Does “1” refer to an object called “a number”? — Luke

No, of course not. That is the mistaken description of how "1" is used which I am trying to expose. If "1" referred to an object called a number, then we could not use "2" to refer to two distinct objects.

That is relevant to this thread because if someone takes it as the basis for an argument about infinity and the infinite, that 1 refers to a mathematical object called a number, it's a false premise. To assume that it does, for the sake of saying something about infinity, is to base what you are saying about infinity in a falsity.


We may disagree on what "the objects studied by metaphysics" refers to. Metaphysics must study all objects in order to have the capacity to distinguish objects existing only within the mind (imaginary and fictional objects) from objects which exist independent of an observer.

The problem is that the existence of each of these types, as a proposed type of object, is difficult to validate, substantiate, or ground, in real principles, without reference to objects of the other type. So there's generally a type of circularity involved in substantiating "existence". That's one reason why dualism has been a prominent metaphysics in the past. My position is that we cannot take the existence of objects, of any type, for granted. Therefore the concept of "object" wherever it is used, must be justified.

Does “1” refer to an object called “a number”?
— Luke

No, of course not. — Metaphysician Undercover

So "1" is not a number? Or, is it specifically that "1" is not an object called "a number"? Who is claiming that "1" is an object?

So "1" is not a number? — Luke

The point is that there is no need to posit "a number" as existing between the symbol "1", and what the symbol refers to in a particular application of mathematics. This is unnecessary obfuscation, like positing "an idea" as existing between my car, and my usage of "car" to refer to it. To posit such an ideal object as existing between the symbol, and what the symbol refers to in application, is a misrepresentation of what is really the case. There is some thinking which occurs as a medium between my use of the word, and the object referred to, but it's incorrect to say that there is a thing, called "an idea" which exists there.

Who is claiming that "1" is an object? — Luke

From your Wikipedia post above:

In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as π or an integer such as 42.

— The value of a variable or a constant is any number or other mathematical object assigned to it.
— The value of a mathematical expression is the result of the computation described by this expression when the variables and constants in it are assigned values.
— The value of a function, given the value(s) assigned to its argument(s), is the value assumed by the function for these argument values.

Once it is established through an axiom, that a number is an object (mathematical object), it becomes a thing which can be counted. In reality, counting numbers is nonsense, as I explained, it's just arbitrary play with symbols.

The point is that there is no need to posit "a number" as existing between the symbol "1", and what the symbol refers to in a particular application of mathematics. — Metaphysician Undercover

Different numerals can represent the same number (or value), such as "4" and "IV". Also, different expressions can represent the same number (or value), such as "2x2" and "1+3". This indicates "a number as existing between the symbol(s)...and what the symbol(s) refer to".

...counting numbers is nonsense... — Metaphysician Undercover

I don't see how you can count (anything) unless you can count numbers.

Different numerals can represent the same number (or value), such as "4" and "IV" — Luke

I have no problem with saying that a numeral represents a value. That is how we establish equality, by giving different things the same value. I do have a problem with saying that a value is an object. Value is something which is predicated. And if we say that value is a predication of the symbol instead of the subject or object which the symbol represents, that is a category mistake. So anytime we say that a numeral represents a value, this is a simplification, and we ought to understand that the value only exists in application. It's like when we state logical expressions using symbols,.to make an example of how to use such symbols. The symbols only have a "value" if they represent something, and the logic is being applied.

Also, different expressions can represent the same number (or value), such as "2x2" and "1+3". — Luke

So in this instance, when "2+2" is applied, and "3+1" is applied, there is an equality between the value of the objects represented by both. Without those objects, which only exist in application, when the numbers actually refer to something, "2+2" and "3+1" are just symbols which cannot be said to have any particular value, or refer to any particular value.

This indicates "a number as existing between the symbol(s)...and what the symbol(s) refer to". — Luke

No such "number" is indicated. "2+2=3+1" is just an expression of symbols, demonstrating how to use mathematical symbols, just like we make demonstrations in other forms of logic, in which the symbols refer to nothing. It is an expression of "form}, which demonstrates how to proceed with the logic. There is nothing referred to by the symbols because the logic is not being applied. It is simply a representation of form. To imagine that there is an object called "a number" referred to by "2", or "3", or another type of mathematical object referred to by "2+2", or "3+1" is just an imaginary fiction, which might be useful for the purpose of teaching, but it is a fiction nevertheless.

