I believe that is called the principle of the identity of indiscernibles. Maybe look into that?

I know that, proving 1+1=2 is hard — Monist

Well, no, it isn't.

With S the successor function, and x+1=S(x), and 1=S(0), we can see that 1+1=S(S(0))=2. You can trivially prove it by using PA's rewrite rules for addition. (PA is standard number theory)

but I do not see anyone trying to prove x=x, because it may seem so simple and obvious — Monist

PA's second axiom:

For every natural number x, x = x. That is, equality is reflexive. — PA

The fact that PA axiomatizes this rule for natural numbers means that it is not provable in PA.

In fact, you can generalize this remark: the defining rule for equality cannot be proven but only be defined in the mathematical theory.

For example in ZFC (=standard set theory), axiom 1, i.e. "the axiom of extensionality", defines the equality of sets:

Thus, what the axiom is really saying is that two sets are equal if and only if they have precisely the same members. The essence of this is: A set is determined uniquely by its members. — ZFC

Again, ZFC's logic for extensionality is an axiomatic starting-point rule that cannot be proven. It can only be used to prove derived rules (=theorems).

The starting-point rules, i.e. the system-wide premises, in a mathematical theory are always arbitrary, unexplained and unjustified beliefs. That is simply the essence of the axiomatic epistemology.

It is difficult to see what is the content of the proposition, x=x is being presented to the reader. You might only say this if somebody was trying to persuade you that x was not x but under what are the conditions might this be a possibility? Of course there is the assumption that x is a constant but even then is it difficult to see how the proposition really carries any meaning.

1+1=2 is a more meaningful proposition than simply 1=1. It tells us something about the definition of each quantity in the relationship and something about the relationship itself. As with the x=x proposition it makes an assumption that numerical values remain constant, that 1 today is the same as 1 tomorrow and unlike the x=x proposition, this is a significant assumption.

Thank you. Appreciated.

"an apple is an apple", but why? I do not get why any certain thing called 'x', should be 'x'. — Monist

"Called" is the act of an intelligence. "Should" is a deeply deceptive and misleading word - there ain't no should. What happens is that as both an idea and as applied to the world, x=x seems to work. And as to the underpinnings of the how or the why it works, it's enough to say that it works because it had better work! Of course there are fancier ways of saying that.

I would ask, how do we know that "apple" means exactly the same thing in both instances of the word.
"An apple is an apple".
For this to mean anything, it would require that each instances of the word "apple" to have a unique meaning.
Or else you are just saying "This is an apple, and its also apple".

What is an apple?
An apple.

Its a way of avoiding the real question. Beyond the appearance of an apple, what is there....what is it really.

Its kind of like saying, "Is reality real"?
Well, if its real reality, then yes. If its fake reality, then no.
The answer is to clarify the question.
"Is what we think of as reality, the real reality, or a fake reality".
I would say, what we think of as reality is a fake reality.

Sorry I want to add one more thing.
Something is axiomatically true, we usually say, if its true by definition.
An apple is an apple because we have defined an apple as an apple.
But how do we know if a definition is correct or complete.
That is a difficult question, I would say.

Sorry if I'm turning the question into something bigger the intended scope.
And sorry for making three separate posts.

The thing you need to get, I think, is that X=X says something about the language we use to describe the world, and not necessarily about the world itself.

In the world nothing is perfectly identical to some other thing. Every apple has some, even if only miniscule, difference compared to another apple. But they are similar enough that we can abstract away from raw sense-data and make up categories and use those to be able think further than mere experience of fleeting moments.

It's not a perfect tool, and that is important to realise so you don't expect things it can't deliver. But at the same time, it's still the best tool we have even if imperfect.

So... to answer your question, this is not about proving x=x, x strictly speaking doesn't equal x. It's about utility, we want x to be equal to x so can get along with whatever it is we want to infer from that.

It's kind of self-evident, isn't it?
Meaning auto-supposes self-identity.
When talking about the soccer match, thinking of the neighbor, etc, we automatically go by identity.
Abandon identity and the posts here are meaningless.

x=x is just setting out the way we use "=".

x=x is just setting out the way we use "=". — Banno

:up:

"an apple is an apple", but why? I do not get why any certain thing called 'x', should be 'x'.

