What is the difference between actual infinity and potential infinity?

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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.

Really? With a child, discussing how the set of 2 pens here plus the set of 2 pens there makes a set of 4?

Wouldn't you want to be ready to climb down from platonist notions or foundations ("2 on the number line", or "the class of all pairs" etc.) and agree that the two separate concrete pairs of objects were being compared and found "equal" in cardinality or size, just as two pens might be found equal in weight, or in length? In other words, equivalent, and in the same equivalence class by this or that mode of comparison (in this case cardinality)? But obviously not identical?

Or would you want to get them with the platonist program straight away, and make sure they understood that 2 on the number line "sends" with itself in a two argument function returning at 4?

Notice they will soon learn to equivocate anyway between identity and equivalence, like any good mathematician not presently embroiled in philosophical or foundational quandary.

Not that @Metaphysician Undercover will be happy with any cavalier embrace of equivocation.

Yes, the irony... that competence in maths should not only involve easy equivocation imputing (with the equals sign) absolute identity here and mere equivalence (identity merely in some respect) there, but then also involve an "identity" (e.g. site menu) sign meaning only a batch-load (for all values of a variable) of cases of "equals", the latter still (in each case) ambiguous between identity and mere equivalence! (The ambiguity removed only by a probably unnecessary commitment to a particular interpretation.)
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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.

I think we're making some progress, and I think this is the heart of the disagreement between us. So maybe we could start with this statement about "symbology". Do you agree with a distinction between the symbol and what is signified by the symbol? The symbols themselves are objects which we read, and these objects, the symbols, have differences between the right and the left, such that they are not the same symbols.

Furthermore, according to the law of identity, two instances of what we call 'the same symbol", are not actually the same. So this "S" is not the same as this "S" (by the law of identity), considering them both as objects. So when we use "same" in this way, to say that they are the same symbol, we are using "same" in a way which is not consistent with the law of identity. This sense of "same" is based in some principle of similarity, not in the principle of identity. As a consequence of this, we have instances where the same word (as "the same symbol") has a different meaning, depending on context. And equivocation can result. This is clear evidence that the phrase "the same symbol", uses "same" in a way which violates the law of identity. Do you agree that this sense of "same", by which we say that this "S" is the same symbol as this "S", is inconsistent with the law of identity?

Please just substitute "identity" or "logical identity" in my argument. My apologies.

I would have to find out what you mean by "logical identity". In general, the sense of "identity" employed by logicians is not consistent with the sense of "identity" described by the law of identity. If you read Stanford on "identity", you'll see two distinct senses. One they call qualitative identity, and the other they call numerical identity. Numerical identity is what is described by the law of identity, identity is specific to each and every thing, a thing is the same as itself. "The same" means one and the same thing, and this is identity. Qualitative identity identifies through qualities, a description. If two things have the same description, they are treated as the same. This sense, qualitative, is the sense generally employed in logic, because logic proceeds with descriptive terms, and predication. But a careful logician will recognize the difference between subject and object, and that predication deals with subjects rather than objects. The law of identity deals with objects rather than subjects, while the logician deals with subject. So when the logician employs "logical identity" it is an identity based in qualitative identity, (the subject is identified through some sort of description), rather than numerical identity which is what the law of identity deals with.

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.

I told you the problem with your proof. The very first premise is false. So it doesn't matter how immaculate your proof is, it is unsound by that false premise. There is absolutely no sense in me wasting my time following all the points of your argument when you start with a premise which is clearly false. Until you fix that premise, or show me an argument which does not require it, there is no point.

Not that Metaphysician Undercover will be happy with any cavalier embrace of equivocation.

That pretty much sums it up.
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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".

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.
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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 know you agreed to that, but I could trust you because of propensity toward lying. And, you've continued to argue that "2+2" is the same (in the sense described by the law of identity) as "4". So your agreement appeared to be worthless in that matter.

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.

OK, so I made my point, you now agree with me, and perhaps we don't have anything more to debate.

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.

I really do not know what you mean by "sending" "two objects to a third object". You seem to think that sending is not a relationship, so what is it? How are Jesus and James "sent" to the Mother, when their true relationship to the Mother is that they are from the Mother?

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.

OK, I can agree with this.

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.

See why I accused you of lying when you were arguing something opposed to this? I knew you were intelligent enough not to actually believe what you were saying.

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.

Why must "2+2" denote an object? Each symbol, "2" is itself an object. The two 2s are distinct symbols, distinct objects. Now the question is what is denoted by each of these symbols. In common language, a word has different meanings depending on the context, and this is how we know, without a doubt, that the different instances of what we call 'the same symbol", are actually different symbols, having different things denoted by each of them. Perhaps, the first "2" denotes something different from the second "2", and the "+" denotes a relation between these.

I agree that "2+2" on its own is not a proposition, but "2+2=4" is. So what is "2+2" on its own? It is just a part of a proposition, terms which need defining. If we add "=4" we complete the proposition and give some meaning to "2+2". Now, consider that as human beings, we require a proposition to identify an object. Yes, it is true that the object has an identity distinct from that given by us, and that is what is stated by the law of identity, (that the identity of the object is proper to itself), but we as human beings also want to give the object an identity, for our sake, and this cannot be done without a proposition. 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.

Now each of the 2s need to be defined, or identified, So we can define one 2 with "1+1=2", and the other 2 with "1+1=2". What defines "1" though? We have four different 1s here. In general, "1" signifies the fundamental unity, an entity, or an object, and each application of the symbol "1" is use to signify a different object. That's how we count, continually adding a new and different "1". Each instance of "1" is itself a different symbol, a different object, also denoting a different object, allowing us to count a multitude of different objects. And this is clear evidence that each instance of a symbol like "1", or "2", or "3", signifies a different thing, otherwise when we count, by continually adding "1", we would just be counting the same object over and over again and the count, the total, or the summation would be invalid because it requires that there are actually that many different things, for the total count to be valid.
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Perhaps, the first "2" denotes something different from the second "2", and the "+" denotes a relation between these.

