# Question of Identity

Conversing with someone trying to convince me that 2 and 1+1 are not the same thing.

His argument was that although 1+1 = 2, they are in fact different because the notation is different. We see "1+1" on one side of the equal sign and "2" on the other. Hence, 1+1 is not equal to 2.

I'm not sure, but this reasoning seems fallacious. I just don't have a strong enough background in philosophy to point out in a precise way exactly why it is.

• Go read Naming and Necessity (Saul Kripke). Seriously, everyone should read it - it's a small book, but packed with so much interesting stuff on this topic. – Benubird Apr 8 '16 at 12:19
• Possible duplicate of Is 1+1=2 true by definition ? – Cort Ammon Apr 9 '16 at 18:41
• This seems to just be a question of semantics. Under number equality, "1+1" and "2" are equal. As bits of texts ("strings" is what programmers call them), they are different, because the characters used are different. So, it depends on context. – James Kingsbery Apr 11 '16 at 18:13

## 4 Answers

This is a good question that reminds me of Frege's monumental paper On Sense and Reference.

Taking Frege's classic view, we say that '2' and '1 + 1' refer to the same thing (i.e. the number 2), but that they have a different sense. So in one respect they are the same thing, but in another they are not.

One of Frege's reasons for this distinction is that asserting an identity such as '1 + 1 = 2' is informative, while '2 = 2' is not. Some less trivial examples are: '47 + 119 = 166', 'Barack Obama = The 44th US president'.

• This is the right approach. For narural languages, the classic example is Hesperus and Phosphorus (also due to Frege, I believe). This difference is obvious computationally - just imagine 2=x, where x is some monstrous expression that takes 10000 years to compute, that results in 2. – user20153 Apr 12 '16 at 20:27

Yes, the famous dichotomy of the equality through intension and equality through extension. The easiest way to explain this dichotomy is thus

• equality intension refers to the meaning of the things you equate, 4/2 is equal to 2 because you want them to be equal in meaning
• equality in extension is equality through computation, it is also called Propositional equality

Be careful because there is also the dichotomy equality-identity. The best part is that these notions differ with the mathematical sub-field wherein you work...

You clearly see that 1+1 is not the same thing at 4/2 which differs equally from 2. The work involved in each of the illustration of 2 is not the same, the reasoning is not the same. Of course, a few people cannot stand that 2, 1+1 and 4/2 differ, because they hate to contextualize the result, through the method of getting the result. These people who hate this situation crave some absolute from which they claim that the method of obtaining the result is irrelevant in obtaining the result. What matters to them is the result and nothing else.

Now, let's look at what happens once you take into account the method. Once you say that the method matters, you attach to the result 2 the method of getting this 2. This brings you the concept of equivalence of proof (of the number 2): the easiest proof of the number is the data of the number 2; you show to yourself 2 and this is the easiest way to prove 2. But you have plenty of other algorithm, other method to obtain the number 2 and these methods lead you to the notion of equivalence of proofs.

I let you dwell in this mess of extension and intension for equality and identity. https://ncatlab.org/nlab/show/equality https://en.wikipedia.org/wiki/Extensionality

See Equality :

In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object.

In mathematics, when we assert the equality: 1+1=2 we are not asserting that the two strings of symbols 1+1 and 2 are equal, but we are asserting that the objects named respectively with "1+1" and "2" are the same object: the number two.

• No. We are asserting that 1+1 and 2 evaluates to the same number. 1+1 refers to an expression, as does 2. But these expressions are not the same, for example 2 is atomic, while 1+1 is not. – Taemyr Apr 8 '16 at 14:23
• @Taemyr - In moder logic, the terms 1 and 2 are individual constanst and the expression 1+1 is a term built from the constant 1 and the binary function +: both denote objects of the domain of the interpretation. The sentence 1+1=2 is true - according to the semantics for f-o logic - exactly when both terms denote the same object. – Mauro ALLEGRANZA Apr 8 '16 at 14:28
• Yes. But the expression is a separate thing from the string. And it's the expression that the string refers to. To get to equality you need to invoke the a model (in the case of your comment FOL arithmetic) in which you can evaluate the expression. – Taemyr Apr 8 '16 at 14:47

Based on the way OP phrased the question: If we consider the equations by literal "symbolic value" instead of implied value, why do we consider that "=" is an equality operator? Do we not need to consider all the symbols in the same context?

I think we should consider all symbols under the purpose for which the symbolic representation is invented i.e. math.

It is just as nonsensical as it is to start writing in English and in the middle of the sentence switch over to Chinese symbols and trying to enforce the grammar rules of one language onto the other.