Numerically identical means one and the same. Clearly 2+2 is distinct from 4, so they are not numerically identical. They are however equivalent, meaning that with respect to some quality (in this case a quantitative value), they are the same. Therefore 2+2 and 4 are qualitatively identical.

Therefore 2+2 and 4 are qualitatively identical. — Metaphysician Undercover

well, quantitatively identical, too, no? Wouldn't you say they're quantitatively identical but not numerically identical?

The accepted distinction, which I know of, is the distinction between numerical identity and qualitative identity. Numerical identity means the same, absolutely, referring to nothing other than the thing itself, the thing is identical to itself. Qualitative identity means the same in some particular way, so it refers to similarity.

I've never heard of "quantitatively identical", but I assume that what you mean by this is the same in reference to quantity. As I said already, this is a form of qualitatively identical. Having the same quantity is a qualitative identity, just like having the same colour, or having the same size, or being made of the same type of material. These are all similarities by which we can identify things, but it does not mean that the things which are classed in these groups are the same absolutely (numerically identical).

They do not reference the same entity though. — Metaphysician Undercover

"2 + 2" and "4" are, usually, different ways of referring to 4. They have different senses, but the same reference. For mathematics and logic, the reference is what matters, so identity is identity of reference, hence we say "2 + 2 = 4." That they have different senses, explains why an equation can be informative. "2 + 2 = 4" does not express the same thought as "4 = 4." (That's Frege's take, and I don't have a really good reason to disagree with him.)

"2 + 2" and "4" are, usually, different ways of referring to 4. They have different senses, but the same reference. — Srap Tasmaner

That's not true. "4" has its own reference, "2" has its own reference, and "+" also has its own reference. Therefore it is false to say that "2+2" and "4" have the same reference. They each mean completely different things, just like "7-3", and "2x2" mean something different, and refer to something different.

Sense is "meaning", reference is what is being "pointed at"

If "4" points at something, then so does "2". Clearly they point at something different.

If "4" points at something, then so does "2". Clearly they point at something different. — Metaphysician Undercover

Sure, "4" refers to 4, "2" refers to 2.

I think Frege construes 2 + 2 as a function. "2" has a sense, and refers to 2. "+" has a sense, but doesn't refer to an object. You can put them together to make a function you could call "... + 2," which also has a sense, composed of the senses of "+" and "2," but no reference because it's incomplete--there's a gap. By putting an object where the gap was, you can get "2 + 2," which has a complex sense, and now has a reference, which is the value of the function, namely 4.

Frege considers 4 a simple object. "4" is a name for 4 with a simple sense, but 4 also has infinitely many names with complex senses, but still the simple reference 4.

Sure, "4" refers to 4, "2" refers to 2. — Srap Tasmaner

If "4" refers to something, and "2" refers to something different, then it is impossible that "2+2" refers to the same thing as "4".

I think Frege construes 2 + 2 as a function. "2" has a sense, and refers to 2. "+" has a sense, but doesn't refer to an object. — Srap Tasmaner

It doesn't make sense to say that "2" and "4" refer to something, but "+" does not refer to anything. Each of these refers to an intelligible object, a concept, and therefore, in your terms, they have a sense. None of them refers to a physical object, and therefore they do not have referents according to this distinction, they only have senses.

So, either "2", "4", and "+", each refer to something, mathematical objects, or they all refer to nothing, and only have senses. It is inconsistent, and therefore illogical to say that one of them has a referent yet another does not.

Frege considers 4 a simple object. "4" is a name for 4 with a simple sense, but 4 also has infinitely many names with complex senses, but still the simple reference 4. — Srap Tasmaner

I don't see how you can justify this claim that "4" has a reference unless you adopt Platonic realism, and allow for the existence of the mathematical object, the idea of 4. If this is the case, then also "2" refers to a different Platonic object, and "+" refers to a different Platonic object.

It seems odd to me that you'd be so eager to say that "2+2" isn't identical to "4," yet you readily say that Joe's subjective experience is identical to Pete's (or at least some part of it is).

