The Wittgensteinian intuition regarding the identity sign, roughly put, and defended here by MU, is that it is the meta-logical expression of synonymy, which upon full analysis of the expression concerned, is eliminated to yield substitution operations among 'non equal' logical terms, each denoting distinguishable objects. This point of view has been shown to be as expressive and as consistent as the logical interpretation of identity, even if on occasion standard theorems of mathematics need amending.

There is no "k" though. What is symbolized is "2+2", two objects and an operator, not one object "k". So this object represented by "k" is not represented by "2+2", it has been wrongly created by you mind, false imagination, nothing here represents it. — Metaphysician Undercover

I introduced the object k as an intermediate clarification step, of course it is not mentioned by 2+2.

I just want you to answer this question

does the expression "2 + 2" denotes two objects or one object?

I know that it contains in it the symbol 2 twice, that is clear, but do you think just because of this containment, then it ought to "denote two objects"

Take the following example: "The planet between planets Venus and Mars"

Obviously this expression contains expressions "Venus", "Mars" and each is denoting an object. So it does contain denotations of two objects in it. BUT it itself denotes ONE object that is the object denoted by expression Earth. What we mean by "denote" here is the object that is the subject of speech of that expression, which is obviously the object denoted by expression Earth in English. This is an example of an expression that contains denotation of more than one object within it, but it itself only denotes ONE object.

In a similar manner 2 + 2 is denoting ONE object.

2 + 2 is equivalent to the expression "The result of summation of 2 and 2"

Or sometimes we express it as "The sum of 2 and 2"

2 + 2 means "the natural number that results from adding 2 to 2"

So '2 + 2' is denoting a single object, although it does contain inside it two occurrences of a denotation, yet it is denoting a single object, similarly 2+3 it contains two distinct denotations, but it itself is denoting one object which is the object that results from adding 2 to 3.

Now whether the SINGLE object denoted by '2 + 2' is itself the same (identical to the) object that is denoted by '4', is something that I personally think it to be the easier and simpler way to formalize. If we say No, the object that 2 + 2 is denoting is different from the object 4 is denoting but it is "equal" to that object, and here equality can be understood as a kind of equivalence relation (a relation that is reflexive, symmetric and transitive), I think this is a more complicated way of looking at it.

Of course the interpretation of = as equality relation is weaker (logically speaking) than interpreting it as identity relation. Many times people prefer or feel more safe with holding weaker assumptions. And so it indeed can be justified as a kind of cautious philosophical approach to the matter. However, I still think that identity, albeit being a stronger interpretation, yet it is much nicer and sharper, and actually much easier formally speaking than the more general equality notion.

Try to formalize PA yourself using "=" as an equality relation. You'll see how cumbersome it would be. Interpreting "=" as identity simplify formalization to a great extent.

meta-logical expression of synonymy, which upon full analysis of the expression concerned, is eliminated to yield substitution operations among 'non equal' logical terms, each denoting distinguishable objects. — sime

Agreed. And I mentioned this to MU. I said that one can indeed interpret the '=' sign as some equivalence relation, no doubt, like that of synonymy, or actually any equivalence relation, of course this can formally work. But the formalization would be more cumbersome, because you are holding to a weaker concept than identity, you'll loose all the merits of identity, which shortens formalization to a great extent. Philosophically speaking one might prefer to hold to the weaker interpretation, but formally speaking, it is not the preferable one. For example how would you DEFINE 4. Using identity I don't need to introduce 4 as a primitive symbol, since I can define it, since it is the unique object that 2 + 2 is denoting. However you cannot define it as such when "=" is just an equivalence relation, you'll need to introduce 4 as a primitive notion, that said you'll need to introduce all naturals as primitive constants of the language, which is in some sense cumbersome.

OK, that's fine. OF course just to make it more precise. I said almost all of mathematics before the era of set theory can be formalized as an extension of first order logic with identity where the symbol "=" is taken to mean "identity" i.e. "being the same as". Actually this is a well known result, actually most of that kind of mathematics can be formalized in second order arithmetic, you can read about it in reverse mathematics which also can be re-formalized as an extension of first order logic with identity. Actually ZFC itself can be formalized as an extension of first order logic with identity, and ZFC is way stronger than almost all of mathematics before the era of set theory. This is a very well known result. — Zuhair

And, of course, this re-formalizing is a logical fallacy, equivocation, as I've demonstrated.

what is that fundamental aspect that enforce us to interpret = sign as some equality relation other than identity. — Zuhair

What is on the right side of the equation is not the same as what is on the left. For example, in "2+2", there is as you've described, an "operator" signified, which is not signified by "4". Therefore it is very clear that what is on the right side is not the same as what is on the left side and so to interpret the "=" as signifying "the same" is wrong.

If there is something fundamental to mathematics against the use of = symbol in it to represent identity, then how PA is formalized as such??? How ZFC is formalized as such and it is generally regarded by many as the official foundation of mathematics? Both are indeed formalized with = in them understood as identity. — Zuhair

I don't believe that PA is actually formulated as such. None of the websites which fishfry referred me to, to support this claim supported that notion. Those websites described PA as based in equality theory, not identity. I think that those people such as yourself who insist on this notion are practising sophistry. I said that way back.

What you are saying is that the current foundational systems of mathematics are committing a fundamental error? (notice that most of those are coined as extensions of first order logic with identity) According to your account they must instead represent the = as an equivalence relation that can hold between distinct objects, and that the object denoted by 2 + 2 must be considered as a distinct object from that denoted by 4. This is strange? why? — Zuhair

No, I think those who interpret, or "coin" these systems as based in identity are committing a fundamental error. We discussed the axiom of extensionality, it defines equality, not identity. It is incorrect to say that this axiom of equality is an axiom of identity.

just want you to answer this question

does the expression "2 + 2" denotes two objects or one object?

I know that it contains in it the symbol 2 twice, that is clear, but do you think just because of this containment, then it ought to "denote two objects" — Zuhair

It is clear that "2+2" must be interpreted as denoting two distinct objects or else the "+" symbol is left meaningless. This is obviously a problem for the ontology of mathematical objects. We commonly think that every instance of the symbol "2" would denote the very same object. This would be a Platonic object, a number, symbolized by the numeral "2". But if this were the case, then we'd have a problem interpreting the "+" symbol. You'd have two instances of the same sign, representing the same thing, "2" with the "+" symbol representing a relationship between an object and itself. You cannot add an object to itself, so what could this "+" sign possibly represent as a relationship between an object and itself, other than identity, "2" is the same as "2"? But this is not what "+" represents. It represents a relation between what is represent by the first "2", and what is represented by the second "2", and if there is a relationship between these two which is not a relationship of identity, they must be distinct objects. Therefore what is represented by the first "2" in "2+2" is necessarily a distinct object from what is represented by the second "2".

