Explain to me then, how this set '2+2', is the same thing as this set, '4'. They look very different to me, and also have a completely different meaning. By what principle do you say that they are the same? — Metaphysician Undercover
They are the same according to the game of identity called as "equality theory". — Zuhair
There is a confusion here between expressions and what they denote, "The Sun" , "The nearest star to Earth" are two DIFFERENT (i.e. not identical) expressions, yes, but they denote the same object! so when we say for example "The Sun = The nearest star to Earth", what we mean is that the object denoted by the expression "The Sun" is Identical to the object denoted by the expression "The nearest star to Earth", — Zuhair
Along this understanding the expression "2+2" is meant to denote some object x, and the expression "4" is also meant to denote some object x, however both expression (though different) denote the SAME object exactly. — Zuhair
then please produce this new law of identity, what you call "equality theory". I've asked fishfry for this principle of identity, to no avail. — Metaphysician Undercover
I've asked fishfry for this principle of identity, to no avail. — Metaphysician Undercover
in a more informal manner, x is equal (identical) to y if every expression true of x is also true of y and vise verse, what we mean by true of is the truth of the denotation of that expression about objects and not the truth of its grammatical structure. — Zuhair
Actually equality is nothing but identity. In first order logic it boils down to substitutivity, as mentioned above. — Zuhair
But you need always to discriminate between what an expression is denoting and what an expression is. I already gave a simple example "The Sun" and "The nearest Star to Earth", in physics those two expressions are referring to exactly the same object but they are indeed two distinct expression! — Zuhair
In the game of arithmetic the expression "2+2" is identical to "4", in the sense that they both denote the same object.. — Zuhair
I have repeatedly explained to you that the axiom of extensionality is directly derived from the logical law of identity. — fishfry
Where's the reference to the "logical law of identity" which you are asserting?Given any set A and any set B, if for every set X, X is a member of A if and only if X is a member of B, then A is equal to B. — Wikipedia
A mathematical equality states that the sets on either side of the equation are the same set. — fishfry
But if you are making a mathematical claim, you're just factually wrong. Mathematical equality is identity of sets. A mathematical equality states that the sets on either side of the equation are the same set. — fishfry
OK, I've said true things about '2+2' which are not true about '4'. Therefore the two are not identical. It's what I've been doing for last number of posts, explaining how '2+2' signifies something different from '4 — Metaphysician Undercover
If this is true, then show me the object which both '2+2' and '4' refer to — Metaphysician Undercover
But that doesn't by itself entail that what they are denoting is not identical! — Zuhair
The expression "The sun" and the expression "Nearest star to earth" are also not identical, the first contains two words, the last contains four words, but they do denote exactly the same object. — Zuhair
Now in PA the symbol 2 is meant to denote the object denoted by the expression S(S(0)), for simplicity let us use the notation || phi || where phi is a functional expression, to denote the OBJECT denoted by phi, so we have:
phi denotes || phi ||.
so according to that 2 is denoting the object || S(S(0)) ||.
Also 4 is denoting the object || S(S(S(S(0)))) ||
Now PA proves that the expression 2 + 2 is denoting the object || S(S(S(S(0)))) ||, which is the same object that expression 4 denotes! So by the meaning given to phi=pi in PA, PA proves that:
2+2=4
The proof of that is present in PA. — Zuhair
However to veer to YOUR side, one can in some sense use a terminology that separates identity from equality, you can stress that identity is full matching, i.e. even with expressions, those would be identical only if every property associated with one of them is also to be associated with the other whether at the language level or the meta-language level, and so you'll demand that everything must match between them even the way how those expressions are written. OK, by this we can say that equality is identity of denotation, and that identity is full matching. If we adopt such terminology then of course 2+2 won't be identical to 4, but 2+2 would be equal to 4, since there is identity of denotation of those expressions. This might be plausible, but it is not often used, well as far as I know of, but it might have its virtues. not sure though. — Zuhair
You seem to have left something out. You've taken the '+' for granted. You've shown me what '2' represents, and you've shown me what '4' represents. Then you claim that '2+2' magically represents the same thing as '4'. — Metaphysician Undercover
Imagine what the regular expression accepts, are expressions like this:
{
{1.2323,343.3333}
,{344.2,0,34343.444,6454.6444}
,{2323.11,834.33}
,{}
,{5 12.1,99.343433}
}
So, it only accepts sets, the members of which must be sets themselves, and these member sets must only contain real numbers.
