In this context, there are two senses of 'count':

(1) A count is an instance of counting. "Do a count of the books."

(2) A count is the result of counting. "The count of the books is five."
— TonesInDeepFreeze

Right,

one is a verb signifying an action, the other is a noun, signifying the result of the action. — Metaphysician Undercover

'count' is also a verb. But here I am mentioning two nouns.

A count(1) implies an ordering and a result that is a cardinality ("quantity", i.e. a count(2)).
— TonesInDeepFreeze

This is what I have been telling you is incorrect. — Metaphysician Undercover

You don't even know what I'm saying. You don't even have the mathematical vocabulary.
You have no standing to tell me what is incorrect in this matter.

A count does not imply an order. — Metaphysician Undercover

I showed you how it does. And less formally, even a child understands that when you count, there's the first item counted then the second item counted ...

you cannot define, or describe counting as ordering — Metaphysician Undercover

And I didn't.

you can weigh a sac of flour and see that it's 5 kg. without ordering each kg of flour — Metaphysician Undercover

There's measuring and there's counting. A measurement might not itself be a (human) count. For example, a digital scale may measure the flour without a person actually counting. On the other hand, counting would be to count the marks on a scale up to the mark where the needle landed. Your argument is grasping at straws.

you can see that there are five books on the shelf without placing them in any order — Metaphysician Undercover

We're not talking about taking in at a glance a quantity. We're talking about counting. You're grasping at straws. I notice you tend to do that after a while in a thread.

On the contrary, sets have no inherent order.
— fishfry

Exactly what I've been arguing, a count is a quantity, not an order, — Metaphysician Undercover

count(2) is a number (quantity, if you like). count(1) is not a number or quantity. We're talking about count(1).

Ordinal numbers are a type of numbers which are used for ordering. — Metaphysician Undercover

That's kind of okay in a very informal sense. But, just to be clear, it is not the definition of 'ordinal'.

Ordering is what defines the "ordinal" aspect, not the "number" aspect. — Metaphysician Undercover

I don't know what you mean exactly by "the ordinal aspect" and "number aspect".

If R is a well ordering of S, then there is a unique ordinal L such that <L epsilon-ordering-on-L> is isomorphic with <S R>. That implies that the cardinality of S equals the cardinality of L.

Anyway, I don't know what point you're trying to make. You disagreed with what fishfry wrote, then he explained how your disagreement is incorrect. You seem not to understand his explanation, though it was eminently clear.

You don't even know what I'm saying. — TonesInDeepFreeze

I know what you said. You said "A count (1) implies an ordering". And I'm telling you that this is false for the reasons I explained. There is more than one way to carry out that action which is counting, and not all ways require ordering. Therefore it is false to say a count (1) implies an ordering.

I showed you how it does. And less formally, even a child understands that when you count, there's the first item counted then the second item counted ... — TonesInDeepFreeze

You showed me one way of counting, which involved ordering, but you also admitted that there are other ways of counting. So clearly you use invalid logic when you say that counting implies ordering. Only that one way of counting, which you demonstrated, implies an ordering, not all ways of counting. You can see that there are five books on the shelf without ordering them at all, just like I can see that there are two chairs in front of me right now, without ordering them at all. That is counting them without ordering them.

A measeurment might not itself be a (human) count. — TonesInDeepFreeze

Why does the action of counting have to be a human count? We have, as humans, devised all sorts of mechanisms to make counting easier, or even do our counting for us. This is the important point here, the essence of counting (what is necessary to the act), is to determine the quantity, no matter how this is done, by machine or whatever. That we commonly do this by ordering is accidental, not an essential aspect of counting.

We're not talking about taking in at a glance a quantity. We're talking about counting. You're grasping at straws. I notice you tend to do that after a while in a thread. — TonesInDeepFreeze

Actually it's you who is grasping at straws. My OED defines "count" definition #1 as "determine the total number or amount of, esp. by assigning successive numbers". Notice that it says "esp.", which means mostly, or more often than not, but it does not mean necessarily. Therefore, to determine the total number or amount of, in a way which is not assigning successive numbers, though it might be a less common use of "count", it is still an act of counting.

Anyway, I don't know what point you're trying to make. You disagreed with what fishfry wrote, then he clearly explained how your disagreement is incorrect. You seem not to understand his explanation, though it was eminently clear. — TonesInDeepFreeze

Right, I don't understand how what fishfry was saying is relevant.

I know what you said. You said "A count (1) implies an ordering". — Metaphysician Undercover

And you don't understand what that means.

There is more than one way to carry out that action which is counting, and not all ways require ordering. — Metaphysician Undercover

Whatever you have in mind, I didn't say that one first declares an ordering. I said that the count itself implies an ordering. The ordering I have in mind is the ordering by the number associated to each item.