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.
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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.

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.
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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.

I think, if the "2" is denoting itself then it is not denoting anything, it is simply a 2. If we say that it is a symbol, and therefore denoting something, then to say that it denotes itself is nothing other than to say that it denotes nothing. A symbol which denotes nothing is not a symbol, so the "2" which is supposed to denote itself is simply an object, denoting nothing.

If we bring this object into a logical operation, it is now a subject. It is a subject because we can move it around at will, use it as we please, it is subject to the will of the logician who uses it. What the subject "denotes", is dependent on how the logician uses it. and this is determined by definitions. As denoting something, the subject is a symbol, and it may denote anything, object, relation, etc., but in logical proceedings it need not represent anything..

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.

A "term" which refers to itself, is not acceptable to me. It is not really a term as per the definition, but a subject employed for trickery, deception. "Term" as per the definition requires that the symbol represent something, and to represent itself is to represent nothing. So the use of a "term" which represents itself is a ploy to avoid the restrictions of the definition, which dictates that the term must represent something.

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.

I see a problem here, the possibility of category error due to confusion between the object which is the symbol (term), and the object which the symbol denotes. You say that relation symbols link terms. Therefore, what is related are the terms, the subjects. And, we must remember that it is not necessary that the subjects represent objects, because the trickery employed which allows that a term represent itself, such that the represented object is really just the subject representing nothing.. This allows that the mathematician may be just playing with subjects, establishing relations between terms which do not represent anything, just like logicians do, but it is in defiance of the definition of "term". The definition of "term" disallows this, but the crafty mathematician has found a loophole.

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.

I agree with this, except that the term must denote an object, as I described above, the mathematician has found a way to employ terms which do not denote objects. Let's just say that the term is a subject, which is a special sort of object, one which is subject to the will of the logician, and it need not denote anything. I'm sure you've seen examples of formal logic, expressed in 'symbols' which do not denote anything. These examples are used in teaching, to demonstrate the logical process. The process is shown using terms which do not denote anything. Mathematics is supposed to be more rigorous, requiring that an object be represented. It is intrinsic to mathematics that objects be represented because if no objects are represented the distinction between numbers is meaningless.

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

So, lets start from the beginning, and enforce the rule of definition, in a rigorous manner. If "2" is a term, it must denote something. What it denotes must be something other than itself. This means that we must assign meaning to "2". That requires a proposition. We cannot just assume that "2" denotes itself, in order that it's properly a term, we must say what it denotes. I suggested "1+1=2" as a proposition which defines what "2" denotes.

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.

Do you realize that this is all one sentence? Maybe you could express it in a more comprehensible way? As I explained in the last post, if "4" is defined by the proposition "4=2+2", and "2" is defined by "2=1+1", then we must turn to the definition of "1+1", as a term, to make "2" intelligible. And, it is necessary that each time the symbol "1" appears it denotes a different object, or else "1+1" is unintelligible.
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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.
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If we bring this object into a logical operation, it is now a subject. It is a subject because we can move it around at will, use it as we please, it is subject to the will of the logician who uses it. What the subject "denotes", is dependent on how the logician uses it. and this is determined by definitions. As denoting something, the subject is a symbol, and it may denote anything, object, relation, etc., but in logical proceedings it need not represent anything..

That's a fantastic explanation of Formalism. I know that you don't like it, well, but by the way its really a nice account explaining my intentions. Yes the whole of arithmetic can be interpreted as just an empty symbol game, and saying that a symbol represent itself is next to saying that it is not representing anything, I agree. You may say an empty symbol is not a symbol, well its a character and that's all what we want, we may call it as "empty symbol", its a concrete object in space and time (even if imaginary) and it serves its purpose of being an "obedient subject" to the wimps of logicians and mathematicians. I really like it.
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Mathematics is supposed to be more rigorous, requiring that an object be represented. It is intrinsic to mathematics that objects be represented because if no objects are represented the distinction between numbers is meaningless

This is indeed a plausible stance! But I think formalists won't agree. I still think that we can have distinctions between numbers even if they are meaningless. There are still meaningful matters that the formalist would hold to, that is the deductions carried in the system, those are non-trivial pieces of knowledge. But again it would be difficult for the formalist to account for the success of some mathematical disciplines in science and various applications, that's where it hurts when it comes to formalism.
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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. 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.

OK, now we're back to the same problem. In "1+1" each "1" is a term denoting a distinct object. Therefore there are two objects denoted. How does it come about that "1+1" denotes a single object, as a term in itself?

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.

I don't see that this is "always the case". Why can't a group of terms be related to another term through the same relation, like in your analogy, a group (Jesus and James) are related to Mary?

The first 1 denotes an object
The second 1 denotes an object
The string 1+1 denotes an object

So the third is the one which needs to be justified. How does "1+1" denote a single object?

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.

I explained why this analogy doesn't work. "Jesus and James" is analogous to "1+1". "The mother of" is a term, denoting an object, a person "the mother", who has an implied relation dictated by the definition of "mother". It does not denote a relation, it denotes a person (object) who has a specific relation.

That's a fantastic explanation of Formalism. I know that you don't like it, well, but by the way its really a nice account explaining my intentions. Yes the whole of arithmetic can be interpreted as just an empty symbol game, and saying that a symbol represent itself is next to saying that it is not representing anything, I agree. You may say an empty symbol is not a symbol, well its a character and that's all what we want, we may call it as "empty symbol", its a concrete object in space and time (even if imaginary) and it serves its purpose of being an "obedient subject" to the wimps of logicians and mathematicians. I really like it.

