Well, the square root operation is closed over real numbers i.e. a square root of a real number has to be a real number — TheMadFool

You've got to be kidding. Think about this statement for, say, five seconds. :roll:

You've got to be kidding. Think about this statement for, say, five seconds. :roll: — jgill

Have I made a boo boo, a big one at that? Expand and explain.

What's the square root of the real number -4?

What's the square root of the real number -4? — jgill

I love Scott Aaronson's comments on this:

Why did God go with the complex numbers and not the real numbers?

Years ago, at Berkeley, I was hanging out with some math grad students -- I fell in with the wrong crowd -- and I asked them that exact question. The mathematicians just snickered. "Give us a break -- the complex numbers are algebraically closed!" To them it wasn't a mystery at all.

But to me it is sort of strange. I mean, complex numbers were seen for centuries as fictitious entities that human beings made up, in order that every quadratic equation should have a root. (That's why we talk about their "imaginary" parts.) So why should Nature, at its most fundamental level, run on something that we invented for our convenience?

Answer: Well, if you want every unitary operation to have a square root, then you have to go to the complex numbers...

Scott: Dammit, you're getting ahead of me!

Alright, yeah: suppose we require that, for every linear transformation U that we can apply to a state, there must be another transformation V such that V^2 = U. This is basically a continuity assumption: we're saying that, if it makes sense to apply an operation for one second, then it ought to make sense to apply that same operation for only half a second. — PHYS771 Lecture 9: Quantum - Scott Aaronson

This is why I think the number 3 can exist but not the 'number' sqrt(2). We never actually work with irrational 'numbers', we only work with their algorithms or rational number approximations. So why do we even need to assume that irrational 'numbers' exist? Why not assume that irrationals are the algorithms that we actually work with? — Ryan O'Connor

Isn't this the difference between an object and a process? We'd say "3" represents a static object, a number, but "sqrt(2)" represents an operation. What would you say about "2+1"? Doesn't that represent an operation rather than an object? The difference between "2+1", and "sqrt(2)", is that the one process adequately resolves to an object. The question I see is what does it mean for a process to resolve to an object, and why does this make the process somehow more valid as a process? So, we say "2+1=3", and we are stipulating an equivalence between the process and the object. But we cannot produce the precise object which "sqrt(2)" is equivalent to.

What validates, or grounds numbers definitionally, is quantitative value "2+1" is equivalent to a definite quantitative value represented by "3". Having a definite quantitative value is what makes the number an object. If we do as you propose, and allow processes which do not have a definite quantitative value, to be "worked with", then we allow indefiniteness into our solutions. The solutions will contain indefinite quantitative values. This is counterproductive because the goal when using mathematics is to measure things, which is to assign to them definite quantitative values.

sqrt(2)" represents an operation — Metaphysician Undercover

sqrt is an operation. sqrt(2) is the object that is the result of the operation applied to the object 2. sqrt is the operation, and 2 is the argument to which the operation is applied.

+ is an operation. 2+1 is the object that is the result of the operation applied to the objects 2 and 1. + is the operation and 2 and 1 are the arguments to which the operation is applied.

what does it mean for a process to resolve to an object — Metaphysician Undercover

Operations are functions. A function is a certain kind of ordered pair. The result of an operation is simply the unique second coordinate for the ordered pair whose first coordinate is the argument.

we cannot produce the precise object which "sqrt(2)" is equivalent to — Metaphysician Undercover

We cannot finitely list the decimal expansion of sqrt(2). But sqrt(2) is a particular object. Also, what is important in this regard is not some object that the sqrt(2) is "equivalent to" (with whatever equivalence relation might be in mind) but rather with sqrt(2) itself.

quantitative value "2+1" is equivalent to a definite quantitative value represented by "3" — Metaphysician Undercover

Your use-mention is inconsistent there. Yes, '3' represents a value. But so also does '2+1'.

3 and 2+1 are values. They are the same value. Exactly the same. 3 = 2+1. 3 equals 2+1. 'equals' is another word for 'identical with'.

'3' and '2+1' are names that represent values. '3' and '2+1' are not equal. They are different names. But they name the same object. They are two different names for the same object.

Look up the subject of 'use-mention'.

Having a definite quantitative value is what makes the number an object — Metaphysician Undercover

A number is an object. If it's not an object, then what is it? If it is something that, according to you, might or might not be an object, then what is that something to begin with if not an object? How can we refer to something that is not an object?

the goal when using mathematics is to measure things — Metaphysician Undercover

Mathematics may be used for purposes other than measuring things.

which is to assign to them definite quantitative values — Metaphysician Undercover

We do mention a definite value when we mention sqrt(2). It doesn't have a finite decimal expansion, but it is a definite value.

sqrt is an operation. sqrt(2) is the object that is the result of the operation applied to the object 2. sqrt is the operation, and 2 is the argument to which the operation is applied. — GrandMinnow

No, if "sqrt" represents an operation, then "sqrt(2)" represents that operation with a qualifier "(2)".

Your us-mention is inconsistent there. Yes, '3' represents a value. But so also does '2+1'. — GrandMinnow

This is incorrect, because "+" represents an operation. So there are two distinct values, "2", and "1" represented, in "2+1", along with the operation represented by "+".

'equals' is another word for 'identical with'. — GrandMinnow

No it isn't, that's a false assumption which I've discussed on many threads. You and I are equal, as human beings, but we are in no way identical with each other. "Equals" is clearly not another word for identical with.

A number is an object. If it's not an object, then what is it? If it is something that, according to you, might or might not be an object, then what is that something to begin with if not an object? How can we refer to something that is not an object? — GrandMinnow

Let's start with numerals, which are symbols. Do you agree that a symbol has a meaning, which is not necessarily an object? So there is no need to assume that "2" or "3" represent objects. We'd have to look at how the symbols were being used, the context, to determine whether they represent objects or not. When I say that there are 6 chairs in my dining room, "6" refers to a number, but this is the number of chairs; the chairs are the objects and the number 6 is a predication. The number is not an object, it is something I am saying about the chairs in my dining room, just like when I say "the sky is blue", blue is not an object.

