Roemer used eclipses of the satellites of Jupiter, and sought how much the event fell behind its prediction. But... this prediction [is] made... by..… - Henri Poincaré

Pure logic could never lead us to anything but tautologies; it can create nothing new; not from it alone can any science issue.

What is zero? It is the number of elements in the class nil. And what is the class nil? It is the class which contains none.
To define zero as nil and nil as none is really an abuse of the wealth of language, and so [Logicists] introduced an improvement into the definition, ... which means in English: zero is the number of the objects that satisfy a condition that is never fulfilled. But as never means in no case, I do not see that any very great progress has been made.

In recapitulation, the mind has the faculty of creating symbols, and it is thus that it has constructed the mathematical continuum, which is only a particular system of symbols. Its power is limited only by the necessity of avoiding all contradiction; but the mind only makes use of this faculty if experience furnishes it a stimulus thereto.