Frege believes that arithmetic is analytical. Why not Frege thinks geometry can be reduced to arithmetic by coordinate system or something else?

  • 1
    Some references would help.
    – Mr. White
    Aug 21 '20 at 6:22
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    "There is accordingly a noteworthy difference between geometry and arithmetic in the way in which their fundamental principles are grounded. The elements of all geometrical constructions are intuitions, and geometry refers to intuition as the source of its axioms. Since the object of arithmetic does not have an intuitive character, its fundamental propositions cannot stem from intuition…", Frege, 1874, see SEP. That one can build arithmetic models of geometry is moot, they are part not of geometry, but of arithmetic.
    – Conifold
    Aug 21 '20 at 6:38
  • Spinoza used a geometric method to present his 'Ethics' and because there are so many interpretations of what precisely geometry consists in there are multiple interpretations of the origin and nature of geometry. Why it may be considered synthetic results from something I term, human geometry. In my Book, 'To Discern Divinity", on page 35, you will find; 'On Human Geometry and the ‘More Geometrica’. It can be downloaded at Academia.edu. The method of human walking and circumlocution daily, involves a barely perceptible gyroscopic sensitivity which guides us. This is the origin of synthetic.
    – user37981
    Aug 21 '20 at 13:39

In Frege's time, geometry had exploded as a discipline. Descartes' invention of the "Cartesian" coordinate system had been analytic, even founded analytic geometry, but over the centuries since then the nature of geometry had broadened vastly.

Leibnitz envisaged a "geometry of position" in which the relations between objects would be captured without any need for Cartesian numbers. Euler's polyhedron formula, concerning the numbers of corners, edges and faces of a polyhedron such as a cube, founded that discipline. In Frege's time, Poincaré greatly advanced it and it became known as topology, sometimes described as "rubber-sheet geometry".

Meanwhile Gauss among others had independently discovered non-Euclidean geometries, in which the angles of a triangle do not add up to 180 deg. Poincaré chipped in with his own ideas there, too. Euclid's axioms came under fire, with each geometry requiring its own set of axioms. These axioms involved things like points, lines and planes, and one thing meeting another. This mode of reasoning is known as synthetic, in contrast to the analytic mode. Klein was systematising the whole thing while Poincaré was doing the same for topology.

All this would have been in Frege's mind when he made his remark.

I do not think he was entirely correct, but that is one for the maths StackExchange.

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