Hey thanks, I'm glad you liked it. Here's the problem though. In logic, we can learn the logical procedures, by playing the "empty symbol game". It is useful for that purpose of teaching procedure, but beyond that mode of practise, it's useless, like an activity not being applied, what Wittgenstein calls language which is idol, or on vacation. To be useful there needs to be substance, subject matter, the symbols must be applied (represent something), to have meaning, and allow the logical proceedings to actually do something.

Arithmetic is different though. It is based in symbols and relations, rather than procedures. So the foundations of arithmetic involve symbols (1,2,3, etc,) which represent objects and the relations between these objects, relations which are determined and inherent within what the symbol represents (its object). This is somewhat different from the foundations of formal logic, which involve what we can and cannot say about an object.

Because of this difference the notion of an "empty symbol game" in mathematics is illogical. Arithmetic and mathematics are structures of coherency, every symbol has its place in the structure according to what it represents, so that an empty symbol would not fit into the structure at all, having no place. Imagine if there really was a symbol in arithmetic, like "2" for example, which represented nothing other than itself. It could have no relationship with any other symbols like "1", "3", "4" etc. because if it did have such relations with other symbols, they would have to be inherent within what is represented by "2". Then '2" would not just represent itself, it would also represent these relations, as "2" actually does.

Think of the object represented by "the mother of...". Not only is a particular object represented, but inherent within that way of representation (how the term is defined) is also its relation to other objects. This is the way it is with the terms of arithmetic, inherent within the object represent by "2" is all the relationships with other arithmetical objects by the coherency of the definitions. This is very important in geometry because some fundamental definitions or axioms, (like the circle has 360 degrees for example) are very arbitrary. If all the definitions which follow, and build up the structure based on this axiom, are not consistent and coherent, then the conceptual structure is useless. The object represented by the symbol "circle" might be completely arbitrary, but it is necessary that there is an object which is consistent with the objects represented by the other terms, thus making the symbol useful, or else everything is incoherent and meaningless.
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So the third is the one which needs to be justified. How does "1+1" denote a single object?

I read the whole of your message about why symbols must denote other objects etc.. Its nice and very DEEEEEEEEEEP, indeed; and I won't differ with you about those points (nor necessarily fully agree with them). I'm concentrating on the conventional terminology specially the syntactical ones.

To many mathematicians (well at the least those who stick to formalism) they are willing to understand mathematics as just a game of empty symbols because that would figure out the deductive streaming in those syntactical games we call as mathematical systems. So we have games with specified rules, we cannot go against them because these rules are fixed, and all of what's in the game is in reality "subject" to these rules.

Now one of the rules is to regard an expression like 1 + 1 to be denoting a single object. There is no justification for that at all, it is a rule of the game, you might object to it being unsubstantial, etc.., yes you may, but that won't change anything, the rules of arithmetic stipulate that 1 + 1 IS a functional expression, and a functional expression always denote an object, whether it does it in a substantial manner (i.e. denote an object other than itself) or whether it does it in vacant manner (like it being denoting itself), it doesn't matter, in both scenarios its a fixed rule of the game that 1 + 1 refers to some object.

The reason is because the "+" operator is stipulated before-hand to be a primitive "binary FUNCTION symbol" And by fixed rules of the game of logic and arithmetic when an n-ary function symbol is coupled with its n many arguments in a formula (which must be terms of course) then the *whole* expression is taken to denote some object (that is besides the objects denoted by its arguments which are shown in the formula). So binary functional expressions for example have 3 object denotations and not two as it appears, in general any n-ary functional expression has n+1 object denotations, although you will only see two objects (arguments) written in the expression. For example lets take the successor function S of peano arithmetic, here S(2) means "The successor of 2" which is a functional expression, you have 2 denoting an object and you have the whole expression i.e. "S(2)" also denoting an object, which is equal to 3. But you don't see it in the expression. Actually this is the main difference between "functional" and "relational" expressions. In a relational expression you see some n-ary relation symbol with n-many terms and this would be a proposition, and there is only n many object denotations, it would be a proposition because its speaking about a relation between those n many objects. While with functional expressions you'll also see an n-ary function symbol with n-many terms, BUT this won't be a proposition, because its not speaking about a relation between n many objects, no, actually its referring to an (n+1)_th object linked to the n many objects symbolized by the n many term symbols you see in the expression, so the total number of denotations is n+1 many object denotations and not n many object denotations as it is the case with relational symbols.

If 2 + 3 was denoting a relation between 2 and 3 and that's it, then it would be a proposition, because either 2 has the relation + to 3 or it doesn't have it, one of these two situations must be true, so it would denote a proposition, but clearly this is NOT the case, we don't deal with 2 + 3 as a proposition at all, we don't say it's true or false, so 2 + 3 must not be something that denotes a relation occurring between two objects, so what it is then? by rules of the game 2 + 3 is short for "the result of addition of 2 to 3" that's what it means exactly, and so 2 + 3 is referring to an object resulting from some "process" applied on 2 and 3 and that process is addition, that's why we call it as a functional expression, because its there to denote something based on a process acting on its arguments, and not to depict a relation between the two objects denoted by its arguments.

I think that's the best I can do in explaining common usage of these expressions.

I'm only explaining the rules of the syntax and its relation to denotations, which is something controlled by the rules of the game. See: rules of syntax of first order logic

You need to review these rule for yourself and see whether I'm telling the truth or not, since I know that you don't trust me!
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I think we're making some progress,

Good news. I'm working on a reply in case it takes a while. I do think you're failing to distinguish between:

* The philosophical question; and

* The mathematical question.