No, if "sqrt" represents an operation, then "sqrt(2)" represents that operation with a qualifier "(2)". — Metaphysician Undercover

I am telling you the terminology and framework of ordinary axiomatic mathematics. 'qualifier' is not the terminology used. Of course, you may set up your own terminology and framework, but the chance that it will make sense for ordinary axiomatic mathematics is slim since you don't know anything about ordinary axiomatic mathematics.

"+" represents an operation. So there are two distinct values, "2", and "1" represented, in "2+1", along with the operation represented by "+". — Metaphysician Undercover

(1) You are still making your use-mention mistake. Yes, '+' represents an operation and '2+1' is a representation of a value, but '2' and '1' are not values, they are representations of values.

Get it straight: The name of the object has quote marks and is not the same as the object. In cases where quote marks are used, we have a term that represents a value. When we mention the value itself, we don't use quote marks.

(2) As I explained, and as you ignored, + is the operation; 2 and 1 are the arguments; and 2+1 is the value of the function for those arguments. Yes, the operation is represented by '+; and the arguments are represented by '2' and '1'. And the value of the operation is represented by '2+1'. And that value is 2+1. Again, you need to learn basic use-mention.

that's a false assumption which I've discussed on many threads — Metaphysician Undercover

Your personal, confused, incoherent and uninformed views in that thread were demolished and shown to be confused, incoherent and uninformed.

You and I are equal, as human beings, but we are in no way identical with each other. "Equals" is clearly not another word for identical with. — Metaphysician Undercover

You are conflating the meaning of the world 'equal' in various other topics, such equality of rights in the law, with the more exact and specific meaning in mathematics. Human equality means that your rights are identical with my rights, and yes, that does not entail that you and I are identical. But in mathematics, which is ordinarily extensional, the equal sign '=' stands for identity. Again, you can use terminology in your own way for your own notions, but the chance that it will make sense for ordinary axiomatic mathematics is slim since you don't know anything about ordinary axiomatic mathematics.

Do you agree that a symbol has a meaning, which is not necessarily an object? — Metaphysician Undercover

Ordinary axiomatic mathematics is extensional. Each n-place operation symbol refers to a function on the domain of the interpretation, and functions are objects. The function might or might not be an object that is a member of the domain, but it is an object in the power set of the Cartesian product on the domain.

there is no need to assume that "2" or "3" represent objects. We'd have to look at how the symbols were being used, the context, to determine whether they represent objects or not. — Metaphysician Undercover

Not in ordinary mathematics where the numerals represent natural numbers. Or, in greater generality, any constant symbols, such as '1' and '2' are either primitive or defined, and in either case they represent members of the domain of interpretation.

As for defined symbols ('1' and '2' are more often defined rather than primitive), it is true that we cannot define the symbol without first proving the existence/uniqueness theorem. That is, we prove that there exists a unique object having a given property, then we define the symbol as standing for that unique object.

So 1 is the unique object that it is equal to the successor of 0. And 2 is the unique object that is equal to the successor of 1.

When I say that there are 6 chairs in my dining room, "6" refers to a number, but this is the number of chairs; the chairs are the objects and the number 6 is a predication. — Metaphysician Undercover

Ordinary mathematics does not view numbers as predictions.

The number is not an object — Metaphysician Undercover

It is the mathematical object that is the number of chairs, and is the number musicians on the album 'Buhaina's Delight', and is the value of the addition function for the arguments 4 and 2 ...

it is something I am saying about the chairs — Metaphysician Undercover

No what you are saying about the chairs is that the set of them has cardinality 6. You're not saying the chairs or the set of them has 6 or is 6. You are saying that the number of them is 6. 6 is not a property of chairs. Rather, a property of a (set of chairs) is that its cardinality is 6.

just like when I say "the sky is blue", blue is not an object. — Metaphysician Undercover

'blue' is an adjective, which is a certain kind of word, which is a linguistic object. It does not alone stand for an object. When we say the sky is blue, we say that the sky has the property of blueness. The sky is the object and blueness is the property. When we say that 2 is even, we mean that 2 has the property of being even. 2 is the object, and evenness is the property. When we say that 2+1 equals 3, we mean that that 2+1 and 3 as an ordered pair <2+1 3> are in the reflexive relation of equality.

'2' is a constant (a kind of "noun") in mathematics.

So you are comparing apples and oranges when you compare the noun '2' with the adjective 'blue'.

The correct analogy is:

With 'the sky is blue' we have the noun 'the sky' that stands for the sky, and 'blue' is an adjective, and 'is blue' stands for the property that holds for the sky.

With 'My dining room has six chairs', we have the noun 'my dining room' that stands for your dining room, and 'has six chair's' stands for the property that holds for your dining room.

With '2+1 = 3', we have the nouns '2+1' and '3', and '=' stands for the 2-place property of equality and indicates in the equation that the property of equality holds for the pair <2+1 3>.

/

I suppose it is an absolute given that you will never look at even page one of a book on the subject of mathematical foundations.

All proofs of the existence irrational numbers (that I'm aware of) are proofs by contradiction. For example, we assume that √2 can only be 1) a rational number or 2) an irrational number. Since we've proved that √2 is not a rational number we conclude that it's an irrational number. Is it possible that this is a false dichotomy? — Ryan O'Connor

Have we really proved the existence of irrational numbers? That's the name of this thread.