I have no problem with saying that a numeral represents a value. That is how we establish equality, by giving different things the same value. — Metaphysician Undercover

Then you must concede that there exists an intermediary between a symbol (numeral) and an object: a value. A value is a number.

Numerals represent numbers which are predicated of objects. But a numeral or a number can also be an object. We can speak of three numerals or four numbers, for example.

"2+2" and "3+1" are just symbols which cannot be said to have any particular value, or refer to any particular value. — Metaphysician Undercover

Both expressions have a value of 4. A child could tell you that.

No such "number" is indicated. "2+2=3+1" is just an expression of symbols — Metaphysician Undercover

Forget those expressions, then. You have accepted that the symbols "4" and "IV" both represent a value of four, and a value is neither an object nor a symbol; it is a number. A number is an abstract concept.

To imagine that there is an object called "a number" referred to by "2", or "3", or another type of mathematical object referred to by "2+2", or "3+1" is just an imaginary fiction — Metaphysician Undercover

You still need to explain how you can count objects without first being able to count numbers.

Then you must concede that there exists an intermediary between a symbol (numeral) and an object: a value. A value is a number.

No, I just explain why this is not the case. Why just go and assert it anyway?
— Luke

Numerals represent numbers which are predicated of objects. — Luke

Hold on here, you're jumping ahead of yourself. An object is one, so you cannot predicate any number other than one of an object. If you have a group or set of objects you can count them, assign a quantity or number to that group or set, but take notice that number, or quantity is actually predicated of the group or set, not of the objects themselves.

But a numeral or a number can also be an object. We can speak of three numerals or four numbers, for example. — Luke

Clearly a numeral is an object, as a symbol. But I do not see how a number can be an object. Number, or quantity is something predicated of a group or set of objects, so how can a number itself be an object?

Both expressions have a value of 4. A child could tell you that. — Luke

This is the ambiguity you tried to introduce earlier. The expression isn't what has the value, it's what the expression refers to that has the value. The numeral "4" does not have the value, of 4, Whatever it is that we refer to with "4", in application, is what is judged to have that value. So "4" is used to refer to that group of objects which is judged to have the value of 4.

You still need to explain how you can count objects without first being able to count numbers. — Luke

We went through this already. 1 refers to one object, add another and it's "2" objects, another, and it's "3" objects. It's how I learned to count, I don't know how you learned to count. Memorizing an order of symbols, 1,2,3,4, etc. is not learning how to count anything, just like learning to recite the alphabet is not learning how to spell anything.

We do say that reciting the numbers in order is counting, but when we learn how to do this it's definitely not a matter of learning how to count numbers, because at that age we are not told that there are abstract numbers, Platonic ideas which the numerals refer to. It's just learning how to count, and this is a distinct meaning of "count" from counting something, which is to determine an amount. Clearly, when we learn how to recite the numerals in order, i.e., learn how to count in this sense of the word "count", we are not told that we are counting (determining the amount of) some abstract ideas called "numbers". This would only confuse the child. In my experience, I learned how to count objects, and recite the symbols, long before I learned that there was supposed to be numbers which the numerals refer to. That would have confused me immensely. It still does.

Hold on here, you're jumping ahead of yourself. An object is one, so you cannot predicate any number other than one of an object. If you have a group or set of objects you can count them, assign a quantity or number to that group or set, but take notice that number, or quantity is actually predicated of the group or set, not of the objects themselves. — Metaphysician Undercover

First you claim that there is no intermediary between symbols and objects, but now you claim that there are both numbers and sets between them? Make up your mind.

Clearly a numeral is an object, as a symbol. But I do not see how a number can be an object. Number, or quantity is something predicated of a group or set of objects, so how can a number itself be an object? — Metaphysician Undercover

Yes, I expressed this poorly. I simply meant that numbers can also be predicated of numerals and numbers themselves.

