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
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
A count does not imply an order. — Metaphysician Undercover
you cannot define, or describe counting as ordering — Metaphysician Undercover
you can weigh a sac of flour and see that it's 5 kg. without ordering each kg of flour — Metaphysician Undercover
you can see that there are five books on the shelf without placing them in any order — Metaphysician Undercover
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
Ordinal numbers are a type of numbers which are used for ordering. — Metaphysician Undercover
Ordering is what defines the "ordinal" aspect, not the "number" aspect. — Metaphysician Undercover
You don't even know what I'm saying. — TonesInDeepFreeze
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
A measeurment might not itself be a (human) count. — TonesInDeepFreeze
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
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
I know what you said. You said "A count (1) implies an ordering". — Metaphysician Undercover
There is more than one way to carry out that action which is counting, and not all ways require ordering. — Metaphysician Undercover
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
Why does the action of counting have to be a human count? — Metaphysician Undercover
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
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
Right, I don't understand how what fishfry was saying is relevant. — Metaphysician Undercover
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
I refuted the argument about seeing things at a glance. — TonesInDeepFreeze
We may infer, by whatever means, that there are a certain number of electrons or volts. — TonesInDeepFreeze
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
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
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
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
Do you accept the OED definition, that to count is to determine the number? — Metaphysician Undercover
"determine the total number or amount of, esp. by assigning successive numbers". — Metaphysician Undercover
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
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
equivocation — Metaphysician Undercover
If there is nothing beyond, why wouldn't it come back to its beginning? — noname
And the sense I have been using is indeed the one that is relevant - assigning successive numbers. — TonesInDeepFreeze
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
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
when you assign "2" indicating the second object, the first object is also implied — Metaphysician Undercover
by what principle do we say that "2" refers to one object, the number 2? — Metaphysician Undercover
If this is the case, then "2" refers to the two objects counted, and a third object, the number 2. — Metaphysician Undercover
the numbers are simply not countable. They are infinite and this renders them as not countable — Metaphysician Undercover
It requires more than innocence to be a saint. — Metaphysician Undercover
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
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
As I said, you can use numbers to order things, but this is not what defines numbers. — Metaphysician Undercover
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
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 don't speak of objects being implied. What are implied are statements (or propositions). — TonesInDeepFreeze
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
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
This involves nothing about "implying objects" or "signifying objects". — TonesInDeepFreeze
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
By the principle of stipulative definition. Anyway, your question doesn't weigh on the mathematical notion of counting. — TonesInDeepFreeze
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
First, there is no general definition of number in mathematics. — fishfry
What is your definition of number? — fishfry
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
You're wrong mathematically, as I've pointed out. — fishfry
I'm arguing against accepted mathematical principles — Metaphysician Undercover
Stipulation does not make truth. — Metaphysician Undercover
First, there is no general definition of number in mathematics.
— fishfry
That's because numbers are not objects — Metaphysician Undercover
How do you feel your campaign is doing?
Has it been worth the struggle?
Have there been casualties?
Are you holding up? — jgill
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
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
the latter is dependent on spatial-temporal relations — Metaphysician Undercover
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
If you are counting books, then aren't books objects? — Metaphysician Undercover
it is necessary that an object has been counted? Therefore an object is implied by any count of 1? — Metaphysician Undercover
When did a "set" enter the picture? — Metaphysician Undercover
When I gave a mathematical representation of a count. — TonesInDeepFreeze
For physical world matters. However, in the mathematics itself, ordinals don't refer to space and time. — TonesInDeepFreeze
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 I gave a mathematical representation of a count. — TonesInDeepFreeze
If mathematics talks about an order which is not temporally, nor spatially grounded, then I think such a mathematics would be nonsensical. — Metaphysician Undercover
do you not agree that the "thing" is an object? — Metaphysician Undercover
Do you agree that the thing is a "unity"? — Metaphysician Undercover
If you simply say "1,2,3,4,5" , you might say "I am counting", but it's not a true count — Metaphysician Undercover
Unless the person knows what the symbols mean they are not really counting to five — Metaphysician Undercover
When I gave a mathematical representation of a count.
— TonesInDeepFreeze
Please, do not jump ahead like that. — Metaphysician Undercover
There is no need to represent (2), the result of the act of counting, as a "set" — Metaphysician Undercover
But it was not the sense in your bookshelf example, which may be represented mathematicaly as the bijection I mentioned. — TonesInDeepFreeze
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 explained to you already why bijection (paring) is an inadequate representation of counting — Metaphysician Undercover
I don't know what a "set" is, you haven't defined it. — Metaphysician Undercover
But you seemed to be using it as if it meant the result of the count — Metaphysician Undercover
Isn't it the case, that the mathematical representation of a count, is the number, which is the result of the count? — Metaphysician Undercover
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. — TonesInDeepFreeze
I have always been completely clear that the bijection represents the count, not the result. You are terribly terribly confused. — TonesInDeepFreeze
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
Do you agree that there is no activity of counting if there is no objects counted? — Metaphysician Undercover
find some agreement or compromise — Metaphysician Undercover
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