jupiviv wrote:Kelly Jones wrote:Only 1+1 = 1+1 in a philosophical sense.
In a mathematical sense, the "=" is an attribution sign, inputting a value to "1+1".
Why are you making the same mistake? 1+1=2 is exactly the same as saying A=A.
1+1=2 is a definition, not a simple identity. The law of identity is
more basic than a definition. For instance, in A=A, A is not defined as anything. It's just saying that A is itself.
I'll break down the mathematical definition "1+1=2", to show what I'm getting at. In the following, I'll use the colon to show assignment of value, to distinguish a definition from a pure identicality.
For example,
= : equivalence symbol of identity (philosophy)
= : equivalence symbol of quantity (maths)
But the two are not identical in meaning, even though the same symbol is used. But, the philosophical equivalence is used in the mathematical equation, prior to the quantity type. It's not two "logics", but one.
(In long-hand, the above are two definitions, and can be read as the following:
The symbol "=" in the law of identity equation is assigned the meaning of "a symbol that indicates equivalence of identity". Note that the mathematical equation also uses the symbol in this way, but it is
implied where the terms are listed, and is not the explicit meaning of the symbol. It's simple once you see it.)
Stage 1: listing of the basic identities, or terms. They are recognised as what they are (A=A), and distinct. It's a very simple step, which many people overlook because of its obvious nature. Yet it is essential. Notice that nothing is defined, or assigned values. It's the same with A=A, in which "A" is never given any finite, specific meaning.
1=1
+=+
1+1=1+1
2=2
=== (It is implied that the "=" in the equation is just a symbol, like the others)
Stage 2: assigning meanings or values to the basic identities, or terms. This step is to make definitions:
+ : addition operator
= : equivalence of quantity, but not equivalence of identity
1 : number, single quantity
2 : number, equivalent quantity to 1+1
1+1 means 2, and two means 1+1(two ones added together.) I'll reiterate - there is no essential difference between the logic used in different fields. If there were, logic would become meaningless.
The "means" gives you the clue that the equation is a definition-type, not an identity-type.
The logic is the same. It may be confusing because the same symbol "=" is used, but the philosophical identity is implied in the mathematical equation. Namely, 1+1=1+1, 1+1:2, 2:1+1.
A proof of this is that 1+1 is mathematically equal to 10-8, but that is not its identity. It's quantity is identical, but its identity isn't.
A=A does not add any new information. It is not a definition.
..