The simpler it gets, the complexer explaining it. — Monist

I really don't get the reason why anyone would ever use that phrase, "An apple is an apple.", unless they're just playing words games, which isn't a complex thing at all.

How is using that phrase different than saying, "An apple" while pointing at an apple? Is your pointing the equivalent of = ?

$X\ne X$

See Banno's Game.

we could do that. What are the consequences? Anything. Not a game worth playing.

That’s why we don’t play that game.

Equivalence classes. Simple example: 1/3 = E = {n/m: n/m = 1/3}. Reflexive: shows 1/3 belongs to E. Symmetry: 1/3 = 2/6 and 2/6 = 1/3. Transitive: 1/3 = 2/6, 2/6 = 4/12 implies 1/3 = 4/12.

The thing you need to get, I think, is that X=X says something about the language we use to describe the world, and not necessarily about the world itself. — ChatteringMonkey

Agreed.

Mathematical sentences, such as X=X, are constituent rules of the mathematical theory that defines them, and are true only in the model(s) for that theory. Such model is never, ever the real, physical world.

For example, if we are talking about natural numbers, we can say that the sentence is true in the standard model for number theory (=PA):

PA $\vDash$ X=X

The models for number theory (=PA) are NOT the physical universe.

The physical universe itself is a model of the elusive and inaccessible theory of everything (ToE). We do not know if the following sentence is true, because for that we would need to know the TOE:

ToE $\vDash$ X=X

We simply have no clue as to whether the sentence mentioned above is an axiom in the ToE, or otherwise a theorem provable from axioms in the ToE.

Mathematics does not prove anything, or asserts any truth, about the physical universe because the model(s) for any available mathematical theory, such as number theory, or set theory, and so on, are not the physical universe.

Just to elaborate on one aspect of the question, in mathematics and logic equality is introduced axiomatically, and self-identity is (usually) part of the definition. See for instance equality in first order logic.

As others have noted, you need to first make clear to yourself what question you are actually asking. Once you do that, the answer may become apparent.

The ability to recognise that ‘x’ is ‘x’, is essential to the ability to ask ‘why’ in any general sense. If you couldn’t recognise and abstract likeness then you couldn't ask 'why anything'? So the answer is: it just is, and asking 'why' it is has no answer.

I believe that The Identity of Indescernibles states that two distinct things do not resemble each other. While my question is why a thing resembles itself.

The thing doesn't necessarily have to resemble itself. I've often mistaken giant squids for snails.

x^2 > x. I get your point.

I know that, proving 1+1=2 is hard
— Monist

Well, no, it isn't. — alcontali

Well, it was much harder for Russell and Whitehead, PA did not satisfy me for many reasons.

The starting-point rules, i.e. the system-wide premises, in a mathematical theory are always arbitrary, unexplained and unjustified beliefs. That is simply the essence of the axiomatic epistemology. — alcontali

Thank you for this reply, it helps me a lot, but does not solve my problem. Aren't axioms, self-evident assumptions? If so, when can we accept self-evident beliefs, just when they are practical? Do we have to analyse the relation between truth and practicality then?

You might only say this if somebody was trying to persuade you that x was not x but under what are the conditions might this be a possibility? — Mike Radford

Under what conditions is x=x true, when we just accept it? I might be persuaded that x=x.

It is difficult to see what is the content of the proposition, x=x is being presented to the reader. — Mike Radford

The proposition is simply: A thing resembles itself. The question is, "what is the proof?"

As with the x=x proposition it makes an assumption that numerical values remain constant — Mike Radford

Lets imagine that x is a variable, then again we end up with x=x. Thank you for your perspective.

Something is axiomatically true, we usually say, if its true by definition.
An apple is an apple because we have defined an apple as an apple.
But how do we know if a definition is correct or complete.
That is a difficult question, I would say. — Yohan

Exactly, I am trying to find a ground to stand on. Starting points known as axioms, simply suck. :-)

Meaning auto-supposes self-identity. — jorndoe

The Law of Identity states that a certain thing is identical to itself, and I ask why.

1+1=2 is a more meaningful proposition than simply 1=1. It tells us something about the definition of each quantity in the relationship and something about the relationship itself — Mike Radford

1=1 may or may not be meaningful, but is seems to be true, but why? (It is meaningful in the sense of understanding the internal relation of any quantity; the Law of Identity)

"an apple is an apple", but why? I do not get why any certain thing called 'x', should be 'x'.