When you send me to SEP and make subtle (and interesting!) points about the nature of identity, that is part of the philosophical problem. About which I have already stipulated that I'm ignorant and open to learning.

But on the mathematical point, you still won't engage and that's still frustrating to me. You reiterate that my first premise is false but I thought I clarified that. Could you please repeat exactly what I said that you think is false? I'd like to respond but I am actually not clear on what you're referring to. I already acknowledged using the word equal when I meant to write identical at some point.

I'll spend some more time reading your interesting post. You make a lot of good points about symbology. I'll give it all some thought.
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Really? With a child, discussing how the set of 2 pens here plus the set of 2 pens there makes a set of 4?

@Metaphysician Undercover mentioned teaching children earlier as well. For the record I'm not speaking of pedagogy, but rather of sophisticated mathematical thinking that has only arisen in the past century and a half. I don't expect to explain the Peano axioms to children; but as adults, we are free to use our most sophisticated mental frameworks.

Wouldn't you want to be ready to climb down from platonist notions or foundations ("2 on the number line", or "the class of all pairs" etc.) and agree that the two separate concrete pairs of objects were being compared and found "equal" in cardinality or size, just as two pens might be found equal in weight, or in length? In other words, equivalent, and in the same equivalence class by this or that mode of comparison (in this case cardinality)? But obviously not identical?

I'm a Platonist when it suits me, and a formalist when that suits me. My derivation of the theorem 2 + 2 = 4 in Peano arithmetic was purely a formal exercise. When doing math it's helpful to think like a Platonist. When doing mathematical philosophy it's often helpful to think like a formalist. I'm a conceptual pluralist in that way.

I'll surely grant your point that two oranges are different than two fish; and that each pair is an instantiation of the abstract concept of two. But I am being careful to not talk about the world at all. Frankly I am not trying to convince anyone that two pens plus two pens is four pens. I take no interest in such mundane applications!

My only hard claim in this thread at this moment is that 2 + 2 = 4 is a theorem of Peano arithmetic. I can base the rest of my argument on that. But in the end its a formalist argument. I have no idea how many pens is two pens plus two pens. Maybe they're four for the price of three. I have no knowledge of such things.

Or would you want to get them with the platonist program straight away, and make sure they understood that 2 on the number line "sends" with itself in a two argument function returning at 4?

We don't burden young minds with higher mathematical abstractions. In fact that was the great failure of the educational fad of "new math," which was coming into vogue around the time I was too old to be scarred by it. After that failed they tried "new new" math, and some other things, and now they've got Common Core about which I hear awful things. I simply am not discussing early math pedagogy. I'm not discussing the subject of what we should teach children about numbers.

Notice they will soon learn to equivocate anyway between identity and equivalence, like any good mathematician not presently embroiled in philosophical or foundational quandary.

No I don't agree at all. Most people will never care one way or another. And you have snuck in a word NOBODY is talking about, equivalence. In math there can be equivalences between very different things. So that's a red herring, a distortion of the argument. No mathematician obfuscates identity versus equivalence. On the contrary, mathematicians are very precise about the distinction.

Not that Metaphysician Undercover will be happy with any cavalier embrace of equivocation.

I am not equivocating anything. In set theory, 2 + 2 and 4 are strings of symbols that represent or point to the exact same abstract mathematical object. That's a fact.

Yes, the irony... that competence in maths should not only involve easy equivocation imputing (with the equals sign) absolute identity here and mere equivalence (identity merely in some respect) there, but then also involve an "identity" (e.g. site menu) sign meaning only a batch-load (for all values of a variable) of cases of "equals", the latter still (in each case) ambiguous between identity and mere equivalence! (The ambiguity removed only by a probably unnecessary commitment to a particular interpretation.)

You're making strawman attacks, assuming facts not in evidence and imputing error to others when in fact your own thinking, or at least your writing, is muddled. I could not track that last paragraph.

Some mathematicians are incredibly careful and thoughtful about these issues. See Barry Mazur's famous essay, When is One Thing Equal to Another Thing?

http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf
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You know that scene with the cinema queue in Annie Hall, where Woody is lucky enough to (ahem) marshal Marshall McLuhan against the bloke quoting him? I'm seriously tempted to ask Professor Mazur to see if he can read my previous post, which has so annoyed you, without recognising it as a passable expression of ideas to be found (to my delight and no surprise at all) in the first section of that pdf (for which I'm grateful).

I feel badly misunderstood, but hey, this is the internet...

Another reason to forego point by point corrections to your post here is that I wanted in the first place to shorten the thread, not lengthen it. As I dared to remark right away, between the vast magisterial tracts talking straight past each other,

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?

IOW, why not be...

a conceptual pluralist in that way.

And, as @Terrapin Station deserves credit for often saying (and on this at least we should take notice), "one thing at a time, please".
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The reason is because the "+" operator is stipulated before-hand to be a primitive "binary FUNCTION symbol" And by fixed rules of the game of logic and arithmetic when an n-ary function symbol is coupled with its n many arguments in a formula (which must be terms of course) then the *whole* expression is taken to denote some object (that is besides the objects denoted by its arguments which are shown in the formula).

Well you'll need to justify this claim. I've never seen it stipulated that the "+" is a binary function symbol. Nor have I seen it stated that when a binary function symbol is used with two terms, that the whole expression must be taken to represent one object. That such and such convention interprets things in this way does not mean that this is a fixed rule of mathematics. It is just one of many possible interpretations, and interpretations are often illogical or incoherent.

Regardless, we seem to agree that the two 1s in "1+1" represent distinct objects, so what needs to be explained is how this function "+" makes these two objects into one. We cannot just stipulate that these two objects are one, because that would be contradiction, so the function must do something to avoid such contradiction. The function must represent a process which makes them into one.

