If "2" denotes "Portnoy's Complaint" — Metaphysician Undercover

It doesn't.

we cannot dispense with the fact that "1" must refer to the object being counted, a book
— Metaphysician Undercover — TonesInDeepFreeze

I'm not saying a number is a book — Metaphysician Undercover

If I'm not mistaken, in another thread, you were using the word 'refer' in the sense of 'denote'. So if not 'denote' what exactly do you mean by 'refer' in this thread?

the number 5 loses its meaning if it does not refer to five of something counted, — Metaphysician Undercover

The numeral '5' has meaning. The number 5 is not the numeral '5'.

5 is the count of a set of five books. 5 is the count of a set of five apples. 5 is the count of the set with two books, one apple, one house, and one person.

The fact that 5 is a count doesn't contradict that 5 also is a number no matter what it happens to count.

5 is the successor of 4. 4 is the successor of 3. 3 is the successor of 2. 2 is the successor of 1. 1 is the successor of 0.

No matter what the numbers count, they exist by virtue of successorship or by being 0.

if we remove "War and Peace", there is no longer two books, and the pairing is invalidated — Metaphysician Undercover

Bijections are not 'validated' or 'invalidated'.

The bijections

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

{<'Portnoy's Complaint' 1>}

of course are different, but nothing is "invalidated". Saying the pairings are "invalidated" is not even sensical.

You might still use "2" to name the book — Metaphysician Undercover

You're doing it again! We do not use '2' to name a book. '2' does not denote a book.

neither Portnoy's complaint nor "War and Peace" need to be paired with either 1 or 2, for there to be a valid count of 2 — Metaphysician Undercover

We can switch them so that we have:

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

But the greatest number in the range is still 2.

We can have a count of 2 [electrons] without establishing the principles required to distinguish one from the other — Metaphysician Undercover

we can talk about 12 volts, without the need to distinguish and label each unit of electrical potential — Metaphysician Undercover

We may infer, by whatever means, that there are a certain number of electrons or volts. That doesn't contradict that when we see discrete objects then we may count them.

Scientific measurement may have its special considerations. Or even everyday situations such as one glass of water having 8 ounces. But the question here is simple counting. How we use the concept of counting is a matter of practical approach, such as putting the water in a beaker with lines and counting the lines in the beaker to the point the water level ends or whatever. Whatever difficulties there may be conceptually with that, they don't negate the more basic notion of counting by bijection.

You look at your bookshelf, number "Portnoy's Complaint" as 2, and bring it in to me, telling me you have two books in your hand, because "Portnoy's Complaint" is identified as two books. — Metaphysician Undercover

I don't do that.

You present as so confused that I wonder whether you are posting as some kind of stunt or dumb cluck character.

At the moment of conception, there is a rapid expansion of cells, like a Big Bang, only on a biological scale if you will. Looking at our observable universe, we see what conception looks like on a cosmic scale. Since space is infinite, there may be an infinite number of Big Bangs, but we’ll never observe them from earth because of the enormous distances involved. The light from a universe 100 billion light years away, won’t arrive on earth for another 86 billion years. — Present awareness

Nothing within the universe is supposed to be able to travel faster than the speed of light - it's called the cosmic speed limit. As galaxies are moving away from us faster than the speed of light, physicists say that as nothing within the universe can move faster than the speed of light, it is the universe itself expanding.

Oh I see I missed your point the other day and this is a good point. Yes even hard science is ultimately nonsense. — fishfry

Yes it is. Unfortunately, most people freak-out when you suggest this, but to come to terms with this idea is the most intellectually liberating thing there is. Imagine not having the pressure of trying to figure everything out, instead, just going with the flow of ideas, allowing them to come and go as do all things.

Nothing within the universe is supposed to be able to travel faster than the speed of light - it's called the cosmic speed limit. As galaxies are moving away from us faster than the speed of light, physicists say that as nothing within the universe can move faster than the speed of light, it is the universe itself expanding. — Down The Rabbit Hole

Well, people say a lot of things and none of it is true (albeit, it might be the best bullshit currently available).