The expression isn't what has the value, it's what the expression refers to that has the value. — Metaphysician Undercover

Right, the expression [e.g. “2+2”] is a set of symbols. I was drawing a parallel between this and numerals.

The numeral "4" does not have the value, of 4, Whatever it is that we refer to with "4", in application, is what is judged to have that value. So "4" is used to refer to that group of objects which is judged to have the value of 4. — Metaphysician Undercover

Exactly my point. So you need to review your claim that “ there is no need to posit "a number" as existing between the symbol "1", and what the symbol refers to”. “4” refers to neither the symbol nor the objects themselves, but instead to an abstract feature/grouping of those objects: a number.

First you claim that there is no intermediary between symbols and objects, but now you claim that there are both numbers and sets between them? Make up your mind. — Luke

You obviously have not understood what I said. Oh well, I'm tired of repeating the same thing over and over, only to have it interpreted in some odd way such that you perceive contradiction. I've already pointed to this problem which you have, and it would be appreciated if you could work on correcting it.

I simply meant that numbers can also be predicated of numerals and numbers themselves. — Luke

As I said last post, i don't see how a number could be predicated of a numeral. The symbol, and what the symbol mean, are two distinct things. You could only predicate a number of a numeral if the number was somehow a quality or property of the symbol.

Exactly my point. So you need to review your claim that “ there is no need to posit "a number" as existing between the symbol "1", and what the symbol refers to”. “4” refers to neither the symbol nor the objects themselves, but instead to an abstract feature/grouping of those objects: a number. — Luke

A group of objects is not an abstract feature. Patterns are real, ontological. The value we give to the group "4", is an abstract feature, but it's a value, therefore a form of quality, not an object.. If you allow the grouping to be arbitrary, mathematics gets lost to randomness. This is why there must be a real ontological difference between two groups of two (2+2), and one group of four (4). Otherwise there would be no reason to refer to the one situation as "2+2", and the other as "4".

...there is no need to posit "a number" as existing between the symbol "1", and what the symbol refers to...

The value we give to the group "4", is an abstract feature, but it's a value — Metaphysician Undercover

A value is a number. Do you acknowledge that?

Given your two claims above, it looks like you now accept that the “abstract feature” of a value/number exists between the symbol and what the symbol refers to. Otherwise you must think that a value/number is a symbol, or that an object is an “abstract feature”.

A value is a number. Do you acknowledge that? — Luke

No! That's exactly what I've been arguing against for this entire thread. How can you not see that that is what I am arguing against, after all this time? There is a specific type of value, a quantitative value, which people have assigned the word "number" to. You have been working hard to disassociate "quantitative value" from all other forms of value, with the intention of claiming that quantitative value is grounded in an object called "a number", rather than the human subject. That argument disguises the true subjective nature of "a value" by making it appear to be an object called "a number".

Given your two claims above, it looks like you now accept that the “abstract feature” of a value/number exists between the symbol and what the symbol refers to. — Luke

I don't at all claim that there is nothing between the symbol and what it represents, that would be ludicrous, and that's why it's ridiculous and extremely frustrating that you would misrepresent what I said, in this way. Of course there is obviously a medium, which is a thinking human being, between the symbol and what it represents, as it is necessary that someone applies the symbol. But a thinking human being is better known as a subject, and is obviously not well represented as an object called "a number".

Your interpretation has taken what I've said about human thinkers applying numerals, and you have misrepresented as an instance of "value" and misrepresented a "value" as a number. So you have made a double misrepresentation.

That's the problem with this sort of Platonism, it takes the human activity of thinking, which is the medium between the numerals and what they refer to, and replaces that with Platonic ideals, "numbers". From this premise you can completely overlook all the mistakes within the principles of mathematics, insisting that there cannot be mistakes because mathematics is objective, the numerals refer directly to mathematical objects, numbers. Therefore it's impossible that there is mistaken value here because that sort of value is based in objects, "numbers", so it is objective. In reality there is the human thinking between the numeral and what it refers to, not an object called "a number", hence mistake is possible within mathematical principles. And that sort of Platonic realism is itself a mistake.