If 2 + 3 was denoting a relation between 2 and 3 and that's it, then it would be a proposition, because either 2 has the relation + to 3 or it doesn't have it, one of these two situations must be true, so it would denote a proposition, but clearly this is NOT the case, we don't deal with 2 + 3 as a proposition at all, we don't say it's true or false, so 2 + 3 must not be something that denotes a relation occurring between two objects, so what it is then? by rules of the game 2 + 3 is short for "the result of addition of 2 to 3" that's what it means exactly, and so 2 + 3 is referring to an object resulting from some "process" applied on 2 and 3 and that process is addition, that's why we call it as a functional expression, because its there to denote something based on a process acting on its arguments, and not to depict a relation between the two objects denoted by its arguments.

I agree that "+" cannot denote a relation. It must denote a process, or function, as you call it. But I disagree that it signifies "the result of addition", it signifies the process of addition itself, not the result of the process. So in your example of "2+3" we have an object denoted by "2" and an object denoted by "3", and the process, "addition" denoted by "+". There is no result of this process denoted, and therefore no third object denoted, just the process. Perhaps the third object you had in mind is "5"? Then wouldn't you say that "2", and "3", along with the process of addition results in "5"? But you really have no result of the process of addition until you state the sum. The process is something carried out by the thinking mind, not the symbol itself.

Good news. I'm working on a reply in case it takes a while. I do think you're failing to distinguish between:

* The philosophical question; and

* The mathematical question.

When you send me to SEP and make subtle (and interesting!) points about the nature of identity, that is part of the philosophical problem. About which I have already stipulated that I'm ignorant and open to learning.

You know this is a philosophy forum don't you? So it's likely that you should expect that we are discussing a philosophical issue. If you want to discuss a mathematical issue, maybe a different forum would be better.

Could you please repeat exactly what I said that you think is false?

This is the false premise you stated:
1.1 We have the law of identity that says that for each natural number, it is equal to itself.

That is not the law of identity. The law of identity is the philosophical principle which states that a thing is the same as itself. In mathematics there are theories of equality, and perhaps axioms of equality, but these are not laws of identity. So what I was asking for was if you know of a law of identity which states that things which are equal have the very same identity.

In set theory, 2 + 2 and 4 are strings of symbols that represent or point to the exact same abstract mathematical object. That's a fact.

It's one thing to make these sorts of assertions, but another thing to justify such claims. This would require showing how this string of symbols "2+2" denotes the exact same abstract mathematical concept as this symbol "4".
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I've never seen it stipulated that the "+" is a binary function symbol. Nor have I seen it stated that when a binary function symbol is used with two terms, that the whole expression must be taken to represent one object. That such and such convention interprets things in this way does not mean that this is a fixed rule of mathematics

Those are present in Peano arithmetic in a very clear manner. You can review a full treatment of them. That they are not fixed rule of mathematics, might be, but they are fixed rules of first order logic that function symbols represent an object and these can take complex form and not just the constant or the unary form.

I agree that "+" cannot denote a relation. It must denote a process, or function, as you call it. But I disagree that it signifies "the result of addition", it signifies the process of addition itself, not the result of the process.

Of course + denote a process, more precisely a "function", of course it doesn't denote the result of a process, I never said that. What I'm saying is the whole expression of "1 + 1" is what is denoting the result of a process, and for that particular string it denotes the result of adding 1 to 1. I'm not sure if you are understanding what I'm saying. I'll try to break it down for you, take the above expression, i.e. "1+1, lets take its parts and see their denotation abilities:

The first symbol 1 : This is a denoting constant (i.e. a zero place function symbol), it denotes an object.
The "+" sign: this doesn't denote an object, but it denotes a process, more specifically a "function"
The second symbol 1: This is a denoting constant, it denotes an object
The string 1 + : This is an "incomplete expression" it doesn't denote an object
The string +1: This is an incomplete expression also, not denoting an object.
The string 1 + 1: This is a functional expression: IT DENOTES AN OBJECT.

So '+' denotes the process of addition itself, but "1 + 1" denotes the object that results from applying the process of addition on two "1" symbols.

It is definitely a rule of the game in logic that the total expression of 1 + 1 (i.e. the three symbols in that sequence) is denoting an object, that's definite, because it represents the result of a functional process. You cannot change this. This is NOT an interpretation of the symbols, to say that they are illogical, equivocal, erroneous, NO! It IS a rule of the game of arithmetic and the underlying logic.

And it makes full sense, because equality is a "binary relation" symbol between OBJECTS, so the symbol for equality, i.e. "=", must LINK two term symbols, i.e. two symbols that denote objects, otherwise the syntactical expression won't copy the semantic content. so you have the structure

"term - binary relation symbol - term"

Here we have 1 + 1 = 2

so = is linking the expression '1+1' to the expression '2', so '1+1' must denote an object. Otherwise the whole expression would be meaningless, it would be equality between what and what?

I tend to think (I'm not sure) that you think that 1 + 1 is an "instance of the process of addition" i.e. the process of addition is itself a big process it doesn't only work on 1's but also on any two naturals, now "1 +1" is just one instance of this addition process. It appears to me that what you have in mind is the following interpretation:

"instance of a process - binary relation symbol - term"

so here possibly you are thinking that 1 + 1 = 2 means that

(the process of adding 1 to 1) = 2

And that this would be a proposition that defines 2 by it being equal to an specific instance of the process of addition.

If I got you right, I think this interpretation can possibly work, I'm not sure, but definitely it is against known rules of the game, because relation symbols link terms and not expressions denoting process instances to terms, unless the "process instances" are understood as some kind of objects? and so expressions denoting them would be "terms"! Well that would be another issue.

