@christianp For this, let's define "big number" very scientifically as a number big enough I have to abstract it away somehow. That is, I can't think directly of N raw objects, I have to think of 𝑥*𝑦 raw objects or 𝑏ᵏ objects.

To comfortably do this, I'll put that number below or equal to 10. Maybe some people can visualize individual things above that, but I think that's generous as I believe the average number is ~4.

Thus, the first counterexample to "half of a big number is also a big number" from the direction of the big numbers would be 20/2, since that's the largest "big number" that could be directly halves into a small number. Coming from the smaller numbers, 11/2 would be the first counterexample.

It may seem odd to define a big number this way, but why not? Every other method is just as arbitrary. Might as well tie it loosely to our psychology.

Perhaps unintuitively, this definition of "big number" causes a number to become big again if it's divided too much, since I doubt many people could visualize complex fractions of things.