Then why do you ask me to repeat myself?

Look, I think it's very important for a rigorous mathematics to distinguish between counting real things, and counting imaginary things. This is because we have no empirical criteria by which we can determine what qualifies as a thing or not, when the things are imaginary. Therefore we can only count representations of the imaginary things, which exist as symbols. So we are not really counting the imaginary things, but symbols or representations of them, and we have empirical criteria by which we judge the symbols and pretend to count the imaginary things represented by the symbols. But this is not really counting because there are no things being counted. We simply assume that the symbol represents a thing, or a number of things, so we count them as things when there really aren't any things there at all.

So counting imaginary things by means of symbols is completely different from counting real things because one symbol can represent numerous things, like "5" represents a number of things. And we aren't really counting things, we are inferring from the symbol that there is an imaginary thing, or number of things represented by the symbol, to be counted. So it's a matter of faith, that the imaginary things represented by the symbol, are really there to counted. But of course they really are not there, because they are imaginary, so it's false faith.

I want to be clear in my mind. Is this your position on the subject? — fishfry

Read my last post.

And that parenthetical is simply to make clear that in this context we're not talking about the technical notion of an empty count. We're talking about counts that start at 1. — TonesInDeepFreeze

We've been talking about what it means to count. And we've determine that the count starts at one. If you know of some other way of counting which is based in something else, let me know please.

If there is a count that reaches 1, then there exists at least one object counted, and if there is a count that reaches 2, then there exist at least two objects counted. — TonesInDeepFreeze

If the count does not reach one, then it is not a count, because one is the beginning of the count. We could count by twos, or fives, or tens, but I don't think you've even accepted this yet, insisting that counting is a bijection with individuals. How do you ever get to the idea that the count "reaches" one when it necessarily starts at one and there is no count prior to one?

Your original and ongoing question regarded the context in which there are books on the shelf. You didn't ask me about the notion of an empty count. — TonesInDeepFreeze

Why do you keep avoiding the question? We're moving on from my original question, because I want to know how you come up with your notion of "countable". This is relevant to the topic of the thread, infinity. How do you proceed from the notion that "a count" is the activity of counting, to the conclusion that zero objects are countable, or that an infinite amount of objects are countable? It seems to me, that to do this you would need to change the definition of "a count".

But about the empty count: It's a technical set theoretical matter. It's not intended that the use of the word 'count' in 'empty count' corresponds to our everyday English senses of 'count'. I happily agree that it's an odd use of the word 'count'. If you don't like the notion, then that's okay in this context, because the representation with a bijection doesn't depend on the notion. — TonesInDeepFreeze

Do you realize, that within a logical system you cannot change the "sense" of a word without the fallacy of equivocation? I think therefore, that we have started with a faulty definition of "a count", your definition (1). If we are going to say that zero objects is a countable number of objects, then we need a definition of "count" which is consistent with this.

Should we try definition (2), the result of a count? How many books are on the shelf? None. We know that there are zero, without counting any. It's an observation, there is nothing which satisfies the criteria for "book", so we make an empirical claim that there is zero books. This is similar to what I said about seeing two chairs, or seeing that there are five books, without pairing them individually with a number (bijection). To derive the number of a specified object, we do not need to count (def 1) the objects. Cleary then, 0 is not the result of an act of counting Can we assume that numbers do not represent "a count" at all, nor do they represent the result of a count, they represent empirical observations? Otherwise, we need a definition of "count" which could be consistently applied, and this doesn't seem possible.

We are not claiming it is a count of actual captains. — TonesInDeepFreeze

You defined "count" with the activity of counting. And we described counting as requiring objects to be counted. I distinguished a true count from a false count on this basis, as requiring objects to be counted. Clearly, if the objects counted are not actual objects, but imaginary objects, it is not a true count.

I think this helps to demonstrate that we cannot define numbers with counting. So, my original assumption that "2" implies a specified quantity of objects, must be false. But now we have the question of what does "2" mean? I think it is a sort of value, and by my statement above, a value we assign to empirical observations. However, if we can assign such a value to imaginary things in a similar way, we need a principle to establish equality, or compatibility, between observed things and imaginary things. This is required to use negative numbers.