Is this a question people are debating for five pages here??? :snicker:

The issue I am looking at, is not how things are viewed by "ordinary mathematics", it is what is meant by the mathematical concepts. If we adhere to how things are viewed by mathematics, as if this is necessarily the correct view of things, as you seem inclined toward, then the discussion is pointless. You'll keeping insisting that I am not seeing things the way that mathematics sees things therefore I am necessarily wrong. I've been exposed to enough of this already, and see no point to it.

(1) You are still making your use-mention mistake. Yes, '+' represents an operation and '2+1' is a representation of a value, but '2' and '1' are not values, they are representations of values. — GrandMinnow

OK, I'll try to adhere to this formality. It is not the convention I am used to, but I'll try it.

2) As I explained, and as you ignored, + is the operation; 2 and 1 are the arguments; and 2+1 is the value of the function for those arguments. — GrandMinnow

No, "2" and "1" signify values. Or do they sometime signify values and other times signify arguments? If so how do we avoid equivocation? Anyway, I see no way that a function, which is a process, could have a value. That's like saying that + has a value.

You are conflating the meaning of the world 'equal' in various other topics, such equality of rights in the law, with the more exact and specific meaning in mathematics. — GrandMinnow

Look, "2+1" means to put two together with one, and 2+1 equals "6-3", which means to take three away from six. You cannot try to tell me that to take three away from six is the exact same thing as putting two together with one, or I'll tell you to go back to elementary school and learn fundamental arithmetic properly.. Your claim is clearly false, equals does not mean identical to, or the same as, in mathematics.

Ordinary axiomatic mathematics is extensional. Each n-place operation symbol refers to a function on the domain of the interpretation, and functions are objects. The function might or might not be an object that is a member of the domain, but it is an object in the power set of the Cartesian product on the domain. — GrandMinnow

I know that a function is a process. And I also know that the concept of process is incompatible with the concept of object. The two are distinct categories. Therefore it is fundamentally incorrect, by way of category mistake, to say that a function is an object.

It is the mathematical object that is the number of chairs, and is the number musicians on the album 'Buhaina's Delight', and is the value of the addition function for the arguments 4 and 2 ... — GrandMinnow

I really don't know what you could possibly mean by this. The number of chairs is referred to by "6". There is a specific quantity and that quantity is what is referred to with "6". I don't see where you get the idea of an object from here. There are six objects which form a group. The group is not itself the object being referred to, because the six are the objects. Therefore the quantity must be something other than an object or else we'd have seven, the six chairs plus the number as an object, which would make seven.

You are saying that the number of them is 6. — GrandMinnow

More correctly, the quantity is six. You assume that this quantity is an object, a number, which is something other than the quantity. But that's not the case, the quantity is six, the number is six, and both "quantity" and "number" mean the same thing here. Why assume that there is something other than a quantity, an object called 6? That makes no sense, where and how are we going to find this object?.

When we say that 2 is even, we mean that 2 has the property of being even. 2 is the object, and evenness is the property. — GrandMinnow

You're just making an imaginary thing, like God, and handing a property, "even " to that thing, like someone might say God is omniscient. When we use the symbol "2", we use it to refer to a group of two things. like chairs or something. When we say that there is an even number of chairs, this means that the group of chairs can be divided into two groups. But the group clearly cannot be divided by three. If you say that "2" refers to some imaginary object, then you can assign to it whatever properties you like. You could make it infinitely divisible if you want. But if you're not adhering to any principles of reality, this is just useless nonsense. What good is assuming an imaginary object which you can attribute any properties to with total disregard for reality?

With '2+1 = 3', we have the nouns '2+1' and '3', and '=' stands for the 2-place predicate of equality, and indicates in the equation that the predicate of equality holds for the pair <2+1 3>. — GrandMinnow

Now you're doing the same thing again, you're claiming two nouns, 2 and 1, are one noun signified as "2+1". Clearly this cannot be the case without equivocation. Either 2 is a noun and it refers to an object, or it is not. But you can't have it sometimes being a noun, and sometimes not without equivocation.



.

No, if "sqrt" represents an operation, then "sqrt(2)" represents that operation with a qualifier "(2)". — Metaphysician Undercover

This is absurd, but then you do see things from an unusual vantage point, inaccessible to many.

How does "sqrt" change from signifying an operation, to signifying an object, simply by qualifying (or quantifying, if you prefer) it with a (2), without equivocation?

It's the same sort of problem which GM has with 2+1. Each of the symbols "2", and "1" refer to a distinct object. But GM claims that in the context of "2+1" there is only one object referred, and "2" and "1" do not each refer to a distinct object. How is this not equivocation?

The issue I am looking at, is not how things are viewed by "ordinary mathematics", it is what is meant by the mathematical concepts. — Metaphysician Undercover

What is meant by whom? What is meant by mathematicians is not what is meant by you.

If we adhere to how things are viewed by mathematics, as if this is necessarily the correct view of things — Metaphysician Undercover

I haven't said here what is necessarily correct. There are formulations of mathematics - classical, constructivist, intuitionist, finitist - and then there are philosophical discussions about them. A formulation is not the same as the philosophical discussions about it. You are free to present a formulation (or at least an outline) of mathematics and then say philosophically what you mean by it. But lacking a formulation, I would take the context of a discussion of mathematics to be ordinary mathematics and not your unannounced alternative formulation.

No, "2" and "1" signify values. — Metaphysician Undercover

You're mixed up as to what I've said. Yes, I agree, and never disagreed, that '2' and '1' denote values.

Or do they sometime signify values and other times signify arguments — Metaphysician Undercover

This is another case in which your nearly total ignorance of mathematics results in your confusions.

Those numbers are both values themselves and also arguments applied to a function that in turn has a value for those arguments.

I see no way that a function, which is a process, could have a value. That's like saying that + has a value. — Metaphysician Undercover

You don't see a lot of things, because you refuse to even look at an introductory book on the subject.

