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
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
OK, that's fine. OF course just to make it more precise. I said almost all of mathematics before the era of set theory can be formalized as an extension of first order logic with identity where the symbol "=" is taken to mean "identity" i.e. "being the same as". Actually this is a well known result, actually most of that kind of mathematics can be formalized in second order arithmetic, you can read about it in reverse mathematics which also can be re-formalized as an extension of first order logic with identity. Actually ZFC itself can be formalized as an extension of first order logic with identity, and ZFC is way stronger than almost all of mathematics before the era of set theory. This is a very well known result. — Zuhair
what is that fundamental aspect that enforce us to interpret = sign as some equality relation other than identity. — Zuhair
If there is something fundamental to mathematics against the use of = symbol in it to represent identity, then how PA is formalized as such??? How ZFC is formalized as such and it is generally regarded by many as the official foundation of mathematics? Both are indeed formalized with = in them understood as identity. — Zuhair
What you are saying is that the current foundational systems of mathematics are committing a fundamental error? (notice that most of those are coined as extensions of first order logic with identity) According to your account they must instead represent the = as an equivalence relation that can hold between distinct objects, and that the object denoted by 2 + 2 must be considered as a distinct object from that denoted by 4. This is strange? why? — Zuhair
just want you to answer this question
does the expression "2 + 2" denotes two objects or one object?
I know that it contains in it the symbol 2 twice, that is clear, but do you think just because of this containment, then it ought to "denote two objects" — Zuhair
2 + 2 is equivalent to the expression "The result of summation of 2 and 2" — Zuhair
2 + 2 means "the natural number that results from adding 2 to 2" — Zuhair
But the formalization would be more cumbersome, because you are holding to a weaker concept than identity, you'll loose all the merits of identity, which shortens formalization to a great extent. — Zuhair
The law of identity doesn't say that a thing is equal to itself, it says that a thing is the same as itself. In formal logic, "the same as" is represented by "=". So when the law of identity is expressed in formal logic as "a=a" or some such thing, the "=" represents "the same as". Zuhair is arguing that all mathematical axioms can be interpreted as "=" representing "the same as", but this is equivocation plain and simple. — Metaphysician Undercover
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
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
I've written plenty to you already that you haven't engaged with, including a proof directly from first principles that 2 + 2 and 4 are identical. — fishfry
all of this is wrong. + is a binary function symbol which means it is a ternary relation symbol, it is a relation between three occurrences of symbols, it relates the first two symbols to a third occurrence of a symbol, here it relates the two occurrences of 2 to a third occurrence of a symbol which is 4, this is explicit when you write it in relational terms as +(2,2,4), but when it is written in functional terms here the confusion would raise since you don't see the third occurring symbol (which is 4) you only see two occurrences of 2 linked by + sign in between, here it means that + is relating the two occurrences of symbol 2 to the symbol '2 + 2', you see here the expression '2 + 2' is acting as a symbol denoting an object of the language. — Zuhair
in your views '2 + 2' represent two distinct objects operated upon by the + operator. While the common view is that '2 + 2' denotes the natural number that results from running the + operator on two occurrences of 2. — Zuhair
Similarly '2 + 2' is an expression that mentions denotations of objects by two occurrences of the symbol 2 and an operator running on them, yet the total expression (i.e. all three symbols in 2 + 2 in that sequence) is denoting non of those, what is denoted by the total expression '2 + 2' is a single object that can be what is denoted by '4' if you interpret '=' as identity, or it can be another object that is related by some equivalence relation to the object denoted by 4, anyway the whole expression of "2 + 2" is not denoting multiple objects, no , it is denoting a single object, because + is a FUNCTION. — Zuhair
We went through this. You provided no such proof, it was just an assertion. — Metaphysician Undercover
We have an operator which expresses a relationship between them. — Metaphysician Undercover
We went through this. You provided no such proof, it was just an assertion. — Metaphysician Undercover
Is it possible you missed this? — fishfry
1.1 We have the law of identity that says that for each natural number, it is equal to itself. — fishfry
The operator + here is a relation between three objects, one expressed by 3 and the other by 5 and the third by the expression "3 + 5". — Zuhair
So the + sign here is understood to be a ternary relation that links objects denoted by the terms of the language which are "3","5","3+5". — Zuhair
and '3+5' is also taken to denote ONE object (because 3 + 5 is a binary function symbol and so it is a term of the language, so it denotes one object (despite having parts of it that denote other objects)). — Zuhair
So to be more precise the operator + in "x + y" means a ternary relation between the object denoted by x in the first role, and the object denoted by y in the second role and the object denoted by "x + y"
so + sign in "2 + 2" means a ternary relation that links the object denoted by 2 in the first role and the same object in the second role and the object denoted by "2 + 2". — Zuhair
does the expression "2 + 2" denotes two objects or one object? — Zuhair
You haven't justified your claim the "2+2" is an object, nor your claim "+" represents a ternary relation. I think you have fallen back into your habit of lying. — 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. — Zuhair
That said the symbol 2 is taken to denote a single object in the universe of discourse because 2 is a constant symbol, while the expression "x" is a term that ranges over many objects of the universe of discourse, this means that x can be substituted by many objects of the universe of discourse. — Zuhair
Any binary function is a ternary relation, please read the syntax and rules of first order logic. — Zuhair
I feel that your problem is that you were thinking of the "+" sign as a binary relation symbol linking two terms of the language and so the expression 2 + 3 would NOT denote an object. Which is wrong!