In nutshell, the conventional way of understanding 1 + 1, well lets say in first order logic extended with the symbols of arithmetic, is for it to be the object that results from the function + operating on the two 1 symbols.

That + is a function, see this
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o in your example of "2+3" we have an object denoted by "2" and an object denoted by "3", and the process, "addition" denoted by "+". There is no result of this process denoted, and therefore no third object denoted, just the process. Perhaps the third object you had in mind is "5"?

No! There is! Please see my message that just precedes this, where I've pin pointed where is that object and I showed you all denotations involved. But to just reply to this here. I'll say:

In "2 + 3" we have an object denoted by "2" and an object denoted by "3", and the process, "addition" denoted by "+", and also we have an object denoted by the total string "2 + 3" itself. I didn't mean 5 at all, since 5 is not shown in the expression "2 + 3". The reason is because "+" is stipulated by the rules of arithmetic and underlying logic to be a FUNCTION, and by rules of the game any function symbol if written with its argument 'terms', then the whole expression of that function symbol and its argument would be denoting of an object. We don't have any mentioning of 5, yet, it is the rules of arithmetic that later would prove to you that the object denoted by 5 is equal to the object denoted by "2 + 3". Remeber equality is a relation between OBJECTs.
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. This would require showing how this string of symbols "2+2" denotes the exact same abstract mathematical concept as this symbol "4".

In set theory, yes that is the case. But of course you can reject set theoretic interpretation of arithmetic. But fishfry is saying the truth about set based implementations of arithmetical concepts. In set theory the symbol 2 denotes the von Neumann ordinal { {}, {{}} }, also the expression "2 + 2" is a functional expression that denotes the von Neumann ordinal { {}, {{}}, { {}, {{}}}, {{{}}} }, and also the symbol 4 is taken to represent exactly the set { {}, {{}}, { {}, {{}}}, {{{}}} }. This follows from the definitions given to those symbols and ZFC, and of course the inference rules and axioms of ZFC. Those are the usual interpretations held in ZFC.
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The law of identity is the philosophical principle which states that a thing is the same as itself. In mathematics there are theories of equality, and perhaps axioms of equality, but these are not laws of identity.

hmmm...., let me think about that, I'm really not sure if "identity" really arise in mathematical system per se. But if you consider first order logic as a kind of mathematical system, since it is a part of "mathematical logic", then of course there is a theory about identity. I'll speak about formal difference between an axiomatic theory of identity and an axiomatic theory of equality if that helps.

An axiomatic theory of equality basically presents equality as an equivalence binary relation, it basically contain the following three axioms:

1. Reflexive: For all x (x = x)
2. Symmetric: Forall x,y (x=y implies y=x)
3. Transitive: For all x,y,z (x = y and y=z implies x=z)

That's all.

Now an axiomatic theory of "identity" stipulate identity as a substitutive binary relation, most of the times it uses the symbol "=" to signify "identity" and not just equality, it basically contain the following axioms:

1. Reflexive: For all x (x = x) [in English: everything is identical to itself]

2. Substitution axiom schema: if phi(x) is a formula in which the symbol x is free and never occur as bound and in which the symbol y never occurs, and if phi(y|x) is the formula obtained by merely replacing some or all occurrences of the symbol x in phi(x) by the symbol y, then all closures of:

for all x,y (x=y implies [phi(x) \iff phi(y|x)]), are axioms.

That's all.

So the axiomatic systems of these two notions are clearly different!

Now identity theory proves all axioms of equality theory, but the converse is not true, i.e. every two identical objects are also equal to each other, since identity relation is an equivalence relation, but not every equality relation is an identity relation. For example the relation "bijection" which means that two sets can have their members linked to each other in a one to one manner, this relation is clearly an example of an equivalence relation (i.e. equality) because it fulfills the three axioms of equality theory, yet bijection is not the identity relation, since we can have two distinct sets that are bijective to each other, and so it violates the substitution scheme.

Identity relation simply states sameness of objects, and it does that by fulfilling the above axioms of identity theory especially the substitution axiom, which mean that any description of the object x also holds if we replace the symbol x and the symbol y, i.e. it is also a description of the object y, and vice verse, so the identical objects are said to be indiscernible from each other using our language, i.e. relative to our language they appear identical.

However in mathematical systems and even in logico-mathematical systems like PA, I see = axiomatized after equality theory! However in more deep formal systems like set theory and Mereology the = symbol is usually taken to represent "identity" and not just equality, and usually ZFC and Mereology are formalized as extensions of first order logic with "identity" rather than with just equality, although most of the time these terms are used interchangeably in set theory and Mereology but vastly to mean "identity" and not just equality, since the axioms about them are those of identity theory and not just of equality theory.
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Those are present in Peano arithmetic in a very clear manner. You can review a full treatment of them. That they are not fixed rule of mathematics, might be, but they are fixed rules of first order logic that function symbols represent an object and these can take complex form and not just the constant or the unary form.

I looked through the Wikipedia page on the syntax of first order logic, which you referred me to above. This is what I think. In the expression "2+2=4", the "2", and "4" symbols are predicate symbols. A predicate symbol represents an "element" with a relational definition. Because of this, expressions like "2+2=4" cannot be considered as terms, and therefore there is no basis to the claim that any objects are represented here.

What I'm saying is the whole expression of "1 + 1" is what is denoting the result of a process, and for that particular string it denotes the result of adding 1 to 1.

