What you are saying is definitely intelligible and sane, there is no problem with it. Although I might disagree with you about self referring terms, but no problem really. We can take terms to be "denoting" objects OTHER than them, no problem at all. So we must have a universe of discourse those expressions are speaking about. Anyhow. That won't change matters so much, since it is agreed that a relation symbol symbolizes a relation between the objects denoted by term symbols that this relation symbol syntactical is coupled with. To give an example of that, lets take the relation symbol "=" denoting equality, here = is a binary relation symbol, so it symbolizes a relation occurring between the objects denoted by the symbols that the = symbol links. Which symbols the = links, the answer is that it links the expression 1 + 1 to the expression 2, so the = sign here represented an equality relation occurring between the objects denoted by these expressions. It is always the case that relations are between objects, and so relation symbols must link terms, because terms are the symbols that denote objects, this is because the symbolization must copy what is symbolized. Since equality is a binary relation between objects, then the symbol for equality, which is "=", must be written as linking symbols that denote objects, since = links 1+1 to 2 then 1+1 must denote an object, and 2 must denote an object. That's why 1+1 must be an expression that denote an object.

In reality 1+1 is a tricky expression, it has many denotations, let me present those

The first 1 denotes an object
The second 1 denotes an object
The string 1+1 denotes an object
[All these three objects denoted can be distinct, since equality is not necessarily identity]
The + sign is denoting a ternary relation that is occurring between the above three objects.
[Imagine that like the the expression "the mother of Jesus and James" here Jesus and James are denoting persons, the whole expression is denoting another person "Mary", "the mother of" is denoting a relation between objects denoted by Jesus, James and by the total expression above.

Now lets take the expression 1 + 1 = 2
Here we have all of the above four denotations, and to it there are the following denotations
2 is denoting an object [which can be distinct from all of the above denoted three objects]
= is denoting an equality relation between the object denoted by 1+1 and the object denoted by 2.

Therefore "2+2" cannot identify an object, because it is not a proposition, but "2+2=4" may be a proposition which identifies "4" as an object. — Metaphysician Undercover

2 is referring to an object (which is itself here), but to identify it in relation other symbols by using the particulars of a certain language (for example in arithmetic those mount to +,x,=,< etc.. symbols) then we'll need propositions, but those can only occur by relating it by a relation symbol to other term symbols so 2= 1+1 won't have any meaning if 1 + 1 was itself not a term of the language denoting some object (which can be taken here to be the string 1 + 1 itself), otherwise if 1 + 1 is not an expression denoting an object (i.e. a term) then how can we related 2 to it via the equality symbol = which is a binary relation symbol (sometimes called two place relation symbol), the whole string of symbols would be meaninging much like writing 2= 1<3 i.e. 2 is equal to (1 being smaller than 3), this is meaningless, it is not a proposition, same if we say 2 = 1 + 1 and envision 1 + 1 as a relational expression expressing a binary relation + occurring between 1 and 1, then we be saying ( 2 is equal to (1 having + relation to 1)) which is meaningless because an object is equal to an object and not to a relation. While if we treat 1 + 1 as a term of the language, lets say it denotes itself, so here 2=1+1 would be (2 is equal to the string 1+1), and this makes sense, since this equality is just a syntactical equivalence.

Perhaps, the first "2" denotes something different from the second "2", and the "+" denotes a relation between these. — Metaphysician Undercover

No problem with two 2's in 2 + 2 being denoting different objects, since they can be interpreted as denoting themselves and they are of course distinct. Now the + sign is just a symbol here that we are seeing it between these two symbols combining all three to get 2 + 2 Now this is by itself as you agreed with me is not a proposition, clearly it is not something that we'd label as true or false. Now generally speaking when we are in a mathematical language we must specify which symbols are taken to refer to objects (even if to themselves) which we call as "terms" and which symbols are taken to refer to "relations between objects" we call them "predicate" or "relation" symbols. Now not every string of symbols constitute a statement of the language, and what we mean by statement of language is actually a proposition, something that can be said of as being true or false. The smallest kind of propositions c are constituted of a relation symbol and all terms that it relates, so if for example you have R being stated as an n-ary relation symbol, so the string of symbols expressing that R would be to concatenate it with n many term symbols in some specific (prefix, infix, etc..) so if R is a binary relation then we concatenate it with two symbols, and so on... which only makes sense because we think of relations as relations between "objects" and objects are represented by "terms" so for a binary relation symbol R we have propositions of the general syntax of:

term R term

In nutshell relation symbols link terms. So for example = is a binary relation symbol, so it must occur between two term expressions, i.e. expressions taken to represent objects. Lets take (2 + 2 = 4)
Now for = to be a relation symbol it must occur between terms, so the totality of whats on the left of it must be a term and so is what's on the right of it, 4 is clearly a term, so 2 + 2 must be a term, otherwise if 2 + 2 doesn't signify a term (i.e. a symbol referring to object) then what = is relating to 4? either 2 + 2 is a relational expression (similar to 1<2) but those are not put next to relation symbols, image the string
1< 2 = 4, it doesn't have a meaning, it is not a proposition, or 2 + 2 might be neither a proposition nor a relation symbol, but this is like for exame 2+ = 4 here "2+" is an example of a string that is neither a term nor a proposition, it even cannot be completed with =4 to produce a proposition.

In order for "2+2" to be completed with "=4" to produce a proposition, then 2 + 2 must be a term of the langauge, and thus denoting an object, even if that object is the string of the three symbols itself!, otherwise we cannot complete it by adding to it a relation symbol and a term after it.