Do you know what a function is? (In mathematics for grownups, there is a more rigorous definition of a function than 'process'.) Do you know how mathematics develops the subject of functions?

Meanwhile, I didn't say the function has a value. I said the function applied to arguments has a value.

It is the mathematical object that is the number of chairs, and is the number musicians on the album 'Buhaina's Delight', and is the value of the addition function for the arguments 4 and 2 ...
— GrandMinnow

I really don't know what you could possibly mean by this. — Metaphysician Undercover

It could not be more clear. 6 is the number of chairs in your dining room, and 6 is the number of musicians on the album 'Buhaina's Delight', and 6 is the number that is the value of the addition function for the arguments 4 and 2.

"2+1" means to put two together with one, and 2+1 equals "6-3", which means to take three away from six. — Metaphysician Undercover

We've gone over this multiple times already. 2+1 is the result of adding 2 and 1. 6-3 is the result of subtracting 3 from 6. The value (result) of adding 2 and 1 is the same exact value (result) as subtracting 3 from 6.

One more try to get through to you. What you get when add 2 and 1 is the same exact thing as what you get when you subtract 3 from 6.

2+1 is not a process. 2+1 is a number. It is the exact same number as 3, and the exact same number as 6-3. And '2+1' is a name of the number 2+1, and it is a name of the number 3, and it is a name of the number 6-3.

Meanwhile, a process in mathematics can be described as a certain kind of sequence of steps. Yes, the sequence of steps in adding 1 and 2 is different from the sequence of steps in subtracting 3 from 6. But the last entry in the sequence - the result - is the same. '2+1' and '6-3' are not names of processes; they are names of a number.

2+1 = 6-3. That is, 2+1 is 6-3.

'2+1' does not= '6-3'. But they are two different names of the same number.

GM claims that in the context of "2+1" there is only one object referred, and "2" and "1" do not each refer to a distinct object. — Metaphysician Undercover

Please do not misrepresent what I said. I said explicitly that '1' and '2' do each refer to a distinct object. My remarks should not be victim to misrepresentation by you.

The number of chairs is referred to by "6". There is a specific quantity and that quantity is what is referred to with "6". I don't see where you get the idea of an object from here. There are six objects which form a group. The group is not itself the object being referred to, because the six are the objects. Therefore the quantity must be something other than an object or else we'd have seven, the six chairs plus the number as an object, which would make seven. — Metaphysician Undercover

The chairs are objects. And the mathematical object that is the number of chairs is the number 6. And the set of chairs also is an object, and it has cardinality 6.

You're just making an imaginary thing, like God, and handing a property, "even " to that thing — Metaphysician Undercover

Mathematical objects and mathematical properties are abstractions. They are not theological claims like the saying that there are gods. Also, properties like 'blueness' and 'evenness' are abstractions. You are free to reject that there are abstractions, but I use abstractions as basic in thought and reasoning.

When we use the symbol "2", we use it to refer to a group of two things. like chairs or something. — Metaphysician Undercover

We (not you though) understand the number 2 as not just the number of chairs or the number of any particular set of objects but also as a number onto itself.

Why assume that there is something other than a quantity, an object called 6? — Metaphysician Undercover

We prove from axioms that there is a unique object having a certain property, and we name it '6'. Why would we want to do that? Because it greatly facilities mathematics. I may refer to 6 itself rather than have to say "the number of chairs in Metaphysician Undercover's dining room".

In mathematics especially, names refer to things. The name '6' refers to something. It refers to a number. It refers to SSSSSS0. Mathematicians are undaunted by the fact that the thing named is an abstract object, not a concrete one.

where and how are we going to find this object?. — Metaphysician Undercover

We don't find it by a physical search. We find it by a mathematical activity in abstract reasoning.

you're claiming two nouns, 2 and 1, are one noun signified as "2+1". — Metaphysician Undercover

You are thoroughly mixed up, not just about mathematics, but about plain and simple things I've just posted.

I did not claim that 2 and 1 are nouns. I did not claim that 2 and 1 are signified by '2+1'.

2 is a number. 1 is a number.

'2' is a noun. '1' is a noun.

2+1 is a number.

'2+1' is a noun.

'2+1' does not denote 1.

'2+1' does not denote 2.

'2+1' denotes 2+1, and '2+1' denotes 6-3, and '2+1' denotes 3.

Learn use-mention, no matter what your philosophy is.

You are free to present a formulation (or at least an outline) of mathematics and then say philosophically what you mean by it. But lacking a formulation, I would take the context of a discussion of mathematics to be ordinary mathematics and not your unannounced alternative formulation. — GrandMinnow

I have no formulation, and no desire to present one. The op asks if something has been proved, therefore we are invited to be critical of formulations which claim to prove that. And there is no need to offer an alternative formulation to point out problems with an existing one.

Please do not misrepresent what I said. I said explicitly that '1' and '2' do each refer to a distinct object. My remarks should not be victim to misrepresentation by you. — GrandMinnow

As I said, you equivocate:

I said explicitly that '1' and '2' do each refer to a distinct object. — GrandMinnow

2+1 is a number. — GrandMinnow

Which is the case, do "1" and "2' each signify distinct numbers, or does "2+1" signify a number? You can't have it both ways because that's contradiction. But I've been trying to go easy on you and settle for the lesser charge of equivocation. If "1" and "2" each signify distinct numbers, then there are two distinct numbers represented by "2+1", so it is contradictory to say that "2+1" represents one number, because there are two numbers represented here.

It could not be more clear. 6 is the number of chairs in your dining room, and 6 is the number of musicians on the album 'Buhaina's Delight', and 6 is the number that is the value of the addition function for the arguments 4 and 2. — GrandMinnow

That the same quantitative value is predicated of the chairs in my dining room, and the musicians on that album, doesn't make that predicate into an object.

