Pondering a slightly obscure factoid got me thinking, what statements that the vast majority of people would think are true either aren’t true or admit some sort of unexpected exception (and I’m not talking about well-worn misconceptions of the Frankenstein/Frankenstein’s monster-type)?
My example is: “The farther away an object (of known dimensions) is, the smaller it looks” . In fact the angular size of object decreases with distance up to a certain distance, but then starts to decrease -a bonus point to anyone who can work out why.
My example would be “If you simultaneously drop two balls of the same size, they will hit the ground at the same time, regardless of their weight.” People often claim this was proved by Galileo.
In fact, this is true only of balls dropped in a vacuum - when there’s air, the heavier ball hits the ground first.
Uh, the apparent size of an object levels off according to 1/4πr^2? Not really sure what you’re asking, an object of known size that is farther away will always appear smaller. Or are you talking about gravitational lensing or something funky?
I don’t think even that is right. The shape of an object falling through air makes a much bigger difference, because the shape largely determines the amount of air resistance. (Unless you’re specifically talking about falling spheres to the exclusion of other shapes.)
For an X-treme X-ample, consider a falling 20-lb cannonball vs. a falling 180-lb person-on-a-parachute.
I presume that he’s talking about pixel resolution. The eye only has so many receptors. Once something is at the 1 receptor size limit, it will continue to be 1 receptor large even as it gets further away. Technically, this makes it stay a constant angular width, but if we continue the logic that “further = smaller”, the viewed size vs. the expected size is increasing as the object gets further away.
(At the sub-receptor level, things no longer shrink, they just start to blend with the surrounding colors - so it’s more of a fade effect than a shrinking to nothingness.)
In the field of mathematics, one can find many paradoxes. A paradox, in most usages, is nothing more than a clearly true fact that isn’t, or possibly a clearly false fact that isn’t – Especially if the contradiction cannot be resolved by any known mathematical logic. Clearly, such cases cannot exist. Except that some such cases do exist.
Consider that a set (in the mathematical meaning of the word) may contain any collection of “objects” (concrete or abstract), and a set may even contain other sets as members. Considering the total abstraction of the concept of sets, most mathematicians agree that a set can even contain itself as a member.
One thing is certain: A set is well-defined if it is clearly the case that each “object” in the “universe” either is or isn’t in the set.
Okay, so any given set either does or doesn’t contain itself as a member, right? So, we can define one set as the set of all sets that contain themselves and its complement, the set of all sets that don’t contain themselves. Surely, every conceivable set must fall into one or the other of these categories.
BUT – does the set of all sets that don’t contain themselves contain itself? Think about that. If it does, then it doesn’t. And if it doesn’t, then it does. (It’s a more elaborate relative of the liar paradox.)
Sure it can. We simply say there is no such thing as a set of all sets. There is, however, a proper class of all sets known as the Von Neumann universe. It’s not a set itself, so it causes no paradoxes.
No it’s not gravitational lensing or anything to do with the limitations of the human eye either (in response to Sage Rat. The angular size if any given object starts to increase with distance after a certain distance- I’d though the distance is very far and I’m fairly certain that it has never been definitively demonstrated observationally, so its more theoretical.
I wouldn’t think of this as a good example as most people will intuitively know that for objects the same size and shape, the heavier object usually falls fastest on Earth.