Notice that not every string of symbols in a language are taken as well formed formulas of that language for example 2 + 2 = is a string of symbols, it is also incomplete, it doesn't represent a term nor a relation, even though it is composed of two terms (the "2") and another term (2+2) and a relation symbol =, but here it doesn't constitute a proposition and it is not itself denoting a term. When you add 4 to it of course it becomes a proposition. So not every part of a proposition is a term or a proposition, examples are 2+, +2, 2=, =4, etc.. all are neither proposition nor terms

2 + 2 is definitely not expressing the occurrence of a binary relation between the two 2's, otherwise it would have been a proposition and we know that 2 + 2 is not a proposition, we know that 2 + 2 = 4 is a proposition and we know that = is a binary relation sign, and we know that it only links terms, so 4 must be a term and 2 + 2 must be a term, and so 2 + 2 must denote an object, now the first 2 denote itself, the second 2 denote itself (those are distinct objects occupying different places in a written expression), and the string 2 + 2 is itself also denoting itself, so the total denotations involved in 2 + 2 is three kinds of denotations each of the 2's denoting themselves and the total expression 2 + 2 also denoting itself, while the expression 2+, +2 , do not denote neither a term nor a relation they are incomplete expressions.

The equality sign = only says here that the object 2 + 2 is equal (i.e. related equivalently) to the object 4, of course they are indeed distinct since 4 is clearly distinct from the symbol 2 + 2, yet they are equal.

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". — Metaphysician Undercover

I thought we got over that point. I agreed with you that "=" is NOT necessarily the identity function, so why you are returning the discussion backwards. I agreed with you that if you interpret "=" just as an equivalence relation (as it is officially formalized in PA for example), then of course the object that the + operator send objects denoted by 3 and 5 to, is NOT necessarily identical with the object denoted by 8. We already passed this point. The debate now is not about that. The debate is about what is the operator +. To me it is nothing but an assignment scheme, i.e. a sending rule, nothing more nothing less, it sends maximally two objects to a third object. Actually although I don't want to go there, one of the intended interpretation of arithmetic is as a closed syntactical system, i.e. non of its expression denotes anything external to it, so for example under that line of interpretation the symbol 2 means exactly that symbol itself, and so for example 2 + 2 has "distinct" symbols on the left and right of the + sign, and although they are "similar" in shape, yet they are two different objects since they occupy different locations on the page, each 2 is denoting itself only. Now also 4 denotes itself only, also to further agree with you 2+2 is denoting nothing but itself (the totality of the three symbols) and so it is NOT the same as 4, not only that every individually written 2 is not the same (identical) to the other, and the equality in 2+2=4 doesn't entail at all identity of what is on the left of it with what's on the right of it, its only an equivalence syntactical rule, and can be upgraded to a substitution syntactical rule without invoking any kind of identity argument at all, and the whole game of arithmetic can be understood as a closed symbolic game nothing more nothing less. This is the extreme that one can go with interpreting equality as just an equivalence relation and not being identity, we'll need to revise our definition of "constants", "functions" to accommodate that. But still we need to maintain that expressions like 2 + 2 denotes an object while expressions like 2 > 1 denotes relations (linkages) between objects and such that expressions like 2 + 2 cannot be labeled as true or false since they are by the rules of the game not propositions, while expressions like 2 > 1 are propositions and they are to be spoken about of being true or false.

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.

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.

Of course, science is different from Religion.

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.

I understand but the practical solution, if you can call it such, is check for the veracity of what is being said rather than the honesty of the person who's saying it. — TheMadFool

Most of the times, you cannot do that, because Religion contain a lot of accounts that cannot be objectively verified.

There is another problem with miracles, that is how can we know that they are outside the capacity of those stranger beings? I mean those beings are more intelligent than humans, have more knowledge, so for them to come up with acts that humans cannot achieve, is no surprise at all. It only proves that those so called angels are more intelligent and capable than humans, that's all, it doesn't mean that those miracles are breaking all rules of nature that only God can break? We ourselves do not know what nature is capable of doing, so completely natural beings that are more intelligent than us can of course bring about deeds in a natural way that we cannot do and so appear miraculous to us, much as our modern technology would appear to a stone age man. And by the way this is no personal verification of the honesty of those beings, it only attests to their higher capabilities than us, which is just a relative point about capabilities and not about honesty! It doesn't mean at all that they are honest. Just because they are more clever, more advanced than us, doesn't at all mean that they are telling the truth about themselves when they made those revelations to us humans. Actually to try to personally verify such beings can only happen if we had exposure with their race and see the signs of lying on those creatures, etc.., which we don't have for the time being. Actually the more intelligent is a being the more its capability of deceiving less intelligent beings, so one must actually be more cautious with such exposures.

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!

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?

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".

I generally agree to the subjectivity matter about religious experience. But here I'm dealing with what I might label as "responsible stance" in front of religious claims. Religious claims can be very loud and very dangerous, it can speak in the name of the most powerful imaginable being, and this puts power into their words among their adherents that they can go to the extreme in abiding by them, which is dangerous if those claims were actually not coming from the real God should that exist. So religions can be the basis for secondary delusions (firmly held erroneous conceptions) and these are very dangerous. As it stands, all major religions of the world are guilty of making such claims.

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!

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.

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.

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?

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.

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.