There is a specific type of value, a quantitative value, which people have assigned the word "number" to. You have been working hard to disassociate "quantitative value" from all other forms of value — Metaphysician Undercover

They are different meanings of the word "value", as demonstrated by the Wikipedia article I posted. If you can't accept this, then I wish you well.

They are different meanings of the word "value", as demonstrated by the Wikipedia article I posted. If you can't accept this, then I wish you well. — Luke

My reply to your Wikipedia quote:

This is what I object to. In no way can a value be an object. — Metaphysician Undercover

That was four days ago. And, it's what I've been arguing for weeks. Did it take you this long to figure out that I really. mean what I say?

This is what I object to. In no way can a value be an object. — Metaphysician Undercover

That was four days ago. And, it's what I've been arguing for weeks. Did it take you this long to figure out that I really. mean what I say? — Metaphysician Undercover

What does any of that have to do with numbers?

I assume you saw the phrase "mathematical object" in the Wikipedia article on value and now you want to argue over the meaning of "objects". No, thanks. Look it up: "A mathematical object is an abstract concept arising in mathematics."

assume you saw the phrase "mathematical object" in the Wikipedia article on value and now you want to argue over the meaning of "objects". — Luke

Clearly our disagreement is not in the meaning of "objects", but in the meaning of "value". You want to disassociate quantitative value from all other sorts of value, claiming that mathematical values are something completely distinct and unrelated to any other type of value. But values do not exist in that way, They exist in hierarchical structures, one type of value receiving its worth from another, like a family tree, until the whole structure is grounded in a material desire or want. Aristotle explained this in his Nichomachean Ethics, one end is for the sake of another end, which is for the sake of another, until there is a grounding. Unless you recognize that values are tied together in this way and it is unrealistic, and a misunderstanding, to separate one type of value (quantitative value) from all others, we will always disagree.

A mathematical object is an abstract concept arising in mathematics." — Luke

Sure, but this does not settle our disagreement concerning the relationship between "value" and "abstract concept". You seem to think that value is a predicate of an abstract concept. I see a concept as grounded in a value, and emergent from that value. Therefore from my perspective the value that a concept expresses is prior to the concept itself. And if there is a new value derived from a concept, it is produced through application of the concept. This means that the value which "mathematical objects", or 'abstract concepts arising in mathematics' is grounded in cannot be a quantitative value at all, because "quantitative value is an abstract concept emergent from the application of arithmetic. Quantitative value has been produced from the application of mathematical concepts, and the sort of "value" which is responsible for the creation of mathematical concepts, and therefore underlying mathematical concepts, is a different type.

Mathematical concepts provide us with quantitative value, but they are produced from another type of value. So we cannot understand mathematical concepts simply through reference to quantitative value because this is circular. To escape this vicious circle which you have been trapped in for weeks now, you need to allow your inquiry to accept the values which lie behind an individual subject's application of abstract concepts, to be relevant in the creation of these concepts. In other words, you need to free yourself from your self-imposed restrictions on "value".

Clearly our disagreement is not in the meaning of "objects", but in the meaning of "value". You want to disassociate quantitative value from all other sorts of value, claiming that mathematical values are something completely distinct and unrelated to any other type of value. But values do not exist in that way, They exist in hierarchical structures, one type of value receiving its worth from another, like a family tree, until the whole structure is grounded in a material desire or want. Aristotle explained this in his Nichomachean Ethics, one end is for the sake of another end, which is for the sake of another, until there is a grounding. Unless you recognize that values are tied together in this way and it is unrealistic, and a misunderstanding, to separate one type of value (quantitative value) from all others, we will always disagree. — Metaphysician Undercover

For god sake, man. There is a meaning of the word "value" which is a synonym for "number". I'm not talking about a type of value, as in the values that people hold or in what people value. It's just another word for a number, or the number represented by an algebraic term. That's it. It has absolutely nothing to do with any other meaning of "value". You can't accept that? Fine. I don't care.

