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. — Per Chance

Off-topic

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

You'll have to excuse me. I'm not well versed in Logarithms or the Law of Identity.

Can you explain why you gave the example of our counting system? I completely miss the point. — Monist

I guess i was just fed up of laymen's evoking of mathematical mysticism in philosophy.

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

Such as the above statement/s for example.

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

Well, what would be the alternative? A thing that is not what it is? An apple that is not an apple?

I appreciate all the answers.

I accepted the risk of being an idiot when starting the topic, and I see the risk is true, as I see answers underestimating the question. Partly it is my fault. My ability to address the problem may be weak because this is not my native language, I will do my best to re-write my question.

+ To establish any knowledge, I have to believe in it
+ I am not able to believe and/or have no ground to build on with certainty.
+ There is some certainty in the language of math itself and is applicable to the world.
+ The basic idea of math is that any value has an identity(is equal to itself). And here the problem for me starts.
+ When applied to the real world: what is, is(even simpler: `being` or `thing`). It seems to me as a good starting point, because to believe anything, you should believe the thing(being) first.

But then, how do I believe a thing, is a thing.

Why x=x ?

If I can explain the necessity of identity, I can prove a thing can exist.

Well, what would be the alternative? A thing that is not what it is? An apple that is not an apple? — litewave

Why not?

Your answers are irrelevant, off-topic and arrogant. The laymens goal is to learn, not to hurt you.

Why not? — Monist

Because if such an alternative existed, it wouldn't exist.

Because if such an alternative existed, it wouldn't exist. — litewave

I see a conclusion, but no premises. How and why?

I think you are not giving due attention to the language angle that was proposed in some of the answers. The very meaning of equality and identity in ordinary language already implies self-identity. To question self-identity is not even a metaphysical move - it is meaningless, like questioning the marital status of bachelors. (In math and logic this has to be set out explicitly though - and indeed it is.)

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

There is no such thing as a thing that has a relationship with itself. Things establish relationships with different things.

Again I don't even understand the point of your question. I dont understand the differece between x and x=x. Why not just say x? In saying x, you are saying what it is. x=x is just redundant information and therefore useless.

There is no such thing as a thing that has a relationship with itself. Things establish relationships with different things. — Harry Hindu

Identity is the relationship one thing bears only to itself.: — The Cambridge Dictionary of Philosophy, 2nd Edition, CUP: 1995

Again I don't even understand the point of your question — Harry Hindu

Why is identity necessary?

I don't understand the difference between x and x=x. — Harry Hindu

Simple, x is an unknown value, x=x is the principle that it is a value.

In saying x, you are saying what it is. x=x is just redendant information and therefore useless. — Harry Hindu

If x=x is useless because it is redundant information, is first-order logic useless as well? Because, it literally depends on this axiom. For every x | x=x

Why is identity necessary — Monist

Does identity exhaust what it is for the thing to be itself? Isn't an identity a label? A label is not the thing. Words are not the thing itself. Are you talking about the thing, or what we call it? Both are different things that have a relationship with each other.

Simple, x is an unknown value, x=x is the principle that it is a value. — Monist

If the value is unknown, how can you say they are equal? It seems that you need to know what x entails for the = to be useful. x does not equal x because both x's are on opposite sides (they occupy different space and are typed at different times on the screen (one is after the other) of the =, so I don't know what you mean for two different x's to be equal.

Does identity exhaust what it is for the thing to be itself? Isn't an identity a label? A label is not the thing. Words are not the thing itself. Are you talking about the thing, or what we call it? Both are different things that have a relationship with each other. — Harry Hindu

Any word can be counted as a label then, that perspective does not help much.

And again, we are not talking about the x`s on your screen, which have different locations :-)

We are talking about a thing, being itself.

I should have used the word variable instead of unknown, in my language, we call that variable idea unknown. Semantic problems...

Maybe the concept of mathematically or algebraically exact equality is produced by a synthesis of the counting and spatial structure modules of cognition, whatever these are. We assign a set of structures such as objects a quantitative label, and then we assign a relatively similar enough set of structures the same quantitative label. The structures are never absolutely the same, but with the reference point of a particular quantitative label, they can be regarded as such, which turns out to be a very practical perspective, though philosophically befuddling.

Well, first things first: X is just a stand-in symbol for any object or number you wish it to be. But I think you got that far yourself already.

