The central problem: what are the objects of geometry? Answer 1: Perceptible objects with shape. Answer 2: Abstractions, mere shapes.

The central problem: what are the objects of geometry? Answer 1: Perceptible objects with shape. Answer 2: Abstractions, mere shapes.

The central problem: what are the objects of geometry? Answer 1: Perceptible objects with shape. Answer 2: Abstractions, mere shapes. Answer 3: Space and its parts.

The central problem: what are the objects of geometry? Answer 1: Perceptible objects with shape. Answer 2: Abstractions, mere shapes. Answer 3: Space and its parts. KANT Geometry is a science which determines the properties of space synthetically and yet a priori Critique of Pure Reason A25/ B40

Before Kant A priori truths Analytic truths Truths of reason Relations of ideas A posteriori truths Synthetic truths Empirical truths

Kant Synthetic a priori Synthetic a posteriori Analytic a priori ---

Two sorts of experience dependence Psychological dependence: we could not form the belief without sense experience. Epistemological dependence: we could not know the proposition believed without experiential evidence. Everything red is coloured is psychologically dependent on experience but not epistemologically dependent on experience. Contrast All blood is coloured.

A Kantian account of a priori knowledge One s knowledge that p is a priori in acquiring or maintaining one s belief that p no experience was used as evidence for p or for anything from which one inferred p.

A Kantian account of a priori knowledge One s knowledge that p is a priori in acquiring or maintaining one s belief that p no experience was used as evidence for p or for anything from which one inferred p.

Kant s argument that geometrical knowledge is a priori (1) Any way of knowing that a geometrical proposition is true is a way of knowing that it is necessarily true. (2) We cannot get knowledge of a necessary truth from the evidence of experience. We cannot get knowledge of a geometrical truth from the evidence of experience.

Kant s argument that geometrical knowledge is a priori (1) Any way of knowing that a geometrical proposition is true is a way of knowing that it is necessarily true. (2) We cannot get knowledge of a necessary truth from the evidence of experience. We cannot get knowledge of a geometrical truth from the evidence of experience. Objections: (1) is false if geometry is about physical space alternatives to the parallels postulate. (2) is false if we have knowledge of physical laws.

But some geometrical knowledge may be a priori. Recall that experience does not seem to be used as evidence in the proof of the angle sum theorem. Experience had other roles, such as to help one grasp the situation described verbally, prompt the mind to form new beliefs or recall old ones, help one hold in mind accumulated information.

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Analytic knowledge Kant: Every F is G is analytic the concept of G is contained in the concept of F. Frege: A truth is analytic it can be known just by means of legitimate definitions and deductive logic. One s knowledge of a truth is analytic one knows it as a definition or just by deduction from definitions.

Some geometric knowledge is analytic, contra Kant. For example A hexagon has exactly six sides. This falls right out of the definition: A hexagon is a polygon with exactly six sides.

Some geometric knowledge is non-analytic. No two straight lines enclose a space. For any three straight line segments, if any two together are longer than the third, there is a triangle with sides equal to those line segments. For any triangle, any two of its sides taken together are longer than the third.

Intuition according to Kant Intuition (anschauung) is immediate awareness. It includes sense perception and imagination, and spatial awareness. An intuition is always (a) of something and (b) immediate, i.e., does not involve interpretation. Mathematics hurries at once to intuition [CPR A715/ B743]

The generality problem How can we reliably reach a general conclusion by reasoning with specific cases? For example In the path to the angle-sum theorem, dissimilar figures conform to the specification of the construction. Hence the path might depend on a feature of the constructed figure not shared by all figures conforming to the specification. Hence the path would be fallacious.