The simpler it gets, the complexer explaining it.
— Monist

I really don't get the reason why anyone would ever use that phrase, "An apple is an apple.", unless they're just playing words games, which isn't a complex thing at all.

How is using that phrase different than saying, "An apple" while pointing at an apple? Is your pointing the equivalent of = ? — Harry Hindu

Instead of 'apple' try 'thing'. Saying "a thing" while pointing at the thing does not explain why the thing identical to the thing. It does not explain the relation between the thing and the thing. x=x does, it simply tells that the thing, is itself. The point is, why? :-)

I know that, proving 1+1=2 is hard
— Monist

Well, no, it isn't.
— alcontali

Well, it was much harder for Russell and Whitehead, PA did not satisfy me for many reasons. — Monist

You might be interested in the idea of why we have a system of counting to ten, switch, then repeat because we have 10 fingers.

The Law of Identity is quite easy to understand.

Instead of 'apple' try 'thing'. Saying "a thing" while pointing at the thing does not explain why the thing identical to the thing. It does not explain the relation between the thing and the thing. x=x does, it simply tells that the thing, is itself. The point is, why? — Monist

The Law of Invoking Sense?

The Law of Invoking Sense? — Per Chance

Which i believe is a natural law.

Aren't axioms, self-evident assumptions? — Monist

Originally, in Greek antiquity, over 2500 years ago, i.e. primarily in Euclid's Elements, axioms were meant to be self-evident. For a long time, classical Greek geometry was the core foundation of mathematics. This is no longer the case. From within mathematics, axioms are nowadays considered arbitrary starting points. Especially the formalist philosophy views it like that.

There may still be a link -- outside the realm of mathematics -- that views our most important axiomatizations, i.e. number theory and set theory, as concepts that are inspired by our innate intuition and by the nature of the universe that surrounds us. From within mathematics, however, it is wrong to view it like that, because the epistemology of mathematics does not allow us to make that kind of claims.

Hence, axioms are best viewed as arbitrary, unexplained, and unjustified beliefs.

If so, when can we accept self-evident beliefs, just when they are practical? — Monist

Mathematics does not seek to be practical. On the contrary, the desire for ever-increasing abstraction leads us to make sure that mathematics is preferably meaningless and and useless:

Abstraction in mathematics is the process of extracting the underlying structures, patterns or properties of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.[1][2][3][4] Two of the most highly abstract areas of modern mathematics are category theory and model theory.[4] — Wikipedia on mathematical abstraction

So, mathematics is preferably unrelated to the physical world and therefore meaningless. It can certainly not be applied directly. It must go through downstream user domains such as science, engineering, and so on, which are empirical and reintroduce the physical universe, with a view on harnessing meaningfulness and usefulness.

Consequently, in and of itself, mathematics is not just meaningless but also useless.

The ontology of mathematics is a set of arbitrary, unexplained, and unjustified core beliefs (=axioms) from which we derive new beliefs (=theorems) that are purposely meaningless and useless.

Do we have to analyse the relation between truth and practicality then? — Monist

The term "truth" in mathematics are facts (=data) in a given structure, i.e. a model, that satisfies a set of logic sentences derivable from a particular theory. It has nothing to do with truth in the physical universe. As I have written before, such mathematical model is never the physical universe, simply because we do not have a copy of the theory of everything.

So, mathematics is not real-world true. As I have mentioned above, mathematics does not seek to be practical either.

Thank you, I will try to fully grasp it.

Mathematics is one of those things social scientists came up with to keep people from getting bored with life in a modern society. So are science, and other branches of story telling. Things to aim for. Go to school, Get a degree. Get a Job. Don't get bored so soon.

You might be interested in the idea of why we have a system of counting to ten, switch, then repeat because we have 10 fingers.

The Law of Identity is quite easy to understand. — Per Chance

What is the relation between the practicality of Base 10 and the necessity of The Law of Identity?

Decimal systems have the use of just naming quantities(switching to Base 12, does not change any quantity), Law of Identity does not deal with the names of a quantity, but the definition, it deals with the property of x.
Can you explain why you gave the example of our counting system? I compeletely miss the point.

For me, it is NOT QUITE EASY to understand, nor is any 'thing' quite easy for me to understand.