To a more specific example with your apples. Everything is what it is and not some other thing. If you have an apple--like a real world apple, it can't be anything else at the exact time you are holding it. It has to be itself. What else could it be? If your apple were to be a not-apple, say a banana, then it wouldn't be apple=banana. It would be banana=banana.

You can even expand the equation: apple=apple=not-banana. Or x=x=~y. Or apple=apple=nothing that is not apple. Or x=x=~(~x).

Five minutes from now it can be mush in your stomach, and five hours from now it can be released as energy and water and air, etc. Five months ago it was in the clouds and the earth and the tree. But right here, now, it is just itself.

Let's try it this way:
Apple=Banana is true, if the properties of apple and banana are completely identical. Certainly, they are not. So Apple≠Banana. It would explain why an apple couldn't be not an apple. But this logic is only true, if the properties of apple are identical to themselves.

If I start counting the properties of Apple now.......yes, they do meet the properties of Apple. As far I could observe the properties of x, they are identical to the properties of x, therefore x=x is true. This is inductive reasoning which does not guarantee anything.

Because if such an alternative existed, it wouldn't exist. — litewave

I see a conclusion, but no premises. How and why? — Monist

It follows from the contradiction "X is not X", because from a contradiction, anything follows:

https://en.wikipedia.org/wiki/Principle_of_explosion

But even without explosion, the original assumption that an object is not what it is, is absurd enough.

Thank you so much!

For now, I am satisfied with the answers. But not with the constitution of beliefs on these given Axiomatic Laws and Logical Principles. So I will continue my journey with questioning them.

And absurd is okay.

https://en.wikipedia.org/wiki/Equivalence_class

If you are using mathematical symbols, then the above should suffice. If you are arguing why a thing is the same as itself, then jump into the deep end of the metaphysical pool and splash about.

:yawn:

Actually the definition of 2 in the dictionary is 1+1. the definition of 10 is 9 + 1

You can actually get good at math just by memorizing definitions.

10 ofcourse to some extent requires knowing the defintion of 1 through 9.

The reason 2 was ever defined is because humans are much deeper thinkers than all known animals.

I'm sure some animals understand what 2 is to some degree.

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

(I suppose your inquiry itself is meaningless, if identity was abandoned; not just your inquiry, every inquiry.)

There cannot be anything in particular prior identity.

Why is identity necessary? — Monist

It's not. (Necessity presupposes identity, if we're talking modal logic at least.)

If I start counting the properties of Apple now.......yes, they do meet the properties of Apple. As far I could observe the properties of x, they are identical to the properties of x, therefore x=x is true. This is inductive reasoning which does not guarantee anything. — Monist

Ah, I think I see a problem here. You're introducing an observer into the mix. Our observations of the world are always (to some extent) inductive. (And, as an aside, people often demand too much of inductive reasoning. You can be justifiable and reasonably sure of something without having to be 100%, infallibly sure of it, but that's a subject for a later day.)

The point that the law of identity simply makes is that any object, like (let's call it, for sake of specificity, and to make clear we're not taking about general apples, but of one specific apple in the time-space continuum) Apple#1 is the same as Apple#1 because Apple#1 DOES have the exact same identical properties, including their location in the space-time continuum as Apple#1. Whether or not you the observer can verify that the apple you're holding is actually the apple you're holding is not as certain, because there is always the off-chance that you're insane/a brain in a vat/under some magicians illusion.

Apple=Banana is true, if the properties of apple and banana are completely identical. Certainly, they are not. So Apple≠Banana. It would explain why an apple couldn't be not an apple. But this logic is only true, if the properties of apple are identical to themselves. — Monist

You just pushed this back from X to properties of X, but that doesn't really change anything. Just do a variable substitution: let Y designate what used to be called X, and let X now be a property of Y. Everything that was said of X would still apply, since it does not actually depend on the meaning of X - it's a purely syntactical exercise. Metaphysics doesn't come into this.

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



I know that, proving 1+1=2 is hard, whilst it is so simple(logically and practically), but I do not see anyone trying to prove x=x, because it may seem so simple and obvious, as it may look pure stupidity to question it, but that is absolutely my point. The simpler it gets, the complexer explaining it.

Can someone help me out please? — Monist

My two cents...

