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6. Mathematics / A. Nature of Mathematics / 2. Geometry

[study of relationships of lines, points, and shapes]

31 ideas
No perceptible object is truly straight or curved [Protagoras]
It is absurd to define a circle, but not be able to recognise a real one [Plato]
Geometry can lead the mind upwards to truth and philosophy [Plato]
Geometry studies naturally occurring lines, but not as they occur in nature [Aristotle]
The essence of a triangle comes from the line, mentioned in any account of triangles [Aristotle]
Euclid's geometry is synthetic, but Descartes produced an analytic version of it [Euclid, by Resnik]
The idea of a triangle involves truths about it, so those are part of its essence [Spinoza]
The sum of its angles follows from a triangle's nature [Spinoza]
Newton developed a kinematic approach to geometry [Newton, by Kitcher]
Circles must be bounded, so cannot be infinite [Leibniz]
Geometry, unlike sensation, lets us glimpse eternal truths and their necessity [Leibniz]
Geometry studies the Euclidean space that dictates how we perceive things [Kant, by Shapiro]
Geometry would just be an idle game without its connection to our intuition [Kant]
Geometrical truth comes from a general schema abstracted from a particular object [Kant, by Burge]
Geometry is not analytic, because a line's being 'straight' is a quality [Kant]
Geometry rests on our intuition of space [Kant]
Bolzano wanted to reduce all of geometry to arithmetic [Bolzano, by Brown,JR]
One geometry cannot be more true than another [Poincaré]
Hilbert aimed to eliminate number from geometry [Hilbert, by Hart,WD]
If straight lines were like ratios they might intersect at a 'gap', and have no point in common [Russell]
Pure geometry is deductive, and neutral over what exists [Russell]
In geometry, empiricists aimed at premisses consistent with experience [Russell]
In geometry, Kant and idealists aimed at the certainty of the premisses [Russell]
Geometry throws no light on the nature of actual space [Russell]
Two points have a line joining them (descriptive), a distance (metrical), and a whole line (projective) [Russell, by PG]
Klein summarised geometry as grouped together by transformations [Quine]
If analytic geometry identifies figures with arithmetical relations, logicism can include geometry [Quine]
The equivalent algebra model of geometry loses some essential spatial meaning [Burge]
You can't simply convert geometry into algebra, as some spatial content is lost [Burge]
Greeks saw the science of proportion as the link between geometry and arithmetic [Benardete,JA]
Modern geoemtry is either 'pure' (and formal), or 'applied' (and a posteriori) [Gardner]