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. — TonesInDeepFreeze

The question was whether there could be a count if there are no books.. If no books are counted, do you consider this to be a count? I think that if no books are counted then there is no activity, of counting, therefore no result of counting either.

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? — TonesInDeepFreeze

I'm asking you if you believe there is such a thing as an empty count. That would be contradiction, obviously, to have an activity of counting when nothing is being counted. Do you agree? You did say that a set could be an empty class. Do you agree, that by your definition of "count" (1) the act of counting, an empty set is not countable? There seems to be discrepancy between how you define the count (1), and and how you say "countable" is defined in the mathematical sense.

I can count the captains of the starship Enterprise even though they're imaginary. — fishfry

That's what I would call a false count, because it's hypothetical. It's like if you look at an architect's blueprints, and count how many doors are on the first floor of a planned building. You are not really counting doors, you are counting hypothetical doors, symbolic representations of doors, in the architect's design. Likewise, if you count how many people are in a work of fiction, these people are hypothetical people, so you are not really counting people, you are counting symbolic representations. We can count representations, but they are counted as symbols, like the architect's representation of a door, may be counted as a specific type of symbol. And when you count captains of the Enterprise, you are likewise counting symbolic representations. If you present this as a true count of actual captains of an actual starship, you'd be engaged in deception. You are not counting captains of a starship, only symbolic representations.

Curious to know: If you deny complex numbers do you likewise deny quantum physics, which has the imaginary unit i in its core equation? — fishfry

Yes, I think quantum physics uses a very primitive, and completely mistaken representation of space and time. That's why it has so many interpretative difficulties.

Problems of interpretation come from trying to explain why the electron sometimes appears as a particle and sometimes a wave. — khaled

I think the issue here is that the electron does not have any real existence as a particle at all. It is a particular quantity of energy, and we, as human beings desire to give that quantum of energy real existence as a unit, making it an entity, called an electron. But there is no real existence of that particle, this is simply how we relate to that energy from our perspective as human beings, with human artifices.

Not sure how that was unclear. — Book273

It was quite unclear. Are you saying that your life will begin anew, at your death? I always thought that death was the end of life, and that is final.

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?

I am not concerned with death. She is an old friend that will call on me as she chooses. When she does, I will hold her hand and walk through that door with her. And it will begin anew. — Book273

"And it will begin anew"? Don't you really mean "It will be finished forever"?

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.

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.

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?

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?

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.

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.

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.

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.

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.

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?

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.

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.

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

People, Trump and co. really could win.Then the fun will be over. — baker

Been there, done that.

It’s uncivilised for sentient beings to undergo involuntary pain and suffering – or any experience below hedonic zero. — David Pearce

Involuntary pain and suffering is most often derived in accidental ways, mistakes and not knowing the potential source, and how to prevent it. I do not think it is possible to eliminate the possibility of such pain and still remain living beings. Nor do I think we ought to attempt to eliminate the possibility of such involuntary pain, as this possibility is what inclines us to think, and develop new epistemological strategies for certainty.

What I think is a far more significant and important issue is the matter of voluntarily inflicting pain and suffering on others. If your goal is to manipulate the human being towards a more civilized existence, then the propensity for human beings to mistreat others is what you ought to focus on, rather than the capacity for pain. See, you appear to be focused on relieving the symptoms, rather than curing the illness itself.

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.

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.

The way I see it is that as human beings, we are first and foremost, animals. That's what defines us, although we like to separate ourselves from the other animals, to say that we're somehow a special type of animal. Let's say that specialness as "civilised". If we're already civilised, then what could it even mean to suggest making us more civilised? If civilised is a general category, then all sorts of particular instances qualify as civilised, and what would make one "more" civilised than another? If we are not yet civilised, then what really does "civilised" mean? Distinguishing us from the other animals, is not even justified now. Unless we answer this type of questions, assuming that such and such is "more civilised", is simply an unjustified assumption.

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

A number (we're talking about natural numbers in this context) is a count in sense (2). That doesn't preclude that a number is a mathematical object. — TonesInDeepFreeze

That's right, it doesn't preclude that the number is a mathematical object. But the point is that your definition (2) stipulates "the result of counting". So correct use of "5" is dependent on the count of the books, that there are five books. Therefore the number 5 loses its meaning if it does not refer to five of something counted, books in this case. Anytime we use "5" regardless of whether you think it refers to a mathematical object or not, it necessarily refers to five distinct units, or else you are using it incorrectly.