I'd be happy to try to explain what I understand of Frege's philosophy of mathematics, if you'd really like me to, but honestly Frege's writings are a way better source than I am. Some of this stuff I struggle with.

I had no intention of converting anyone or "winning an argument." I stopped by because you were talking about something there's prior art for. Just offering a way of thinking about this stuff that you might find helpful. If you don't, no harm no foul.

If you find these sorts of questions interesting, then you really ought to read Frege. (If, on the other hand, you find them an annoying waste of time that gets in the way of doing more interesting stuff, then probably not.)

It seems odd to me that you'd be so eager to say that "2+2" isn't identical to "4," yet you readily say that Joe's subjective experience is identical to Pete's (or at least some part of it is). — Terrapin Station

You're maybe not "eager," but let's say "comfortable" concluding that if Joe and Pete both assert that 2 + 2 = 4, then there must be something similar about the states of their respective brains.

It seems odd to me that you'd be so eager to say that "2+2" isn't identical to "4," yet you readily say that Joe's subjective experience is identical to Pete's (or at least some part of it is). — Terrapin Station

You seem to be missing the point Terrapin. I believe that Joe's subjective experience is similar to Pete's. And, according to this similarity, we can identify them both as having "subjective experience", through the principles of what is called qualitative identity. Likewise, we can say that "2+2" is similar to "4" through the principles of qualitative identity.

You on the other hand, have claimed that subjective experience is particular material involved in particular relations and particular processes. According to your claim, it is impossible that both Joe and Pete have subjective experience, because they each have different material in different relations and distinct processes. Joe and Pete are distinct.

Are you ready to dismiss this idea, that subjective experience is particular matter involved in particular relations, and particular processes, or have you figured out another way to justify the idea that both Pete and Joe are have subjective experience?

So you're not saying that Joe and Pete have subjective experience that's numerically identical.

What I don't get is why you have a "qualitative identity" that can't obtain via material.

I don't recall what you said about noses earlier. Do you believe that noses aren't "qualitatively identical" (I'm putting that phrase in quotation marks partially because I don't use it), or do you believe that there's something about noses that isn't material?

What I don't get is why you have a "qualitative identity" that can't obtain via material. — Terrapin Station

Say you have two distinct material entities. What's to say that they are similar except a mind making that judgement? You might say "they just are similar", but that's just you making an assertion. In reality they are distinct and therefore different.

I don't recall what you said about noses earlier. Do you believe that noses aren't "qualitatively identical" (I'm putting that phrase in quotation marks partially because I don't use it), or do you believe that there's something about noses that isn't material? — Terrapin Station

No, noses clearly are qualitatively identical, that's why we can call each one a nose. The different noses are not numerically identical, by the very fact that they are different.

It's not that there is something about noses themselves, which isn't material, it's the fact that they can be readily identified by the same name "nose", which is immaterial. Since "nose" does not refer to one particular nose, there is no particular material which "nose" refers to. Therefore "nose" does not refer to any particular material thing. If it did refer to a material thing, it would refer to that particular material thing and nothing else. It does not, it can be used to refer to many different instances of material things. Therefore it is false to say that "nose" refers to a material thing.

It sounds to me like you're getting extremely confused by language/by how language works.

At any rate, so things that are material can be qualitatively identical in your view.

Also, similarity is simply things being relatively more alike than different in some respect. That's an objective quality.

At any rate, so things that are material can be qualitatively identical in your view. — Terrapin Station

Of course material things can be qualitatively identical, it's the identity which is immaterial, not the thing itself. That's the difference between numerical identity and qualitative identity. With numerical identity, the identity is nothing other than the thing itself. With qualitative identity, the identity is necessarily something other than the thing itself, because numerous things have the same identity.

Also, similarity is simply things being relatively more alike than different in some respect. That's an objective quality. — Terrapin Station

Whether the quality is said to be subjective or objective is irrelevant. It still requires a mind to make the comparison, and determination, that two distinct things are objectively similar, classifying them together, giving them the same name (qualitative identity). The identity which the thing has, in the case of the qualitative identity, is a property of the mind which assigns that identity.