2 + 2 is equivalent to the expression "The result of summation of 2 and 2" — Zuhair

But "the result of summation of 2 and 2", refers to two objects. There cannot be a summation without a multitude. You just want to say that "2+2" means "4", but it doesn't, it means something different than "4" and you don't seem to be capable of respecting this.

2 + 2 means "the natural number that results from adding 2 to 2" — Zuhair

You're so ridiculous that it's becoming funny. "The natural number that results from adding 2 to 2" is "4". If someone wanted to say "the natural number that results from adding 2 to 2" they would just say "4". Instead, they say "2+2", because this means something different from "4".

But the formalization would be more cumbersome, because you are holding to a weaker concept than identity, you'll loose all the merits of identity, which shortens formalization to a great extent. — Zuhair

I think I now see why you lie about this matter. Identity is seen by you as a stronger position. You want your position to appear stronger. So you will continue to lie in the claim that you are in the stronger position, attempting to deceive people into believing that your mathematics is stronger than it actually is.

The law of identity doesn't say that a thing is equal to itself, it says that a thing is the same as itself. In formal logic, "the same as" is represented by "=". So when the law of identity is expressed in formal logic as "a=a" or some such thing, the "=" represents "the same as". Zuhair is arguing that all mathematical axioms can be interpreted as "=" representing "the same as", but this is equivocation plain and simple. — Metaphysician Undercover

I take no responsibility for and neither endorse nor necessarily agree with anything written by anyone on this site but myself; nor do I necessarily disagree. I have no idea why you are quoting some other poster's thoughts to me on this subject. I've written plenty to you already that you haven't engaged with, including a proof directly from first principles that 2 + 2 and 4 are identical.

You claimed the other day that the law of identity does not apply to the number 3, or numbers in general. At that point I assumed you've simply given up rational debate and/or recognized the impossibility of your own position. Talk me down please.

But this is not what "+" represents. It represents a relation between what is represent by the first "2", and what is represented by the second "2", and if there is a relationship between these two which is not a relationship of identity, they must be distinct objects. Therefore what is represented by the first "2" in "2+2" is necessarily a distinct object from what is represented by the second "2". — Metaphysician Undercover

all of this is wrong. + is a binary function symbol which means it is a ternary relation symbol, it is a relation between three occurrences of symbols, it relates the first two symbols to a third occurrence of a symbol, here it relates the two occurrences of 2 to a third occurrence of a symbol which is 4, this is explicit when you write it in relational terms as +(2,2,4), but when it is written in functional terms here the confusion would raise since you don't see the third occurring symbol (which is 4) you only see two occurrences of 2 linked by + sign in between, here it means that + is relating the two occurrences of symbol 2 to the symbol '2 + 2', you see here the expression '2 + 2' is acting as a symbol denoting an object of the language.

in your views '2 + 2' represent two distinct objects operated upon by the + operator. While the common view is that '2 + 2' denotes the natural number that results from running the + operator on two occurrences of 2. It is like the expression "The planet between planets Venus and Mars", it does mention two distinct denotations those are the planets Venus and Mars, and it does mention an operator running on them which is the "between" operator. However what it denotes is non of those, what it denotes is the planet Earth, which is ONE object. Notice that there is no symbol or word inside that phrase that symbolize what the total phrase is denoting, however the total phrase itself does denote planet Earth. Similarly '2 + 2' is an expression that mentions denotations of objects by two occurrences of the symbol 2 and an operator running on them, yet the total expression (i.e. all three symbols in 2 + 2 in that sequence) is denoting non of those, what is denoted by the total expression '2 + 2' is a single object that can be what is denoted by '4' if you interpret '=' as identity, or it can be another object that is related by some equivalence relation to the object denoted by 4, anyway the whole expression of "2 + 2" is not denoting multiple objects, no , it is denoting a single object, because + is a FUNCTION.

Just because an expression contains (mentions) inside it different denotations, operators, relations, etc.. doesn't mean that it is denoting those, or that it denotes multiple objects, no it can be using those to denote a single object that is non of them (as it is the case of 2 + 2).

None of the websites which fishfry referred me to, to support this claim supported that notion. Those websites described PA as based in equality theory, not identity. — Metaphysician Undercover

Yes I agree, many of them do it that way. But definitely there are formalizations of PA as an extension of first order logic with identity, but they often don't mention the axioms for identity since they consider it as part of the underlying logic, which in this case it is usually taken to be "first order logic with identity".
For ZFC, it is usually formalized in first order logic with identity, but sometimes formalized using one primitive that is the membership symbol. However most formulations of ZFC are extensions of first order logic with identity. And that suits set theory, since if = doesn't represent identity why should we define 'singleton' sets after the equality relation then?

I've written plenty to you already that you haven't engaged with, including a proof directly from first principles that 2 + 2 and 4 are identical. — fishfry

We went through this. You provided no such proof, it was just an assertion.

all of this is wrong. + is a binary function symbol which means it is a ternary relation symbol, it is a relation between three occurrences of symbols, it relates the first two symbols to a third occurrence of a symbol, here it relates the two occurrences of 2 to a third occurrence of a symbol which is 4, this is explicit when you write it in relational terms as +(2,2,4), but when it is written in functional terms here the confusion would raise since you don't see the third occurring symbol (which is 4) you only see two occurrences of 2 linked by + sign in between, here it means that + is relating the two occurrences of symbol 2 to the symbol '2 + 2', you see here the expression '2 + 2' is acting as a symbol denoting an object of the language. — Zuhair

Why the change? We were talking about objects, now you switch to symbols. Let's maintain consistency. We were talking about what the symbols denote. Tell me what "+" denotes, not how it relates one symbol to another. If it denotes a relation, then what is it a relation between, distinct objects, or one and the same object?

in your views '2 + 2' represent two distinct objects operated upon by the + operator. While the common view is that '2 + 2' denotes the natural number that results from running the + operator on two occurrences of 2. — Zuhair

If we have two occurrences of 2, then 2 here is a symbol. That symbol represents one object, the number two. If we have two occurrence of the same natural number, 2, then the only natural number represented is 2. What does your operator do, show that 2 is the same as 2?

Similarly '2 + 2' is an expression that mentions denotations of objects by two occurrences of the symbol 2 and an operator running on them, yet the total expression (i.e. all three symbols in 2 + 2 in that sequence) is denoting non of those, what is denoted by the total expression '2 + 2' is a single object that can be what is denoted by '4' if you interpret '=' as identity, or it can be another object that is related by some equivalence relation to the object denoted by 4, anyway the whole expression of "2 + 2" is not denoting multiple objects, no , it is denoting a single object, because + is a FUNCTION. — Zuhair

This all depends on what your operator "+" represents. You cannot jump to conclusions without explaining what the symbol represents. We have two occurrences of the same object, 2. We have an operator which expresses a relationship between them. As I told you already, the only reasonable relationship between the same object is that of identity. So is the operator "+" expressing a relationship of identity between the two occurrences of the same object? If not, then I think that the two occurrences of "2" are not symbolizing the same object. If there is an expressed relationship between two objects, which is not an expression of identity, then the two objects must be distinct objects.

