I was in conversation with my brother many years ago and he was talking about an experiment where social pressure could make people believe that the longer line was the shorter, and the shorter the longer. I believe his point was that social reality can not correspond to physical reality, and in that case, the social reality is wrong.
I pointed out that other contradictory ideas, for example, that two things can be both alike and different, so a triangle and a star can be alike in being polygons, but different in how many angles they have, is also a contradiction, much like the longer line being the shorter and the shorter, longer. Admittedly they are slightly different because “like” is a two-way relationship while “larger” only goes one way, from the larger to the shorter, but that there is another contradictory relationship that we find acceptable is enough.
Much later my brother argued that paradoxical relationships of likeness and difference can be resolved by making them a matter of degree. Two things are “like” 30% and “unlike” 70%, for example.
The problem with this is that = and ≠ are related to likeness and difference, they are types of likeness and difference. Identity is a type of likeness, and ≠ is a type of difference, and they are used in analysis books to build the concept of number. We can’t have the natural numbers without the idea that 1+1=2. So = and ≠ are prior to matters of degree. To be clear, if = means strictly identity, so two apples are not equal to each other (if they were, they would be both = and ≠). One apple is equal to itself. Now, evoking Wittgenstein, we would not bother to say, or be interested in the slightest in, 1+1=2, if 1+1 and 2 were not slightly different from each other. The same way we never say “This apple is this apple”. One could say that 1+1 is the act of grouping two apples, and 2 is the apples already grouped. Looked at in this way 1+1 ≠ 2 ; they do not have the same identity.
Because the same problem of likeness and difference arises with = and ≠, we cannot use degree to solve the problem and use = and ≠ to solve what we mean by degree, that is circular.
It may be that this is related to Buddha’s idea of anatta (not-self). The parts of one’s self do not add up to the self we perceive, yet those parts, we say, are what the self is. Buddha uses exactly this argument about a cart– that no-where in the parts of the cart do we find the essence of the cart: 1+1 does not contain the idea of 2, or as Western philosophers including Kant have noticed, 1+1=2 is synthetic.
This is a good thing. Remember Buddha’s talk of the “Deathless” is beyond duality, beyond death and life, self and not-self. If the concept of identity were defensible, or a duality such as = and ≠ could create a system of numbers that could fully describe reality, there would be no escape, no ascendance to the Deathless.