We better dispense with that notion. It's nuts. A number is not a book. — TonesInDeepFreeze

I'm not saying a number is a book, that's nonsense. But when we use "5" it is necessary that there are five distinct units indicated in that usage or else you are using "5" in an unacceptable way. Do you agree?

So the numeral does not denote a book, but rather it denotes the number that is paired to the book in the bijection (or, in everyday terms, in the pairing off procedure we call 'counting'). — TonesInDeepFreeze

Strictly speaking this (bijection in the way you describe it) is not a valid count. Suppose we say that there are two books. "War and Peace" is numbered as 1, and "Portnoy's Complaint" is numbered as 2. The relation between "Portnoy's Complaint" and the number 2 is not a simple pairing. This is evident from the fact that if we remove "War and Peace", there is no longer two books, and the pairing is invalidated. You might still use "2" to name the book, but it is not a valid count of two, because there is only one book.

So we cannot say that "Portnoy's Complaint" is paired with 2. That is a false representation because it does not include the necessary requirement of another book. "Portnoy's Complaint" can only be paired with 2 in a valid count, if there is another book paired with one. Furthermore, 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. Do you recognize this point? There is no need for a pairing to have a valid count. We can have two objects, and say that there are two, without naming either as one or two, they are simply two.

This latter point is something which is very important to understand, especially when we count things like electrons which are difficult to distinguish from one another. We can have a count of 2 without establishing the principles required to distinguish one from the other. We can say that there are two electrons in the same orbit, without the need of distinguishing one from the other. We have principles which say they are distinguishable, but we need not distinguish them. Likewise, we can talk about 12 volts, without the need to distinguish and label each unit of electrical potential, as 1,2,3, etc..

So it is very clear that your method of representing "a count", as pairing a number with a unit (bijection) is a totally inadequate representation of what a count really is.

We don't say "''1' denotes 'War And Peace' and '2' denotes 'War And Peace' together with 'Portnoy's Complaint'". That's crazy. — TonesInDeepFreeze

You think it's crazy, but it's what's required to have a valid count. If "2" denotes "Portnoy's Complaint", unconditionally, and you have no other books, then obviously your count of 2 books is invalid. If you deny this requirement them you allow for invalid counts. 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. That's what's really crazy.

Right, but do you agree that it is necessary that there is a thing counted, a book in this case? So as much as the numeral "1" "denotes" what you call the number 1, which is a property of "the count", it must also refer to the one book, or else the count is not a true count. For "the count" to qualify as a true count, there must be something which is counted. If "1" does not refer to the book, as well as what you call the number, then there is nothing being counted, and therefore no count, because if there is nothing being counted, this does not qualify as a count.

Therefore, we cannot dispense with the fact that "1" must refer to the object being counted, a book, as well as what you call the number 1, or else we have annihilated "the count" as false because we cannot have a count with nothing being counted. But we cannot annihilate the count, because that is what gives logical coherency to the numbers.

If this is not clear to you, imagine that you go to count the books, and you count the same book over and over again, 1, 2, 3, 4, etc., such that you could have an infinite number of books, by counting the same book over and over again. That is not a valid or true count. 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.. If we remove the need to have distinct objects being counted, then the count is not a valid or true count.

But to return to the earlier example of playing chess, one can fanatically aspire to improve one's game and play to win even though one will invariably lose. — David Pearce

I don't agree. One cannot play to win if the person knows that winning is impossible. In a similar way one cannot get the same enjoyment from winning if the person knows that losing is impossible. So there must be the real possibility of losing (suffering) if winning is to be enjoyable.

Two invincibly happy (trans)humans can play competitive chess against each other and both improve their game. Honestly, I don't see the problem! — David Pearce

The point is that one must lose, and suffer from the loss, if the other is to win and obtain the enjoyment of winning. If we remove the winning and losing from the game, we can't call it a competitive game.