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. I am arguing that no mathematical axioms can be interpreted in this way because it is fundamental to mathematics that the two sides of the equation represent distinct things, while the law of identity indicates that "the same" refers to one and only one thing. — Metaphysician Undercover

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.

You say that it is fundamental to mathematics that the two sides of 2 + 2 = 4 must represent distinct objects.

I say that if = stands for identity, then it would mean that 2 + 2 denotes (represents) exactly the same object that 4 represents (denotes). Obviously you object to that, you say that there is something fundamental against this.

what is that fundamental aspect that enforce us to interpret = sign as some equality relation other than identity. Notice that identity relation is a kind of equality relation, but the converse is not true, you can have an equality relation that is not identity. OK. But why = as used in mathematics, for example in arithmetic, why it is not reducible to identity in your understanding?

Notice that Peano arithmetic which is a very famous theory of arithmetic, is indeed formalized nowadays as an extension of first order logic with identity, of course with the understanding that "=" is taken to represented identity relation and not any other kind of equality.

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.

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?

Didn't you just say "all" older mathematical systems can be formalized as systems where "=" represents identity? And now you ask about those which cannot. Oh what a tangled web we weave when first we practise to deceive. — Metaphysician Undercover

I didn't ask about those which CANNOT, I asked about those which are not. I mean are not presented in a formal manner as an extension of first order logic with identity. Of course they can be formalized as an extension of first order logic with identity, but I'm asking about when we don't do that and leave it un-formalized. Then in this case how are we to understand "=" symbol in them. I'd say that it is not necessarily the identity symbol. Yes that is correct of course.

Therefore "equal" in ZFC cannot mean "same" as determined by the law of identity. — Metaphysician Undercover

if ZFC is presented as an extension of first order logic with identity, then of course "=" would stand for identity. IF we don't do that, then of course it would not necessarily stand for identity.

Only specific mathematical systems are based in first order logic, perhaps ZFC is one of them — Metaphysician Undercover

Not only ZFC, you have PA (peano arithmetic) nowadays presented as an extension of first order logic with identity. And there are many other systems also so presented, you can read about reverse mathematics. Anyhow almost all of traditional mathematics before the era of set theory and modern mathematical logic, nearly all of it can be re-formalized as extensions of first order logic with identity systems, and of course the "=" in them would be understood to represent identity.

It is very clear that ZFC derives its meaning of "equal" from the traditional meaning of "equal", and not from the law of identity, because ZFC does not cite the law of identity, and as we've seen, it allows that two distinct things are "equal". Therefore "equal" in ZFC cannot mean "same" as determined by the law of identity. — Metaphysician Undercover

As I said above, this depends on how you formalize ZFC, if you formalize it as extension of first order logic with identity then the = symbol in it would stand for identity. If not then it can stand for some other equivalence relation.

There you go, continuing with your lies. You are fully aware that this is not true, being the well-educated individual that you are. Yet you assert it anyway! Why lie? What's the purpose? — Metaphysician Undercover

I don't know why you keep assuming that I'm lying? Anyhow. The fact that nearly all of traditional mathematics can be formalized as extension of first order logic with identity is well known, you can see reverse mathematics for that. And you can serf the web for Harvey Friedman's grand conjecture, etc..

Exactly, an "ordered pair". And an ordered pair is two objects. Why say that this is false? Your propensity for lying never stops amazing me. — Metaphysician Undercover

The ordered pair of the two objects, here in your example (2,2) is not what is denoted by "2 + 2", I'm trying to tell you that but you keep refusing to listen, the expression "2 + 2" is the object that the + operator send the ordered pair (2,2) to. To clarify this: the + operator is sending the pair (2,2) to some object call this object k, to represent that for you by an informal sketch:


(2,2) ---+---> k

Now "2 + 2" is that object k, in other words "2 + 2" is not denoting the ordered pair (2,2), No! '2 + 2' is denoting the object that the operator + send the pair (2,2) to, and that object, i.e., k is exactly the natural number denoted by the symbol 4. In other words "2 + 2" is denoting exactly the same object that 4 is denoting. That's the easiest way to understand it.

You may say No. not necessarily, 2 + 2 is denoting an object k, and 4 is denoting an object L, where L is not identical to k, but L is equal to K, i.e. L is possessing some relation R to k where R is some equivalence relation that can occur between distinct (non-identical) objects. So according to this 2 + 2 is denoting an object that have the relation R to the object denoted by 4 where R is some equivalence relation that is not necessarily the identity relation, of course the intention is that the = sign stand for that equivalence relation R. OK this is a possible case of course, but this is more complicated! It is much easier to stipulate that R is the identity relation itself.

Anyhow as I said before if you present arithmetic or any mathematical theory that contain the symbol = as an extension of first order logic with identity, then = would be taken to symbolize identity itself. if not then it can stand for some other equivalence relation.

I hope that settles matters.

One object but two digits — Shamshir

Yes! On the informal level I would agree, but formally NO. It only symbolizes a number that the operator + is sending the pair (2,2) to. It doesn't speak of anything of that number having two or more digits. In reality the best representation of four is as four strokes, but this is besides the formal system of arithmetic actually.

