Where there is life there is a pattern, and where there is a pattern there is mathematics. Once that germ of rationality and order exists to turn a c… - William Rowan Hamilton

We should not believe... that commensurability is a quality of every magnitude as of all the numbers; and whoever has not investigated this subject, shows a gross and unseemly ignorance of what the Athenian Stranger says in the seventh treatise of the Book of the Laws, [namely], "And besides there is found in every man an ignorance, shameful in its nature and ludicrous, concerning everything which has the dimensions, length, breadth, and depth; and it is clear that mathematics can free them from this ignorance. For I hold that this [ignorance] is a brutish and not a human state, and I am verily ashamed, not for myself only, but for all Greeks, of the opinion of those men who prefer to believe what this whole generation believes, [namely], that commensurability is necessarily a quality of all magnitudes. For everyone of them says: "We conceive that those things are essentially the same, some of which can measure the others in some way or other. But the fact is that only some of them are measured by common measures, whereas others cannot be measured at all".

[E]ven in the most precise part of science, in mathematics, we cannot avoid using concepts that involve contradictions. ...[I]t is well known that the concept of infinity leads to contradictions that have been analyzed, but it would be practically impossible to construct the main parts of mathematics without this concept.

[...] I who do not even dare to say, when one is added to one, whether the one to which the addition was made has become two, or the one which was added, or the one which was added and the one to which it was added became two by the addition of each to the other. I think it is wonderful that when each of them was separate from the other, each was one and they were not then two, and when they were brought near each other this juxtaposition was the cause of their becoming two. And I cannot yet believe that if one is divided, the division causes it to become two; for this is the opposite of the cause which produced two in the former case; for then two arose because one was brought near and added to another one, and now because one is removed and separated from other. And I no longer believe that I know by this method even how one is generated or, in a word, how anything is generated or is destroyed or exists, and I no longer admit this method, but have another confused way of my own.