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Immanuel Kant, Geometry and Space Prolegomena to Any Future Metaphysics

+Descartes review: Imagination vs. understanding

n  Triangles… n  can be imagined. n  can be understood.

n  Chiliagons, myriagons… n  cannot be imagined. n  can be understood.

n  So, understanding ≠ imagination.

n  Imagination represents spatial extension.

+Concepts and intuitions

n Concepts: representations of the pure understanding

n Intuitions: sensory representations, including representations of the imagination

n synthetic: the concept of the predicate isn’t contained in the concept of the subject

+ Geometrical knowledge

n  Things you think you know about geometry: 1.  Parallel lines don’t intersect.

2.  The interior angles of a triangle sum to 180 degrees

3.  The shortest path between two points is a straight line.

4.  Space has three dimensions.

n  Note: Given further assumptions, 1, 2, and 3 are equivalent.

n  Two questions: i.  How do you know these things?

ii.  What do you know, when you know these things?

n  Kant: satisfactory answer to i implies surprising answer to ii

+How do you know it?

n  Kant: It cannot be a posteriori. n  No possible experience could show it is false.

n  If no possible experience could show it is false, then no experience shows it is true.

n  Kant: It cannot be analytic.

n  concept shortest path is quantitative

n  concept of straight line is qualitative

n  So, the concept of the subject cannot contain the concept of the predicate.

n  So, this knowledge is a priori and synthetic.

*The shortest route between two points is a straight line.

+How do you know it?

n  Kant: Also must be a priori. n  No possible experience

could show that it is false...

n  Equivalent to synthetic-sounding claim that no more than three lines can meet at right angles

n  Also equivalent to claim that your left hand can’t fit in your right glove.

*Space has no more than three dimensions.

+Background on Geometry

+Historical background: Euclid

n  Ancient Greeks were pioneers in geometry. Algebra came later.

n  Euclid was believed to have given five postulates from which all the theorems of geometry could be derived.

n  Rationalist philosophers saw Euclidean geometry as paradigm of human knowledge.

+Postulates and theorems

n  Euclidean geometry: all theorems allegedly follow from five postulates (aka, axioms) n  Euclid supplemented postulates with a few definitions

n  a few minor holes that Euclid did not notice

n  Assuming the postulates and denying a theorem is therefore a contradiction. n  ‘If these axioms are true, then this theorem is true’ is analytic.

n  Are the postulates analytic or synthetic? How do we know that they are true?

+Euclid’s five postulates

1.  Any two points can be joined by a straight line.

2.  Any straight line segment can be extended indefinitely in a straight line.

3.  Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

4.  All right angles are congruent.

5.  If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

+What do you know when you know Euclid’s postulates?

n  Realism: Geometry is the branch of mathematics concerned with space. Euclid’s postulates are true statements about space.

n  Formalism: Geometry is a formal system. Euclid’s postulates are empty definitions/stipulations, whose validity doesn’t depend on their applicability to space.

n  Transcendental Idealism: Stay tuned...

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Realism Geometrical theorems = truths about real space

Problem: How could we know them a priori?

+Formalism

n  “Every theory is only a scaffolding or schema of concepts together with their necessary mutual relations, and the basic elements can be conceived in any way you wish. If I take for my points any system of things, for example, the system love, law, chimney-sweep, … and I just assume all my postulates as relations between these things, my theorems—for example, the theorem of Pythagoras—also hold of these things.”

David Hilbert

+Review: How do you know it?

n  Kant: You could not know it a posteriori. n  No possible experience could show it is false, so no experience shows it is true.

n  Kant: Your concept shortest path is quantitative—i.e., it involves the quantitative concept of the path’s length.

n  Kant: Your concept of straight line is qualitative—i.e., it involves the qualitative concept of the path’s straightness.

n  So, the concept of the subject cannot contain the concept of the predicate.

n  So, this knowledge is a priori and synthetic.

*The shortest route between two points is a straight line.

+ Non-Euclidean geometry §  Hyperbolic and elliptic

geometries remove “parallel postulate.” §  interior angles of

triangles either always greater or always less than 180 degrees.

§  straightest not a straight line

+ Geodesics The shortest routes between two points on a sphere

+Review: How do you know it?

n  Kant: Also must be a priori. n  No possible experience

could show that it is false...

n  Equivalent to synthetic-sounding claim that no more than three lines can meet at right angles

n  Also equivalent to claim that your left hand can’t fit in your right glove.

*Space has no more than three dimensions.

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2-D projection of a rotating 3-D cube Still frames with shading

+Recap: Formalism vs. Realism

n  Formalism: Euclid’s postulates are analytic definitions. n  Pros: explains a priori knowledge of geometry

n  Cons: geometry not about real space, all geometries on equal footing

n  Realism: Euclid’s postulates are truths about space. n  Pros: geometry concerns real space, some

n  Cons: doesn’t explain a priori knowledge

+Kant’s Prolegomena, Transcendental Idealism First Part

+Transcendental idealism

n  Kant’s transcendental idealism: middle ground between formalism and realism

n  His middle ground concedes to realism: n  Geometrical truths are truths about space.

n  3-D Euclidean geometry is uniquely accurate.

n  His middle ground concedes to formalism: n  Our knowledge of geometrical truths is a priori.

n  This a priori knowledge couldn’t give us any insight into reality.

n  Rough idea: Space is a construct of your mind.

+Form vs. matter of a floor plan

n Lines on a page = form of a floor plan

n Configuration of lines = matter of a floor plan

n What can you know about a floor plan before looking?

+Space: the form of intuitions of sensibility

n  Kant’s “intuitions” = Hume’s “impressions”

n  Kant’s “Sensibility” = Locke’s “sensation,” the faculty of “outer sense”

n  Kant: sensory impressions have common form, despite different matter

n  Spatial relations are part of the form of our sensory intuitions.

n  We can know the form of intuitions independently of the matter of particular intuitions.

+Kant’s Prolegomena, The Copernican Revolution First Part

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Noumenal vs. phenomenal Noumenal world: reality “as it is in itself”

Phenomenal world: reality as it appears to us

Kant: Our knowledge restricted to phenomenal world only.

+Kant’s “Copernican Revolution”

n  Kant: “Thus far it has been assumed that all our cognition must conform to objects.”

n  Kant: Instead, we should “assume that objects must conform to our cognition.”

n  Why is this a “Copernican Revolution”?

Nicolaus Copernicus

+20th century developments

n  Empirical evidence supports: n  Space is non-Euclidian

(general relativity). n  Space is not 3-D (string

theory).

n  Reaction 1: Kant was vindicated! He realized that space as we know might not correspond to reality.

n  Reaction 2: Kant was refuted! He thought that our capacity to understand reality was more limited than it really is.

Albert Einstein