Your site provides the terminology of first order logic, not mathematics. The use of "=" is not the same in first order logic as it is in math. To equate these two is to equivocate and that is a fallacy of logic — Metaphysician Undercover

Well PA is a mathematical system. Most formal mathematical systems nowadays are stipulated as extensions of logical systems, in particular first order logic with identity. And it is about those mathematical systems that I was speaking. Even older mathematical systems like ordinary math, all those can be recaptured more effectively as systems extending first order logic with identity. I've shown you the axioms of first order logic with equality and you replied that the equality sign in them is not about identity, when I showed you that this is just a terminology preference, and that it is also named as first order logic with identity and I showed you the rationale behind those axioms and its relationship to the informal notion of identity, you replied that this is not mathematics. In reality all older mathematical systems that you know of can be formalized as extensions of first order logic with identity, and in those systems the symbol = is taken to represent identity.

Now the question is what about older systems that are not formalized as extensions of first order logic with identity, can we understand the = in them as something other than identity, the answer is yes of course, it can be taken to be an equivalence relation, and I agree you'd better name them as equality, since it is not necessarily the identity relation. But formal recapturing of them as extensions of first order logic with identity with the = sign taken to represent identity, is by far a much sharper and more well defined an rigorous approach.

But anyway your argument that the expression '2 + 2' is taken to represent two objects is outright false, even in ordinary math the expression '2 + 2' is taken to denote a single natural number that is sent to by the + operator from the pair {2,2} [more precisely one must write it as (2,2) since it is an ordered pair], it doesn't denote two natural numbers as you think, because + is a FUNCTION.

possibly I don't agree with R&W on that. + is not a logical operator, it is a mathematical operator, but as you know we can speak logically about extra-logical concepts, we can add them to any logical system, but of course the result is not a purely logical system, but a logically compatible system you may say, usually refereed to as logically extended system. The trivial complete system that I've depicted is not a pure logical system, it is a logico-mathematical system. I think it's complete. i.e. not subject to Godel's incompleteness theorems, but I'm not sure really.

But '2+2' denotes two objects, each with a value of two. What do you think the '+' sign is there for, decoration? — Metaphysician Undercover

Here is your error, you think that '2 + 2' denotes TWO objects. This is wrong. You are not understanding the operator "+", this is a two place FUNCTION symbol, you need to read some logic related to mathematics, i.e. foundational work on mathematics. "+" is a two place FUNCTION, it means that it is a ternary relation that sends a pair of objects to ONE object for that particular pair, so suppose you are summing 9 and 8 here the addition "+" function would send the pair {9,8} to ONE number that is 17, in other words view addition as some process that at each time it has TWO INPUTS and ONE output such that whenever you input the same values again you get the same output again. Now it is important to understand what the expression "2 + 2" means, it means the OUTPUT of summing 2 with 2. In other words the expression "2+2" denotes the object that the operator + would send the pair {2,2} to. I hope this is clear. So '2 + 2' by definition of functionality of "+" cannot denote two objects. The appearance of two symbols in it, i.e. the symbol "2" appearing twice, doesn't mean that "2 + 2" is denoting two objects at all, "2 + 2" is the VALUE of the function + for the pair {2,2}, and it is ONE object. You are not discriminating between "denotation" and 'information "predication" accompanied with that denotation', '2+2' denotes ONE object and only ONE object which is the value of the + operator on the pair {2,2}, but '2 + 2' carries information [this is not denotation] related to that denotation, that the single object denoted by '2+2' can be split in half, i.e. it is the value of a pair having identical projections, that doesn't mean that it is denoting two objects at all. On the other side the expression '4' is a constant symbol, it also denotes a single object, but a constant symbol is a zero function symbol, so it does NOT carry with it any additional information about what it denotes, but at the same time it doesn't denote absence of any kind of information about what it denotes, so it doesn't denote an object that is not divisible in half, i.e. cannot be the value of + function from a pair with identical projections, it cannot assert that negative information about what its denoting because it is a ZERO place function symbol.

Your main error is that you think that "2 + 2" is denoting two objects.

By showing parts, '2+2' indicates a particular division of the object, unlike '4' which indicates no such difference. So '2+2' denotes an object divided in a particular way, in half, whereas '4' denotes no such division. Therefore '2+2' denotes a different object from '4'. — Metaphysician Undercover

You say '2+2' denotes an object divided in half. Well I'd say: OK no problem.

You continue saying whereas '4' denotes no such division.

Yes the correct wording is that '4' doesn't denote such a division, this is clearer. However it doesn't deny it? You seem to be confusing : Not denoting phi , for denoting not phi. So you seem to be arguing that since '4' is not denoting that the object it denotes is an object that is divided in half, then it follows according to your reasoning that 4 is denoting an object that is not divided in half. This is an error. Not claiming something doesn't mean that you are claiming its negation. I'm not claiming that my son would pass the exam, it doesn't follow from this that I'm claiming that my son will not pass the exam.

So 4 not denoting that what it denotes is dividable in half, doesn't mean that 4 is denoting an object that is not divisible in half.

Absence of denotation doesn't mean denotation of absence.

Absence of denotation just signal incompleteness of information.

We are not claiming that expressions supply FULL information about what they are denoting.

2 + 2 only shows some extra-information about what it denotes more than the constant symbol 4 shows about what it denotes. That doesn't mean that what they are denoting is not the same object. I can say that Barack Obama is one of the presidents of the united states. Another time I can say that Barack Obama is one of the presidents of the united states that has a Nobel price. The first expression did NOT denote that Barack Obama had a Nobel price, yet I didn't deny it! It is only the case that the second sentence had more information, but both are speaking exactly of the same person. In a similar manner 2+2 and 4 are denoting exactly the SAME object, but 2+2 is denoting more information about that object than 4 does, but again 4 is not denying what 2+2 is denoting.