Imagine x = God
Then, by definition, x is all good, all powerful and all knowing

If x is NOT = x then x is NOT God and so x is not all good, not all powerful and not all knowing

So we get the following contradictions:

x is all good AND x is not all good
x is all powerful AND x is not all powerful
x is all knowing AND x is not all knowing

These contradictions arise because we assumed x is not = x

Therefore, x = x

In more general terms every x has a set of properties and let's call it P so Px (x has properties P) is true

If x is not x then x will not have properties P i.e. ~Px

Px & ~Px is a contradiction

Therefore x is not = x is false i.e. x = x

In argument form:

1. If x = x is false then x has property P AND x doesn't have property P
2. It is false that x has property P AND x doesn't have property P (contradiction)
Ergo
3. x = x is true (modus tollens)

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



I know that, proving 1+1=2 is hard, whilst it is so simple(logically and practically), but I do not see anyone trying to prove x=x, because it may seem so simple and obvious, as it may look pure stupidity to question it, but that is absolutely my point. The simpler it gets, the complexer explaining it.

Can someone help me out please? — Monist

Whatever an x is, x has a set of finite properties, say Ax, Bx and Cx.

If x is not x then one of the following will be true: ~Ax or ~Bx or ~Cx or all.

We then have a contradiction: Ax & ~Ax or Bx & ~Bx or Cx & ~Cx.

So, it's false that x is not x or it's true that x = x

An argument:

Px = x has all properties that x has
Ox = x doesn't have at least one of the properties x has
x = x the law of identity
~(x = x) the law of identity is false

1. Px
2. ~(x = x) > Ox
3. Ox > ~Px
4. ~(x = x).....assume for reductio ad absurdum
5. Ox.....2, 4 MP
6. ~Px.....3, 5 MP
7. Px & ~Px....1, 6 conj (contradiction)
8. x = x 4 to 7 reductio ad absurdum

An argument:

Px = x has all properties that x has
Ox = x doesn't have at least one of the properties x has
x = x the law of identity
~(x = x) the law of identity is false

1. Px
2. ~(x = x) > Ox
3. Ox > ~Px
4. ~(x = x).....assume for reductio ad absurdum
5. Ox.....2, 4 MP
6. ~Px.....3, 5 MP
7. Px & ~Px....1, 6 conj (contradiction)
8. x = x 4 to 7 reductio ad absurdum — TheMadFool

Awesome, but excuse me please, can you explain why Ox > ~Px ?

Awesome, but excuse me please, can you explain why Ox > ~Px ? — Monist

Well, if x lacks one property from a set of properties P that defines x then it follows that it's false that x has all the properties P. Right?

Any word can be counted as a label then, that perspective does not help much. — Monist

Why? Words are things. If x is a variable, then x can be anything, including a word.

And again, we are not talking about the x`s on your screen, which have different locations :-)

We are talking about a thing, being itself.

I should have used the word variable instead of unknown, in my language, we call that variable idea unknown. Semantic problems... — Monist

I doesn't make a difference. If you don't know what x is, then how can you say it is equal?

Why talk about a thing being itself? What problems do you hope to solve, or answers you expect to get?

Why not just talk about the thing, as opposed to the thing being itself? How is that any different than talking about the thing being itself?

And absurd is okay — Monist

No it's not. Philosophy is rife of absurd questions. Some questions just aren't worth asking.

Any word can be counted as a label then, that perspective does not help much.
— Monist
Why? Words are things. If x is a variable, then x can be anything, including a word. — Harry Hindu

x=x being true, can be put in words as 'the necessity of identity.'
If that is understood, now;

A. I did not understand the underlying reason of your statement:''words are not the thing itself'', how did you come to the conclusion that I am talking about the words. And how did you relate that conclusion with the word 'identity', without considering the concept of identity.
B. Words are things. Agreed. x can be anything, including a word. Agreed. What has this to do with the context here?
C. Where did you see that I am defending that x=x is true, I do not know. I ask why it is necessary to be so, since axioms are based on this very idea.
D. It is important to talk about a thing being itself, to understand what constitutes the being of a thing. For example; if identity is false for things, nothing may even exist. I try to catch what is going on, and why I do that is explained in my other reply.
E. You can talk about many things about the thing, one of them is their identity.
F. The absurdity of rational thought constituted on axioms is okay, but me questioning it is not... I prefer being free.

Some questions just aren't worth asking. — Harry Hindu

G. It reminds me of a Greek tossing Socrates on the street :-) I wish I could have that certain statements, without knowing the answers :-)

Thank you,

Congrats! You just discovered the property of identity.