You may pick up 'War And Peace' and say (or think) '1', then 'Portnoy's Complaint' and say '2' etc. The ordering implied by that count is {<'War And Peace' 'Portnoy's Complaint'>} because 1<2.

or

You may pick up 'Portnoy's Complaint' and say '1', then 'War And Peace' and say '2'. The ordering implied by that count is {<'Portnoy's Complaint' 'War And Peace'>} again because 1<2.

You can see that there are five books on the shelf without ordering them at all, just like I can see that there are two chairs in front of me right now, without ordering them at all. — Metaphysician Undercover

You see my response in this post about ordering. And in my previous post I refuted the argument about seeing things at a glance. But you skip what I said. Again, immediate instantaneous impression of a quantity is not at stake in a discussion about counting. Counting is different from immediate instantaneous impression of a quantity. You really make yourself look like a dishonest interlocutor with such a grasping at straws argument.

Why does the action of counting have to be a human count? — Metaphysician Undercover

It doesn't. Indeed I mentioned a purely mathematical formulation of counting that doesn't require consideration of human features. But when you get into certain kinds of measurements of quantities, it may be murky whether it's what we mean by 'count'. By 'count' in this context, we mean consideration of discrete objects that are recognized each one at a time, one after another, just as your original example of books on a shelf.

the essence of counting (what is necessary to the act), is to determine the quantity, no matter how this is done — Metaphysician Undercover

"determine the total number or amount of, esp. by assigning successive numbers". Notice that it says "esp.", which means mostly, or more often than not — Metaphysician Undercover

But this is how you started this tangent on counting:

Count the books on the shelf for example. "Book" signifies the type of unity being counted, "1" signifies that a unity called "a book" has been identified, and a first one has been counted , "2" signifies two of these units, etc.. — Metaphysician Undercover

That's talk about "a first" and "units". That sets a context that is a far cry from the far broader "determine the total number". You can't blame me for addressing the kind of counting that you mentioned yourself (counting chairs, one after another, or books, one after another) and then switch the context to something far wider.

Either you are being intentionally sneaky or you just forgot the context that you set up yourself.

Right, I don't understand how what fishfry was saying is relevant. — Metaphysician Undercover

Right, it is common that you lose your place in the discussion.

I said that the count itself implies an ordering. The ordering I have in mind is the ordering by the number associated to each item. — TonesInDeepFreeze

Clearly, to see that there are two chairs in front of me, does not require that I associate a number to each of them. Therefore "the count", the determination that there are two chairs, does not imply an order. We can count (determine the number) without associating a number with each item. Therefore associating a number with each item is not an essential aspect of counting, or the count itself.

I refuted the argument about seeing things at a glance. — TonesInDeepFreeze

All you said is "We're not talking about taking in at a glance a quantity". That's your idea of a refutation? The definition of counting is to determine the number, clearly "taking in at a glance" qualifies.

From my experience with you, your mode of argument is to define the term in an unacceptable, false way (in the sense of correspondence with how the word is actually used), which begs the question. So, you define counting in a way which excludes any form of determining the quantity without any ordering, to support your conclusion that counting implies ordering. Obviously your so-called refutation is fallacious because you're just begging the question.

Do you accept the OED definition, that to count is to determine the number? And do you accept the fact that we can determine the number without ordering as you said here?

We may infer, by whatever means, that there are a certain number of electrons or volts. — TonesInDeepFreeze

The important point, which I'll return to, is that when we have a count, it is necessary that there are as many objects as the count indicates, but it is not necessary that any object is paired with any number. When you recognize this, you'll see that the act, which is counting (determining the count), is not necessarily an ordering, or pairing. Counting, the act which produces a count, is not necessarily an ordering.

That's talk about "a first" and "units". That sets a context that is a far cry from the far broader "determine the total number". — TonesInDeepFreeze

I was giving an example of counting. Did you or did you not agree that there is more than one way to count? If so , then you ought to be able to understand that a count does not imply an ordering.

Now that we have somewhat of an idea about what each other thinks about this matter, let's return to the issue at hand. Let's look at the numeral "2", and see if we can agree on the valid use of it. When we use "2" within the act of counting, do you agree that it signifies that a quantity of two objects have been counted. or do you believe that the numeral pairs with one particular object as "the second"?

If you choose the latter as the use of "2", then I would argue that you are talking about an act of ordering, not an act of counting and these two are distinct. Do you recognize the difference between such ordering, and counting? When we say or write "2" it is implied that there is a quantity of objects, two, which is referred to. When we say "second", it is not necessary that there is such a quantity, because when we say "second", the first may have already disappeared, like counting the hours. So "second" refers directly to one object, and there is no necessity that the prior object still exists, because we are not saying that there are two objects. But when we say "2" it implies that there is a quantity of two objects, or else it's not a valid use of "2".

So when we are counting the hours, and we assign "2" to the second hour, what is really being said is that it is the second hour, not that there are two hours. And these two ways of "counting" one being determining the number or quantity, the other being assigning numerals to an order of things, are very distinct and ought not be conflated by reason of equivocation.