Rather, what needs questioning is the widespread assumption that the "raw feels" of suffering are computationally indispensable. If the indispensability hypothesis were ever demonstrated, then this result would be a revolutionary discovery in computer science: — David Pearce

The issue, in my mind, is not whether suffering is indispensable, but the question of whether we can have gain without the possibility of suffering. If it is the case, as I believe it is, that all actions which could result in a gain, also run some risk of loss, and loss implies suffering, then to avoid suffering requires that we avoid taking any actions which might produce a gain. But if gain is necessary for happiness, and this is inevitable due to biological needs, then the goal of happiness cannot include the elimination of suffering. Therefore the goal of eliminating suffering must have something other than happiness as its final end. What could that final end be? If eliminating suffering is itself the final end, but it can only be brought about at the cost of eliminating happiness, then it's not such a noble goal.

Maybe contemplating the pain of a defeated opponent sharpens the relish of some winners today. Let's hope such ill-will has no long-term future. — David Pearce

It's not the pain of the opponent which I am talking about here, it's the aspect of the pleasure derived by the winner, which is produced by knowing that the pain of losing has been avoided. So the winner does not wish ill-will on the loser, only attempting to avoid the potential of the pain for oneself. That's "sportsmanship", you do not intend ill-will on the opponent, only the best for yourself. But the game is designed such that there is a loser. In the competition, all competitors know that someone (or team) will suffer the pain of lose. In good sportsmanship, it is the goal of the competitors to win and avoid such pain. It is not their intent to inflict pain on others. The joy in winning is intensified not by the thought that others are in pain, but by knowing that the pain of lose, for oneself, has been avoided.

So, competing against earlier iterations of oneself or an insentient AI does not address the issue, because we still must allow for the possibility that one loses, and therefore suffers from the lose. Replacing the opponent with an AI does not remove the necessity for the possibility of lose, and the consequent pain and suffering. The issue here is that much joy and happiness, and the drive, motivation, or ambition for success, comes from the desire to avoid the pain and suffering caused by failure. If we remove that pain and suffering, extinguish the possibility of failure, make the AI always lose no matter what, or whatever is required to negate the possibility of suffering, then there is no drive or ambition to better oneself.

But as I said, emphasizing hedonic uplift and set-point recalibration over traditional environmental reforms can circumvent most – but not all – of the dilemmas posed by human value-systems and preferences that are logically irreconcilable. — David Pearce

So what would be the point to continually inducing the joy and pleasure of winning in a person, without requiring the person to actually compete and win, or even do anything, to receive that pleasure? If it is not required to do the good act, to receive the pleasure of doing a good act, then when is anyone ever going to be doing anything good?

1 is the count at the first member of the set, a particular unity (whatever it is). 2 is the count at the second member of the set. Etc. And '1' and '2' name different individual numbers. And 1 is the count of the members of the set with one unit. And 2 is the count of the members of a unity that is a set with two members. And a set with one member is a different kind of unity from a set with two members. — TonesInDeepFreeze

I do not assume any sets, or numbers, to begin with. 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..

'2' denotes the number 2. The number 2 is the count of a set with two members. And a set of two members is itself a unity as a set. But '2' does not denote a unity; it does not denote the set that it counts. It denotes the COUNT of a set that is itself a unity. When we say that a set is a unity, we mean that it is one set, while we recognize that the number of members of the set may be greater than one. — TonesInDeepFreeze

The inconsistency arises now, if we say that numerals signify numbers rather than the things being counted. Let's call the number, "the count" which seems acceptable to both of us. Let me look at the difference between a count of one, and a count of two.

To have a count of one, there must be an object which is counted. In order for the count to be a valid count, there must be something which is counted. This is not the number 1 which is counted. It is something independent, an object like a chair, or a car, one of the things which is going to be counted. What validates the count of one, is an independent object, what I call a fundamental unity, which is counted.

Now let's consider a count of two. The count of two is justified by the existence of two such objects. But you want to say that "the count" itself is an object, the number two. So we have two distinct types of objects referred to with "a count of two". We have the two material objects, which have been counted, justifying the count of two as a valid count, and also we have the count itself, as an abstract object, which is called the number.

So, if we assume the reality of abstract objects, numbers, then when we use "2", there is always, if it is a valid use of "2", two distinct types of objects referred to. There is a number, 2, which as a unified object, as "the count", and there is also two of another type of unity, being the things counted, in order that the count is a true and valid count. In the case of "1" however, we can say that the number is the fundamental unity, the thing being counted, and also the abstract unity, represented as "the count", because they are each one simple unity. Therefore we would have consistency saying that the number 1 is both the thing being counted, making a valid count, and an abstract object itself.