If I'm not mistaken, in another thread, you were using the word 'refer' in the sense of 'denote'. So if not 'denote' what exactly do you mean by 'refer' in this thread? — TonesInDeepFreeze

"Refer" is more general than denote, such that to denote is a specific type of referring. So when we say that a word refers to something, whether that something is a thing, an activity, an idea, a concept, or whatever, it means that we must direct our attention toward whatever it is which is referred to, in order to understand the use of the word.

The numeral '5' has meaning. The number 5 is not the numeral '5'. — TonesInDeepFreeze

The number 5 is a concept, therefore it has meaning, like any other concept.

The fact that 5 is a count doesn't contradict that 5 also is a number no matter what it happens to count. — TonesInDeepFreeze

My point is that 5 must count something, or else we forfeit its meaning. There is no sense to saying that there is a count of five which does not have five things.

5 is the successor of 4. 4 is the successor of 3. 3 is the successor of 2. 2 is the successor of 1. 1 is the successor of 0.

No matter what the numbers count, they exist by virtue of successorship or by being 0. — TonesInDeepFreeze

This is simply not true. Numbers are defined by quantity, not order. If you want to define numbers by order, then you assign temporality as the difference between 1 ,2,3 and 4. But this is not at all how numbers are used. We might assign numbers to units of time, like first second, third, fourth, but it's really not true to say that numbers derive their value from order or succession, rather than from quantity.

of course are different, but nothing is "invalidated". Saying the pairings are "invalidated" is not even sensical. — TonesInDeepFreeze

As I explained, what is invalidated is your representation of the count as a pairing. The pairing you described is not a valid representation of a count, for the reasons explained.

You're doing it again! We do not use '2' to name a book. '2' does not denote a book. — TonesInDeepFreeze

If your count is nothing but a pairing, then that is all you are doing, assigning a number to a book, naming a book, with a number. This is why your representation of a count, as a pairing, or bijection, is incorrect. That is not what a count is.

We can switch them so that we have:

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

But the greatest number in the range is still 2. — TonesInDeepFreeze

If you switch them, then your original pairing is invalidated. Which pair is the true representation of the count? It can't be both at the same time. But in a true count, neither book is paired with 1 or 2, because a count is not a paring. There are two books, and neither one is number 1 or number 2, they are equivalent as books.

That doesn't contradict that when we see discrete objects then we may count them. — TonesInDeepFreeze

Sure, we can count discrete objects (units), that's what I've been arguing is necessary for a count, to have discrete units which are counted. What is incorrect, for the reasons explained, is your representation of a count, as an act of pairing a discrete unit with a number. Do you understand those reasons given?

How we use the concept of counting is a matter of practical approach, such as putting the water in a beaker with lines and counting the lines in the beaker to the point the water level ends or whatever. Whatever difficulties there may be conceptually with that, they don't negate the more basic notion of counting by bijection. — TonesInDeepFreeze

Again, this is completely untrue. If we want to know what a number is, within a count, then we must produce a true representation of what a count is. To simply produce a false representation of a count, for the sake of supporting your claim of what a number is, is to just beg the question with a false premise.

You present as so confused that I wonder whether you are posting as some kind of stunt or dumb cluck character. — TonesInDeepFreeze

You present yourself as someone who has not yet learned how to count.

"Refer" [...] means that we must direct our attention toward whatever it is which is referred to, — Metaphysician Undercover

Then:

(1)

You might still use "2" to name the book — Metaphysician Undercover

doesn't belong here. We do not use '2' to name a book.

(2) It seems your 'refer' might be close to what I mean by 'to pair with' or, more everyday, 'to associate with'.

In everyday understanding, when we count, we associate one thing with 1, then the next thing with 2, etc. Literally. We say the numbers, one for each object as we count the objects. Mathematically. this is expressed as a function from the set of things counted to a set of numbers:

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

That's a mathematical rendering of picking up 'Portnoy's Complaint' and saying '1', then picking up 'War And Peace' and saying '2', and if those are the only books, then saying 'The count is 2'.