This is proof of your's and fishfry's mistake. You cite "equality axioms". Equality axioms are not identity axioms. You and fishfry both arbitrarily replace "equality with identity. Sophistry rules! — Metaphysician Undercover

No! Equality rules are spoken as Identity rules by mathematicians, it just happens that equality is used more: see this site on terminology:

Glossary of First-Order Logic

Just use the find function on your browser, and search for "identity" and read all of what it says about it in that site.

The "=" symbol is used to symbolize identity, so x=y actually means that x and y are exactly the same object, i.e. they are identical, and not that they are having the same value and remain discriminate at the same time.

To be more precise, due to shortages of formal languages, it is better to call identity as indiscernibility, because under that theory in question we say that x and y are identical if the theory in question cannot have an expression phi(x) (written in the language of the theory and in which x occur) and an expression phi(y|x) [which is the expression obtained by merely replacing all occurrences of x by y in formula phi(x)] such that phi(x) is true of x and phi(y|x) is not true of y. So we say that x and y are indiscernible under the language of that theory. Of course that doesn't necessarily mean that they are in reality identical, it just means that the theory in question cannot discriminate between them and so it see them as "identical", i.e. it says that they are identical.

The indiscernibility of identicals is a famous law, and in first order logic it is the law that I wrote (and that ironically you said it is not about identity??] see:

Leibniz's law

About the first law of identity which is reflexivity law, i.e. that every thing is identical to itself, this is just a trivially true statement about identity, there is no dispute about that.

So the theory that fishfry and I are mentioning is about "identity", yes its known as equality theory, other sources name it as identity theory, but basically it is about 'identity" as indiscernibility under substitutivity, and it is certainly not about equality as common reference (which is what you think it is about), it doesn't make sense to think of it as being about common reference, why should we have a law about indiscernibility of objects that has common value under certain functions??

However, as I said you can "technically" speaking have some theories that see some objects as identical, but other theories can discern between them, yes this can happen, much as we human can see a star and think its one while in fact it is two or more stars.

On the other hand if we are to understand Equality in YOUR sense as assignment to a common object, like in having a fixed function F over a certain domain D, so we'll say that all elements of D are equal under F, to just mean they are assigned the same value (image) under F. Note here that D can have many members. This use of 'equality' is not perfect, it is mentioned in common languages like that, yes, but it is imprecise, it hides a lot of details, and certainly it is NOT what is meant by equality which is symbolized by "=" in mathematics. In mathematics when = is used it is meant to symbolize "identity", i.e. sameness of objects, and not assignment to a common value as you think.

Equality as used in PA and in ZFC, and generally in first order logic with equality, that is symbolized by "=", in those contexts it exactly means identity or sameness of objects, more precisely speaking that the theory in question cannot discriminate between x and y if it proves that x=y.

I just wanted to add, that we can actually have a very simple system in which 2 + 2 = 4, that of first order logic and add to it primitives of identity (equality) symbolized as "=" which is a binary relation symbol, and of "+" denoting addition which is a two place function symbol, and of "1" denoting what we customarily know as one, which is a constant symbol. I'll try to coin a system in which 1 is the first number, i.e. doesn't have zero in it.

Axioms:

Equality axioms:
1. for all x (x=x)
2. if phi(x) is a formula in which x occur free, and never occur as bound, and y doesn't occur, and phi(y|x) is the formula obtained from phi(x) by merely replacing each occurrence of the symbol x in phi(x) by the symbol y, then all closures of

for all x,y (x=y -> [phi(x) <-> phi(y|x)])

are axioms

Addition axioms:
x + y =/= 1
x + y =/= x
x + y = y + x
(x + y) + z = x + (y + z)

Define: x=2 iff x=1+1
Define: x=3 iff x=2+1
Define: x=4 iff x=3+1

Theorem: 2 + 2 = 4

Proof:
By definition of 4 we have: 3+1=4
By definition of 3 we have 2+1=3, use identity axioms to replace this and get:
(2+1)+1=4
By associative law we have (2+1)+1 = 2+(1+1), use identity axioms and replace to get:
2+(1+1) = 4
then by definition of 2 we have 2= 1+1, so by identity axioms replace to get:
2 + 2 = 4
QED

Actually what is used in the above proof is only the definitions of 2,3,4, and the identity and associative laws.

I think theory of addition is complete as far as I know.

You seem to have left something out. You've taken the '+' for granted. You've shown me what '2' represents, and you've shown me what '4' represents. Then you claim that '2+2' magically represents the same thing as '4'. But all I see is a claim that S(S(0)) +S(S(0)) represents the same thing as S(S(S(S(0)))). — Metaphysician Undercover

Yes the reason is because I'm holding PA, and it shows you the rules about +, so I didn't want to go to all of that technical side. So I just mentioned that the proof is present in PA, and I didn't want to go to this technical detail. But if you follow the axioms of PA you will begin with S(S(0)) + S(S(0)) which denotes || S(S(0)) + S(S(0)) || (you won't see this explicitly written in references about PA, but that's what PA is actually saying, I'm just clarifying it), to just end up with:

S(S(0)) + S(S(0)) denoting || S(S(S(S(0)))) || .

So just go to PA to fill in the missing part, you'll see that for yourself.