We went through this. You provided no such proof, it was just an assertion. — Metaphysician Undercover

I provided a direct proof from first principles. You might have questions or specific objections. But to claim I did not provide a proof means you happened to not see my lengthy post, or ... well I just don't know. It would be helpful if you'd specifically engage with the proof I gave; not claim I didn't give one.

We have an operator which expresses a relationship between them. — Metaphysician Undercover

No the operator + doesn't express a relationship between the objects those symbols are denoting, for example lets take the expression "3 + 5" you seem think that "+" here is representing a binary relation between the object denoted by 3 and the object denoted by 5, which is wrong. The reason is because + is NOT a binary relation, it is a "TERNARY relation". Every binary FUNCTION is in reality a ternary RELATION, and + is a binary function. The operator + here is a relation between three objects, one expressed by 3 and the other by 5 and the third by the expression "3 + 5". Let me try to give a helpful analogy that of the relation "son of", when we say for example "Issac is the son of Abraham and Sarah" now the relation "Son of" is a ternary relation, it links an individual to his two parents, so three people are involved in this relation, so for the example above Son is linking Sara, Abraham to the son of Sarah and Abraham. Similarly when we see "3 + 5" we are seeing THREE terms of the language, those are "3", "5" , "3+5", you know that the symbol + is not a term of the language. So the + sign here is understood to be a ternary relation that links objects denoted by the terms of the language which are "3","5","3+5". Each of those terms is denoting a single object, so 3 is denoting one object, 5 is denoting one object, and '3+5' is also taken to denote ONE object (because 3 + 5 is a binary function symbol and so it is a term of the language, so it denotes one object (despite having parts of it that denote other objects)).

With the case of 2 + 2 matters becomes more confusing, here + sign is relation that links the object denoted by 2 to itself and to the object denoted by "2 + 2", so it doesn't just link the former object to itself, No! it links it to itself and to the object denoted by 2+2 which is not equal to 2. This is a little bit confusing. To give an analogy is a little bit more complex. Suppose a country X only allow adoption of a child to an adult if one adult has a job (earns an income) and an adult that know how to work at house (cook, clear, wash,etc..), so in this case it allows it between maximally two adults and one child, "adopted son in country X" is a ternary relation, but it can also occur between two objects sometimes, if a single adult has a job and also is capable of doing house work, so you can have an adopted son of Mrs J and Mrs J, it means he is the son of Mrs. J that earns a job and of Mrs.J that can do house work.

So to be more precise the operator + in "x + y" means a ternary relation between the object denoted by x in the first role, and the object denoted by y in the second role and the object denoted by "x + y"
so + sign in "2 + 2" means a ternary relation that links the object denoted by 2 in the first role and the same object in the second role and the object denoted by "2 + 2".

That's why I was saying that '2 + 2' is a FUNCTIONAL expression of the language, it denotes a single object (because it is a functional expression) even though parts of it (which are 2 in first role and 2 in second role) are denoting other kind of an object, still what "2 + 2" is denoting is something else other than what any of its two terms shown in the expression are denoting.

I think you tend to think that denotation of an expression is the sum total of all denotations of its parts, for example the denotation of expression "The planet between Venus and Mars" in your sense is the total denotations made by all parts of that sentence, now Venus and Mars are parts of that sentence and each is denoting a separate object. Now the total expression (all the six word words in that sequence) is definitely denoting a SINGLE object which is of course planet Earth. However in your sense you take the denotation of the above phrase to mean the set of objects denoted by expressions Venus, Mars, "The Planet between Venus and Mars", so in your sense denotation of an expression is the total denotations made by all denoting parts of that expression. While in my sense I take the denotation of the sentence to mean what the total expression is denoting, which in this case it would be Planet Earth, which is a single object. According to that line of terminology the total denotation of 2 + 2 is of course not the same as the total denotation of 4, that's obvious, because the first 2 is denoting an object but in the first role, the second 2 is denoting another situation which is the same object but in a second role, and the whole expression "2 + 2" is denoting a third object. While 4 is only making one denotation, i.e. of a single object, because it is an atomic expression, it has no denoting proper parts, it has only itself as a denoting part. But here when we say 2 + 2 = 4, we are not speaking about the total denotations involved in 2 + 2 , no we mean what is denoted by '2 + 2', and here it means a single object that is related by the = relation to the object denoted by 4.

In nutshell the + operator in the functional term 'x + y' is a ternary relation between the object denoted by x in the first role and the object denoted by y in the second role and the object denoted by 'x + y'.

The problem was that you were treating "+" as a binary symbol, so you thought that the expression "x + y" doesn't denote an object that might differ from the object denoted by x and that denoted by y. No we have THREE objects in play, and not two.

There is another way of interpreting the + sign which is as a kind of a relation between ordered pairs of the input objects to an output object, so + in the expression x + y only means a relation that sends the ordered pair of the object denoted by x and the object denoted by y, lets symbolize that pair by (x,y), this ordered pair is an object of course, now + sends (x,y) to the object represented by 'x + y'. Now we come to what does the ordered pair means, I'll use the originally posed set theoretic ordered pair of Wiener

(x,y) = {{{x},0}, {{y}}}

So for the case of 2 + 2 = 4, the + operator is the relationship that sends the object
{ {{2},0}, {{2}} } to the object denoted by '2 + 2'.

You see here the + operator is interpreted as a binary relation between an ordered pair of two objects and some output object. But even here it doesn't mean that it is a binary relation between the two input objects, so it is not the binary relation between objects denoted by 3 and 5 in the expression 3 + 5. But it can be interpreted as the binary relation between (3,5) and the single object denoted by 3+5.

We went through this. You provided no such proof, it was just an assertion. — Metaphysician Undercover

Is it possible you missed this?

https://thephilosophyforum.com/discussion/comment/328116

Also you claimed that the law of identity does not apply to numbers; that for example 3 is not the same as 3. Please clarify or retract. Thank you.

ps -- Originally you claimed that in ZFC there are equalities that are not identities. I know of no such instance nor have you presented a single such example. Clarify or retract please. The only examples I know are natural injections, such as identifying the integers with the copy of the integers contained in the real numbers; and in casual contexts where we call two isomorphic groups "the same" when we know that we mean isomorphic. Other than those two contexts, I know of no instance in which mathematical equality is anything other than set identity and logical identity. I have challenged you on this point and found your responses lacking in specificity.