To summarize now. Let's say that "1" refers to the number 1, which represents the count, and is also the thing counted, abstract numbers. We cannot use "2" in the same way. "2" might refer to a number, which represents "the count", as an object, or it might refer to the two distinct objects which are counted. It cannot refer to both, due to the inconsistency of one being one object, and the other two. My contention is, that if we use "2" to refer to "the count" itself, as the number 2, an abstract object, and this is what you are doing in your post, then the count itself is rendered false or invalid, because "2" cannot refer to both one object and two objects at the same time without contradiction.

"It's not enough to succeed. Others must fail", said Gore Vidal. “Every time a friend succeeds, I die a little.” Yes, evolution has engineered humans with a predisposition to be competitive, jealous, envious, resentful and other unlovely traits. Their conditional activation has been fitness-enhancing. In the long run, futurists can envisage genetically-rewritten superintelligences without such vices. After all, self-aggrandisement and tribalism reflect primitive cognitive biases, not least the egocentric illusion. Yet what can be done in the meantime? — David Pearce

So this "intensely rewarding experience" which we get from succeeding in competition, you designate as seated in a vice, or vices, This would mean that it is a bad rewarding experience which ought to be eliminated. But on what principles do you designate some rewarding experiences as associated with vices, and some as associated with virtues? I would think that if you want to eliminate some such intensely rewarding experiences, and emphasize others, you would require some objective principles for distinguishing the one category, vice, from the other, virtue.

If society puts as much effort and financial resources into revolutionising hedonic adaptation as it's doing to defeat COVID, then the hedonic treadmill can become a hedonistic treadmill. Globally boosting hedonic range and hedonic set-points by biological-genetic interventions would certainly be a radical departure from the status quo; but a biohappiness revolution is not nearly as genetically ambitious as a complete transformation of human nature. And complications aside, hedonic uplift doesn't involve creating "losers", the bane of traditional utopianism. — David Pearce

I think I've already mentioned the problem with this perspective. That is the divisiveness that such a proposal (which you admitted might be unethical) would induce. Global cooperation is not facilitated without consistent belief. Look at the issue of climate change for example, and even an immediate threat to the lives of many, like COVID, does not obtain unanimous consent to the designated required response. You might find a good example of global cooperation with the issue of CFCs and the ozone layer. That was a serious issue which seemed to obtain global cooperation.

However, it appears to me like such cooperation is more likely to be obtained in the face of serious evil, rather than the effort to obtain some designated good. So I feel like the challenge to you would be to demonstrate that failing to follow your proposed program would be a great threat to humanity. I perceive three levels of attitude toward action, or inaction, in relation to such a proposal. There are those who say "do it", and may start such an action, those who say "do nothing" (status quo), and those openly opposed to doing it. It seems like those who say "do it" have a huge task to persuade the others, and bring them onboard, which must be carried out prior to starting any such action. This would require a huge effort of education and some very strong principles. That is because starting any action without first persuading the others, logically would shift those in the "do nothing" group over to the "openly opposed" group.

Participating at TPF has necessitated that I become an expert at grade school principles, because many people here do not seem to understand these very basic principles, like what "=" signifies. And so, I have to explain over and over again, the same principle, in as many different ways as possible, in an attempt to dispel the misunderstandings which these people hold. It seems to be much easier to teach young children these principles than it is to teach adults who have already developed bad habits of misunderstanding, by accepting contrary principles. So the teacher of adults, must become an expert, rather than just an average teacher, requiring not only to instill good habits of understanding, but first needing to dispel bad habits of misunderstanding.

But neither P nor Q are stated coherently by you. And there's no reason to think anyone wants P or Q anyway. — TonesInDeepFreeze

Then I conclude that what needs to be discussed is clarification of P and Q.