That's clearly not the case. A function, or process is indicated by "+". There is nothing to indicate "the result" of the function. Consider for example cause and effect. The cause may be represented or described without representing the effect. When only the cause is represented, there is nothing to indicate what the effect is. Though it is true that "effect" is implied by "cause", unless one already knows that x cause has y effect, y would not be indicated by stating "the cause is x". "1+1" represents a process, it does not represent the effect or result of that process. This is evident from the fact that one might state "1+1" without knowing that the result is "2". And of course this is the natural process of summation, we write down the numbers to be summed before we know the result of the summation.

So '+' denotes the process of addition itself, but "1 + 1" denotes the object that results from applying the process of addition on two "1" symbols.

No, absolutely not. It may be true that "+" denotes the process, but there must be something which is active in the process, or else you just have a type of process indicated, with no specifics. So the two "1" symbols denote the elements involved in the specified process, and there is nothing to indicate what is caused by the process, "the result" of the process.

It is definitely a rule of the game in logic that the total expression of 1 + 1 (i.e. the three symbols in that sequence) is denoting an object, that's definite, because it represents the result of a functional process. You cannot change this. This is NOT an interpretation of the symbols, to say that they are illogical, equivocal, erroneous, NO! It IS a rule of the game of arithmetic and the underlying logic.

It is an interpretation, a faulty one. There is nothing in the rules to say that the expression represents "the result" of a process. You are making that up, back to your old habit of bullshitting again.

Here we have 1 + 1 = 2

so = is linking the expression '1+1' to the expression '2', so '1+1' must denote an object. Otherwise the whole expression would be meaningless, it would be equality between what and what?

According to the definitions on the referred page, these numerals are predicate symbols, therefore they denote elements. The expression is not meaningless though, it demonstrates a relation.

In "2 + 3" we have an object denoted by "2" and an object denoted by "3", and the process, "addition" denoted by "+", and also we have an object denoted by the total string "2 + 3" itself. I didn't mean 5 at all, since 5 is not shown in the expression "2 + 3". The reason is because "+" is stipulated by the rules of arithmetic and underlying logic to be a FUNCTION, and by rules of the game any function symbol if written with its argument 'terms', then the whole expression of that function symbol and its argument would be denoting of an object. We don't have any mentioning of 5, yet, it is the rules of arithmetic that later would prove to you that the object denoted by 5 is equal to the object denoted by "2 + 3". Remeber equality is a relation between OBJECTs.

Until you demonstrate how numerals like "2" and "3" denote objects rather than elements, as predicate symbols, which is clearly explained in the rules, I think we ought to stop saying that these denote objects.

hmmm...., let me think about that, I'm really not sure if "identity" really arise in mathematical system per se.

That's the point. Fishfry keeps trying to switch out "equality" for "identity", as if the two have the same meaning. But "identity" has a very specific, well defined meaning in philosophy, and no such definition in mathematics. So Fishfry's use is either an attempt at equivocation, to smuggle the philosophical meaning into mathematics as if "equality" means the same as philosophical "identity", Or else he just brings a non-defined word into a mathematical usage which would leave it meaningless.

Now an axiomatic theory of "identity" stipulate identity as a substitutive binary relation, most of the times it uses the symbol "=" to signify "identity" and not just equality, it basically contain the following axioms:

Perhaps you can do for me what I've asked of fishfry. Show me an axiomatic theory of identity which is proper to mathematics.

However in more deep formal systems like set theory and Mereology the = symbol is usually taken to represent "identity" and not just equality, and usually ZFC and Mereology are formalized as extensions of first order logic with "identity" rather than with just equality, although most of the time these terms are used interchangeably in set theory and Mereology but vastly to mean "identity" and not just equality, since the axioms about them are those of identity theory and not just of equality theory.

I haven't yet been shown these identity axioms of ZFC. The one which fishfry steered me to, the axiom of extensionality is clearly stated as an equality axiom. So if it is taken to represent identity, I think that's a faulty interpretation. It presupposes some sort of identity with the use of "same", but it doesn't stipulate what "same" means.
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The one which fishfry steered me to, the axiom of extensionality is clearly stated as an equality axiom. So if it is taken to represent identity, I think that's a faulty interpretation.

I wrote to you the identity relation in ZFC. I already wrote that explicitly it is the reflexive and substitution axiom schema, those are the identity theory of first order logic, and ZFC is *usually* formulated as extension of the rules of first order logic and those axioms of identity theory. However *sometimes* ZFC defines "identity" in the following manner "it also uses the symbol = to mean "identity" and not equality"

Define (=): x = y \iff \forall z (x \in z \iff y \in z)

However, this approach is not favored a lot, the majority would define axioms of ZFC as an extension of identity theory (i.e. the reflexive and substitution scheme).

This is what I think. In the expression "2+2=4", the "2", and "4" symbols are predicate symbols

No this is wrong! It seems you didn't read it well, the expression 2, 4 are called zero placed function symbol, or simply constants, and those are TERMS of the language and they denote objects. That's the usual presentation in Peano arithmetic and most mathematical system. However, we CAN formalized 2 and 4 as predicates that's not a problem at all, this can be done. But it is not the usual thing.
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already wrote that explicitly it is the reflexive and substitution axiom schema, those are the identity theory of first order logic, and ZFC is *usually* formulated as extension of the rules of first order logic and those axioms of identity theory.

All the information I've seen shows that the reflexive axiom and the substitution axiom are equality axioms. Why do you think they that are identity axioms?

It seems you didn't read it well, the expression 2, 4 are called zero placed function symbol, or simply constants, and those are TERMS of the language and they denote objects.

I saw no such rule, to dictate that "2" and "4" are "zero placed function symbols", on the page you referred. I think you're making this up. Any way "zero" would indicate an absence of objects.

However, we CAN formalized 2 and 4 as predicates that's not a problem at all, this can be done. But it is not the usual thing.