Two distinct things may be equal. For example, distinct human beings are said to be equal. — Metaphysician Undercover

Thanks for this account and the two points after it. But in mathematics when we are speaking about equality we don't mean this really. Equality in mathematics which occurs between expressions, especially when it occurs between functional expressions then it meant to be identity of denotation by those expressions.

You think 2 + 2 is denoting a process that involves a combination of two units to form another unit, which is wrong. 2+2 is "describing" such a process, but NOT denoting such a process, it is denoting what results from that process, that's your error. 2+2 is a functional term in mathematics, it denotes ONE and just ONE particular object, the + is a two place function symbol, it is an assignment that sends pairs of objects to single objects per each pair, so x +y = z is meant to be an assignment that sends single substitution of x and single substitution of y to a SINGLE substitution of z. so it sends the ordered pair (x,y) to a single z for each particular substitution of x,y. so 2+2 is meant to be the object that + sends the pair (2,2) to. When we way 2+2 = 1+3 we (in mathematics) mean that the single object that 2+2 denotes is "identical" to the single object that 1+3 denotes, that's what is meant. It means identity of denotation, that's all.

I can exactly mirror you argument to say that "The Sun" and "The nearest star to Earth and Jupiter" do not denote the same object? since the first is just involving one object, while the later is involving a process of two things being near to a third object, and it involves the meaning of star, earth, and Jupiter, so it is speaking of TWO entities with a relation from them (near) towards a third entity that at the end points to that third object, so the denotation of those two expressions is distinct, which is WRONG.

We need first to agree on what constitutes a "denotation" of an expression, and then we can argue its identity.

There is a difference between the details involved in an expression and what that expression is denoting. Denotation of expressions is determined by definition of the rules of the language in which that expression is meaningful. So I agree that there is a lot of say information going on in the expression 2+2, much more than the simple reference involved in the expression 4, that's right, but that doesn't affect their denotation, because their denotation is set by the rules of arithmetic and not by these aspects, by the rules of arithmetic 2+2 is a term of the language and it can ONLY be substituted by a SINGLE object of the universe of discourse, which is as single as 2 is and as single as 4 is, it is as single as any number is (which are the singular objects of the universe of discourse of PA), this is a rule of the language of PA, that + is a function. This is a stipulation, consider it an axiom. And by rules of arithmetic (say PA) it PROVES that the single object denoted by 2+2 is exactly identical to (i.e. the same as) the single object denoted by 4. Much as physics say that the single object denoted by "The Sun" is exactly identical to the single object denoted by "The nearest star to Earth and Jupiter", even though there are particular differences in details of those expressions including differences in syntax (particular wordings, number of them, grammatical differences etc..) and difference in semantics (information involved in these sentences), still both sentences are "denoting" the same object. So definitely different expression can convey different set of information to just denote the same object, that's obvious, and 2+2=4 is just once case of that situation.

In nutshell in mathematics the denotation of 2+2 is already stipulated to be a single object that is as single as 2 is and as single as any natural number is, and the denotation of 4 is of course a single object since its a number, and that that single object denoted by 2+2 is exactly the same( is identical to) the single object denote by 4, and that's what 2 + 2 = 4 exactly means.

OK, I've said true things about '2+2' which are not true about '4'. Therefore the two are not identical. It's what I've been doing for last number of posts, explaining how '2+2' signifies something different from '4 — Metaphysician Undercover

The two expressions are of course not identical, they are indeed distinct expressions, I already said that, that's clear because the expression 2+2 contains three symbols in it, while the expression 4 contains only one symbol, of course they are not identical. But that doesn't by itself entail that what they are denoting is not identical! There is a difference between "identity of expressions" and "identity of what expressions are denoting". The expression "The sun" and the expression "Nearest star to earth" are also not identical, the first contains two words, the last contains four words, but they do denote exactly the same object. Now the identity symbol "=" between any two expressions phi, pi , i.e. the expression phi = pi , means phi and pi are denoting the same object, it doesn't mean that phi and pi are identical expressions. You are confusing identity of expressions and identity of what they denote.

If this is true, then show me the object which both '2+2' and '4' refer to — Metaphysician Undercover

Your wish is my command! First one must note that expressions 2+2, 4 , 2+2=4, all of those doesn't have any innate meaning by themselves, we need to assign meaning to those symbols, otherwise they are just blind string of characters. For example the string of symbols "0 + 0 + 0 = 10" is true in Arabic language, since 0 denote number 5 in English, and 10 denote number 15 in English, while obviously it is false in English. So symbols by themselves are blind, they only acquire meaning by conventional definitions. So 2+2=4 is only true relevant to the context that assigns meaning to its symbols, for example in the system of arithmetic. Now lets take some arithmetical system, for example PA (peano arithmetic) as our background system which assigns meaning to symbols 0, 2, + , =, 4. Now in Peano arithmetic 0 is a constant symbol, it means it is an expression denoting a single object of the domain of discourse of PA. Now the expression S(x) is a functional expression it means its a term of the language of PA that denotes only one object for each particular substitution of x, similarly the expression x + y for any particular substitution of x and y, denotes a single object because + is a two place function symbol. The meaning of "phi = pi" when phi and pi are functional expressions in the language of PA means " phi denotes the same object pi denotes".

Now in PA the symbol 2 is meant to denote the object denoted by the expression S(S(0)), for simplicity let us use the notation || phi || where phi is a functional expression, to denote the OBJECT denoted by phi, so we have:

phi denotes || phi ||.

so according to that 2 is denoting the object || S(S(0)) ||.