Of course the notion of 'one' is related to that of a unity. But even aside from parsing, I don't know who in particular you think holds that "The "2" represents two of those individuals together, and "3" represents three, etc". It would help if you would cite at least one particular written passage by someone that you think is properly rendered as "the numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc" and "'2" and "3" represent some kind of unity". — TonesInDeepFreeze

Whenever we count something it is like this. 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..

t would help if you would cite at least one particular written passage by someone that you think is properly rendered as "the numeral "1" represents a basic unity. an individual. — TonesInDeepFreeze

I assume you know how to use Google or some other search facility. You could simply search this if you need such a confirmation, instead of asking me to do your research for you. Here is the first paragraph from the Wikipedia entry on "1":

1 (one, also called unit, and unity) is a number and a numerical digit used to represent that number in numerals. It represents a single entity, the unit of counting or measurement. For example, a line segment of unit length is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer.[1] It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0. — Wikipedia

Also, if you actually are interested (which you don't seem to be by your half-hearted replies, and refusal to do any research yourself, and near complete denial of the relation between one and unity), you could look into number theory, and the reason why 1 is generally determined as not a prime number. Here's the first entry I get when I Google that question, is 1 a prime number: https://blogs.scientificamerican.com/roots-of-unity/why-isnt-1-a-prime-number/
Here's a passage from that article:

In the very most basic example, we can ask whether the number -2 is prime. The question may seem nonsensical, but it can motivate us to put into words the unique role of 1 in the whole numbers. The most unusual aspect of 1 in the whole numbers is that it has a multiplicative inverse that is also an integer. (A multiplicative inverse of the number x is a number that when multiplied by x gives 1. The number 2 has a multiplicative inverse in the set of the rational or real numbers, 1/2: 1/2×2=1, but 1/2 is not an integer.) The number 1 happens to be its own multiplicative inverse. No other positive integer has a multiplicative inverse within the set of integers.* The property of having a multiplicative inverse is called being a unit. The number -1 is also a unit within the set of integers: again, it is its own multiplicative inverse. We don’t consider units to be either prime or composite because you can multiply them by certain other units without changing much. We can then think of the number -2 as not so different from 2; from the point of view of multiplication, -2 is just 2 times a unit. If 2 is prime, -2 should be as well.

*This sentence was edited after publication to clarify that no other positive integer has a multiplicative inverse that is also an integer. — * reference above

What does it mean? — fishfry

"2+2=green" means that whatever is represented by "2+2" is equal with whatever is represented by "green". Isn't that the way we use logic? We learn to apply the rules without regard for what the particular symbols represent.

When I say that 2 + 2 = 4 has meaning, it's because I have defined '2', '4', '=', and '+' according to the standard mathematical conventions, either within the Peano axioms or ZF set theory. In other words from my viewpoint 2 + 2 and 4 and '=' all refer to something. The somethings that they refer to are abstract mathematical objects. And I will stipulate that when you challenged me to define exactly what I mean by those, I was stuck. I admit that! But at least by saying what these expressions refer to (in my mathematical ontology), I can thereby assign meaning and value to them. The meaning and value of these expressions derive from the referents I have assigned to them. — fishfry

You are missing something in your interpretation of "2+2=4". The "+" signifies an operation, not an object. Do you understand that an operation, as an action, is something other than an object?

But you say that 2 + 2 and 4 don't refer to anything. So it is now incumbent on you -- not just for me, but for working out your own thoughts for yourself -- to figure out how to define the meaning and value of syntax tokens that you claim don't refer to anything at all! Do you take my point here? — fishfry

Sure, I see your point. It's not difficult, the task you ask of me; "2" signifies a quantity, "+" signifies an operation of addition, "=" signifies 'has the same quantitative value as', and "4" signifies a quantity. So, "2+2=4" signifies that a quantity of two, added to another quantity of two, through that operation of addition, has the same quantitative value as the quantity of four. See how easy it is? Grade school stuff.