This tangent on counting arose from my response to your comment:

There is a fundamental problem with the concept of numbers. The numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc. But then we want "2" and "3", each to represent a distinct unity as well. So we have to allow that "1" represents a different type of unity than "2" does, or else we'd have the contradiction of "2" representing both one and also two of the same type of unity. — Metaphysician Undercover

And later you have said:

To have a true count, "1" must refer to the first book, "2" refers to the first and second together, "3" refers to those two with a third, etc. — Metaphysician Undercover

Count the books on the shelf for example. "Book" signifies the type of unity being counted, "1" signifies that a unity called "a book" has been identified, and a first one has been counted , "2" signifies two of these units, etc.. — Metaphysician Undercover

Numerals are used fundamentally for counting things, objects like chairs, cars, etc.. There is no such thing as "the count", without things that are counted. So in that situation "1" signifies the existence of one object counted, "2" signifies two, etc. — Metaphysician Undercover

To have a true count, "1" must refer to the first book, "2" refers to the first and second together, "3" refers to those two with a third, etc.. — Metaphysician Undercover

Ordinarily, when someone says "I counted the books on the shelf", we understand that he used numbers (indeed as the positive natural numbers are sometimes called 'the counting numbers'), numbering in increasing order as he looked individually at each book, and not that just that he immediately perceived a quantity. That is the ordinary sense of counting I have been talking about.

Also, for example, if I see an 8 oz glass and that it's full of water, then I may say that the quantity of water is 8 ounces, without counting in the sense of numbering each ounce one by one. But that's not what people ordinarily mean by 'counting'.

Again, if you mean some wider sense, then of course certain of my remarks would not pertain.

to see that there are two chairs in front of me, does not require that I associate a number to each of them. — Metaphysician Undercover

That's the case where there is immediate recognition of the number of objects, but when immediate recognition is not possible, then we count with numbers. Counting in the sense of numbering is what would be understood in the context of this discussion.

Do you accept the OED definition, that to count is to determine the number? — Metaphysician Undercover

I don't doubt that you quoted part of an OED definition:

"determine the total number or amount of, esp. by assigning successive numbers". — Metaphysician Undercover

And the sense I have been using is indeed the one that is relevant - assigning successive numbers. You are only retroactively saying that the sense we should use is the widest sense. That widest sense is not what one would ordinarily and fairly understand by "count the books on the shelf".

I suggest, going forward, that if you wish to use 'count' in the wider sense, you would say 'count(wide)'.

So, as you understand that by 'count' I mean in the sense of 'successive numbering', you may see that my mathematical representation of it is correct and that indeed an ordering is induced. That is not question begging. I am telling you the sense of the word 'count' I mean, then going on to provide a mathematical representation, then showing how, in both the everyday context and in the formal mathematical context, an ordering is induced.

He brings the example of a spaceship flying into space and asks what would happen if it went on and on. Is there an end point or does one eventually loop back to the starting point? These possibilities seem rather implausible. — spirit-salamander

If there is nothing beyond, why wouldn't it come back to its beginning?

When we use "2" within the act of counting, do you agree that it signifies that a quantity of two objects have been counted. or do you believe that the numeral pairs with one particular object as "the second"? — Metaphysician Undercover

Both. That is entailed by remarks in an earlier post of mine.

equivocation — Metaphysician Undercover

There is no equivocation. The second book is mapped to 2. And 2 is also the greatest element in the range of the mapping,

If there is nothing beyond, why wouldn't it come back to its beginning? — noname

What would that look like exactly?

In a small bubble surrounded by nothing (or not surrounded by anything), no matter which way you looked you would see the back of your head. If you threw something away from you, it would hit you in the back.

If you threw something away from you, it would hit you in the back. — noname

In a way this describes this thread. :roll:

And the sense I have been using is indeed the one that is relevant - assigning successive numbers. — TonesInDeepFreeze

OK, let's proceed using your sense of counting, "assigning successive numbers". Do you agree with me, that when you assign "2" indicating the second object, the first object is also implied, as necessary to make your assignment of second a valid and truthful assignment? The "2" does not simply pair with the second object, because "second" implies that there was a first, so this is more than a straight pairing, because there is necessarily implied another pairing between "1" and the first object. Therefore "2", in this count, of assigning successive numbers, refers to or signifies, two objects, the first and the second.

So, as you understand that by 'count' I mean in the sense of 'successive numbering', you may see that my mathematical representation of it is correct and that indeed an ordering is induced. — TonesInDeepFreeze

It is your representation of counting as a simple pairing which I objected to. Even when restricted to a "successive numbering", counting is not a simple pairing. This is because, as I explained, when you pair the second, the pairing of the first is also implied, therefore referred to within the mention of "second". To say "second" refers to the first pairing and the second paring, as two distinct pairings.

Ordinarily, when someone says "I counted the books on the shelf", we understand that he used numbers (indeed as the positive natural numbers are sometimes called 'the counting numbers'), numbering in increasing order as he looked individually at each book, and not that just that he immediately perceived a quantity. That is the ordinary sense of counting I have been talking about.