Also 4 is denoting the object || S(S(S(S(0)))) ||

Now PA proves that the expression 2 + 2 is denoting the object || S(S(S(S(0)))) ||, which is the same object that expression 4 denotes! So by the meaning given to phi=pi in PA, PA proves that:

2+2=4

The proof of that is present in PA.

What you had in mind is an example of theoretic x meta-theoretic confusion.Which is something that almost everyone passes through!

However to veer to YOUR side, one can in some sense use a terminology that separates identity from equality, you can stress that identity is full matching, i.e. even with expressions, those would be identical only if every property associated with one of them is also to be associated with the other whether at the language level or the meta-language level, and so you'll demand that everything must match between them even the way how those expressions are written. OK, by this we can say that equality is identity of denotation, and that identity is full matching. If we adopt such terminology then of course 2+2 won't be identical to 4, but 2+2 would be equal to 4, since there is identity of denotation of those expressions. This might be plausible, but it is not often used, well as far as I know of, but it might have its virtues. not sure though.

Actually the somewhat acceptable definition would be as the Creator of the Universe. This assumes that existence of the universe begs a maker. This is of course just a matter of belief, it is not proven. On the other hands some might entertain a definition of God as the source of Good things in the universe like Love, beauty, kindness, empathy, etc.. The second definition need not be confused with the first, for a belief in ONE maker of the universe who has absolute dominion on it, might raise ethical points against him, like allowing evil to happen at such a great scale that is not explainable on the basis of testing, or any reasonable basis really. The second God might not have that dominion but it is more beautiful in the sense it only give rise to what is good, so there are no ethical points against him, but he looks weak in comparison with the first God. I think the nature of God and the relation of its dominion to what's happening in the universe is almost impossible question to solve, that we might as well be agnostic about it. The real point is to seek HOPE in existence being something more than just an emotionless stream of stuff. It confers more value to existence. However on the other side, we notice that the known revelations are doing very bad job in drawing an image about that God. Unfortunately the bible depicts God ordering some of his prophets to kill thousands of people including children, and their animals, burn whole cities, but take their GOLD to the treasury of God? Others picture God as sending you to an EVERLASTING TORTURE for not believing that an apparently human being called Jesus who lived some couple of thousand years ago is in reality God himself impersonated in a human form? What a sin? Other religions would also picture God sending the MAJORITY of people he created to an EVERLASTING burning in fire, for not believing in a revelation made by Angles for which we have no evidence whatsoever of them being really angles coming from him? Clearly those religions are just human ways of trying to solve the religious philosophical questions about existence, they failed, they need to be improved. That doesn't mean that there is no possible source of Goodness in this universe which is a rational being, it only means that we failed to approach him in the right way. It doesn't mean he is not helping us, however we cannot prove that he did. On the other hand It would be a mystery to explain his apparent silence?

then please produce this new law of identity, what you call "equality theory". I've asked fishfry for this principle of identity, to no avail. — Metaphysician Undercover

There is no new law at all. It is a schema of statements, in first order logic it would be expressed as: that x = x, i.e. everything is equal (identical) to itself, and that if phi(x) is an expression in which x occur and if phi(y) is obtained from phi(x) by merely substituting all occurrences of x in phi(x) by the symbol y, then the law is:

x=y implies [phi(x) iff phi(y)]

in a more informal manner, x is equal (identical) to y if every expression true of x is also true of y and vise verse, what we mean by true of is the truth of the denotation of that expression about objects and not the truth of its grammatical structure.

Actually equality is nothing but identity. In first order logic it boils down to substitutivity, as mentioned above.

But you need always to discriminate between what an expression is denoting and what an expression is. I already gave a simple example "The Sun" and "The nearest Star to Earth", in physics those two expressions are referring to exactly the same object but they are indeed two distinct expression! The former has two words the latter had five! So truly they are distinct expressions but they are denoting exactly the same object. In an exactly similar manner in mathematics the expression "2+2" is nothing but a functional term, it denotes an object that is exactly the same (identical) object that the constant term "4" is denoting. This is no deception, much as the different statements in the first example are no deception. The idea is about what can be called a "consequential truth" here. In the game of arithmetic the expression "2+2" is identical to "4", in the sense that they both denote the same object. i.e. this is a consequence of the axioms and rules of inference of that game, this is a theorem, and consequential fact, we need to determine exactly when two different expressions in the language of arithmetic denote the same object, just because they are different it doesn't mean that they can't denote the same object. We have a formal game here, and we want to know which of those expressions denote the same object and what are not, this is no deception, it is not even trivial, that's what we want to know.

All the rest of your account on trying to a kind of prove that "2+2" must be an expression denoting something that is different from the expression "4", is NOT correct, neither conceptually nor formally.

By the way the objection you stated that if they are indicating the same object then 2+2=4 would be equal to 4=4 and therefore would be vacant, this objection is already a property that Kant had spoken about when he defined analytic truths, i.e. its just repetition of what has been already said. Which is correct.

Consequential results are not that easy to figure out, they turn to be very tricky, that even if at the very conceptual root they are repetitions of statements, yet the recognition about which statements boils to be repetitions of which other statements is not that easy to determine and sometimes its even impossible to know relevant to a fixed set of axioms in arithmetic.