So, it only accepts elements from the power set of real numbers. (Correct?) — alcontali
What occurs to me is that you only have rational numbers in your sets. — joshua
How will you represent irrational numbers with a finite number of symbols, especially those that aren't computable? — joshua
So any subset of those strings is at most countably infinite. — joshua
You seem to have left something out. You've taken the '+' for granted. You've shown me what '2' represents, and you've shown me what '4' represents. Then you claim that '2+2' magically represents the same thing as '4'. But all I see is a claim that S(S(0)) +S(S(0)) represents the same thing as S(S(S(S(0)))). — Metaphysician Undercover
Two distinct things may be equal. For example, distinct human beings are said to be equal. — Metaphysician Undercover
1.1 We have the law of identity that says that for each natural number, it is equal to itself. — fishfry
This puts the matter to rest. The expressions 2+2 2+22 + 2 and 4 44 refer to the same number. — fishfry
These are strings of symbols manipulated by formal rules. — fishfry
On the math there is no question. 2+2=4 2+2=42 + 2 = 4 is an identity derived directly from the law of identity, the Peano axioms, and the definitions of the numbers and of + ++. As I say it's practically a definition. — fishfry
So just go to PA to fill in the missing part, you'll see that for yourself. — Zuhair
the + is a two place function symbol, it is an assignment that sends pairs of objects to single objects per each pair — Zuhair
When we way 2+2 = 1+3 we (in mathematics) mean that the single object that 2+2 denotes is "identical" to the single object that 1+3 denotes, that's what is meant. It means identity of denotation, that's all. — Zuhair
I can exactly mirror you argument to say that "The Sun" and "The nearest star to Earth and Jupiter" do not denote the same object? since the first is just involving one object, while the later is involving a process of two things being near to a third object, and it involves the meaning of star, earth, and Jupiter, so it is speaking of TWO entities with a relation from them (near) towards a third entity that at the end points to that third object, so the denotation of those two expressions is distinct, which is WRONG. — Zuhair
And by rules of arithmetic (say PA) it PROVES that the single object denoted by 2+2 is exactly identical to (i.e. the same as) the single object denoted by 4. — Zuhair
We need first to agree on what constitutes a "denotation" of an expression, and then we can argue its identity. — Zuhair
Equality axioms:
1. for all x (x=x)
2. if phi(x) is a formula in which x occur free, and never occur as bound, and y doesn't occur, and phi(y|x) is the formula obtained from phi(x) by merely replacing each occurrence of the symbol x in phi(x) by the symbol y, then all closures of — Zuhair
This is proof of your's and fishfry's mistake. You cite "equality axioms". Equality axioms are not identity axioms. You and fishfry both arbitrarily replace "equality with identity. Sophistry rules! — Metaphysician Undercover
By showing parts, '2+2' indicates a particular division of the object, unlike '4' which indicates no such difference. So '2+2' denotes an object divided in a particular way, in half, whereas '4' denotes no such division. Therefore '2+2' denotes a different object from '4'. — Metaphysician Undercover
No! Equality rules are spoken as Identity rules by mathematicians, it just happens that equality is used more: see this site on terminology: — Zuhair
So the theory that fishfry and I are mentioning is about "identity", yes its known as equality theory, other sources name it as identity theory, but basically it is about 'identity" as indiscernibility under substitutivity, and it is certainly not about equality as common reference (which is what you think it is about), it doesn't make sense to think of it as being about common reference, why should we have a law about indiscernibility of objects that has common value under certain functions?? — Zuhair
In mathematics when = is used it is meant to symbolize "identity", i.e. sameness of objects, and not assignment to a common value as you think. — Zuhair
So you seem to be arguing that since '4' is not denoting that the object it denotes is an object that is divided in half, then it follows according to your reasoning that 4 is denoting an object that is not divided in half. This is an error. — Zuhair
Not claiming something doesn't mean that you are claiming its negation. I'm not claiming that my son would pass the exam, it doesn't follow from this that I'm claiming that my son will not pass the exam. — Zuhair
So 4 not denoting that what it denotes is dividable in half, doesn't mean that 4 is denoting an object that is not divisible in half. — Zuhair
Absence of denotation doesn't mean denotation of absence.
