It is very revealing as Poincaré distinguishes between mathematical and physical continuity1, but it is even more interesting how he views the continuum. He introduces the concept of different orders of continuity; first-order continuity is what is provided by adding rational numbers to the number line, and second-order continuity is, which appears when irrational munbers also appear on the number line.2
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1 As he described in his writing titled "Science and hypothesis", which is more an essay of philosophy of science, rather than a mathematical writing.
2 Quoting Poincaré: "For brevity let me call the set of members which form under the same laws as the sequence of rational numbers as the first-order mathematical continuity. If we insert now new members according to the laws of forming the incommensurable numbers, the resulting set hereinafter will be called the second-order continuity." (This is not a translation from the original language but from Hungarian language.)
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