from the glacier

While reading Science and Method the other night, Poincare puts the versus in the field of mathematical speculative theory as a contradiction between Kant and Leibniz. Or, Bertrand Russell relying on logic alone versus an idea of induction and basis in common thought processes as the background for mathematical reasoning. Basically, the logicians have said that there can be no synthetic judgments a priori in mathematical reasoning. It has to be logic and the rules of inference (not including induction) simply. But, as Poincare notes, one logician say that a conjunction combines TWO things. This is all before he comes to a definition of ONE.

But I guess my question is, stated by example, does the statement " all circles are similar" [in Euclidean Geometry, because we all know they aren't in other Geometries] follow from the definition of circle, or from the definition of similarity? We could say that because circularity is defined as equidistant from a point, and also show that any other similar figures have a defined ratio to a point of all sides that circles are defined as similar figures. But so also with squares, but this doesn't seem to be as great of a revelation. Maybe because we can line up squares meeting at one vertex and the two sides will correspond (at least to the extension of one side) whereas when when have two circles line up at one point, there is not the same "overlapping" . . . Perhaps it is evident from induction that all rectilinear similarly equiangular and equilateral figures are similar, but it is not so evident with curves. Curves are "trickier".

I for one, would like to lay down the following LAW: (Beitia's first law of mathematics)

1: Induction is necessary.

2: Mathematical objects exist prior to the science of mathematics.

3: Mathematics truly is (with the Philosopher) the study of magnitude qua magnitude.

3a: not a relationship of formal symbols (an arithmetic formal calculus)

4: Man does not make mathematics, he discovers it.

Thus, math is as much art as science, and a view toward the beautiful guides a mathematician as sure as a poet. Re-read Euclid II. 14. He proves the anunciation without even constructing that which is his goal. Beautiful. Genius. Examine II. 11 - the golden ratio. He constructs the golden ratio along the vertical side to cut the horizontal in such ratio - all without having even defined ratio yet. Beautiful. Genius. Examine Godel's inconsistency proof (for those with some training in mathematical logic). The way he turns Russell on his head and forces the sentencial calculus to talk about itself - Beautiful... Genius. . . [for those who cannot read mathematical logic, he constructs within system X a statement that translates "I cannot be proved in system X" - pure genius]

Beauty guides all science