Absence of denotation just signal incompleteness of information. — Zuhair
2 + 2 only shows some extra-information about what it denotes more than the constant symbol 4 shows about what it denotes. That doesn't mean that what they are denoting is not the same object. — Zuhair
I can say that Barack Obama is one of the presidents of the united states. Another time I can say that Barack Obama is one of the presidents of the united states that has a Nobel price. The first expression did NOT denote that Barack Obama had a Nobel price, yet I didn't deny it! It is only the case that the second sentence had more information, but both are speaking exactly of the same person. In a similar manner 2+2 and 4 are denoting exactly the SAME object, but 2+2 is denoting more information about that object than 4 does, but again 4 is not denying what 2+2 is denoting. — Zuhair
It does not state that the sets are the same, it states that if the members are the same, then the sets are equal. Therefore the sets remain distinct, as two equal sets, not one and the same set. — Metaphysician Undercover
I just wanted to add, that we can actually have a very simple system in which 2 + 2 = 4, that of first order logic and add to it primitives of identity (equality) symbolized as "=" which is a binary relation symbol, and of "+" denoting addition which is a two place function symbol, and of "1" denoting what we customarily know as one, which is a constant symbol. I'll try to coin a system in which 1 is the first number, i.e. doesn't have zero in it. — Zuhair
1.1 We have the law of identity that says that for each natural number, it is equal to itself.
— fishfry
This is our point of disagreement. The law of identity does not say this, you are claiming this. — Metaphysician Undercover
But '2+2' denotes two objects, each with a value of two. What do you think the '+' sign is there for, decoration? — Metaphysician Undercover
Your site provides the terminology of first order logic, not mathematics. The use of "=" is not the same in first order logic as it is in math. To equate these two is to equivocate and that is a fallacy of logic — Metaphysician Undercover
One object but two digits — Shamshir
Well PA is a mathematical system. Most formal mathematical systems nowadays are stipulated as extensions of logical systems, in particular first order logic with identity. And it is about those mathematical systems that I was speaking. — Zuhair
I've shown you the axioms of first order logic with equality and you replied that the equality sign in them is not about identity, when I showed you that this is just a terminology preference, and that it is also named as first order logic with identity and I showed you the rationale behind those axioms and its relationship to the informal notion of identity, you replied that this is not mathematics. — Zuhair
In reality all older mathematical systems that you know of can be formalized as extensions of first order logic with identity, and in those systems the symbol = is taken to represent identity. — Zuhair
Now the question is what about older systems that are not formalized as extensions of first order logic with identity... — Zuhair
But anyway your argument that the expression '2 + 2' is taken to represent two objects is outright false, even in ordinary math the expression '2 + 2' is taken to denote a single natural number that is sent to by the + operator from the pair {2,2} [more precisely one must write it as (2,2) since it is an ordered pair], it doesn't denote two natural numbers as you think, because + is a FUNCTION. — Zuhair
Didn't you just say "all" older mathematical systems can be formalized as systems where "=" represents identity? And now you ask about those which cannot. Oh what a tangled web we weave when first we practise to deceive. — Metaphysician Undercover
Therefore "equal" in ZFC cannot mean "same" as determined by the law of identity. — Metaphysician Undercover
Only specific mathematical systems are based in first order logic, perhaps ZFC is one of them — Metaphysician Undercover
It is very clear that ZFC derives its meaning of "equal" from the traditional meaning of "equal", and not from the law of identity, because ZFC does not cite the law of identity, and as we've seen, it allows that two distinct things are "equal". Therefore "equal" in ZFC cannot mean "same" as determined by the law of identity. — Metaphysician Undercover
There you go, continuing with your lies. You are fully aware that this is not true, being the well-educated individual that you are. Yet you assert it anyway! Why lie? What's the purpose? — Metaphysician Undercover
Exactly, an "ordered pair". And an ordered pair is two objects. Why say that this is false? Your propensity for lying never stops amazing me. — Metaphysician Undercover
Of course they can be formalized as an extension of first order logic with identity, — Zuhair
Anyhow almost all of traditional mathematics before the era of set theory and modern mathematical logic, nearly all of it can be re-formalized as extensions of first order logic with identity systems, and of course the "=" in them would be understood to represent identity. — Zuhair
I don't know why you keep assuming that I'm lying? — Zuhair
(2,2) ---+---> k
Now "2 + 2" is that object k, in other words "2 + 2" is not denoting the ordered pair (2,2), No! '2 + 2' is denoting the object that the operator + send the pair (2,2) to, and that object, i.e., k is exactly the natural number denoted by the symbol 4. In other words "2 + 2" is denoting exactly the same object that 4 is denoting. That's the easiest way to understand it. — Zuhair
Perhaps you can clarify this point for me then. The law of identity is that a thing is equal to itself. — fishfry
So when the law of identity is expressed in formal logic as "a=a" or some such thing, the "=" represents "the same as". Zuhair is arguing that all mathematical axioms can be interpreted as "=" representing "the same as", but this is equivocation plain and simple. I am arguing that no mathematical axioms can be interpreted in this way because it is fundamental to mathematics that the two sides of the equation represent distinct things, while the law of identity indicates that "the same" refers to one and only one thing. — Metaphysician Undercover
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