The value (result) of adding 2 and 1 is the same exact value (result) as subtracting 3 from 6. — GrandMinnow

Sure, the resulting value of each is 3, but that's not the issue. Your claim is that "=" signifies identical to. 6-3 equals 2+1, but what is signified by "6-3" is not the same as what is signified by "2+1". You agree about this. Therefore it should be very clear to you that "=" does not signify identical to.

If you say that they have the exact same value, then we are using "equal" in the way I suggested. You and I have the exact same value in the legal system, therefore, as human beings we are equal, just like 6-3 has the same value as 2+1 in the mathematical system, but in neither case are the two equal things identical.

One more try to get through to you. What you get when add 2 and 1 is the same exact thing as what you get when you subtract 3 from 6. — GrandMinnow

OK, let's go with this then. If you do something, and derive a result, this is necessarily a process. So you are very clearly talking about two distinct processes represented by "2+1", and "6-3". Two distinct and different processes can have the same end result, and so those processes can be said to be equal. Does this imply, that in mathematics you judge a process according to the end result? If so, then how do you propose to judge an infinite process, which is incapable of producing an end result, like those referred to in the op?

Mathematical objects and mathematical properties are abstractions. They are not theological claims like the saying that there exists a God. Also, properties like 'blueness' and 'evenness' are abstractions. You are free to reject that there are abstractions, but I use abstractions as basic in human reasoning. — GrandMinnow

I don't see the difference. You are invoking an imaginary object represented by "2", just like a theologian might invoke an imaginary object represented by "God". Each of you will try to justify the claimed existence of your imaginary object. You are not showing the necessity required, which the theologians show, so you are not doing a very good job of it.

We prove from axioms that there is a unique object having a certain property, and we name it '6'. — GrandMinnow

This is so contradictory to what you've been arguing. You've been arguing that 4+2 is 6, and 10-4 is 6, and that there is potentially an infinite number of different things which are 6. And it isn't just a matter of different names for the same thing, because "4" and "2" must each name a unique thing, so it's impossible that "4+2" is just a different name for "6". How can you now claim to be able to prove that there is a unique object named "6", when you've been arguing that all these different things are the same as 6, by virtue of equality. You are getting yourself so tangled up in a web of deceit, that's it's actually becoming ridiculous.

we are invited to be critical of formulations — Metaphysician Undercover

But you don't know anything about the formulation of classical mathematics.

there is no need to offer an alternative formulation — Metaphysician Undercover

But your account of the meaning of mathematics is not compatible with the ordinary formulation of mathematics, so if your account were to have any consequence, then it would need to refer to some other formulation.

As I said, you equivocate: — Metaphysician Undercover

You misrepresented me when you wrote that I said that '1' and '2' do not refer to distinct objects. I said that they do refer to distinct objects. I'll say again, please do not misrepresent what I've said. I don't expect you to have the intellectual honesty to retract your claim about what I said, but I ask that at least you don't do it again.

And I have not equivocated.

Which is the case, do "1" and "2' each signify distinct numbers, or does "2+1" signify a number? You can't have it both ways because that's contradiction. — Metaphysician Undercover

A contradiction is a statement and its negation. You have not shown any contradiction in what I said. The fact that '1', '2' and '2+1' each denote distinct numbers is not a contradiction.

it is contradictory to say that "2+1" represents one number, because there are two numbers represented here. — Metaphysician Undercover

'2+1' has '1' and '2' as parts in the term. The term '1' denotes the number 1, the term '2' denotes the number 2, and the term '2+1' denotes the number 2+1, which is 3, which itself is also denoted by the term '3'. That is not a contradiction.

that the same quantitative value is predicated of the chairs in my dining room, and the musicians on that album, doesn't make that predicate into an object. — Metaphysician Undercover

I didn't say 6 is an object on account of 6 being the number of chairs or musicians. That was not my argument at all. You're confused about the structure of this discussion itself.

what is signified by "6-3" is not the same as what is signified by "2+1". You agree about this. — Metaphysician Undercover

No I do not. You are completely confused. (1) I would put aside 'signify' since you might have some special sense for it. I have mentioned 'refer to' or 'denote'. (2) And I do say that '6-3' and '2+1' denote or refer to the same number. I've said that about a dozen times already and you still don't recognize that that is what I say.

If you say that they have the exact same value, then we are using "equal" in the way I suggested. — Metaphysician Undercover

You are confused even about what you are saying yourself!

Saying that they have the exact same value is to say that they are identical.

Again '2+1' and '3' are not identical. They are different terms. But 2+1 and 6-3 are identical. They are the exact same number and they are the exact same referent of the terms '2+1', '6-3', and '3'.

you are very clearly talking about two distinct processes represented by "2+1", and "6-3". — Metaphysician Undercover

Again, 2+1 and 6-3 are not processes. They are numbers. They are the same number. They are the number 3. You skipped entirely my explanation that a process may be expressed as a certain kind of sequence of steps. The sequence of the steps is not the same as the last step itself, which is the result of the process. 2+1 and 6-3 denote the result. It is the same result.

Two distinct and different processes can have the same end result, and so those processes can be said to be equal. — Metaphysician Undercover

You have nearly everything backwards. A process is a sequence of steps. If one sequence of steps is different from another sequence of steps, then those are different processes even if their results are the same. Sequences are identical to each other if and only if every step in the sequences is the same. The sequence of steps in adding 1 to 2 is different from the sequence of steps in subtracting 3 from 6. But the results are the same for both. '1+2' and '6-3' do not denote processes; they denote numbers.

Does this imply, that in mathematics you judge a process according to the end result? — Metaphysician Undercover

No. Decidedly not. I just explained above, now for the second time.

