are violet" would do just fine. — Srap Tasmaner

Absolutely. Let's keep in mind that it does not mean the same thing as countable as the word is commonly understood.

I don't think anyone in this thread had forgotten, or that anyone was confused. Some people reject talking about infinite collections, I think, or reject talking about performing operations on them. We who accept and they who reject disagree, but we all agree on what we're talking about.

I don't think anyone in this thread had forgotten, or that anyone was confused. — Srap Tasmaner

I think I could find cases of it in this thread. I'm not going to mine it to find them though

Some people reject talking about infinite collections — Srap Tasmaner

And since you bring that up, let's look at the difference between a collection, an extensional definition, and a set. Just because I think we need to stuff that difference down this thread's throat. :blush:

The extension of an idea need not be thought of as an abstract object. A set has to be thought of that way. There's no choice. The people who invented set theory knew that.

You left out classes, often in this context called "proper classes," I believe (since the word "class" has many uses), collections that are too big to be a set, for example.

I used to know a lot more of the technical side of this stuff than I do now, but I don't think where people have disagreed it was primarily about technical issues anyway. It looks to me like even our differences regarding mathematics are not primary, but result from broad differences in outlook.

:up:

Hyperobjects, that's another hip new member of club.

Supertask. It's the reason Zeno's paradox stands.