Also, for example, if I see an 8 oz glass and that it's full of water, then I may say that the quantity of water is 8 ounces, without counting in the sense of numbering each ounce one by one. But that's not what people ordinarily mean by 'counting'.

Again, if you mean some wider sense, then of course certain of my remarks would not pertain. — TonesInDeepFreeze

OK, I agree that this is the "ordinary way" that a person counts, so we have a pretty good understanding between us as to what counting is, so let's go back to the fundamental problem I mentioned in the first place. When you say "2" if you are counting (ordering in your sense), and there are two objects referred to by "2", the fist and the second (the first is necessary to validate the notion of "second"), by what principle do we say that "2" refers to one object, the number 2?

I think you agree with me on the necessity of having two objects to make the use of "2" or "second", a true or valid use. So if we say that "2" also refers to one object, a number, then this type of object must be completely distinct from the other type of object, or else we'd have contradiction, because now there are three objects indicated, the first, the second, and the number 2. If this is the case, then "2" refers to the two objects counted, and a third object, the number 2.

Now, do you see the need to say that the number 2, if it is to be considered an object, must be a distinct type of object, or else we'd have three objects being referred to by "2"? If you see this need, to say that the number 2, if it is supposed to be an object, must be a very distinct type of object from the type of objects which we count, or order when counting, then you ought to also see the need to ask whether it is even possible to count this type of object. I think it is impossible to count these so-called objects because the fact that they are the count, rather than what is counted, is what distinguishes them from the objects which are counted. Therefore, as "the count" , and distinguished from what is counted as "not what is counted", they are by definition not countable. So the simple solution (I offered already), is to recognize that they are not really objects and therefore not countable.

That the numbers, proposed as objects, are not countable, is also evident from the problem of infinite regress. If we wanted to count the numbers, as objects, it would require a different numbering system from the one we use to count ordinary objects, to avoid equivocation. For example, when we have two ordinary objects, we have the number 2 which is another object that would be counted as 1,object if numbers are counted. So we cannot have both "1" and "2" describing how many objects are there unless the "1" was part of a distinct numbering system from the "2". However, then these numbers in the distinct system would be proposed as objects as well, and we'd want to count them alao, so we'd need another numbering system to count them. then we'd proceed toward an infinite number of numbering systems, in the attempt to count all the numbers which count the numbers which count the numbers, ad infinitum..

The simple solution again, is to recognize the truth of the fact, that the numbers are simply not countable. They are infinite and this renders them as not countable, by definition. So we ought not even attempt to count them as this is known to be impossible. Also, we can clearly see that the numbers are not objects, and so they are not something which is countable.

when you assign "2" indicating the second object, the first object is also implied — Metaphysician Undercover

I don't speak of objects being implied. What are implied are statements (or propositions). And the mathematical representation I have mentioned doesn't even need to involve such things as "it is implied that there exist [insert the members of the field of the bijection here]."

As I mentioned, there are two senses of 'count' here:

(1) A count is an instance of counting. "Do a count of the books."

(2) A count is the result of counting. "The count of the books is five."

In order not to have to continually specify which sense I mean, I'll use 'count' in sense (1) and 'result' for sense (2).


Again, here is the mathematical representation I have told you about:

A (non-empty) count is a bijection form a set onto a set of natural numbers (where 1 is in the set and there are no gaps). The result is the greatest number in the range of the count.


Here is a count:

{<'War And Peace' 1> <'Portnoy's Complaint' 2>}

The result of that count is 2.

The ordinary order induced by that count is <'War And Peace' 'Portnoy's Complaint'>


Here is another count:

{<'Portnoy's Complaint' 1> <'War And Peace' 2> }

The result of that count is 2.

The ordinary order induced by that count is <'Portnoy's Complaint' 'War And Peace''>


This involves nothing about "implying objects" or "signifying objects".

Of course, though, it is already assumed that there are objects (books on a shelf in this case) named 'War And Peace' and 'Portnoy's Complaint'. But that's not a mathematical concern. It's just a given from the physical world example.

by what principle do we say that "2" refers to one object, the number 2? — Metaphysician Undercover

By the principle of stipulative definition. Anyway, your question doesn't weigh on the mathematical notion of counting.

If this is the case, then "2" refers to the two objects counted, and a third object, the number 2. — Metaphysician Undercover

You are using 'refer' without specifying in what you sense of the word you mean. Here is what obtains:

2 is the cardinality of a set of two objects.

'2' names 2.

'2' names the cardinality of a set of two objects.

2 is the cardinality of {'War And Peace' 'Portnoy's Complaint'}.

'2' names the cardinality of {'War And Peace' 'Portnoy's Complaint'}.

In the bijection, 'Portnoy's Complaint' is mapped to 2.

'2' names the number that 'Portnoy's Complaint' is mapped to in the bijection.

The result of the count is 2.

'2' names the result of the count.