There is nothing simple about your point of view. Nor have you explained "what '=' signifies" in the least. I haven't seen you do it. — fishfry

Come on fishfry I've said over and over again that "=" signifies having the same value. I even quoted Wikipedia in the last post:: "In an equation, it is placed between two expressions that have the same value, or for which one studies the conditions under which they have the same value." Now don't come off saying that I haven't explained what "=" signifies. This is how I ended the last reply to you:

Do you accept that there is a difference between "is the same as" and "has the same value as"? The former phrase is the phrase used by the law of identity. The latter phrase is what is signified by "=", as the Wikipedia entry indicates. — Metaphysician Undercover

It's funny. You can't answer the question I put to you: If 2 + 2 has no referent, how does it obtain its meaning or value? — fishfry

That question is simple too. A word can derive its meaning through examples, like "green", without referring to any particular thing. It can derive meaning from a definition, like "square", and "circle" do, without referring to any particular thing. And there is a number of other ways, by which usage hands meaning to a word, which does not refer to any particular thing. In this case, we use "2" to signify a quantity, and "+" to refer to the operation of addition, and the symbols derive their meaning from that usage.

But I have a perfect understanding of what the meaning and value of 2 + 2 are. — fishfry

Clearly you do not have a "perfect understanding of the meaning of "2+2", because your interpretation does not include the operation of addition, which is signified by "+". You cannot simply leave out the meaning of some symbols in the phrase, then claim to have a perfect understanding of the phrase.

I DO have a crystal clear understanding of how the meaning and value of 2 + 2 derive from the mathematical REFERENT of the expression. Whereas you DENY there is a referent, so you are STUCK trying to figure out how to derive the expression's meaning and value. Why don't you work on this and let me know if you have any fresh ideas on the matter. — fishfry

OK, if you're so convince that you are correct in your crystal clear understanding, interpret the expression for me, "2+2", symbol by symbol, and show me how that expression signifies the object signified by "4".

Agreed on this point. But note that I can define what the value of 2 + 2 is, and you can't. Because you deny that 2 + 2 has any referent. — fishfry

Tell me please, in your mind, how is a value an object?

But you deny the expressions have any referents at all, so I don't see how you're in a position to claim that they have the same value, or different values, or any values at all. How can we know their values if they have no referents? — fishfry

A value is not a thing, or object, it is what a mind assigns to a thing, as a property, just like "big", "heavy", "green", etc. So a value is the product of a judgement. There is no referent because we assign the same property to multiple things, due to the abstract nature of properties. And, we assign the same value to multiple things, so there cannot be an object as a referent. "Green" doesn't refer to any particular thing, because many things are green, so there is no referent for "green". It is a judgement we make.

It is a similar situation with "2", we assign that value to many different situations, as the property of them, but it has no particular referent. We can start with, 'what a thing is worth' as a defining feature of "value", and see that a value exists in relation to a purpose. A thing is worth something only to the extent that it is desirable for some purpose, useful toward some goal or something like that. So we know the value of "2" by its usefulness. That is what the judgement is based in.

I, on the other hand, have a perfectly sensible way to define their values, based on the referents I have assigned them in PA or ZF. I can do this from first principles. — fishfry

One big problem here, your interpretation leaves out the operation signified by "+". And, it is by means of these various operations that the numerals obtain their signified values. They are useful for these operations. So your way, is really not at all sensible, because you completely neglect the operations by which the numerals get the values which are associated with them.

Not only that, but your way creates an unnecessary layer of separation between the numeral and the represented value, which is commonly called a number. This medium, or separation obscures the true meaning, and value represented by the numeral, making it much more difficult to understand the nature of quantities.

* You claim 2 + 2 has no referent, and since it has no referent, you can't tell me how to determine its value. — fishfry

You have obviously misunderstood. I have no problem telling you how we determine the value of "2+2". We simply look at how the symbols are used, just like when we determine the meaning of "green". That's why I said it's a matter for grade school, which you took as an insult. We see that in common use 2+2=4, so 2+2 clearly has the same value as 4, that's what the "=" tells us.

What I can't get past is that physicist have used General Relativity to derive a size for the universe, and pretty much agree on the result; in doing so they relied on the relativistic versions of the equations you refer to.

And yet, without showing us the calculations, you insist that they are wrong.

I don't think there is more to say here. That the velocity of light is a constant, fixed for all observers, is fundamental to physics. — Banno

I explained this to you already. What results from the application of general relativity, is the conclusion that space is expanding. This separation of things which is accounted for by the concept of spatial expansion is a type of motion of things relative to each other, which does not qualify as "motion" within the precepts of the general theory of relativity. Therefore we can conclude that there are motions of material things in the universe, to which general relativity is not applicable. This is regardless of whether the principles of relativity theory, special or general, are fundamental to physics.

