Quotes on Mathematics

Mathematics is a field in which much that appears obviously true is in fact false. … For example, nothing could seem more reasonable than the assertion that every surface – every page of a book, say – has two sides. It is reasonable, but it is false. The Mobius band, a figure produced by twisting a strip of paper and then gluing the ends together, has only one side. This is no trick, no joke, no semantic sport. It is an aspect of reality that cannot be dismissed, and therefore no proof of anything can ever make use of the falsehood that all surfaces have two sides.

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Goedel’s Theorem … proves that there exist meaningful mathematical statements that are neither provable nor disprovable, now or ever – neither provable nor disprovable, that is, not simply because human thought or knowledge is insufficiently advanced but because the very nature of logic renders them incapable of resolution, no matter how long the human race survives or how wise it becomes. There is no way to escape this conundrum. It is not a question of sophistry of any kind. The theorem itself was proved some decades ago… So there it stands – an assertion that to the end of time cannot be proved and yet cannot be disproved.

(both quotes are from a 1972 essay written by the German mathematician Alfred Adler)