There is no equivocation or contradiction in any of that.

the numbers are simply not countable. They are infinite and this renders them as not countable — Metaphysician Undercover

Setting aside your other confusions, I will address the term 'countable' as used in a mathematics, to prevent misunderstanding that might arise:

'countable' is a technical term in mathematics that does not adhere to the way 'countable' is often used in non-mathematical contexts.

In non-mathematical contexts, people might use 'countable' in the sense that that a set can be counted as in a finite human count.

But in mathematics 'countable' doesn't have that meaning. Instead, in mathematics the definition of 'countable' is given by:

x is countable iff (there is a bijection between x and a natural number or there is a bijection between x and the set of natural numbers).

It requires more than innocence to be a saint. — Metaphysician Undercover

I like that! Or as the late, great Howard Cosell once said to a reporter: "If ignorance is bliss, you, my friend, must be ecstatic!"

That's what I meant, and though you can use numbers in ordering, it is not what defines them, quantity does. — Metaphysician Undercover

OK, so doesn't this support my point, order is not what defines a number? If not, then I really don't know what you are trying to demonstrate, and how it is relevant. Perhaps you could explain. — Metaphysician Undercover

First, there is no general definition of number in mathematics. We can define real numbers, rational numbers, p-adic numbers, hyperreal numbers, and so forth, but there is no general definition by which we can say what a number is. Therefore it's possible that you are working from an entirely different definition; in which case you could well be correct via your definition, but not correct mathematically.

What is your definition of number?

In math, there are ordinal numbers and cardinal numbers. In fact in modern set theory, the cardinals are defined in terms of ordinals. That is, a cardinal number is a special type of ordinal. So in math, ordinals are logically prior to cardinals.

But as I say, if you are working from a different definition, your point of view could be consistent with that. But not with the usage in math.

Exactly what I've been arguing, a count is a quantity, not an order, hence what I said "numbers are defined by quantity, not order". — Metaphysician Undercover

Not in math. After all, some numbers have neither quantity nor order, like $3 + 5i$ in the complex numbers. No quantity, no order, but a perfectly respectable number. You take this point, I hope. And are you claiming a philosopher would deny the numbertude of $3 + 5i$? You won't be able to support that claim.

As I said, you can use numbers to order things, but this is not what defines numbers. — Metaphysician Undercover

There is no general definition of number; and complex numbers have neither order nor quantity.

Here's an example by analogy. Ordinal numbers are a type of numbers which are used for ordering. Ordering is what defines the "ordinal" aspect, not the "number" aspect. — Metaphysician Undercover

You're wrong mathematically, as I've pointed out. But what is your definition of number then? And how do you account for $3 + 5i$? What about familiar real numbers like $\pi$? No quantity except by stretching the term. You're using an extremely restrictive concept of number.

In a similar way, human beings are a type of animal said to be rational. Rational defines the human aspect but it does not define the "animal" aspect. — Metaphysician Undercover

I'm sure you don't need to explain that to me. But number and order are not an instance of this phenomenon. And as I noted, a cardinal is actually a special kind of ordinal; not the other way 'round.

I don't speak of objects being implied. What are implied are statements (or propositions). — TonesInDeepFreeze

The statement is not implied, it is explicit, stated as "first", "second", etc... What is implied, in order that your count be a true count, is that there are objects counted . Otherwise, as I said it is not a true or valid count. You can state "first", "second", "third", "fourth", but unless there is something referred to, you are not counting anything and it's not a true or valid count.

In order not to have to continually specify which sense I mean, I'll use 'count' in sense (1) and 'result' for sense (2). — TonesInDeepFreeze

I like that, instead of calling (2) the count, we'll call it the result of the count. We might even call it the conclusion, Then I can say that the conclusion is unsound if there aren't any objects counted, because to say "that is the second", or "there are two", is not true unless there are objects which have been counted. To count "1", or "first", without counting anything is to make a false statement.

A (non-empty) count is a bijection form a set onto a set of natural numbers (where 1 is in the set and there are no gaps). The result is the greatest number in the range of the count. — TonesInDeepFreeze

As I explained in my last post, we ought not consider that a number is a countable object, for the reasons I described. So I consider such a count to be a false count.

This involves nothing about "implying objects" or "signifying objects". — TonesInDeepFreeze

Of course it implies objects. You have mentioned things being counted. I deny that natural numbers are things which can be counted. Therefore I conclude that your result is unsound, by this false premise that natural numbers are things which can be counted.

Of course, though, it is already assumed that there are objects (books on a shelf in this case) named 'War And Peace' and 'Portnoy's Complaint'. But that's not a mathematical concern. It's just a given from the physical world example. — TonesInDeepFreeze

Truth and falsity may not be a mathematical concern, but it is a philosophical concern.

By the principle of stipulative definition. Anyway, your question doesn't weigh on the mathematical notion of counting. — TonesInDeepFreeze

Stipulation does not make truth.

Setting aside your other confusions, I will address the term 'countable' as used in a mathematics, to prevent misunderstanding that might arise:

'countable' is a technical term in mathematics that does not adhere to the way 'countable' is often used in non-mathematical contexts.