Is it possible you missed this? — fishfry

No, I didn't miss it, we went through it already, your first premise is false:

1.1 We have the law of identity that says that for each natural number, it is equal to itself. — fishfry

The law of identity states that a thing is the same as itself, not that a natural number is equal to itself. This is the problem, you keep asserting that the law of identity says something about equality, when it does not. It says something about identity. So you wrongly proceed to claim that mathematical principles which say something about equality are base in the law of identity. That is why I kept asking you to back up this claim, that there is a law of identity which states something about equality.

The operator + here is a relation between three objects, one expressed by 3 and the other by 5 and the third by the expression "3 + 5". — Zuhair

So it's now becoming clear to you that "3" and "5" in "3+5" denote separate objects, just like "2" and "2" denote separate objects in "2+2".

So the + sign here is understood to be a ternary relation that links objects denoted by the terms of the language which are "3","5","3+5". — Zuhair

Right, we have a multiplicity of objects denoted.

and '3+5' is also taken to denote ONE object (because 3 + 5 is a binary function symbol and so it is a term of the language, so it denotes one object (despite having parts of it that denote other objects)). — Zuhair

You are claiming that a "binary function" is an object. I see no justification in this. A function is an activity, or a relation. Each of these may be a property of an object, or a relation between objects, but is not an object itself.

So to be more precise the operator + in "x + y" means a ternary relation between the object denoted by x in the first role, and the object denoted by y in the second role and the object denoted by "x + y"
so + sign in "2 + 2" means a ternary relation that links the object denoted by 2 in the first role and the same object in the second role and the object denoted by "2 + 2". — Zuhair

You haven't justified your claim the "2+2" is an object, nor your claim "+" represents a ternary relation. I think you have fallen back into your habit of lying.

does the expression "2 + 2" denotes two objects or one object? — Zuhair

It is a language expression. In principle, it has nothing to do with real-world "objects".

There are rewrite rules that allow for carefully reducing this language expression, "2 + 2" to other language expressions e.g. "1 + 1 + 2". The complete sequence of permissible rewrite operations that demonstrates that the symbol stream "2 + 2" is extensional with "1 + 1 + 2" is called a "proof".

The expression "2 + 2 = 1 + 1 + 2" does not necessarily mean anything outside the system of basic rules that define that system. Seriously, it is completely self-contained with no reference to the real, physical world.

Simply put, potential infinite means unknown possibilites while actual infinity means known actualization of possibilities.

The main problem here, mostly at the term infinite, which is nmpossible for me. It is not unlimited, but it's not limited by. "Not limited by" is much more make sense than "unlimited".

God is infinite? Unlimited? Can do anything? God is unlimited, therefore God can be less/more than God's self? It's impossible. It against axiom: "thing can't exceed beyond thing itself (without additional from outside) - from a glass of water can't be poured into a gallon of water.

I prefer say, God is not limited by.

Agreed. But for those who request some kind of referential interpretation for the symbols, i.e. semantics, it would be nice to try figure that out as I was doing with MU. But essentially you are right. The matter is that arithmetic is nothing but a game played with symbols.

You haven't justified your claim the "2+2" is an object, nor your claim "+" represents a ternary relation. I think you have fallen back into your habit of lying. — Metaphysician Undercover

Those are not my claims. Please read about the syntax of first order logic which is the background logic used in foundational systems of mathematics. Please read what it means to be "terms" of the language, and also read about "functional terms" in particular and how to differentiate it from relational expressions. You'll see that there is a difference between a relational expression for example x R y, which means that x bears the the relation R to y, for example 1 precede 2, this is a relational sentence you see two "terms" linked by a relation symbol, here that expression is not a term of the language. The usual interpretation is that terms of the language range over OBJECTS (i.e. elements) of the universe of discourse, while relational symbols do not range over elements of the universe of discourse. That said the symbol 2 is taken to denote a single object in the universe of discourse because 2 is a constant symbol, while the expression "x" is a term that ranges over many objects of the universe of discourse, this means that x can be substituted by many objects of the universe of discourse. On the other hand the relation symbol = is not substituted by any object in the universe of discourse, because it is a relational (predicate) symbol and it is not a term of the language. Now the expression "2 + 2" is by definition of the syntax of first order logic, is considered to be a "term" of the language, because the + sign denotes a FUNCTION, and the rule is that when you have a function symbol F and you have a string of terms (x_1,x_2,...,x_n), then the expression F(x_1,x_2,..,x_n) is considered as a "TERM" of the language, which means that it denotes OBJECTs in the universe of discourse, and moreover if each of the variables x_1,x_2,..,x_n is replaced by a "constant" symbol like for example c_1,c_2,..,c_n , then F(c_1,c_2,..,c_n) is taken to denote a SINGLE object in the universe of discourse. Now addition "+" is considered as a binary FUNCTION symbol, so +(2,2) is considered as a term of the language that denotes only ONE object in the universe of discourse. Now we often write +(2,2) using the infix notation 2 + 2. so 2 + 2 is a term of the language, and so it denotes an object in the universe of discourse. Those are the rules of first order logic.

Any binary function is a ternary relation, please read the syntax and rules of first order logic.

I'm speaking about matters that are standard definitions of syntax of first order logic, its non of my manufacture.

Now if you have two terms x,y, and you have a binary relation symbol R, then the expression "x R y" does NOT denote an object of the universe of discourse, now if you substitue x by some constant a and y by some constant b, then you have the expression a R b, now this expression is something that can either be true or false, i.e. its a proposition, so a R b doesn't denote an object because truth or falsehood is of propositions and not of objects.

But if you have two terms x,y, and you have a binary function symbol F, then the expression "x F y" does indeed denote an object of the universe of discourse, and it denotes ONE object for each substitution of x,y by constant symbols, now substitute x by a constant a and y by a constant b, then the expression a F b (which is the infix form of the prefix form F(a,b)) will denote an object and it is not a proposition, i.e. the expression a F b is not something that can be true or false, notice that for example 2 + 2 is not a proposition since it is not something that we'd say about it being true or false, while for example 1 < 2 is a proposition because it is a relational expression of two constants liked by one relation symbol. Also notice that 2 + 2 = 4 is also a proposition because it contains a binary relation symbol "=" that links TERMs of the language, so 2 + 2 is the term on the left side of = and 4 is the term on the right side of =. IF 2 + 2 was not a term, suppose for example it was a binary relation expression and + is expressing a binary relation between 2 and 2, then 2 + 2 = 4 won't be a proposition because the left side is not a term and any proposition involving a binary relation symbol must have the left and right side of that symbol being "terms" of the language, because relation symbols are symbols that symbolize links between OBJECTs of the universe of discourse, and those objects can only be denoted by TERMs of the language.