Which pair is the true representation of the count? — Metaphysician Undercover

It is the very point that you can count more than one way.

You can count 'War And Peace as the first, then 'Portnoy's Complaint' as the second. Or you can count 'Portnoy's Complaint' as the first, then 'War And Peace' as the second. In either case, both counts show that there's a first and second, thus there are two.

Everybody knows that but you.

The sequence a,b,c,d,e is a sequence of five letters. e is letter five.

Imagine not having the pressure of trying to figure everything out, instead, just going with the flow of ideas, allowing them to come and go as do all things — synthesis

Like a leaf in a stream, floating quietly in sluggish waters, but skimming past whirlpools to be on its way, frivolous and ethereal. :cool:

In everyday understanding, when we count, we associate one thing with 1, then the next thing with 2, etc. Literally. We say the numbers, one for each object as we count the objects. Mathematically. this is expressed as a function from the set of things counted to a set of numbers: — TonesInDeepFreeze

We might say the numbers "one for each object as we count the objects", but that does not mean that we associate "2" solely with the object pointed to when "two" is said. In reality we associate "2" with having counted two objects, so the first object is also associated with the spoken "2". It is imperative to the count that the other object counted is remembered, and is an integral part of the meaning of "2'" when it is spoken. If the other object is not remembered as a part of the 2, then we could go back to the first object and say "3", but that's not a valid count.

It is the very point that you can count more than one way.

You can count 'War And Peace as the first, then 'Portnoy's Complaint' as the second. Or you can count 'Portnoy's Complaint' as the first, then 'War And Peace' as the second. In either case, both counts show that there's a first and second, thus there are two. — TonesInDeepFreeze

Right you can count the objects in any order that you want. Therefore "pairing", or bijection, which represents the count as assigning a specific order to the objects is a false representation of counting. In instances when there is a small number of objects we can look at them and see the number of objects, without giving them any order at all. So ordering them, or "pairing" them is accidental to the count, it is not an essential aspect of counting. We simply do it as an aid, to ensure that we are not making a mistake and producing a false count.

You ought to accept and understand this fact, because it is fundamental to many forms of measurement, and how we actually count something in reality. When we weigh something, we do not pair a different part of the object with each gram counted, and when we measure the electrical potential we do not pair each part of it with a volt. This is clear evidence, that in general practice, counting something is not a matter of pairing objects with numbers. In modern practice, we deal with billions, trillions, and numbers so high, that if counting something was a matter of pairing, we'd never get done counting any of these astronomically high numbers which we deal with.

The sequence a,b,c,d,e is a sequence of five letters. e is letter five. — jgill

That's an arbitrary designation, dependent on a stipulation that there is a left to right order to the sequence. "a" could just as easily be letter five, or we could assume an ordering which makes any of the letters number five. The point being that even though we order things when counting, (first, second, third, counted, etc.) because it facilitates distinguishing between what has been counted, and what has not been counted, helping to ensure certainty, ordering is not essential to counting. We can count things without ordering them.

You just don't get it.

The sequence a,b,c,d,e is a sequence of five letters. e is letter five. — jgill

That's an arbitrary designation, dependent on a stipulation that there is a left to right order to the sequence. "a" could just as easily be letter five, or we could assume an ordering which makes any of the letters number five — Metaphysician Undercover

No. I have implied the order of the sequence. It's not arbitrary. A more mathematical format would be
(a,b,c,d,e). This omission may have confused you. :roll:

Again, arbitrary. That you designate "a" as first in that sequence, is arbitrary.

The tuple notation is defined in mathematics.

And, of course, for sequences of length at least 2, there are different permutations. That there are permutations does not affect the count, since the count is the greatest number in the range, which remains constant under permutation.

You don't know anything.

The point is that to describe a count as a tuple is not a correct description of a count. You just don't get it.

This is simply not true. Numbers are defined by quantity, not order. — Metaphysician Undercover

You're failing to distinguish between cardinals and ordinals.

Let me give you a standard example. Consider the positive integers in their usual order:

1, 2, 3, 4, 5, 6, 7, 8, ...