By convention the "+" sign is a binary function symbol linking two terms of the language, and so the expression 2 + 3 would BE denoting an object. — Zuhair
The problem is that you have repeated stated that "2" denotes an object. Unless the "+" annihilates the existence of the object denoted by "2", to create a new object, then "2+2" cannot denote an object as well as "2" denoting an object at the same time, without contradiction. So if "2+2" denotes an object, by what means is the object denoted by "2" annihilated in favour of this new object denoted by "2+2"? And, if "2" no longer denotes an object its meaning is lost, such that "2+2' can no longer be equal to "4". — 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". — Zuhair
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
"+" 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
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
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
The law of identity states that a thing is the same as itself, not that a natural number is equal to itself. This is the problem, you keep asserting that the law of identity says something about equality, when it does not. It says something about identity. — Metaphysician Undercover
But 8 also refers to a completely different object from objects referred to by "3" and "5", and "8" is also not at all equivalent or the same as "3 and 5". I see the analogy is perfect! Why you say it fails? — Zuhair
(notice that 3 and 5 is not the same as 3 + 5, 3 and 5 is the totality of the objects referred to by 3 and 5, it is not what the + operator sends 3 and 5 to. The totality of the object referred to by 3 and the object referred to by 5 is NOT equivalent to the object referred to by 8, those are different objects, the latter is refers to an individual object, the former refers to a totality of two separate objects, so they are not the same nor are they equal). — Zuhair
The mother of Jesus and James = Mary — Zuhair
Yes + sends objects denoted by the symbols it occurs between, to some object. The objects denoted by the symbols the symbol of + is written in between (in infix notations) would be sent by the + operator to an object as specified by the rules of arithmetic. Just because + occur between two symbols doesn't mean that the objects those two symbols are referring to are distinct objects No. For example "2 + 2" here the first and second "2" which are linked by + symbol, both of those do refer to exactly the same object, why? because 2 is a "constant" term of the language, so it can only refer to a single object in the universe of discourse. — Zuhair
Now the + operator would refer that single object (symbolized by "2") into the object referred to by the symbol "4", that's it. — Zuhair
If I use the word same instead of equal does that satisfy you? — fishfry
The fact that you say you read my post and this is your complaint means we're done. You have no substantive reply? I showed a proof from first principles that 2 + 2 and 4 are identical. You ignore it? — fishfry
There is no case in math of an equality meaning anything other than identity; whether of abstract objects (logical identity) or sets (set identity or equality). Set identity is the same as set equality. — fishfry
If you don't agree that's your right, even if you haven't and can't show a single example to support your claim. But in all this time you have not presented an argument. And you have never engaged substantively. And from me to you, you're factually wrong. All the best. — fishfry
Equal is the same as same. — fishfry
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
It is some logical principle which allows us to speak of multiple things as one object. — Metaphysician Undercover
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
There is nothing in "3+5" to take the place of "mother", there is just Jesus and James. — 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: — Zuhair
2 is Defined as the object that the + operator would send the single object 1 to. (this is: 1 + 1 = 2)
,,,
This is the principle of "Set", a set is what turns multiple objects into one entity, it turns Jesus and James — Zuhair
So for example we can view "Mother" in the above sentence as a "sending rule" it sends the pair {Jesus, James} to another object which here happens to be Mary. EXACTLY a similar thing is happening here the "+" operator is sending the pair {3,5} to another object which by rules of arithmetic and arbitrary definitions this other object is enforced to be 8. — Zuhair
I didn't say that + would make 3 and 5 into one object, I said it will send them to one object, if I did say that it makes them into one object, then I only meant that it would send them to one object. + is not a merging process, it is an assignment scheme. — Zuhair
There is (implicitly) "sum of". (Not that the analogy follows through completely, as Zuhair points out.) — bongo fury
But some theories I think would fare far better if they do that, for example Set Theory, here to say that the set X defined for example as: for all y ( y in X if and only if y=empty set ), this is usually symbolized as {{}} or as {0}. — Zuhair
No, that's the point you cannot validly substitute "same" for "equal". It will produce equivocation. You don't seem to understand this — Metaphysician Undercover
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