To demonstrate this with an example: Take the expression "Mary is the Mother of Jesus and James" this sentence itself is not denoting an object, its denoting the relationship of Mary to Jesus and James, so it's denoting a ternary relation between a mother and two of her sons, and this relationship is itself not an object, and it is indeed a proposition that can either be true or false, so this sentence is an example of a relational expression, it doesn't by itself denote an object. That's very clear. But on the other hand take the sentence "The mother of Jesus and James", here you are seeing a functional expression, now this expressing is DENOTING an object which is Mary, here "the mother of" is a function symbol, and at the same time it is a ternary relation from "the mother of Jesus and James" to Jesus and James. Notice that "The mother of Jesus and James" is not an expression that can be false or true? No, it is a functional expression that is denoting an object (which is Mary) and not a relation between the two objects Jesus and James order for it to be true or false.

I feel that your problem is that you were thinking of the "+" sign as a binary relation symbol linking two terms of the language and so the expression 2 + 3 would NOT denote an object. Which is wrong!

By convention the "+" sign is a binary function symbol linking two terms of the language, and so the expression 2 + 3 would BE denoting an object.

Hope that helps!

Those are not my claims. Please read about the syntax of first order logic which is the background logic used in foundational systems of mathematics. Please read what it means to be "terms" of the language, and also read about "functional terms" in particular and how to differentiate it from relational expressions. — Zuhair

Sorry but you've lost my interest, too much lying and too much nonsense..

That said the symbol 2 is taken to denote a single object in the universe of discourse because 2 is a constant symbol, while the expression "x" is a term that ranges over many objects of the universe of discourse, this means that x can be substituted by many objects of the universe of discourse. — Zuhair

I explained the problems with this position. If the numeral 2 always denotes the same object then the only relation between "2" and "2" is identity. So when I write "2+2" both 2s denote the same object and the expression is meaningless unless "+" symbolizes identity. Then "2+2" would say 2 is identical to 2. But "+" does not denote identity, so either the 2's denote distinct objects or the "+" is nonsense.

Can you explain what the "+" denotes in a way which is not nonsensical, adhering to your stated principle that the symbol 2 always denotes the very same object? You now have claimed that "+" is a function which denotes an object. So with "2+2" you have one object, another object, then the first object again. That's meaningless nonsense. Or, "+(2,2), which denotes one object and two instances of a different object, still meaningless nonsense. how could there be two instances of the same object?

Any binary function is a ternary relation, please read the syntax and rules of first order logic. — Zuhair

I hate to have to bring this to your attention, but you are getting further and further from showing that "2+2" is the same as "4", because you are making "2+2" more and more complex, while "4" is simple. Clearly they are not the same object, and your demonstrations are simply proving this.

I feel that your problem is that you were thinking of the "+" sign as a binary relation symbol linking two terms of the language and so the expression 2 + 3 would NOT denote an object. Which is wrong!

By convention the "+" sign is a binary function symbol linking two terms of the language, and so the expression 2 + 3 would BE denoting an object. — Zuhair

The problem is that you have repeated stated that "2" denotes an object. Unless the "+" annihilates the existence of the object denoted by "2", to create a new object, then "2+2" cannot denote an object as well as "2" denoting an object at the same time, without contradiction. So if "2+2" denotes an object, by what means is the object denoted by "2" annihilated in favour of this new object denoted by "2+2"? And, if "2" no longer denotes an object its meaning is lost, such that "2+2' can no longer be equal to "4".

The problem is that you have repeated stated that "2" denotes an object. Unless the "+" annihilates the existence of the object denoted by "2", to create a new object, then "2+2" cannot denote an object as well as "2" denoting an object at the same time, without contradiction. So if "2+2" denotes an object, by what means is the object denoted by "2" annihilated in favour of this new object denoted by "2+2"? And, if "2" no longer denotes an object its meaning is lost, such that "2+2' can no longer be equal to "4". — Metaphysician Undercover

That's really strange. Just see the example of 'The mother of Jesus and James", this sentence is denoting a single object that is Mary, also Jesus in it is denoting an object and James too and those objects are different from Mary. Just because the whole sentence is denoting a different object from what some of its parts are denoting, it doesn't mean that it annihilates the existence of the objects denoted by its part. This is like saying if the above sentence denotes Mary the it annihilates the existence of an object denoted by "Jesus", and an object denoted by "James".

That's really strange. Just see the example of 'The mother of Jesus and James", this sentence is denoting a single object that is Mary, also Jesus in it is denoting an object and James too and those objects are different from Mary. Just because the whole sentence is denoting a different object from what some of its parts are denoting, it doesn't mean that it annihilates the existence of the objects denoted by its part. This is like saying if the above sentence denotes Mary the it annihilates the existence of an object denoted by "Jesus", and an object denoted by "James". — Zuhair

I can't see how the analogy is relevant. Jesus and James denote distinct objects. The two 2's denote one object. Even if we take "3+5" which clearly denotes two distinct objects, and say that "+" makes this into one object, this is completely different from "Jesus and James", because "Mary" does not make Jesus and James into one object. "Mary" refers to a completely different object with a relation to both Jesus and James. So if "+" makes "3+5" refer to a single object with a relationship to the objects "3" and "5", like the relationship which "Mary" has to "Jesus and James" how could this object be the same as "8"? Mary is a completely different object from Jesus and James, and not at all equivalent or the same as "Jesus and James". So the analogy really fails.

Do you see what I mean? It's very clear that if "+" makes "3+5" into a single object with the same relation to the two mentioned objects "3",and "5", that "Mary" has with "Jesus and James", this newly created object cannot be said to be the same as "8", because "8" has a completely different relationship to "3+5" from the relationship which "Mary" has with "Jesus and James"..

But the situation here is much more difficult, with a deeper, more fundamental layer of complexity. We were talking about "2+2", which you claim indicates two instances of the same object. This contradicts the law of identity already. So, before there is any point to discussing how "+" makes two objects into one, you need to demonstrate how it is consistent with your principles to treat two occurrences of "2" as denoting two different objects. According to the law of identity, there must be two distinct objects denoted here, but you claim that mathematical principles deny this. So, before you can talk about the "+" making two objects into one, we need to deal with whether "2+2" denotes two objects or one.

So if "+" makes "3+5" refer to a single object with a relationship to the objects "3" and "5", like the relationship which "Mary" has to "Jesus and James" how could this object be the same as "8"? Mary is a completely different object from Jesus and James, and not at all equivalent or the same as "Jesus and James". So the analogy really fails. — Metaphysician Undercover

Finally you are nearly getting what I mean. Yes exactly I'll re-iterate what you wrote because it captures what I said in a very good manner, but I'll add my words in betwen in two brackets

"+" makes "3+5" refer to a single object with a relationship to the objects (referred to by) "3" and "5", like the relationship which "Mary" has to "Jesus and James" — Metaphysician Undercover

That's exactly what I mean.