If so, then how do you propose to judge an infinite process, which is incapable of producing an end result, like those referred to in the op? — Metaphysician Undercover

An infinite process that does not terminate is one that computes a recursively enumerable function. As to the notion of an infinite process that does terminate, I am not aware that this is a rigorous mathematical notion, though we are familiar with the philosophical notion of a supertask. I explained that also in a previous post to another poster.

You are invoking an imaginary object represented by "2", just like a theologian might invoke an imaginary object represented by "God". — Metaphysician Undercover

One difference is that mathematics works with algorithmically checkable axioms and rules of proof. Also, mathematics itself does not opine as to the ontological status of abstract objects but instead recognizes that we carry out mathematics with abstract reasoning regarding abstract objects - whatever we take such abstractions to be. Also, I tend to think that, strictly speaking, mathematicians could dispense even with the notion of objects, though it would make mathematical discussion clumsy.

Also, you have not answered how other abstractions could be acceptable, such as blueness or evenness or the state of happiness, etc.

You've been arguing that 4+2 is 6, and 10-4 is 6, and that there is potentially an infinite number of different things which are 6. — Metaphysician Undercover

No they are not different things. '4+2' and '10-4' and '6' are different names for the same thing.

You just don't get the difference between names and the things named. That is a critical failure of understanding. You can't get past your confusions until you grasp the very simple distinction between a name and the thing that is named.

When we say that there is an even number of chairs, this means that the group of chairs can be divided into two groups. — Metaphysician Undercover

When we say that n is even we mean that n is a natural number such there exists a natural number k such that n = 2k. But, yes, that does imply that a set with even cardinality has a partition of 2 sets with equal cardinality.

But the property, in everyday discussion (not even necessarily mathematics), of evenness of numbers is itself an abstraction. Properties are not things that are physical objects. Yes, physical objects have certain properties. But the properties themselves are abstractions and are not physical objects. You can't point to the property of blueness as a physical object. You can only point to certain blue things as having that property, but then you are pointing to those particular objects and not to the property itself.

Aside from your lack of understanding of use/mention, I suspect that another big obstacle for you is that you don't understand that usually mathematics is extensional, not intensional.

'2+1' and '6-3' are different terms, so, even though extensionally they name the same number, the terms themselves have different intensional meanings. But ordinary mathematics works extensionally.

There have been proposals for formulating intensional mathematics. But I don't know whether your remarks would be pertinent to such formulations. Meanwhile, your remarks are completely off-base when it comes to mathematics as it is ordinarily studied.

'2+1' and '6-3' are different terms, so, even though extensionally they name the same number, — GrandMinnow

I don't understand intensional versus extensional with respect to math (or anything else for that matter) but I've been trying to explain to @Metaphysician Undercover for two years that 2 + 2 and 4 refer to the exact same mathematical object. Without success, of course, but never mind that.

Can you briefly explain to me what that means? How you can use the concepts of intensional and extensional to make the point that 2 + 2 and 4 are two names for the exact same thing?

the terms themselves have different intensional meanings. — GrandMinnow

Ah maybe this is a clue. @Meta keeps saying to me (at least before we stopped discussing the issue) that 2 + 2 refers to the process of adding two things together; and 4 refers to a single thing. Therefore they don't have the same meaning. So that must be "intensional" as you say.

I've heard these two terms for years without understanding. And also intentionality with a 't', that's Searle's point that the Chinese room doesn't have intentionality, or "aboutness." I gather this is an entirely different concept than intensionality with an s?

https://plato.stanford.edu/entries/logic-intensional/

And a classic brief introduction to the subject is:

Introduction To Mathematical Logic, pages 1-9, by Alonzo Church

/

Most simply, ordinary mathematics is extensional, so substitutability of terms holds. That is, the principle of "substitute equals for equals" holds. That is, roughly put, for any terms T and S, and formula F, from T=S we may infer F[x|S] from F[x|T]. For example:


from

4 is even

we may infer

2+2 is even


But consider intensional contexts, such as this:


4 is even

and

4 = (((182/2)-1)/2)-66

and

Bob knows that 4 is even

therefore

Bob knows that (((182/2)-1)/2)-66 is even


The premises are true, but the conclusion is false if Bob doesn't know that 4 = (((182/2)-1)/2)-66.

Putting in 'knows' throws us into an intensional context where substitutability may fail.

/

I am not well versed beyond such basics as that, so for more on the subject I recommend the Stanford article and the passages in the Church book.

But you don't know anything about the formulation of classical mathematics.

...

But your account of the meaning of mathematics is not compatible with the ordinary formulation of mathematics, so if your account were to have any consequence, then it would need to refer to some other formulation. — GrandMinnow

As I said, if this point is of relevance then the discussion is pointless.

A contradiction is a statement and its negation. You have not shown any contradiction in what I said. The fact that '1', '2' and '2+1' each denote distinct numbers is not a contradiction. — GrandMinnow

I can't believe that you do not understand the contradiction. Let' take the expression "2+1". Do the symbols "2" and "1" refer to distinct objects. If so, then there are two objects referred to by "2+1", and it is impossible, by way of contradiction, that "2+1" refers to only one object. Do you understand this?

A process is a sequence of steps. — GrandMinnow

This is false. A process may be described as a sequence of steps. The sequence of steps is not the process, it is the description of the process. That this is an important distinction is evident from the fact that the very same process may be described in different ways, different steps, depending on how the process is broken down into steps. That's why different people can use different methods to resolve the same mathematical equation.

Also, you have not answered how other abstractions could be acceptable, such as blueness or evenness or the state of happiness, etc. — GrandMinnow

I don't see any need to consider an abstraction as an object. Abstraction is simply how we interpret things, and there is no need to assume objects of meaning as a fundamental part of the interpretive process.

