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Single Idea 17222
[filed under theme 6. Mathematics / A. Nature of Mathematics / 2. Geometry
]
Full Idea
It follows from the nature of a triangle that its three angles are equal to two right angles.
Gist of Idea
The sum of its angles follows from a triangle's nature
Source
Baruch de Spinoza (The Ethics [1675], IV Pr 57)
Book Ref
Spinoza,Benedict de: 'Ethics', ed/tr. White,WH/Stirling,AH [Wordsworth 2001], p.204
A Reaction
This is the essentialist view of mathematics, which I take to be connected to explanation, which I take to be connected to the direction of explanation.
The
31 ideas
with the same theme
[study of relationships of lines, points, and shapes]:
1553
|
No perceptible object is truly straight or curved
[Protagoras]
|
9867
|
It is absurd to define a circle, but not be able to recognise a real one
[Plato]
|
8726
|
Geometry can lead the mind upwards to truth and philosophy
[Plato]
|
9790
|
Geometry studies naturally occurring lines, but not as they occur in nature
[Aristotle]
|
12372
|
The essence of a triangle comes from the line, mentioned in any account of triangles
[Aristotle]
|
6297
|
Euclid's geometry is synthetic, but Descartes produced an analytic version of it
[Euclid, by Resnik]
|
17197
|
The idea of a triangle involves truths about it, so those are part of its essence
[Spinoza]
|
17222
|
The sum of its angles follows from a triangle's nature
[Spinoza]
|
18079
|
Newton developed a kinematic approach to geometry
[Newton, by Kitcher]
|
13163
|
Circles must be bounded, so cannot be infinite
[Leibniz]
|
13008
|
Geometry, unlike sensation, lets us glimpse eternal truths and their necessity
[Leibniz]
|
8739
|
Geometry studies the Euclidean space that dictates how we perceive things
[Kant, by Shapiro]
|
8740
|
Geometry would just be an idle game without its connection to our intuition
[Kant]
|
16899
|
Geometrical truth comes from a general schema abstracted from a particular object
[Kant, by Burge]
|
16930
|
Geometry is not analytic, because a line's being 'straight' is a quality
[Kant]
|
16919
|
Geometry rests on our intuition of space
[Kant]
|
9618
|
Bolzano wanted to reduce all of geometry to arithmetic
[Bolzano, by Brown,JR]
|
10245
|
One geometry cannot be more true than another
[Poincaré]
|
13472
|
Hilbert aimed to eliminate number from geometry
[Hilbert, by Hart,WD]
|
14442
|
If straight lines were like ratios they might intersect at a 'gap', and have no point in common
[Russell]
|
14151
|
Pure geometry is deductive, and neutral over what exists
[Russell]
|
14153
|
In geometry, empiricists aimed at premisses consistent with experience
[Russell]
|
14154
|
Geometry throws no light on the nature of actual space
[Russell]
|
14152
|
In geometry, Kant and idealists aimed at the certainty of the premisses
[Russell]
|
14155
|
Two points have a line joining them (descriptive), a distance (metrical), and a whole line (projective)
[Russell, by PG]
|
16949
|
Klein summarised geometry as grouped together by transformations
[Quine]
|
8994
|
If analytic geometry identifies figures with arithmetical relations, logicism can include geometry
[Quine]
|
16901
|
The equivalent algebra model of geometry loses some essential spatial meaning
[Burge]
|
9159
|
You can't simply convert geometry into algebra, as some spatial content is lost
[Burge]
|
3332
|
Greeks saw the science of proportion as the link between geometry and arithmetic
[Benardete,JA]
|
21444
|
Modern geoemtry is either 'pure' (and formal), or 'applied' (and a posteriori)
[Gardner]
|