In non-mathematical contexts, people might use 'countable' in the sense that that a set can be counted as in a finite human count.

But in mathematics 'countable' doesn't have that meaning. Instead, in mathematics the definition of 'countable' is given by:

x is countable iff (there is a bijection between x and a natural number or there is a bijection between x and the set of natural numbers). — TonesInDeepFreeze

Obviously, I do not accept this stipulative definition of "countable", for the reasons explained in my last post. Principally, if we use numbers to count numbers, the numbering system which counts numbers will need to be different than the numbers being counted (by the reasons explained), then we'll want another numbering system to count those numbers, and another to count those numbers, etc', ad infinitum.

There is really no reason to attempt to count the natural numbers, when we know that this is impossible because they are infinite. And numbers are not even countable objects in the first place, they are imaginary, so such a count, counting imaginary things, is a false count. Therefore natural numbers ought not be thought of as countable.

First, there is no general definition of number in mathematics. — fishfry

That's because numbers are not objects, and therefore they cannot be described or identified as such. And since they cannot be identified, they cannot be counted.

What is your definition of number? — fishfry

It is a value representing a quantity.

Not in math. After all, some numbers have neither quantity nor order, like 3+5i3+5i in the complex numbers. No quantity, no order, but a perfectly respectable number. You take this point, I hope. And are you claiming a philosopher would deny the numbertude of 3+5i3+5i? You won't be able to support that claim. — fishfry

Yes, that's a symptom of the problem I explained to TIDF. Once we decide that numbers are objects which can be counted, then we need to devise a numbering system to count them. So we create a new type of number. Then we might want to count these numbers, as objects as well, so we need to devise another numbering system, and onward, ad infinitum. Instead of falling into this infinite regress of creating new types of imaginary objects (numbers), mathemajicians ought to just recognize that numbers are not countable, and work on something useful.

You're wrong mathematically, as I've pointed out. — fishfry

Of course I'm wrong mathematically, I'm arguing against accepted mathematical principles. But the question is one of truth and falsity. Are numbers objects which can be counted, rendering a true result to a count, or are they just something in your imagination, and if you count them and say "I have ten", you don't really have ten, a false count is what you really have?

I'm arguing against accepted mathematical principles — Metaphysician Undercover

How do you feel your campaign is doing?
Has it been worth the struggle?
Have there been casualties?

Are you holding up? :chin:

:grin:

When I say 'P is implied', then P is a statement, not an object.

So I don't say

'War And Peace' is implied.

But I do say

That 'War And Peace' is on the bookshelf is implied.

This is just a matter of being very careful in usage that may be critical in discussions about mathematics.

Regarding this example of counting, I take it as a given assumption that

'War And Peace' is on the bookshelf and 'Portnoy's Complaint' is on the bookshelf.

I am not deriving ''War And Peace' is on the bookshelf and 'Portnoy's Complaint' is on the bookshelf' as implied by anything other than the initial assumption of the example.

And, of course, I am not showing an example of a non-empty count on the empty set. It is a given assumption of the example that:

the set of books on shelf = {'War And Peace' 'Portnoy's Complaint'}

/

Stipulation does not make truth. — Metaphysician Undercover

I knew you would respond in a way that would evince that you don't understand the concept of definition.

First, there is no general definition of number in mathematics.
— fishfry

That's because numbers are not objects — Metaphysician Undercover

No, your belief that numbers are not objects is not the reason that mathematics doesn't provide a definition of 'is a number'.

How do you feel your campaign is doing?
Has it been worth the struggle?
Have there been casualties?

Are you holding up? — jgill

Actually, I'm starting to get a real feel for the problem now, and I sincerely want to thank TIDF and fishfry for helping me come to this realization. I now see that there is a fundamental difference between using numerals to signify quantities, and using them to signify orders. The former requires distinct entities, objects counted, for truth in the usage, while the truth or falsity of the latter is dependent on spatial-temporal relations. So the truth of a determined quantity depends on the criteria for what qualifies as an object to be counted, while the truth of a determined order is dependent only on our concepts of space and time. So, in the case of quantity, truth or falsity is dependent on the truth of our concept of distinct, individual objects, but in the case of ordering, truth or falsity is dependent on the truth of our concepts of space and time. Since we think of space and time as continuous, non-discrete, we have two very different, and incompatible uses of the same numerals.

When I say 'P is implied', then P is a statement, not an object.

So I don't say

'War And Peace' is implied.

But I do say

That 'War And Peace' is on the bookshelf is implied.

This is just a matter of being very careful in usage that may be critical in discussions about mathematics. — TonesInDeepFreeze

Sorry, I don't follow this at all. If you count "1", then it is implied that there is one thing (an object) counted. Do you, or do you not agree with this? If you do not agree, then what are you counting when you count "1"? If you are counting books, then aren't books objects? And you could be counting any type of objects, or maybe just objects in general. But don't you agree that if you count "1", it is necessary that an object has been counted? Therefore an object is implied by any count of 1?