No they are not different things. '4+2' and '10-4' and '6' are different names for the same thing. — GrandMinnow

You agreed that they are different things which have the same result, or the same value. If they are different things, then having the same result, or the same value does not justify calling them the same thing.

Here's what you said:

We've gone over this multiple times already. 2+1 is the result of adding 2 and 1. 6-3 is the result of subtracting 3 from 6. The value (result) of adding 2 and 1 is the same exact value (result) as subtracting 3 from 6.

One more try to get through to you. What you get when add 2 and 1 is the same exact thing as what you get when you subtract 3 from 6. — GrandMinnow

Are you taking that back now? Why do you want to say that adding 2 to 1 is the exact same thing as taking 3 from 6, instead of what you already agreed, that they are distinct things with the same end result? You know the truth in this matter, why try to deny it?

Properties are not things that are physical objects. — GrandMinnow

Then why treat properties as if they are any sort of object? You treat numbers as if they are some sort of objects, when really they are a property of the thing which is numbered.

I suspect that another big obstacle for you is that you don't understand that usually mathematics is extensional, not intensional. — GrandMinnow

I've argued elsewhere that the axiom of extensionality is a falsity. It is the means by which you say that two equal things are the same thing, which is obviously false. So it's not necessarily that I do not understand extensionality, but I apprehend it as based in false premises.

That is, the principle of "substitute equals for equals" holds. — GrandMinnow

In other words, equal things may be considered as the same thing. And that's clearly false.

But your account of the meaning of mathematics is not compatible with the ordinary formulation of mathematics, so if your account were to have any consequence, then it would need to refer to some other formulation.
— GrandMinnow

As I said, if this point is of relevance then the discussion is pointless. — Metaphysician Undercover

You are repeating yourself without arguing specifically to the point I made. You argue by mere assertion. My point stands.

A contradiction is a statement and its negation. You have not shown any contradiction in what I said. The fact that '1', '2' and '2+1' each denote distinct numbers is not a contradiction.
— GrandMinnow

I can't believe that you do not understand the contradiction. — Metaphysician Undercover

Given your pattern of ignorance and confusion I can believe that you don't understand that you haven't shown a contradiction in my remarks. A contradiction implies both a statement and its negation. You have not shown how anything I've said implies both a statement and its negation.

Let' take the expression "2+1". Do the symbols "2" and "1" refer to distinct objects. If so, then there are two objects referred to by "2+1", and it is impossible, by way of contradiction, that "2+1" refers to only one object. — Metaphysician Undercover

You skipped the answer I gave to that already.

The sequence of steps is not the process, it is the description of the process. That this is an important distinction is evident from the fact that the very same process may be described in different ways, different steps, depending on how the process is broken down into steps. — Metaphysician Undercover

Since you have not given a mathematical definition of 'process', I am taking 'process in the sense of 'algorithm' or 'effective procedure'.

Mathematics addresses your point by recognizing that different processes may compute the same function.

Abstraction is simply how we interpret things — Metaphysician Undercover

There are two different senses, e.g. (1) "I think by means of abstraction" and (2) "My thinking has resulted in arriving at the abstract concept of blueness."

No they are not different things. '4+2' and '10-4' and '6' are different names for the same thing.
— GrandMinnow

You agreed that they are different things which have the same result, or the same value. — Metaphysician Undercover

No, I definitely did not agree with that.

Indeed, I have explained for you that '4+2', '10-4' and '6' are not things that have results. They are NAMES, not numbers and not processes. Then, 4+2, 10-4, and 3 are the same number. What I do recognize is that, e.g., the process of adding 2 to 4 is different from the process of subtracting 4 from 10.

For about the fifth time:

4+2 is a number.

'4+2' is not a number; it is the name of a number.

If you simply refuse to understand the use/mention distinction, then you are doomed to continue in confusion.

Why do you want to say that adding 2 to 1 is the exact same thing as taking 3 from 6 — Metaphysician Undercover

Please stop ignoring the distinctions I have said multiple times already.

I said the RESULT of adding 2 to 1 is the same as the RESULT of subtracting 3 from 6. I do not say that the processes are the same.

why treat properties as if they are any sort of object? — Metaphysician Undercover

The abstraction called 'blueness' is an abstract object. 'Blueness' can be the subject of a sentence in the manner of a subject that refers to an object. For example, the previous sentence itself is one in which 'blueness' is the subject. And 'blueness' refers to the abstraction blueness.

I suspect that another big obstacle for you is that you don't understand that usually mathematics is extensional, not intensional.
— GrandMinnow

I've argued elsewhere that the axiom of extensionality is a falsity. — Metaphysician Undercover

The extensional nature of mathematics does not depend on the axiom of extensionality. They are different things. You just jump to the conclusion that because 'extension' is found in describing both things that one must depend on the other. You don't know what you're talking about.

In other words, equal things may be considered as the same thing. And that's clearly false. — Metaphysician Undercover

It's true by definition. You are welcome to define terminology in your own way, but I'm telling you in the meanwhile how the terminology is defined in mathematics.

I am not well versed beyond such basics as that, so for more on the subject I recommend the Stanford article and the passages in the Church book. — GrandMinnow

Thank you for the references. Your post actually gave me a glimmer of understanding as to what @Metaphysician Undercover is talking about. And for that matter, what I'm talking about.

The denotation of 'the father of Jane Fonda and Peter Fonda' is Henry Fonda. The denotation is not Jane Fonda nor Peter Fonda nor the sibling relation nor the process of fatherhood.

'the father of Jane Fonda and Peter Fonda' refers to one specific person, and that person is Henry Fonda.

The denotation of '2+1' is 3. The denotation is not 2 nor 1 nor the process of adding 1 to 2.

Another way of saying '2+1' is 'the sum of 2 and 1'.

And the sum of 2 and 1 is 3. It is not 2 nor 1 nor the process of adding 1 to 2. Rather the sum is the RESULT of the process of adding 1 to 2. The sum is not a process; it is a number.