There is a meaning of the word "value" which is a synonym for "number". — Luke

Sure there is, and that's your vicious circle, which I just pointed out. Define "value" as number, and define "number" as value. Claim that they are synonyms and live in your little bubble withou having a clue as to what a value or a number is. If you read my reply to jorndoe, a week or so back, I explained how a value is defined by a scale. The number just indicates a position relative to the scale.

I'm not talking about a type of value, as in the values that people hold or in what people value. — Luke

This is incorrect. Any specified value is defined by a specific scale. So there are many different types of value, all relative to different scales. Therefore any value is a type of value and the type is determined by the scale. There is no such thing as a value which is independent from a scale of evaluation, and the scale determines the type of value. So unless you are talking about "value" in the most general sense, which you clearly are not, because you've been rejecting my talk of "value" in the most general sense, you are necessarily talking about a type of value. Therefore your claim that you are not talking about a type of value is false. You very clearly are talking about a type of value.

It's just another word for a number, or the number represented by an algebraic term. — Luke

I already explained how the idea of "number" on its own, as a supposed medium between a numeral and things to be counted, does not make sense. When we count things, we apply the numerals right to the things being counted, in the process of counting. If you want to ground the existence of "number", in some type of value, then we need to refer to the evaluation scale to see what type of value it is. I already suggested that it is a quantitative value. Do you agree?

Therefore any value is a type of value and the type is determined by the scale. There is no such thing as a value which is independent from a scale of evaluation, and the scale determines the type of value. — Metaphysician Undercover

It's as though I am talking about the bank of a river and you keep telling me that I must be talking about a financial institution.

So unless you are talking about "value" in the most general sense, — Metaphysician Undercover

What is ""value" in the most general sense"? A word can have more than one meaning. What basis is there for assuming that the word "value" can have only one meaning? I mean, just look in the dictionary.

It's as though I am talking about the bank of a river and you keep telling me that I must be talking about a financial institution. — Luke

I don't see the analogy. You appear to be avoiding the points I made.

What basis is there for assuming that the word "value" can have only one meaning?

I never suggested that it did. I said there are all different types of values. Obviously, each different type entails a different meaning for the word.

What basis is there for claiming that there is a meaning for "value" which refers to something completely independent from all other types of value, such that it cannot be called a type of value?

I said there are all different types of values. — Metaphysician Undercover

This implies the same meaning of the word "value" across all "types of values".

What basis is there for claiming that there is a meaning for "value" which refers to something completely independent from all other types of value — Metaphysician Undercover

The dictionary for one thing. My knowledge of different meanings/uses of the word "value" for another.

Google offers these different meanings of the word "value":

1.
the regard that something is held to deserve; the importance, worth, or usefulness of something.
"your support is of great value"

the material or monetary worth of something.
"prints seldom rise in value"

face value
the worth of something compared to the price paid or asked for it.
"at £12.50 the book is good value"

2.
principles or standards of behaviour; one's judgement of what is important in life.
"they internalize their parents' rules and values"

3.
the numerical amount denoted by an algebraic term; a magnitude, quantity, or number.
"the mean value of x"

4.
Music
the relative duration of the sound signified by a note.

5.
Linguistics
the meaning of a word or other linguistic unit.

the quality or tone of a spoken sound; the sound represented by a letter.

6.
the relative degree of lightness or darkness of a particular colour.

These aren't different "types of values"; they are different meanings of the word "value". Note that not all of these are synonymous with "the desirability of a thing".

This implies the same meaning of word "value" across all "types of values". — Luke

That's nonsense, meaning is relative to context, usage. For example, there are different types of animals. When I talk about this animal here, my cat, "animal" has a completely different meaning from when I talk about that animal over there, my dog. The fact that all the things called "animal" can be classed together in one group, as animals, does not mean that whenever someone refers to one of those animals, "animal" has the same meaning. I am talking about this animal here now, my cat, do you see how the meaning of "animal" is completely different from when I am talking about that animal over there, my dog.

We can say the same thing about "value". There is a reason, or reasons why we categorize something as a value, just like there are reasons why we categorize something as an animal. I suggested, that a "value" is related to a scale, But this does not mean that when I talk about the value of a dollar, or the value of zero degrees Celsius, "value" has the same meaning. That would be ridiculous. The meaning is determined by the scale being referred to, just like the meaning of "animal" in my example, is determined by the creature being referred to.