Now you made the quesiton

how could this object be the same as "8"? Mary is a completely different object from Jesus and James, and not at all equivalent or the same as "Jesus and James". So the analogy really fails. — Metaphysician Undercover

But 8 also refers to a completely different object from objects referred to by "3" and "5", and "8" is also not at all equivalent or the same as "3 and 5". I see the analogy is perfect! Why you say it fails?

(notice that 3 and 5 is not the same as 3 + 5, 3 and 5 is the totality of the objects referred to by 3 and 5, it is not what the + operator sends 3 and 5 to. The totality of the object referred to by 3 and the object referred to by 5 is NOT equivalent to the object referred to by 8, those are different objects, the latter is refers to an individual object, the former refers to a totality of two separate objects, so they are not the same nor are they equal).

You seem to confuse the pair of 3 and 5 which is usually written as (3,5), with (3 + 5), No! these are of course two distinct objects, much as the pair (Jesus, James) is different from (the mother of Jesus and James) are different. The analogy about this point is perfect really!

To complete the analogy:

The object referred to by "The mother of Jesus and James" is Equivalent (or the same as) the object referred to by "Mary".

Permit me to write it using = as:

The mother of Jesus and James = Mary

Now, the object referred to by "3 + 5" is equivalent (or the same as) the object referred to by "8".

so we have 3 + 5 = 8.

I see a perfect analogy, where do you see it fail?

So, before there is any point to discussing how "+" makes two objects into one, you need to demonstrate how it is consistent with your principles to treat two occurrences of "2" as denoting two different objects. — Metaphysician Undercover

Yes + sends objects denoted by the symbols it occurs between, to some object. The objects denoted by the symbols the symbol of + is written in between (in infix notations) would be sent by the + operator to an object as specified by the rules of arithmetic. Just because + occur between two symbols doesn't mean that the objects those two symbols are referring to are distinct objects No. For example "2 + 2" here the first and second "2" which are linked by + symbol, both of those do refer to exactly the same object, why? because 2 is a "constant" term of the language, so it can only refer to a single object in the universe of discourse. Now the + operator would refer that single object (symbolized by "2") into the object referred to by the symbol "4", that's it. So it referred one object (even though it had double reference, i.e. referred to by two symbols) to an outcome object. Like in saying that "Mary is the mother of Jesus and Jesus", its only saying that Mary is the mother of Jesus, the double presence of Jesus didn't change anything, it doesn't mean that there are two sons of Mary by the name Jesus.

But of course for the analogy of Mary being the mother of Jesus and James, with 3 + 5 = 8, here the analogy in some sense breaks because the mother of Jesus and Jesus is still Mary, while 3 + 3 doesn't equal 8. So regarding this point the analogy fails. Well we don't expect analogies to agree on all points anyway because we already know that the relation mother is not identical with the relation addition.

Hope that helps!

The law of identity states that a thing is the same as itself, not that a natural number is equal to itself. This is the problem, you keep asserting that the law of identity says something about equality, when it does not. It says something about identity. — Metaphysician Undercover

If I use the word same instead of equal does that satisfy you?

The fact that you say you read my post and this is your complaint means we're done. You have no substantive reply? I showed a proof from first principles that 2 + 2 and 4 are identical. You ignore it?

You're not debating in good faith.

You could have said, "Oh I categorically reject the work of Giuseppe Peano and everything he stands for." Or, "I didn't understand the chain of equalities." Or SOMETHING. Anything. But you simply will not engage substantively.

You're asserting a falsehood. There is no case in math of an equality meaning anything other than identity; whether of abstract objects (logical identity) or sets (set identity or equality). Set identity is the same as set equality.

If you don't agree that's your right, even if you haven't and can't show a single example to support your claim. But in all this time you have not presented an argument. And you have never engaged substantively. And from me to you, you're factually wrong. All the best.

ps -- I apologize if this comes off rude or confrontational. I'm genuinely frustrated that you won't engage on the substantive technical points I made. I presented a proof from the Peano axioms that 2 + 2 and 4 are the same thing. Same as in same. Same as in equal. Equal is the same as same. You simply chose not to engage. I find that too frustrating to continue the convo. It's not you, it's me.

But 8 also refers to a completely different object from objects referred to by "3" and "5", and "8" is also not at all equivalent or the same as "3 and 5". I see the analogy is perfect! Why you say it fails? — Zuhair

It fails because you are arguing that the "+" makes "3" and "5" into one object, an object which is the very same as "8", but there is no way to make "Jesus" and "James" into one object which is the very same object as "Mary". Do you see what I mean?

(notice that 3 and 5 is not the same as 3 + 5, 3 and 5 is the totality of the objects referred to by 3 and 5, it is not what the + operator sends 3 and 5 to. The totality of the object referred to by 3 and the object referred to by 5 is NOT equivalent to the object referred to by 8, those are different objects, the latter is refers to an individual object, the former refers to a totality of two separate objects, so they are not the same nor are they equal). — Zuhair

I know that, but it's not the issue. The issue is how does the "+" turn two distinct objects into one? And further, how is that created object the very same as "8"? Your analogy does not turn Jesus and James into one object. And even if we could somehow understand Jesus and James unified as one object, this object would in no way be the same as Mary.

The mother of Jesus and James = Mary — Zuhair

There is nothing in "3+5" to take the place of "mother", there is just Jesus and James. If you remove the "and" from "Jesus and James", and replace it with "mother", you have "Jesus mother James". You have nothing to show whether the mother is of Jesus, James, both, or neither. So it is really the "and" in "Jesus and James" which acts as the function, to show that one object "mother" is related to both Jesus "and" James. So let's replace the "+" with "and", then we have "three and five", just like "Jesus and James", and the analogy is good. What unites Jesus and James into one object, like you claim three and five are united as one object? Clearly it is not "mother". It is some logical principle which allows us to speak of multiple things as one object.

By applying this logical principle we can create objects through some sort of synthetic unity. Consider "2" for example. It denotes an object, a number, but within the meaning of "2", there is two distinct things signified, united within one object, 2. This is a synthetic unity, the object 2, is created by a principle of union which unites two other objects..

Yes + sends objects denoted by the symbols it occurs between, to some object. The objects denoted by the symbols the symbol of + is written in between (in infix notations) would be sent by the + operator to an object as specified by the rules of arithmetic. Just because + occur between two symbols doesn't mean that the objects those two symbols are referring to are distinct objects No. For example "2 + 2" here the first and second "2" which are linked by + symbol, both of those do refer to exactly the same object, why? because 2 is a "constant" term of the language, so it can only refer to a single object in the universe of discourse. — Zuhair

You don't see the problem here? Let's say each "2" in "2+2" denotes the same object. Also, "+" sends objects denoted to some object. We have only one object denoted, "2". There are no other objects denoted, just "2" and the operator "+". If "+" sends "2" to any object other than "2", it could be any object, randomly selected. There are no rules of arithmetic which would allow that the object "2", and the operator "+", could give us an object other than "2".