We agree then, that there are no objects denoted by "2+2=4"? On what basis would you claim that "2+2" is identical to "4" then?
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We agree then, that there are no objects denoted by "2+2=4"? On what basis would you claim that "2+2" is identical to "4" then?

No of course we don't agree. The usual formulation is for 2 to be a constant (zero place function symbol), that's the usual convention. Now there are some second order logic theories that can interpret arithmetic in a manner that 2, 1, etc.. are predicate symbols, but those do have equality of predicates axioms in them. I'm not willing here to discuss these versions because they are un-important. The usual ones especially for peano arithmetic is for 1,2,.. etc to be constants and so they are terms of the language denoting an object in the universe of discourse. You need to read PA and first order logic very well. from my discussion with you, it is clear that you are so ignorant of even very well known conventions. You are simply discussing matters that you don't know much about, for example you don't know that constants are zero place function symbol, which tell a lot about how much you know of commonly held syntax. You need to read some foundation of mathematics book, then you can come a speak about it.
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In my opinion, the concept of infinity is dubious, because it is a mixture of an operand and an operator. If you ask why, the answer is the following. As a quantity infinity is always undefined, since there is no specific quantity in infinity. Therefore the meaning naturally shifts to the point of view of the operator and an unlimited operation of division into mathematical units. Thus, the result is that the operation of unlimited repeated mathematical action is converted into a quantitative operand. Actual infinity exists only in universe, if there are unlimited quantities expressed by parameters. I believe that the difference between actual and potential is only our ability to measure and prove unlimited quantities. Though in the real world this is hardly possible. And to sum up, I think infinity is a useless concept that needs to be rid from mathematics and physics.
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Now there are some second order logic theories that can interpret arithmetic in a manner that 2, 1, etc.. are predicate symbols, but those do have equality of predicates axioms in them.

Are you still unwilling to accept a difference between equality and identity? I thought we agreed to that difference a long time ago.

You need to read some foundation of mathematics book, then you can come a speak about it.

Why would I want to waste my time doing that, when I find inconsistencies and contradictions in the conventional interpretations of the very first principles?
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Why would I want to waste my time doing that, when I find inconsistencies and contradictions in the conventional interpretations of the very first principles?

What? that's really strange. You need to first read it and then know about it then you should decide whether its worthy or not. You need to get a good book on mathematical logic like Mendelson's, or Shoenfield's, or Suppes's logic, then you can read Peano arithemtic, and then Set theory. You need to read them carefully, solve the exercises , etc.. it is not something that you'll manage to know on one glance or so.

After you manage to learn about mathematical logic, Peano arithmetic and Set theory, then you can start discussing matters about them, or matters that they are used to be a foundation of, which is most of known mathematics. Otherwise the discussion would be really very poor.
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this is also a good book. Look how does he use "identity" interchangeably with equality.

https://www.math.uni-bielefeld.de/~frettloe/papers/wikibuch.pdf
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Are you still unwilling to accept a difference between equality and identity? I thought we agreed to that difference a long time ago.

Equality is in the least sense an equivalence relation and that's it, it doesn't necessarily satisfy the substitution schema. However when logicians are speaking about equality in the sense of satisfying the substitution schema, then in reality they mean "identity", so for example if you formalize peano arithmetic on top of axioms for = that are the reflexive axiom and the substitution scheme, then in here what you mean by = is exactly "identity" and 2 + 2= 4 exactly means the OBJECT denoted by expression '2 + 2' is the object denoted by expression 4. If you don't formalize Peano arithmeetic on top of substitution scheme but you keep the reflexive axiom and add to it symmetry and transitivity axioms, then here you are using a weak notion of equality which is just some non-specific equivalence relation, here the substitution scheme doesn't work because its not there, and you can have the object denoted by 1+1 not being identical to the object denoted by 4.

Most mathematical logicians would like equality to be interpreted as identity because the substitution scheme makes matters easier.

And again and again symbols like 0,1,2,3,... all these symbols are CONSTANTs of the language, they are defined constants (except 0 which is a primitive constant symbol) they are TERMS, they are in reality zero place function symbols (i.e., function symbols without arguments), each of those denote ONE object in the universe of discourse. = is a relation symbol, and + is a two place function symbol, which is in turn a process that sends object(s) referred to by two occurrences of symbols to an object referred to by a third expression. And it is always the case that functions and relations are associating objects denoted by terms of the language.

That's the convention, it is simple and crystal clear, it is not involved in any contradiction at all.

Arithmetic is in the LEAST just an game played with empty symbols, and so symbols are just subjects to the rules of the game that's it; in the MOST they are about something other than those symbols, like about some platonic world of mathematical objects.

Mathematics as a whole range between strict formalism to outright Platonism.

I personality hold that the formalist side is the purest form of mathematics, when mathematical systems are trying to capture some particular subject matter, here is it becomes what I call as "applied mathematics" in the most general sense, and most mathematics is coined as applied mathematics to a platonic realm, however usually those are considered as pure forms of mathematics and what is considered as applied mathematics is reserved to their application in empirical sciences.

Good Luck!
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Otherwise the discussion would be really very poor.

Then why did you waste so much of your time discussing this with me?

However when logicians are speaking about equality in the sense of satisfying the substitution schema, then in reality they mean "identity",

I know enough but the substitution axiom to know that it deals with equality not identity. The difference is that two distinct things may be equal, but they cannot have the same identity. The substitution axiom allows that one thing may be substituted by another equal thing, so it clearly accepts that these are two distinct things, not one and the same thing, with one identity. The substitution axiom allows that two distinct things, with differences between them, which don't make a difference to the purpose of the logician, may be substituted as equals. But clearly, that there are differences between them means that they are not one and the same thing, as required by the law of identity.
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