When I talk about this animal here, my cat, "animal" has a completely different meaning from when I talk about that animal over there, my dog. The fact that all the things called "animal" can be classed together in on group, as animals, does not mean that whenever someone refers to one of those animals, "animal" has the same meaning. I am talking about this animal here now, my cat, do you see how the meaning of "animal" is completely different from when I am talking about that animal over there, my dog. — Metaphysician Undercover

These are the same meaning of the word "animal", with a definition such as: "a living organism that feeds on organic matter, typically having specialized sense organs and nervous system and able to respond rapidly to stimuli."

I suggested, that a "value" is related to a scale, But this does not mean that when I talk about the value of a dollar, or the value of zero degrees Celsius, "value" has the same meaning. That would be ridiculous. The meaning is determined by the scale being referred to, just like the meaning of "animal" in my example, is determined by the creature being referred to. — Metaphysician Undercover

It's not the same. Your example of "animal" uses the same definition and has the same meaning whether it's a cat or a dog. The Google definitions I provided for "value" are not the same and do not all have the same meaning.

These are the same meaning of the word "animal", with a definition such as: "a living organism that feeds on organic matter, typically having specialized sense organs and nervous system and able to respond rapidly to stimuli." — Luke

In this discussion, all we've done is digressed, from the meaning of "number" to the meaning of "value", to the meaning of "meaning". I now think I see why it's so difficult for us to agree on anything, or make any progress in discussion, and all we do is digress.. We have a very deep difference in how we relate to what a word means.

If I'm talking about that animal lying on the floor over there, I believe that "animal" refers to the dog I am talking about, and that reference gives meaning to the use of "animal" in that context. You seem to think that when I am talking about that animal, there is some abstract idea, defined as you described above, which exists as an intermediary between the word, and my use of the word, constituting "the meaning" of the word. This is exactly what you've been arguing with numerals, that there is an abstract idea, "a number", which constitutes "the meaning" of the numeral, and is intermediate between the numeral and its usage. I thought you supported Wittgenstein, who dismissed all that Platonic idealism as nonsense.

Let's consider what really exists between the word "animal" and the way that I use it. I have had some education, and have developed some habits of usage, and I might refer to a dictionary or other sources like Wikipedia now and then. There is no specific definition, such as the one you offered, which constitutes "the meaning" of the word for me, which I reference every time I use that word. Sure, you might argue that every time I use the word "animal", it is consistent with your proposed definition, therefore your proposed definition is "the meaning" which the word has when I use it, but that is not a valid conclusion. That a thing is consistent with a description does not necessitate the conclusion that it is the described thing because the identity of the particular is within the particular itself (law of identity), and cannot be represented through universal terms which refer to more than one thing. Therefore the identity of the particular cannot be concluded necessarily through reference to the universal, or more general. The meaning of any particular instance of usage of symbols is specific to that particular instance, and cannot be expressed in universal terms. We must refer to the material substance of what is referred to in that particular instance of use, to determine the true meaning.

I believe that this attitude which you have toward meaning is the reason why you have such a hard time discussing these philosophical issues with me, and continually misinterpret me. You see my use of certain words, and instead of referring to the context of my usage, to derive the meaning of those words, you refer to some "idea" of "the meaning" of the word, which you have for that word, and this provides for you, the wrong interpretation, or meaning. Then you take this incorrect meaning (faulty interpretation) based in some supposed idea of "the meaning" of that word, instead of the meaning meant by me, revealed by the context of my usage, and insist that I've contradicted myself.

In this discussion, all we've done is digressed, from the meaning of "number" to the meaning of "value", to the meaning of "meaning". — Metaphysician Undercover

That's probably because you tried to argue that the word "value" has only one meaning, then I provided several other different dictionary definitions, and then you tried to change the subject.

I'm done with your twisting of the discussion. You're wrong.

That's probably because you tried to argue that the word "value" has only one meaning, — Luke

Another example of pathetic interpretation.

I'm done with your twisting of the discussion. — Luke

It's not me who never learned how to read philosophy.