Now the + operator would refer that single object (symbolized by "2") into the object referred to by the symbol "4", that's it. — Zuhair

This is what I mean, that selection of "4" is a random choice. Why not "5", or "8", or any other of an infinity of possible objects? Why does that + operator send the single object "2" into the object "4", and not some other object?

If I use the word same instead of equal does that satisfy you? — fishfry

No, that's the point you cannot validly substitute "same" for "equal". It will produce equivocation. You don't seem to understand this.

The fact that you say you read my post and this is your complaint means we're done. You have no substantive reply? I showed a proof from first principles that 2 + 2 and 4 are identical. You ignore it? — fishfry

You showed no such proof. You showed that they are equal, according to equality theory, and you claimed that equality is based in the law of identity which it is not. Therefore you have no proof. Do you understand? Your proof requires that the law of identity says something about equality, which it does not. Therefore you have no proof.

There is no case in math of an equality meaning anything other than identity; whether of abstract objects (logical identity) or sets (set identity or equality). Set identity is the same as set equality. — fishfry

We're just rehashing the same thing. It may be the case, that in math equality means identity as you claim, but as I explained to you, this is not "identity" as defined by the law of identity. So you (and all other mathematicians who say this) are using "identity" in a way which is inconsistent with the law of identity. Do you understand that the law of identity states that a thing is identical to itself? So to use "identity" as equality and then claim that equality is supported by the law of identity in that way, is equivocation, plain and simple.

If you don't agree that's your right, even if you haven't and can't show a single example to support your claim. But in all this time you have not presented an argument. And you have never engaged substantively. And from me to you, you're factually wrong. All the best. — fishfry

My example is the law of identity. It is very simple, and very explicit. A thing is the same as itself.

So for example, let's define equal to, as identical to, as you are wont to do. Then we'll adhere to the law of identity for our definition of identical. Clearly "2+2" is not identical "4" according to the law of identity, therefore "2+2" is not equal to "4" by your own definition of equality. That is a substantive argument if I've ever seen it. Now, Zuhair has wasted pages trying, to no avail, to demonstrate that "2+2" actually is identical to "4". But Zuhair is just digging a deeper hole, recognizing the truth that "2+2" is not identical to "4", neither is what "2+2" signifies identical to what "4" signifies, so Zuhair tends to lie in an attempt to get out of the hole.

Equal is the same as same. — fishfry

How can you say this? All human beings are said to be equal, but no two are the same. Many things are said to be equal which are not the same. Do you recognize this? If so, how can you say that "equal is the same as same"?

Zuhair thinks that mathematics would be so much better, stronger, if instances of "equal to" really meant "identical to", so Zuhair will forever argue that this is actually the case, in order to create the illusion that mathematics is stronger than it really is.

This is what I mean, that selection of "4" is a random choice. Why not "5", or "8", or any other of an infinity of possible objects? Why does that + operator send the single object "2" into the object "4", and not some other object? — Metaphysician Undercover

Because the rules of arithmetic and and the arbitrary definitions dictates that! I showed you how formally this can run in a prior comment on a system that is by far much easier than PA. I'll re-present it here:

2 is Defined as the object that the + operator would send the single object 1 to. (this is: 1 + 1 = 2)
4 is Defined as the object that the + operator would send objects 3 and 1 to. (this is: 3 + 1 = 4)
3 is Define as the object that the + operator would send objects 2 and 1 to. (this is: 2 + 1 = 3)

Notice that the choice of symbols in those definitions is really arbitrary, there is no control whatsoever on choosing 2 as the symbol for what the + operator would send the object 1 to, nor there is any control on choosing the symbol 4 to represent the object the + operator would send the objects 3 and 1 to, and same arbitrariness apply to choosing the symbol 3 to represent what the operator + is sending the object 3 and 1 to. Yes these are arbitrary. But once made, then we cannot change them, since + is a function, it permits only one outputs per specific input objects.

Now from those definitions and from the rules of arithmetic, it is here where arbitrariness would stop, because we'll be enforced here to say that 2 + 2 = 4, in other words the object the operator + is sending the object 2 to is 4.

Proof:
by definition of 4 we have: 3 + 1 = 4 [this is an arbitrary definition as you said]
by definition of 3 we have: 2 + 1 = 3 [this is also arbitrary definition]
by rule of identity (the substitution schema) we can substitute identicals! so we have:
(2 + 1) + 1 = 4
By associative law we have
(2 + 1) + 1 = 2 + (1 + 1)
by identity (substitution schema) we substitute identicals to have:
2 + (1 + 1) = 4
but by definition of 2 we have 1 + 1 = 2 [arbitrary definition of 2]
by identity (substitution schema) we substitute identicals to have:
2 + 2 = 4
QED

So arbitrariness ends after we have made the choice of the definitions and the choice of the axioms and the choice of the inference rules, after making those a machinery would set in and it dictate how the arbitrarily chose symbols would related to each other.

So YES, definitely part of it is indeed arbitrary, the starting part!

It is some logical principle which allows us to speak of multiple things as one object. — Metaphysician Undercover

Yes, there is a big debate whether this is to be called as a "logical" principle. Nowadays it is generally held to be a mathematical principle. This is the principle of "Set", a set is what turns multiple objects into one entity, it turns Jesus and James into one entity which is the pair {Jesus, James}. But still you are confusing the + operator for "and", you think that the + operator is that joining logical principle that turns multiple objects into a single entity, you are confusing the + operator for the set operator, which is not correct. View the + operator just and a "sending" rule, a rule that sends objects to objects that's all. So for example we can view "Mother" in the above sentence as a "sending rule" it sends the pair {Jesus, James} to another object which here happens to be Mary. EXACTLY a similar thing is happening here the "+" operator is sending the pair {3,5} to another object which by rules of arithmetic and arbitrary definitions this other object is enforced to be 8. The rules of arithmetic (after making the arbitrary definitions of each number) would "control" this assignment (sending) of objects, it would control which object the + operator would send the pair {3,5} to.

Think of the + operator as a sending scheme, that's all.

It fails because you are arguing that the "+" makes "3" and "5" into one object, an object which is the very same as "8", but there is no way to make "Jesus" and "James" into one object which is the very same object as "Mary". Do you see what I mean? — Metaphysician Undercover

I didn't say that + would make 3 and 5 into one object, I said it will send them to one object, if I did say that it makes them into one object, then I only meant that it would send them to one object. + is not a merging process, it is an assignment scheme.