This is just a matter of being very careful in usage that may be critical in discussions about mathematics.

Regarding this example of counting, I take it as a given assumption that

'War And Peace' is on the bookshelf and 'Portnoy's Complaint' is on the bookshelf.

I am not deriving ''War And Peace' is on the bookshelf and 'Portnoy's Complaint' is on the bookshelf' as implied by anything other than the initial assumption of the example.

And, of course, I am not showing an example of a non-empty count on the empty set. It is a given assumption of the example that:

the set of books on shelf = {'War And Peace' 'Portnoy's Complaint'} — TonesInDeepFreeze

I don't see how this is relevant. You seem to have changed the subject. We were not talking about sets. We were talking about (1) the act of counting, and (2) the result of this act. When did a "set" enter the picture?

the latter is dependent on spatial-temporal relations — Metaphysician Undercover

For physical world matters. However, in the mathematics itself, ordinals don't refer to space and time.

If you count "1", then it is implied that there is one thing (an object) counted. Do you, or do you not agree with this? — Metaphysician Undercover

Agree.

If you are counting books, then aren't books objects? — Metaphysician Undercover

Yes.

it is necessary that an object has been counted? Therefore an object is implied by any count of 1? — Metaphysician Undercover

I just told you that I don't use the 'implied' that way.

In your post you said, "it is implied that there is one thing". And that is how I use 'imply' too. I use 'imply' to say 'It is implied that [fill in statement here].

Then you said, "an object is implied".

I don't use 'implied' to say '[fill in noun phrase here] is implied'.

When did a "set" enter the picture? — Metaphysician Undercover

When I gave a mathematical representation of a count.

When I gave a mathematical representation of a count. — TonesInDeepFreeze

For your next trick, do one of an earl. :cool:

I'll do one of Earl Hines's "Blues In Thirds".

For physical world matters. However, in the mathematics itself, ordinals don't refer to space and time. — TonesInDeepFreeze

I was talking about truth and falsity in the use of mathematics, and I use these terms in the sense of correspondence with reality. So it's not necessarily the "physical world" we are talking about, it's "reality" in general. If mathematics talks about an order which is not temporally, nor spatially grounded, then I think such a mathematics would be nonsensical. I've seen some people argue for a "logical order" which is neither temporal nor spatial, but this so-called logical order, which is usually expressed in terms of first and second, is always reducible to a temporal order.

In your post you said, "it is implied that there is one thing". And that is how I use 'imply' too. I use 'imply' to say 'It is implied that [fill in statement here].

Then you said, "an object is implied".

I don't use 'implied' to say '[fill in noun phrase here] is implied'. — TonesInDeepFreeze

When you agree that "it is implied that there is one thing", do you not agree that the "thing" is an object? Can we go to my original term, a "unity". Do you agree that the thing is a "unity"? I mean, we could stick to calling it a "thing", as you seem to agree that there is something which is referred to as "thing" here, but why quibble about terms? Like I said in the last post, what we call the thing is irrelevant; we could call it "object", "entity", "unity", "particular", "individual", "book", "War and Peace", whatever, so long as there is something counted. What is important is that this name refers to something or else you are not truly counting. Do you agree? Even if you are counting names or titles, "War and Peace", etc., those are still "things" which are being counted

If you simply say "1,2,3,4,5" , you might say "I am counting", but it's not a true count, because nothing is counted, therefore the symbols actually refer to nothing whatsoever, and the count itself is invalidated because that sequence of symbols does not have any meaning at all. Suppose someone memorizes that sequence of symbols, 1-5, and repeats them saying "I can count to five". Unless the person knows what the symbols mean they are not really counting to five, they are just repeating symbols. If they know what the symbols mean, then they know that there must be five things (objects, unities, individuals, or whatever you want to call them), or else the count is false. Do you agree? If not how do you validate the meaning of the symbols?

When I gave a mathematical representation of a count. — TonesInDeepFreeze

Please, do not jump ahead like that. You spent days differentiating between (1) the act of counting, and (2) the result of that act. As far as I can see, the "mathematical representation" of both (1) and (2) consists of numerals, "1", "2", "3", etc.. There is no need to represent (2), the result of the act of counting, as a "set", or whatever your intent is. Let's just adhere to these defined principles, and maintain clarity.

If mathematics talks about an order which is not temporally, nor spatially grounded, then I think such a mathematics would be nonsensical. — Metaphysician Undercover

The mathematics of ordering and ordinals may be applied to study of space and time, but the mathematics itself doesn't mention space and time.

do you not agree that the "thing" is an object? — Metaphysician Undercover

I agree that things are objects. In my previous post, I answered essentially the same question, when I said 'Yes'.

Do you agree that the thing is a "unity"? — Metaphysician Undercover

I already shared my thoughts about 'unity' earlier in this thread.

