What is Leibniz's Law? (The Identity of Indiscernibles) 

So.. there is only one electron?

These problems arise if we assume, as Leibniz did, that space (as well as time) is relational, not absolute. Otherwise spatial location would seem to enter into questions of identity. Aside from Kant's argument from incongruous (or incommensurate) symmetrical counterparts as a proof of the absolute reality of space, just consider a possibly practical example of how spatial location may be taken to be a property (whether impure, extrinsic, or otherwise) which matters for identity questions. Forgetting about magnetic properties, and just considering the earth's poles as geometric markers, if you know something is located at the North Pole, then you know it is not identical with something located at the equator, regardless the qualitative identity of the objects. In the two spheres example, we are considering spatial objects, so there is space in this universe. If there is space, it can presumably be represented by a coordinate grid. If one ball occupies one set of points, and the other a different set of points, that would seem to go a long way toward demonstrating their mutual discernibility, since even if you assume space is merely relational, it does not follow that the coordinate model of space is invalid. No matter what set of points you claim the first ball occupies, the second ball must occupy a different set, otherwise they are apparently not spatial, and indeed not even purely geometric objects.

How does this work with Leibniz's view that 1=1, though they have the same numeric value, the spatial distinction between the first notation of "1" and the second one leads to some sort of "difference"?

4:24 I don't understand why we have to conclude that there is only one object in the universe. Each one of the spheres have the same pure and intrinsic properties such as roundness and let's say color. And we could say that all of them have the pure property of being close to other objects (it does not specify which means it is a pure property). Thus we can conclude that there are three spheres in the universe.

Thank you.

What are the pure intrinsic properties shared commonly between two things-in-themselves?

I absolutely oppose the idea at 3:24 that Leibniz's law holds for pure intrinsic properties. This is because in logic, the idea of "properties" is expressed by predicates and relations, and "a = b" implies that "a" and "b" can be substituted for each other salva veritate. If two things managed to be identical but did not sharing the same relations, then substituting them would lead to invalid inferences. Therefore, sharing both "pure intrinsic properties" (predicates) and "pure extrinsic properties" (relations) should be a necessary condition (though maybe not sufficient?) for identity.

If we take abstractions to be true, real things, then Leibniz’s Law can be applied. Two abstractions can have different underlying realities but abstract out to be the same thing. All the matter making up all the golf balls in the world can be abstracted out to represent the abstraction “Golf Ball”. And since each abstraction is indiscernable, they are actually the “same” abstraction. They are all golf balls. But that’s not to say all of the underlying realities are the same, all the matter making up these golf balls is different. It still makes sense with intuition. Leibniz’s Law, the identity of indiscernables, (or the 2=2 principle in Abstract Realism) comes in handy anywhere you have a macro state which can be represented by any one of multiple micro states. Here’s an example: someone simulates this planet in a computer. The abstraction of “you” is identical in the simulation and in real life. This proves that you would be fully conscious in the simulation. There are many other implications, but that’s just an example.

Nice & stimulating .4 me the objects must share in the time & locality to be the same otherwise they r replicates.

There are alternative formulations to set theory were the identity property is weakened. Namely the quasi-set theory, which has inspiration on quantum physics.

the philosophy equivalent to the one electron postulate.

"If 2 objects are indiscernible (have all the same properties) then they are the same object. Lmao, layman Extensionality! If we require ALL properties to be identical, then no unobvious objects are indiscernible, because we can take "distance from London", "distance from Warsaw" and "distance from Berlin" as properties, and after triangulation, 2 objects become indiscernible ONLY if they are in the exact same place. Interesting things begin to happen when we limit this requirement to Pure properties. Then, in an universe where there exist only 2 metal balls, they are indiscernible, because they have the same size, mass, weight, distance from another ball, etc. So that universe contains only one object. In conclusion, a mildly interesting topic, but seems rather useless in comparison to Extensionality.

If x and y are identical, then the property x=x is true, and because x equals x, y will also have to equal x. So the law is true when given a broad definition, but question begging, because if two objects share all the same properties, they share the property of being identical to the same object, itself, which is the question we are trying to answer with this law.