The area of a triangle with a horizontal base = half its height x its base. Fallacious path: 21

The area of a triangle with a horizontal base = half its height x its base. Fallacious path: 22

The area of a triangle with a horizontal base = half its height x its base. Fallacious path: 23

This procedure depends on a feature not shared by all triangles: the base angles are not obtuse. 24

The generality problem How can we reliably reach a general conclusion by reasoning with specific cases? The individual drawn figure is empirical, and yet serves to express the concept without damage to its universality, for in the case of this empirical intuition we have taken account only of the action of constructing the concept. [CPR A714/ B742]

The generality problem How can we reliably reach a general conclusion by reasoning with specific cases? [T]hat which follows from the general conditions of the construction must also hold generally of the object of the constructed concept. [CPR A715/ B743]

General conditions for constructing a triangle (?) 1. Construct a straight line segment. 2. From one end of that line segment construct another line segment at an angle to it. 3. Construct a third line segment joining the unattached ends of the first and second.

General conditions for constructing a triangle (?) 1. Construct a straight line segment. 2. From one end of that line segment construct another line segment at an angle to it. 3. Construct a third line segment joining the unattached ends of the first and second. Objection: the general conditions come from our concept of triangle; so knowledge deduced from the general conditions would be (covertly) analytic, not synthetic.

Conditions for reliable generalizing Distinguish between the initial figure and the elaboration of the figure. For a theorem about all of kind K (1) The type T of construction that elaborates the specific initial figure of kind K must be applicable to all of kind K; (2) The reasoning must be applicable to all figures of kind K with an elaboration of type T. 29

... and (3) The generalizing must depend on the holding of both (1) and (2): if, in similar visual reasoning, (1) or (2) failed to hold, one would detect the failure and refrain from generalizing. 30

The Space Problem Geometrical truths are spatial facts. Space is external to and independent of the mind. How can we have a priori knowledge of spatial facts?

The Space Problem Geometrical truths are spatial facts. Space is external to and independent of the mind. How can we have a priori knowledge of spatial facts? Kant s answer : Space is not independent of the mind! if we remove the subject or even only the subjective constitution of the senses in general, all the constitution, all relations of objects in space and time, indeed space and time themselves, would disappear [CPR A42/ B59]

Kant s answer (?) Space is the framework imposed by the mind on experience of objects as outer and extended. We can get knowledge of space merely by becoming aware of the nature of this mental framework.

Kant s metaphysics TRANSCENDEN TALLY IDEAL TRANSCENDEN TALLY REAL EMPIRICALLY REAL Real phenomena, e.g. stars. Forms of intuition e.g. space EMPIRICALLY UNREAL Illusory phenomena, e.g. dream images, rainbows, mirages. Noumena

How did Kant reach the surprising conclusion that space is mind-dependent? 1. Euclidean geometry is the necessary truth about space. 2. Our knowledge of geometrical truths is synthetic a priori. 3. The only explanation of 2 given 1 takes space to be the framework imposed on outer experience by the mind. 4. So space just is the framework imposed on outer experience by the mind.

The space problem: an alternative answer Geometrical truths are spatial facts. How can we have a priori knowledge of spatial facts? Distinguish between (a) claims about the geometry of actual space, (b) claims about the geometry that space would have if space were as the mind represents it. 36

How the mind represents space Singular and infinite [A 25/ B39-40] Without gaps Constant zero curvature at any scale..... This representation does not depend on geometrical concepts or beliefs. It is an intellectual task to uncover the geometry of this kind of space. 37

The space problem Geometrical truths are spatial facts. How can we have a priori knowledge of spatial facts? Answer We cannot know a priori the geometrical facts of actual space. But we can know a priori geometrical facts about the kind of space that is represented by the mind. 38

Two ways to go post-kant (1) Keep the view that geometry is about actual physical space; reject the view that geometrical knowledge is a priori; OR (2) Reject the view that geometry is about actual physical space; keep the view that geometrical knowledge is a priori.

Elaboration of the second way 1. Geometrical theories are theories about the way space could be (possible spaces). 2. Euclidean geometry is about those possible spaces that conform to our mental representation of space. 3. Our knowledge of truths of Euclidean geometry is a priori, and often synthetic.