In the critique of pure reason, according to my reading, Kant is positing that propositions of mathematics are true because they can be situated in space and time, i.e, they can be conceived in space and time (geometry and algebra). For example, I can conceive of a triangle in space. Since space and time are pure intuitions, making judgements of triangle is legitimate (for eg sum of its angles are 180 degrees).

Now, what if I posit a unicorn? I can definitely conceive of a unicorn in space, just like I can do with a triangle. Someone suggested to me that you need empirical data like color - but then you need empirical data for triangle as well (dots and lines and its colour or width) - but this doesn't look right to me either.

Some would also suggest that I need a concept of a horse with a horn, but all these things separately I can conceive of without any experience. I can literally start from a shape akin to triangle and points in spaces and create a unicorn in my head, just like I can create any shape in space and any particular kind of movement in time.

So will Kant admit that either (1) Triangle must empirically exist for our mathematics to be legitimate or (2) It is legitimate to have a discourse about unicorn - perhaps it is just useless to do that - and the same would stand true for 'God' existing in space and time.

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    As long as "unicorns" refer to unicorn shapes only they are no different than triangles, geometry studies shapes of any complexity. But if we wish to adorn triangles or unicorns with colors, and reason about those, that would require empirical input. Facts about colors are not a priori, they are beyond mere forms of intuition. – Conifold May 26 '20 at 23:10
  • Is Kant suggesting you can think of a triangle without the color black and width of lines? – Rajan Aggarwal May 27 '20 at 5:35
  • One can not even visualize a triangle without imagining specific angles and side lengths. But mental imagery makes no difference to reasoning about them. It is only the schemata of triangles that contribute to the synthetic a priori reasoning, not the accompanying mental imagery. And schemata discount angles, sizes, let alone colors. Kant's transcendental schematism isn't naive visualization, it is modeled on dealing with Euclid's diagrams, which are also full of concrete information not meant to be salient. – Conifold May 27 '20 at 5:59
  • The properties of any triangle are the properties of space itself as pure form of intuition. – Mr. White May 27 '20 at 6:11
  • I am still not convinced about how mental imagery can be visualised in our minds without lines and points of intersection - something akin to colours (i mean lines themselves have colors). I will grant you that derivations of angles have nothing to do with colors or width of lines, but you can't have an angle without two lines intersecting - and line for me is in no way pure space, it's an empirical object, much like colors. – Rajan Aggarwal May 27 '20 at 6:59

The question asked deals with the following part of the CPR : Analytic of Principles, Chapter II, Section 3, §4 " Postulates of empirical thought in general".

These postulates give the " critical ( non-dogmatic) " version of modal notions such as " possibility" , " existence" , " necessity" ( and their opposites).

  • I think your reasoning can be represented as follows :

(1) If a concept can be represented in pure imaginary space, then it is a legitimate concept ( after Kant).

(2) I can represent to myself a unicorn in pure imaginary space.

(3) So the concept of a unicorn is legitimate.

(4) But, as anyone knows, the concept of a unicorn is not legitimate, for unicorns simply do not exist.

(5) So (1) must be false.

  • One could question premise (2) : as Conifold points out, what I can represent to myself in pure imagnary space is not a unicorn but the shape of a unicorn; and in fact if this shape as a shape represents a concept, it will not be the concept of a unicorn, but a geometrical/mathematical concept ( maybe a general equation of the form f(x,y) = z ), and Kant has an explanation for this.

  • One could question premise (4) : after all, the fact that there is no unicorn does not rule out the concept of unicorn as illegitimate; a concept is legitimate as soon as its object is simply possible ( not necessarily real / actual) ; the triangle concept is legitimate because it agrees with the formal conditions of the possibility of experience ( namely space); the concept of unicorn is also legitimate ( a unicorn is a possible object of experience)

  • So (1) need not be rejected, but simply more precisely phrased : "legitimate" is ambiguous inasmuch as it can refer either to the " possibility" of the object or to its " actuality" ( existence)

  • I think it is also worth noting that even if I can represent to myself a unicorn in pure imaginary space, that does not confer to that concept the same epistemological status as the concept of triangle. For, the pure representation of a triangle in space will allow me to state synthetic propositions regarding triangles, for example : " for all triangle, the sum of its angles is equal to 180°". But what synthetic proposition regarding unicorns shall I base on my unicorn pure / imaginary representation?

For example , will it allow me to state things such as

  • the DNA of a unicorn has such-and-such property ?

  • the average cardiac rythm of a unicorn is above such and such value?

The fact that the pure representation of a unicorn ( if it exists) is totally sterile epistemologically is, to me, the main reason not to put the unicorn concept on a par with the triangle concept.

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    Thank you for the explanation. It seems to me then, that the lines and intersection of lines (necessary for an angle) is presumed to be purely a priori and not empirical. This is is not convincing to me since lines and intersection are objects of experience. Otherwise, about the unicorn being an epistemologically sterile thing, I am completely convinced by you. This would mean, then, if we couldn't see a triangle, doing math on it is also sterile, but we do have triangles in the phenomenal world. This piece of information then, that triangles have been experienced, is necessary for legitimacy. – Rajan Aggarwal May 27 '20 at 10:29
  • Kant aims at finding an intermediary path between 2 thesis : (1) [ old dogmatic metaphysics] being logically/ conceptually possible is enough for a concept to have objective validity ( that is, for its object to be possible) (2) being actually given in experience is necessary for the object of a concept to be possible [ empiricism] . Kants intermediary path is : the possible is what agrees with the possibility of experience. – user37859 May 27 '20 at 10:44
  • What I mean is that Kant takes experience into account in his definition of possibility. But the mistake of empiricists ( according to him) is to reduce possibility to actual experience ( what is perceived). You may have a look at the section of CPR I refer to in the answer : Kant explains why old metaphysics cannot explain the impossibility of a figure enclosed in only 2 lines. – user37859 May 27 '20 at 10:46

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