Now consider the exact same set of positive integers, but with the number 3 moved to the very end:

1, 2, 4, 5, 6, 7, ..., 3

We can implement this new order relation by defining the "funny less than" symbol $\prec$ as follows:

* If $m$ is any positive integer and $m \neq 3$, we say that $m \prec 3$. So $2 \prec 3$ and $47 \prec 3$ and so forth.

* If $m,n$ are positive integers we say that $m \prec n$ if $m \neq 3$ and $m \lt n$ where $\lt$ is the usual less-than relationship.

Now the quantity of positive integers is exactly the same in either case, since the ordered set $(\{1, 2, 3, \dots \}, \lt)$ and the ordered set $(\{1, 2, 4\dots, 3 \}, \prec)$ have the exact same elements, just slightly permuted. There is a one-to-one correspondence between the elements of the two ordered sets.

But the two ordered sets have a different order type, since 1, 2, 3, ... has no last element, and 1, 2, 4, ..., 3 does.

Both sets have cardinality $\aleph_0$. But 1, 2, 3, ... has order type $\omega$; and 1, 2, 4, ..., 3 has order type $\omega + 1$.

Numbers that denote quantity are called cardinals; and numbers that denote order are called ordinals. This insightful distinction goes back to Cantor in 1883.

In grade school they teach this distinction as "one, two, three ..." versus, "first, second, third, ..." The distinction turns out to have more substantive content in the transfinite domain.

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

This is simply not true. Numbers are defined by quantity, not order. — Metaphysician Undercover

You're failing to distinguish between cardinals and ordinals. — fishfry

MU knows so little of mathematics and yet is so confident. It's almost an admirable trait . . . but not quite.

The point is that to describe a count as a tuple is not a correct description of a count. You just don't get it. — Metaphysician Undercover

We don't describe a count as a tuple. You don't even know what it is that you don't get.

You're failing to distinguish between cardinals and ordinals.

Let me give you a standard example. Consider the positive integers in their usual order: — fishfry

The point is that we were talking about a count, which is a measure of quantity, not an order. To use numbers to indicate an order is a different matter. So to demonstrate the use of numbers in ordering now, is to equivocate, because an order does not necessarily imply a count

Now the quantity of positive integers is exactly the same in either case, since the ordered set ({1,2,3,…},<)({1,2,3,…},<) and the ordered set ({1,2,4…,3},≺)({1,2,4…,3},≺) have the exact same elements, just slightly permuted. There is a one-to-one correspondence between the elements of the two ordered sets. — fishfry

That is not true. These sets do not have the same elements. If "..." implies an infinite extension of the order, then 3 does not exist in the second set. Therefore they do not have the same elements. The symbol "3" is there, but the number is excluded by the infinite order which must occur prior to it. That's an obvious problem with your mode of equivocation, and conflating counting and ordering, it allows for contradiction. You can describe an order which is never ending (infinite), then say that there is a 3 after the end of it. And for you, that 3 is there. But of course you've just accepted the contradiction.

You don't even know what it is that you don't get. — TonesInDeepFreeze

Well of course. If I knew what it is was that I didn't get, that would mean I was getting it.

Try again, maybe after you explain an infinite number of times, I'll get it.

It's almost an admirable trait . . . but not quite. — jgill

That's like when the judge hands down the guilty verdict and thinks: 'that guy was so persistent in his claims of innocence, that I almost feel like letting him go free'. But in this case lack of knowledge is innocence, so there's no guilty verdict to be handed out. Why not just pure admiration then?

maybe after you explain an infinite number of times, I'll get it — Metaphysician Undercover

I doubt it.

we were talking about a count, which is a measure of quantity, not an order — Metaphysician Undercover

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."

A count(1) implies an ordering and a result that is a cardinality ("quantity", i.e. a count(2)). Different orderings may determine the same cardinality, so different counts(1) whose result is the same count(2) may imply different orderings .

the number is excluded by the infinite order which must occur prior to it — Metaphysician Undercover

R as defined below is a well ordering of the set of natural numbers:

R = {<n m> | (~n=3 & m=3) v (n<m & ~n=3 )}

And let 'w' stand for the set of natural numbesrs. w+1 is the ordinal of <w R>.

lack of knowledge is innocence — Metaphysician Undercover

If lack of knowledge is innocence, then you are a saint.