Of course material things can be qualitatively identical, it's the identity which is immaterial, not the thing itself. That's the difference between numerical identity and qualitative identity. With numerical identity, the identity is nothing other than the thing itself. With qualitative identity, the identity is necessarily something other than the thing itself, because numerous things have the same identity. — Metaphysician Undercover

Right, so you can't object that consciousness can't be material because if so, it can't be "qualitatively identical."

(Again, not that I agree with the notion of "qualitative identity" as you're presenting it. I'm just stating the above under the umbrella of your views.)

Whether the quality is said to be subjective or objective is irrelevant. It still requires a mind to make the comparison, — Metaphysician Undercover

Which makes whether it's subjective or objective hardly irrevelant. If it requires a mind it's subjective. Per my usage, that's the definition of subjective.

Right, so you can't object that consciousness can't be material because if so, it can't be "qualitatively identical." — Terrapin Station

What I objected to is you saying that consciousness is particular material, involved in particular relations, and particular processes. Such particularity denies the possibility of qualitative identity.

Which makes whether it's subjective or objective hardly irrevelant. If it requires a mind it's subjective. Per my usage, that's the definition of subjective. — Terrapin Station

Then I will conclude that per your usage, all such similarities are subjective. There is no such thing as an objective similarity, because the fact that two things are similar, not the same thing. implies that they are different. Only a mind can say that they are similar, and therefore if we adhere to your usage, you produce inconsistency with your claim that there are objective qualities. There is no such thing as objective qualities if we adhere to your usage.

What I objected to is you saying that consciousness is particular material, involved in particular relations, and particular processes. Such particularity denies the possibility of qualitative identity. — Metaphysician Undercover

So you believe that noses are not particular material, in particular relations, undergoing particular processes?

So you believe that noses are not particular material, in particular relations, undergoing particular processes? — Terrapin Station

That's right, "nose" is defined as "an organ above the mouth...". To define "nose" as "particular material, in particular relations, undergoing particular processes" is kind of ridiculous don't you think?

That's right, "nose" is defined as "an organ above the mouth...". To define "nose" as "particular material, in particular relations, undergoing particular processes" is kind of ridiculous don't you think? — Metaphysician Undercover

Why would we suddenly be talking about definitions per se?

We are talking about "what" something is. I am just demonstrating how ridiculous your description of subjective experience, or noses, or whatever, as particular material, in particular relations undergoing particular processes, is.

We are talking about "what" something is. — Metaphysician Undercover

Right, that's ontology. So why would you bring up definitions all of a sudden and whether something would make a good definition? We're supposed to be doing ontology. We're not pretending that we're writing a dictionary.

Do you believe that noses are material?

Noses are partially composed of material, but material does not make up the relationships, nor the processes which that material is involved in. These are immaterial. So a nose is both material and immaterial.

Noses are partially composed of material, but material does not make up the relationships, nor the processes which that material is involved in. These are immaterial. So a nose is both material and immaterial. — Metaphysician Undercover

Insofar as they're composed of material, it's not particular material, in your view? Is it some sort of general material?

Actually, I don't believe there is such a thing as particular material. I think "matter" is purely conceptual, and therefore by nature, a universal, general. It is a concept initiated by Aristotle in an effort to account for the observed continuity of temporal existence.

So to answer your question, since all physical objects are changing with each passing moment of time, exchanging material with their environment, it is impossible that any object, noses included, could consist of particular material. We would have to be able to stop time at a particular moment, and say that the nose consists of this particular matter, at this particular time, in order to conclude that such an object is composed of particular matter. Since it is impossible to stop time, time keeps passing with each moment, and the material keeps changing with each moment, it is impossible that such an object is composed of particular matter.

Actually, I don't believe there is such a thing as particular material. I think "matter" is purely conceptual, — Metaphysician Undercover

Do you believe there are any particulars?