Now, Gary would prefer not to apply the general theory of relativity, and therefore avoid the conclusion that space is expanding. That's a valid starting point. From this perspective we can take all the observed motions, and class them together, and see that there is good reason not to apply the principles of relativity, as they are inadequate. So we ought to seek a better theory which can account for all the motions in the universe as "motion".

I understood that; I thought you meant that you do want to take '2' and '3' as representing a type of unity, while you think that that is contradicted by 'the numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc" so that it needs correction .

Am I not correct that that is your view? — TonesInDeepFreeze

That's pretty close, except I do not necessarily want to take "2" and "3" as representing unities, that's why I said "if" we want to. I see the numeral as representing a group with a specific number of things in that group, but the unity of that group is questionable.

.

More basically, I don't know why one would fret over any of this, since I don't know anyone who claims "the numeral "1" represents a basic unity. an individual. — TonesInDeepFreeze

I find that very strange I hear them used that way all the time. I suppose I didn't explain very well. Isn't this how we count? One represents one unit, two represents two of those, etc.. If you're put off by the terminology, "unity", "represent", etc., that's understandable, but why don't you just relax and enjoy the simplicity of the terms. It seems to me like there's always some people who get really flustered, and then have difficulty understanding simple terms, as soon as you mention any sort of problems within the systems of mathematics.

In sum, I can't make sense of what you're trying to say. — TonesInDeepFreeze

Yes, I can see that. You haven't really ever thought about such fundamental issues as how we use numerals, and you don't really understand why anyone else would. Why did you engage me, if what I was saying appeared so foreign to you?

Suggestion: You could reference some actual piece of mathematical or philosophical writing that you disagree with and show how you think you can correct it. — TonesInDeepFreeze

I've addressed particular pieces of mathematical writing which I disagree with before, in the past, but I cannot think of any way to correct these issues. So people have told me that if I don't have a solution, then don't point out a problem. But I think that's nonsense. I think we have to find the problems, and get a good clear understanding of why and how they are problems, before we can move toward an adequate solution. Solutions don't come easily, they require a thorough understanding of the problems.

Yes, as I thought, you find that there is a problem with the notion (whatever it means) that 'the numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc". — TonesInDeepFreeze

No, I see no problem with that in itself. The problem is when we want to say that, and also that "2" and "3" represent a type of unity.

But (aside from even trying to parse the broken phrases) I don't know who says anything along the lines of 'the numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc". So I don't see why you think it is a problem that needs to be addressed. — TonesInDeepFreeze

I don't know about you, but I always use "1", "2", and "3" in that way. If you don't ever talk about 1 chair, 2 or 3, or any number of other things like that, then I guess you don't use them the same way. But if someone asks you how old you are, do you answer with a number?

I thought you meant that there is a fundamental problem with:

"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."

And that your supposed solution to the supposed problem is:

"[...] we have to allow that "1" represents a different type of unity than "2" does [...]" — TonesInDeepFreeze

There is no proposed solution. The issue was stated as a fundamental problem with numbers, without a solution.

Or perhaps you would make clear which parts of your passage are ones you are critiquing and which parts are ones you are claiming. — TonesInDeepFreeze

I am not critiquing anything, the whole thing is what I am claiming. I am claiming that there is a fundamental problem with numbers. If "1", "2", "3", etc. , are used to represent unities, then "2" and "3" must represent a different type of unity from "1", for the reason I explained.

Now here is a proposal for a solution. If "2" and "3" are said to represent numbers, then maybe we ought to say that "1" represents something other than a number.

Perhaps consider the most intensely rewarding experiences of human life. They are experienced as intensely significant by their very nature. — David Pearce

Let's take an example then, competition. Winning a competition is one of the most intensely rewarding experiences for some people. Even just as a spectator of a sport, having your team win provides a very rewarding experience. But we can't always win, and losing is very disappointing. How do you think it's possible to maintain that intensely rewarding experience, which comes from success, without the possibility of disappointment from failure? It seems like a large part of the rewarding feeling is dependent on the possibility of failure. We can't have everyone winning all the time because there must be losers. And there would be no rewarding experience from success, without the possibility of failure. How could there be if success was already guaranteed?