Ok then if one side of the square has distance 1, then the diagonal has distance sqrt(2). What else would you call the distance?
Post by twetch#1832
You're working within Euclidean geometry. Space is discrete, not continuous, so there are no "squares" that have a distance of 1 unit.
Create a square on your computer screen. You'll see it's composed of a countable number of units (pixels).
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If your geometric investigations have only been within Euclidean geometry, or only working within continuous space, then discrete space can be unintuitive or shocking.
But alas, all space that we encounter, mental and physical, is actually discrete.
Space is not discrete in Euclidian geometry.
I know. I'm saying you are working within Euclidean geometry when you ask about the diagonal of a square with a side the length of 1.
I'm saying that no spatially-extended square has a side the length of 1, since actual squares are composite objects.
In discrete space (in the real world), units of length refer to spatially-extended points. When we say "That square has a side whose length is 9", that means "there are 9 units which compose its side."
So, you can't have a composite object made of 1 unit.
One more time, just to rephrase it:
"The diagonal of a square whose side has a length of 1" is a Euclidean object.
There are no Euclidean objects in non-Euclidean space.
All actually-existing space is non-Euclidean.
Therefore, there is no such diagonal
So either points are distributed in orthogonal grid and diagonal has equal length to the side of square (same amount of collinear points) or points are distributed in different fashion and squares don't exist?
It's self-evident.
Yes it is, if you think about it thoroughly enough.
Self-evident doesn't always mean "immediately obvious." It means "simply by examining the concepts, you can know it."
Multiple professors have told me that logical contradictions exist. They are self-evidently wrong but haven't thought enough about it yet