However the reality of 2+2=4 is not only linked to the above formal consequential game reasoning. One might say that all of that consequential game is just vacant, and that mathematics is not vacant as analytic reasoning is, so there must be a kind of truth to 2+2=4, something more akin to synthetic truth Kant was speaking about. The answer is that the truth of 2+2=4 is inherited from the truth of the axiomatic system in which 2+2=4 is a theorem of. The axioms themselves are not analytically derived from prior sentences, if they are consistent, then they are true of some model, and the truth of the axioms whatever it might be is inherited down to all of theorems derived in the system axiomatized by those axioms. On can say that all theorems are just repetitions of what's in the axioms, so the truth of 2+2=4 is related to the background of the axioms of the axiomatic system in which it is proven, which is CORRECT! That doesn't prevent 2+2=4 being the same as 4=4 at all, it doesn't make it trivial because it is only an aspect exposing the truth of what's in the axioms.

Again to sum it up, although "2+2" and "4" are two distinct expressions, yet they both denote the same object, much as how expressions "the Sun" and "The nearest star to Earth" are distinct and yet denote the same object.

Explain to me then, how this set '2+2', is the same thing as this set, '4'. They look very different to me, and also have a completely different meaning. By what principle do you say that they are the same? — Metaphysician Undercover

They are the same according to the game of identity called as "equality theory". There is a confusion here between expressions and what they denote, "The Sun" , "The nearest star to Earth" are two DIFFERENT (i.e. not identical) expressions, yes, but they denote the same object! so when we say for example "The Sun = The nearest star to Earth", what we mean is that the object denoted by the expression "The Sun" is Identical to the object denoted by the expression "The nearest star to Earth", that's very clear, it is identity of the denoted and not of the denoting expression, of course the denoting expressions are different. That an object can be denoted by different expressions is well known, and it poses no problem whatsoever. Along this understanding the expression "2+2" is meant to denote some object x, and the expression "4" is also meant to denote some object x, however both expression (though different) denote the SAME object exactly.

With Actual infinity you have a set that contains EVERY natural number as an element, while with potential infinity you can only have finite sets of naturals but without having a limit to formation of bigger finite sets of naturals, so when you have a set of say x,y,...,z naturals i.e. when you have the set {x,y,..,z} which is finite you can simply form the still FINITE set {x,y,...,z, k} where k is not an element of the first set, this process of always being able to add an extra-element to a finite set to get a bigger finite set is what is called as Potential Infinity. The difference is that Actual infinity poses an additional claim that is the existence of a set that contains ALL natural numbers as elements of it and that that set is an infinite set of naturals. And so this set would witness actual infinity.

there are no heresies here. Mathematics is not religion. And what you said was indeed harsh. It is not persons that we are discussing here, it is the subjects they propose. So we need to be objective here. Brouwer is certainly a great monument in mathematical foundation and philosophy no doubt, but he need not be true in everything he say.

It establishes science and mathematics as epistemic domains instead of subject matters. They are indeed not about anything. They are knowledge-justification methods — alcontali

Exactly! Very nicely put!

take any of lets say the meaningless rule following games. Add a rule that assigns to each symbol in that game a "particular meaning", this is a rule! Right? so it can be one of the rules in a rule following game, call it a meaning assignment rule. For example assign to the extra-logical symbols in the language of geometry some particular spatial concept, like for example a point being a kind of region in space, or being an atom in space, etc.., and then run the whole game of geometry in connection with this particular meaning. By then the result is a meaningful rule following game. This is a legitimate rule following game, but it incorporates meaning within it. Of course you can strip away the meaning by removing the meaning assignment rules, and you get the bare empty symbol rule manipulating game, OK, but that's ANOTHER game.This kind of games (the meaningful ones) is important in concept analysis, and in applied mathematics in general. For example applied mathematics in physics, can only be played in a meaningful manner, same in chemistry, ethics, etc... those are usually only played with meanings incorporated within them. Even arithmetic can be played in mind with some meaning incorporated to its symbols like for example of naturals being indices of the quantity of elements in finite sets, of summation being the indices of disjoint unions of indexed sets, etc.. this makes playing those games easier form the human side. But as you said they are not indispensable, you can dispense with any such connotation and proceed in a meaningless manner like a machine. No problem.

Formalism is the point of view that mathematics is about Non-meaningful rule following games. More precisely put: mathematics is about string (of empty symbols,i.e. meaningless) manipulation rules While here mathematics is about rule following games which can involve meaning as part of those rules, so it is not necessarily about manipulating strings of empty symbols.

what I wanted to say is that "logical truth" is a relative concept, when you say that a statement s is logically true, I'll respond by saying "in relation to which system?", there is no absolute logical truth, it is always a consequence of some rule following game. Now back to Godel's sentence "G". Lets say we have a theory (a rule following game) T, now Godel demonstrated that there is a sentence "G^T" that corresponds to "G^T is not provable in T", now this sentence G^T is NOT a logical truth in T, because it is not a consequence of the rules of T, but it is a logical truth of T+Con(T). Of course any sentence s if it is a logical truth in T then it is a logical truth T+Con(T), but the opposite doesn't follow, you can have a sentence s that is a logical truth in T+Con(T), but it is not a logical truth in T, and Godel's sentence G^T is an example of such a sentence. What I wanted to say is that logical truth is "consequential truth" or you can call it "provability", so a logical truth in T is exactly what's provable in T. You cannot show a sentence that is a logical truth in T that is not provable in T, that's a contradiction, but you can show a sentence that is a logical truth in T+Con(T) that is not provable in T, because simply T+Con(T) is ANOTHER rule following game that is different from just T.