For those who are unaware, at least 1 and 2 are thanks to Georg Cantor and his invention of set theory. He had to endure a lot of resistance from the mathematical luminaries of his day - Kronecker and apparently, even Poincaré! Like Nietzsche, he died in a mental institution, the vicious attacks on his ideas (mostly by Kronecker) having supposedly played a part in Cantor's breakdown.
His mathematical ideas also ran afoul of some religious "philosophers" who, as is usually the case, are never ever open minded enough to consider truly groundbreaking ideas (virtually by definition, seeing as they are religionists). because "many mathematicians agreed with Kronecker that the completed infinite may be part of philosophy or theology, but that it has no proper place in mathematics".
Even the great Gauss, according to the wikipedia article was saying that "Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics", e.g. "the only access we have to the infinite is through the notion of limits, and hence, we must not treat infinite sets as if they have an existence exactly comparable to the existence of finite sets."
The irony, of course, is that when the notion of infinitesimals/limits were first introduced, they were ALSO subject to a very similar kind of criticism! (And as usual, religionists figured in such critcisms, most notably Bishop Berkeley).
Our perceptions can tell us "instantly" that all the chairs are taken, but beyond a certain single-digit number, we cannot conclude instantaneously how many chairs there are or people sitting on them.
Such a perceptual limitation seems to be what spurs humans to come up with the whole machinery of formal computation just as our physical limitations spurs us to invent the universe of devices we make use of in modern civilization. There are at least two "computers" in our brain. The visual one and the one that does...
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