Well, that is not completely right. I understand very well that "=" can be understood as equivalence relation, I concede to that. I personally would prefer it to be understood as "identity" relation, since this "at least to me" would make matters easier to control from the formal workup side, identity is a the sharpest kind of equivalence relation. But anyhow it's not necessary to interpret "=" as how it is used in mathematics as the strong notion of "identity". But some theories I think would fare far better if they do that, for example Set Theory, here to say that the set X defined for example as: for all y ( y in X if and only if y=empty set ), this is usually symbolized as {{}} or as {0}. Now to say that this set is a "SINGLETON" set is to say that it has only ONE member, but if we just stipulate that "=" is an equivalence relation that can occur between distinct (non identical) objects, then it would be absurd to label it as singleton since there is no guarantee for it having just one element, it can indeed have MANY elements all being "equal" to the empty set, I would have called it "EQUALTON". So I think in the context of set theory, the equality sign is better to be understood as identity relation. By the way I need to conceded that this is also not necessary 'technically speaking' since indeed we can take equality to be just an equivalence relation, but to me that would make matters murky on both the informal intuitive and the formal technical workup accounts, i.e. I mean as far as set theory is concerned.

For the more mathematical looking foundational theories like PA, of course it suffices to interpret equality as just being an equivalence relation, and that it can hold between distinct objects, that might give us some room of freedom in making some interpretations, actually PA is presented officially with its equality symbol just stipulated as an 'equivalence relation' plus some simple closure principle on naturals, so indeed it is officially treated just as an equivalence relation closed on naturals.

There is nothing in "3+5" to take the place of "mother", there is just Jesus and James. — Metaphysician Undercover

There is (implicitly) "sum of". (Not that the analogy follows through completely, as @Zuhair points out.)

But anyway, instead of trying to clear up his actual or perceived misunderstandings about out-and-proudly platonic math concepts, shouldn't the mathematicians offer the finitist (especially since he objects to the identity of the 2's in 2+2) cardinal arithmetic and see if he is satisfied with that?

Because the rules of arithmetic and and the arbitrary definitions dictates that! I showed you how formally this can run in a prior comment on a system that is by far much easier than PA. I'll re-present it here: — Zuhair

No you did not show me that. You showed me "ternary relation". You did not show me how one object and the "+" function makes a completely different object.

2 is Defined as the object that the + operator would send the single object 1 to. (this is: 1 + 1 = 2)
,,,
This is the principle of "Set", a set is what turns multiple objects into one entity, it turns Jesus and James — Zuhair

See, you have inconsistency here. First you are talking about sending one object to a completely different object, then you are talking about turning two objects into one object.

So for example we can view "Mother" in the above sentence as a "sending rule" it sends the pair {Jesus, James} to another object which here happens to be Mary. EXACTLY a similar thing is happening here the "+" operator is sending the pair {3,5} to another object which by rules of arithmetic and arbitrary definitions this other object is enforced to be 8. — Zuhair

OK, but this does not satisfy what is required in order to say that "3+5" denotes the same object as "8", which is what you are arguing, that these two are identical objects. All it shows is that the two objects "3" and "5" are related to a third object "8", like Jesus and James are related to Mary. It does not make Jesus and James identical to Mary. Your example introduces a fourth object, " the mother of". Mother is not a sending rule at all, it is a named object.

I didn't say that + would make 3 and 5 into one object, I said it will send them to one object, if I did say that it makes them into one object, then I only meant that it would send them to one object. + is not a merging process, it is an assignment scheme. — Zuhair

The debate is whether "2+2" denotes the same object as "4". If the sending rule only "sends" "3+5" to the third object "8", and does not make those two objects into the object denote by "8", it does not fulfil the requirement of saying "3+5" denotes the same object as "8".

There is (implicitly) "sum of". (Not that the analogy follows through completely, as Zuhair points out.) — bongo fury

The analogy is ill, and unacceptable. What acts as the sending operator in "Mother of Jesus and James", is the :"and". It sends both "Jesus" and "James" to the same object, "the mother of". Zuhair wants to make "mother" the sending operator, sending "Jesus" and "James" to "Mary". In reality though, "Mary" is just another name for the object called "the mother of" and the analogy is way off base.

But some theories I think would fare far better if they do that, for example Set Theory, here to say that the set X defined for example as: for all y ( y in X if and only if y=empty set ), this is usually symbolized as {{}} or as {0}. — Zuhair

What are you saying, that an empty set is the same as a set having only one member, "0"? You know that 0 is a mathematical object, just like any other integer don't you? therefore the set is not empty.

As I say, though, at what point does arithmetic become a philosophical puzzle for you, or your child?

Have addition as union of disjoint sets. Is that ok?

No, that's the point you cannot validly substitute "same" for "equal". It will produce equivocation. You don't seem to understand this — Metaphysician Undercover

It was a typo, a Freudian one if you like since I agree that this particular error makes you totally right and me totally wrong. I get that.

From my end it was a totally honest error. As a math person I'm so used to talking about equality that I never think about logical identity. I'm learning a lot from this thread. I typed "equal" because I've done it a million times in math; and the only time I talk about an identity is something like $e^{i \pi} + 1 = 0$. a specific numeric and/or algebraic identity meaning it's true by virtue of syntax. It's a logical truth. But as I think of it, that's logical identity too.

More than ever I see that mathematical equality is the same thing as logical identity. The same morally and the same technically in any mathematical framework you like.

By the way if called on to do so, I could drill that symbology down to an identity of sets. The thing on the left and the thing on the right are the same thing. It was true even before Euler discovered that fact in 1740. At that moment it became new knowledge; but it did not suddently become a new truth about the world. It's a mathematical truth. It was always true.

And I do acknowledge that my fingers typed equality when in the context of the discussion I should have used the word identity; and that this purely random or perhaps Freudianly determined; either way, it was one hell of a bad typo. I see that it generated a lot of confusion. Please just substitute "identity" or "logical identity" in my argument. My apologies.

To sum up my view I would say that

* On the philosophical aspect, I suspect I have much to learn. I'd be surprised to find out that logical identity is somehow different than mathematical equality; but I would not be surprised to be surprised.

* On the math aspect, you're just wrong. But neither of us has said anything new for quite some time, and I have nothing to add. Only that I'm disappointed at a personal level that I took the trouble to work out an immaculate technical proof; and you are just totally disinterested in actually following and engaging with the argument. It's your privilege not to engage, but that is definitely a disappointment at my end. I'd say to myself "Pearls before swine" but I'd never say such an uncharitable thing in public. But the phrase did pop into my head, and that does about sum up how I feel about the mathematical aspect.

I actually just popped in tonight to mention that someone, possibly a member of this forum, posted the following to philosophy.SE today:

https://philosophy.stackexchange.com/questions/67227/is-there-a-difference-between-equality-and-identity

Some interesting thoughts there.