structure for 'Mathematics'    |     alphabetical list of themes    |     expand these ideas

6. Mathematics / A. Nature of Mathematics / 5. Geometry

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

30 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]
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]
Geometrical truth comes from a general schema abstracted from a particular object [Kant ,by Burge]
Geometry would just be an idle game without its connection to our intuition [Kant]
Geometry is not analytic, because a line's being 'straight' is a quality [Kant]
Geometry rests on our intuition of space [Kant]
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]
Euclid's geometry is synthetic, but Descartes produced an analytic version of it [Resnik]
Bolzano wanted to reduce all of geometry to arithmetic [Brown,JR]