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Single Idea 8994

[filed under theme 6. Mathematics / A. Nature of Mathematics / 2. Geometry ]

Full Idea

Geometry can be brought into line with logicism simply by identifying figures with arithmetical relations with which they are correlated thought analytic geometry.

Gist of Idea

If analytic geometry identifies figures with arithmetical relations, logicism can include geometry

Source

Willard Quine (Truth by Convention [1935], p.87)

Book Ref

Quine,Willard: 'Ways of Paradox and other essays' [Harvard 1976], p.87

A Reaction

Geometry was effectively reduced to arithmetic by Descartes and Fermat, so this seems right. You wonder, though, whether something isn't missing if you treat geometry as a set of equations. There is more on the screen than what's in the software.

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]
14152 In geometry, Kant and idealists aimed at the certainty of the premisses [Russell]
14154 Geometry throws no light on the nature of actual space [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]