In Prolegomena to Any Future Metaphysics, Kant argues that space (and time) are not qualities of objects, but a priori intuitions that allow the concepts of objects in our minds.

To argue in favor of this idea, he writes that

those who cannot yet get free of the conception, as if space and time were actual qualities attaching to things in themselves, can exercise their acuity on the following paradox

This paradox is exemplified with

two spherical triangles from each of the hemispheres, which have an arc of the equator for a common base, can be fully equal with respect to their sides as well as their angles, so that nothing will be found in either, when it is fully described by itself, that is not also in the description of the other

What I don't understand is, why the "hemisphere", or alternatively the coordinates of the point that does not touch the equator, is not considered as one of the qualities of the object, just as the common base and their sides were. How would Kant explain this?

1 Answer 1


A coordinate is not an intrinsic property of a given point of the spheric triangle. It is a means that we use to locate points on the sphere.

The intrinsic properties of a spheric triangle are the angles between the sides and the sides themselves, also expressed as angles. Both spheric triangles of Kant's example have the same angles and sides, hence the same intrinsic properties.

  • 1
    What is an intrinsic property? Why is a side intrinsic but not a coordinate? What is the original German term for intrinsic?
    – gsmafra
    Commented Apr 27, 2022 at 21:48
  • 1
    A property of a geometric object is 'intrinsic' if it depends only on the object but not on the way the object is embedded into an ambient space. Intrinsic = intrinsisch (German). I do not know if Kant already has this technical term.
    – Jo Wehler
    Commented Apr 27, 2022 at 22:02
  • Document search didn't return "intrinsisch" to me in either the original Prolegomena or in Caygill's Kant Dictionary, so I assume Kant does not use that term. Could "innere" be interpreted the same way? About the argument it seems that: 1. Space is not a quality of objects because we disconsider an object's coordinates to ascribe "sameness". 2. We disconsider a coordinate because it is not an intrinsic property to it. 3. To be instrinsic means it does not refer to position in space. Looks circular to me. Not that I find circular arguments fallacious, but is there anything else to it?
    – gsmafra
    Commented Apr 27, 2022 at 22:16
  • Yes: 'innere'='intrinsisch'. ad 3) and 2): I agree. Then the coordinate of a point of the triangle is not intrinsic because it is just a means to locate the point in space. ad 1) I agree too. Two triangles in the plane are congruent if and only if they agree with respect to sides and angles. Kant carries over this criterion to spheric triangles. Why do you consider the chain 1)2)3) circular?
    – Jo Wehler
    Commented Apr 27, 2022 at 22:27
  • I guess to be really circular we would need something like "If space is not a quality of objects, then its 'properies' cannot be intrinsic". But maybe there's no reason to assert that? If not, then 3) is left as an axiomatic proposition, but I can't see that as obvious
    – gsmafra
    Commented Apr 27, 2022 at 22:35

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