Henry Fonda = the father of Jane Fonda and Peter Fonda. Henry Fonda IS the father of Jane Fonda and Peter Fonda.

'Henry Fonda' and 'the father of Jane Fonda and Peter Fonda' both refer to one identical person. Henry Fonda is identical with the father of Jane Fonda and Peter Fonda. There are not two different people - Henry Fonda vs. the father of Jane Fonda and Peter Fonda. There is one identical person with two ways of referring to him.

There are not two different numbers 2+1 vs. 3. There is one identical number with two ways of referring to it.

2+1 and 3 are symbols referring to the same mathematical entity. The symbol 2+1 also contains descriptive information about the process used to arrive at the mathematical entity (which can also be denoted by the alternate symbol 3). The symbols are different but the abstract entity they refer to are surely the same.

It seems that @Metaphysician Undercover is mistaking the symbols for the entity they refer to.

The denotation of '2+1' is 3. The denotation is not 2 nor 1 nor the process of adding 1 to 2. — GrandMinnow

In general, "2" denotes a number, and "1" denotes a number, but in this particular circumstance, "2" does not denote a number, and "1" does not denote a number. Therefore you equivocate.

I did not say there is a circumstance in which '2' and '1' do not denote numbers.

'2+1' is a compound term made from the constants '2' and '1' and the operation symbol '+'.

'2' denotes a number. '1' denotes a number. '+' denotes an operation. '2+1' denotes the result of the operation + applied to the numbers 2 and 1. That result is a number. Therefore, '2+1' denotes a number.

2' denotes a number. '1' denotes a number. '+' denotes an operation. '2+1' denotes the result of the operation + applied to the numbers 2 and 1. That result is a number. Therefore, '2+1' denotes a number. — GrandMinnow

Ok, we we have the numbers 2 and 1 denoted, and the operation + is denoted. Where is the result of the operation denoted? It seems to me like you're jumping the gun. Jumping to the conclusion, assuming that the some result of the operation, 3, is already denoted when clearly it is not denoted

That's the reason why we need to denote = 3, if we want to denote some result, because "2+1" on its own does not say 3. Otherwise there would be absolutely no purpose to the "=" because everything which 2+1 equals would already be said simply by saying "2+1". Therefore "2+1" would denote an infinite number of things, and that would make interpretation impossible. Furthermore, equations would be absolutely useless because the right side would just be saying the exact same thing as the left side, along with all the infinite other things that are equal. What would be the point to an equation in which the right side represented the exact same thing as the left? You'd never solve any problems that way, because the problem would be solved prior to making the equation. If you didn't know that the two sides signified the exact same thing already (meaning the problem is solved) you could not employ the equals sign.

I see that you are confused about the most basic aspects of mathematics, language and reasoning. On certain points, your understanding is not even at the level of a six year old child. I'm offering you help here, though I doubt you'll take it in.

Where is the result of the operation denoted? — Metaphysician Undercover

You just now quoted me with the answer to that question:

'2+1' denotes the result of the operation

already denoted — Metaphysician Undercover

In a rigorous context, things are denoted by the method of interpretation of a language. In the usual interpretation, by the recursive method, the denotation of '2+1' is determined from the denotations of '2', '1' and '+'.

Otherwise there would be absolutely no purpose to the "=" because "2+1" on its own does not say 3. — Metaphysician Undercover

'2+1' does not have '3' in it, but '2+1' and '3' name the same object. To express that '2+1' and '3' name the same object we write:

2+1 = 3

Or, put another way, to assert that 2+1 equals 3 ('equals' also said as 'is identical with', also said as 'is the same object as') we write:

2+1 = 3

Otherwise there would be absolutely no purpose to the "=" because everything which 2+1 equals would already be said simply by saying "2+1". — Metaphysician Undercover

The purpose of '2+1 = 3' is to assert that 2+1 equals 3.

Therefore "2+1" would denote an infinite number of things — Metaphysician Undercover

You got it exactly backwards. Our method does not lead to '2+1' denoting infinitely many things. '2+1' denotes exactly one thing. On the other hand, 2+1 is denoted infinitely many ways:

2+1 is denoted by '2+1'

2+1 is denoted by '3'

2+1 is denoted 'sqrt(9)'

2+1 is denoted by '((100-40)/3)-17'

etc.

and that would make interpretation impossible — Metaphysician Undercover

The method of interpretation in mathematics does quite fine, thank you (nothwithstanding wrinkles such as Lowenheim-Skolem).

equations would be absolutely useless because the right side would just be saying the exact same thing as the left side — Metaphysician Undercover

The equation is the statement that the left side stands for the exact same thing as the right side. That is very useful.

If we want to know how much a company did in sales, the accountant starts by seeing that the company got 500 dollars from Acme Corp., and 894 dollars from Babco Corp, and 202 dollars from Champco Corp. Then the accountant reports:

500+894+202 = 1596

It's useful to know that '500+894+202' names the same number as named by '1596'.

You'd never solve any problems that way, because the problem would be solved prior to making the equation. — Metaphysician Undercover

No, usually the problem is to find out or prove that the left side and right side are equal or not equal. Or to find another term to more easily represent, say, the left side.

From my sales reports I have 500+894+202. Then I follow a procedure to eliminate '+' and arrive at just one numeral: '1596;. And I conclude: 500+894+202 = 1596. So I then see that '500+894+202' and '1596' name the same number.

If you didn't know that the two sides signified the exact same thing already (meaning the problem is solved) you could not employ the equals sign. — Metaphysician Undercover

One wouldn't honestly claim to know that the equation is true until one worked it out that it is true. Or to find a right side without '+' in it, then first one might have to perform the addition on the left side. This doesn't vitiate anything I've said.