(3)

If you simply say "1,2,3,4,5" , you might say "I am counting", but it's not a true count — Metaphysician Undercover

That would be another sense of the English word 'count', and it may be represented mathematically as

<1 2 3 4 5>

But it was not the sense in your bookshelf example, which may be represented mathematicaly as the bijection I mentioned.

Unless the person knows what the symbols mean they are not really counting to five — Metaphysician Undercover

I don't have an opinion about that.

When I gave a mathematical representation of a count.
— TonesInDeepFreeze

Please, do not jump ahead like that. — Metaphysician Undercover

What? I'm not jumping ahead. I'm referring back. You asked me where the notion of 'set' came from in this discussion, so I told you.

There is no need to represent (2), the result of the act of counting, as a "set" — Metaphysician Undercover

You are critically confused on the very point here, and one that previously you even said you understood. That point is that the result is different from the count. I didn't represent the result as a set*. I explicity said (several times) that the result is a number. Meanwhile I represented the count (not the result) as a bijection, which is a certain kind of set.

(* Putting aside the technical sense of all objects as sets in formal set theory.)

But it was not the sense in your bookshelf example, which may be represented mathematicaly as the bijection I mentioned. — TonesInDeepFreeze

I explained to you already why bijection (paring) is an inadequate representation of counting, as defined by you (1). This effort required a number of posts. I assume you didn't understand.

I'm sorry Tones, but you've really lost me now. You don't seem to be directly addressing any of the points I make, and we do not seem to be understanding each other at all, at this point.

You are critically confused on the very point here, and one that previously you even said you understood. That point is that the result is different from the count. I didn't represent the result as a set*. I explicity said (several times) that the result is a number. Meanwhile I represented the count (not the result) as a bijection, which is a certain kind of set. — TonesInDeepFreeze

I don't know what a "set" is, you haven't defined it. But you seemed to be using it as if it meant the result of the count, i.e. the number. I asked where did the notion of a set come from, and you said "When I gave a mathematical representation of a count." Isn't it the case, that the mathematical representation of a count, is the number, which is the result of the count? Or, you might give a mathematical representation of the activity of counting as "1+1+1+1...". However you've already agreed that there's more than one way to count, so there is probably a number of different acceptable mathematical representations of counting. Bijection though, as described by you as pairing, is not an acceptable representation, for the reasons I already explained.

I explained to you already why bijection (paring) is an inadequate representation of counting — Metaphysician Undercover

And at every juncture I pointed out where you are wrong or confused.

I don't know what a "set" is, you haven't defined it. — Metaphysician Undercover

x is a set iff (x is the empty class or (x is a non-empty class and there is a y such x is a member of y)).

Or, the sets are objects that satisfy the set theory axioms.

Or, the sets are the objects that the quantifier ranges over.

But you seemed to be using it as if it meant the result of the count — Metaphysician Undercover

No, I did not. I have always been completely clear that the bijection represents the count, not the result. You are terribly terribly confused.

Isn't it the case, that the mathematical representation of a count, is the number, which is the result of the count? — Metaphysician Undercover

No! And I've told you this already. What is wrong with you? The mathematical representation of the count is a representation of the count, not of the result. The representation of the count is the bijection. The result of the count is a number.

I don't know what a "set" is, you haven't defined it. — Metaphysician Undercover

In my naivete I once thought of a set as a collection of things called elements. Then I learned the error of my ways. Now I try to avoid thinking of them at all. It's a refreshing experience, like standing at the beach with the soft winds off the ocean caressing your body. :cool:

x is a set iff (x is the empty class or (x is a non-empty class and there is a y such x is a member of y)).

Or, the sets are objects that satisfy the set theory axioms.

Or, the sets are the objects that the quantifier ranges over. — TonesInDeepFreeze

So, a set is a class. How's that relevant? Say we're counting books, the set is called "books" then. Do you agree that there must be some of these things (objects) which are classed as "books", for us to have a true count. If there aren't any books, we do not have any counting of books at all.

I have always been completely clear that the bijection represents the count, not the result. You are terribly terribly confused. — TonesInDeepFreeze

And I've been completely clear, that bijection is unacceptable as a representation of counting. Therefore one or both of us misunderstands what the activity of counting is, so we are stuck here, unable to proceed until we find some agreement or compromise on this. Do you agree that there is no activity of counting if there is no objects counted?

Do you agree that there must be some of these things (objects) which are classed as "books", for us to have a true count. — Metaphysician Undercover

I've answered that already a few times. To have a non-empty count, of course there exist the objects counted, and in you example, these objects are books.

Do you agree that there is no activity of counting if there is no objects counted? — Metaphysician Undercover

Now I'm answering yet again, there is no no-empty count if there are not objects counted.

Now, are you going to continue asking me this over and over again?

find some agreement or compromise — Metaphysician Undercover

I don't seek agreement or compromise. I'm interested in showing where your remarks are incorrect, especially ignorant and/or confused, and sometimes also to add explanations about mathematics, whether you ever understand them.