The point is that we were talking about a count, which is a measure of quantity, not an order. To use numbers to indicate an order is a different matter. So to demonstrate the use of numbers in ordering now, is to equivocate, because an order does not necessarily imply a count — Metaphysician Undercover

I only replied to what you said and did not investigate the context. You wrote: "Numbers are defined by quantity, not order ..." If you didn't mean that you should not have written that.

That is not true. These sets do not have the same elements. — Metaphysician Undercover

My God, you wield your ignorance like a cudgel. I could have just as easily notated the two ordered sets as:

* $(\{1,2,3,4, \dots \}, \lt )$ and

* $(\{1,2,3,4, \dots \}, \prec )$

which shows that these two ordered sets consist of the exact same underlying set of elements but different linear orders. Remember that sets have no inherent order. So {1,2,3,4,...} has no inherent order. The order is given by $\lt$ or $\prec$.

More concisely:

* $(\mathbb N, \lt )$ and

* $(\mathbb N, \prec )$

Tell me again how you think these two ordered sets don't have the same elements.

If "..." implies an infinite extension of the order, then 3 does not exist in the second set. Therefore they do not have the same elements. The symbol "3" is there, but the number is excluded by the infinite order which must occur prior to it. That's an obvious problem with your mode of equivocation, and conflating counting and ordering, it allows for contradiction. You can describe an order which is never ending (infinite), then say that there is a 3 after the end of it. And for you, that 3 is there. But of course you've just accepted the contradiction. — Metaphysician Undercover

Crap sandwich. Nonsense. Garbage. And I'm being restrained. In the two formulations $(\mathbb N, \lt )$ and $(\mathbb N, \prec )$, explain to me how $\mathbb N$ has two different meanings. I really want to hear this.

On the contrary, sets have no inherent order. Given a set, we can put many different orders on it; just as if we have a class of school kids we can line them up by height or we can line them up by alphabetic order of their last name. Two different orders on the same underlying set. Of course both these orders have the same order type, as all orders on a finite set do. In the transfinite case, the same underlying set may have multiple distinct order types imposed on it.

I will agree with you that the notation {1,2,4,...,3} is meant to be suggestive. But in fact this is exactly the same set as {1,2,3,...} since sets have no inherent order. Order is something imposed on top of an existing set; and given a set, many different orders and order types may be imposed on it.

Why don't you have a look at the Wiki page on ordinal numbers and learn something instead of continually arguing from your lack of mathematical knowledge?

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

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.

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. A count does not imply an order. You might order things to facilitate your activity of counting, but as you agreed, there's more than one way to count, and as I've been telling you, they are not all necessarily instances of ordering. Therefore you cannot define, or describe counting as ordering. That's why you can weigh a sac of flour and see that it's 5 kg. without ordering each kg of flour. And, you can see that there are five books on the shelf without placing them in any order. "A count" only implies a quantity, five, and there is no necessity of any particular order, or any order at all, only a quantity.

If lack of knowledge is innocence, then you are a saint. — TonesInDeepFreeze

It requires more than innocence to be a saint.

You wrote: "Numbers are defined by quantity, not order ..." If you didn't mean that you should not have written that. — fishfry

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

My God, you wield your ignorance like a cudgel. I could have just as easily notated the two ordered sets as:

* ({1,2,3,4,…},<)({1,2,3,4,…},<) and

* ({1,2,3,4,…},≺)({1,2,3,4,…},≺)

which shows that these two ordered sets consist of the exact same underlying set of elements but different linear orders. Remember that sets have no inherent order. So {1,2,3,4,...} has no inherent order. The order is given by << or ≺≺. — fishfry

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.

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

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".

Why don't you have a look at the Wiki page on ordinal numbers and learn something instead of continually arguing from your lack of mathematical knowledge? — fishfry

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

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. 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.