In my reading of Kant's CPR (I mention this because I don't want an answer according to his other critiques), I don't seem to understand on what basis is Kant distinguishing statements in math and statements in theology.

For instance, it is a synthetic a priori judgement to say that sum of all angles of a triangle amount to 180 degrees. To arrive at this, one has used pure concepts of understanding and applied them to a triangle (consistently), and one can do this without needing posteriori experiences since the concept of triangle can be purely a priori. In this specific example one has utilized the concept of space, for example, and made a thesis - this Kant would call legitimate (it's how science and math operate).

However, he then becomes critical of metaphysics which applies concepts of understanding in a way that he says transgresses the limit of reason. My question is if all we use are concepts of understanding (we don't have any other way of discourse) to establish anything, given that the derivation remains consistent with these concepts, why is he critical of these metaphysical statements? I understand, for example, how a specific thesis in metaphysics is erroneous, for instance the ontological proof is erroneous because it assumes existence to be a necessary predicate. However, how can he say that reason necessarily leads us to error? If it did lead us to error, we could just realize the error that we made using the same concepts of understanding, correct?

Kant however is not saying that this particular or that particular argument is fallacious, he is saying it was inevitable that they would be fallacious because they were using the concepts of understanding outside their scope - this is the point I am not able to grasp. What is the scope exactly? How is thesis on God outside the scope and angles of a triangle inside the scope? Or perhaps I misread his argument entirely.

In summary, what distinguishes the synthetic a priori judgements of Math (180 degree rule), and other metaphysical discussions of God (like in Aquinas for instance).

NOTE: It'll be great if you could answer in reference to CPR only. I understand there will definitely be philosophies that would repudiate the presuppositions of Kant itself, but I want to understand his specific viewpoint as of now.

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    Mathematics is about things in space (geometry) and time (arithmetic). So they are subject to these forms of our intuition, and can be reasoned about synthetically, through pure intuition, as far as purely formal aspects are concerned. God and other subjects of metaphysics are beyond our experience in space and time, so we can only reason about them analytically, and that is as barren as analytic reasoning in mathematics without synthesis. Hence any substantive metaphysical arguments are fallacious.
    – Conifold
    May 25, 2020 at 17:56
  • "Mathematics is about things in space (geometry) and time (arithmetic)." Yes, but the concepts of understanding aren't limited to space and time. Are you saying only things that can be thought of in terms of space and time are legitimate? Moreover, since Kant himself declares that we cannot think of anything without these concepts of understanding, how is substansive metaphysics fallacious then? Are you implying that space and/or time are necessary a priori concepts to make any judgement, i.e, theyr retain a special status over things like causality or modality? May 25, 2020 at 18:11
  • Only things confined to space and time can be reasoned about synthetically (and only some formal sides of those things can, in addition, be reasoned about a priori). Concepts of understanding can be applied beyond space and time, this is how Kant justifies talking about noumena, but all one can do with them legitimately are logical trivialities, coming from applying identity, non-contradiction and excluded middle. Metaphysics is an attempt at synthetic reasoning about noumena, which are not subject to the forms of intuition that enable it.
    – Conifold
    May 25, 2020 at 18:31
  • The direct sources to answer the part on metaphysics are A254|B310 and Prol.,4:373f., fn.: metaphysics are problematic since our reason stretches its deductions beyond possible experience while using concepts (and objects) coming from experience, ie. beyond their due ground. Mathematics are a priori to start with, so reason cannot work beyond due boundaries here. Maybe I'll find the time for a proper answer tomorrow.
    – Philip Klöcking
    May 25, 2020 at 20:09
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    You need to construct the triangle in imagination to deduce anything interesting about it (beyond what is plain in its definition). And it will conform to the conditions of possible (empirical) experience because the same productive imagination used to construct it is also used to frame perceptions based on sensations.
    – Conifold
    May 26, 2020 at 4:03

1 Answer 1


Short answer : metaphysical propositions deal with " the Absolute" and the Idea of the Absolute ( the Soul, the World, God) is ( in virtue of its genesis explained by kant in the beginning of the Transcendental Dialectics ) an illusory idea, a pseudo-concept ( at least from a theoretic standpoint).

CPR, Transcendental Dialectics, Book I, Section II " On transcendental Ideas"

  • In order a judgment to be legitimate, if it has to be synthetic judgment , you need a ground that links the predicate to the subject. And this ground has to be non conceptual ( not purely logical), otherwise the judgment would be analytic. Simply analyzing the subject to find a predicate that was already involved in it yields an analytic judgment. For example : a material object is spatially extended.

  • But besides concepts ( intellectual representations) we have nothing else than intuitions ( sensible representations). So, only intuition ( be it pure/a priori or empirical/ a posteriori ) can provide the ground for synthetic judgments ( that is, for the linking of the predicate to the subject).

  • Mathematical judgment are legitimate, because mathematical concepts can be " constructed" in pure intuition. Due to my ( pure a priori) representation of space, it is impossible for me to imagine a path from point A to point B that is shorter than the straignt line segment from A to B : I " see" intuitively that the proposition " the straight line is the shortest path from A to B " is necessarily true ( and this necessity is not a logical one , for the proposiition is not analytic).

  • But in metaphysics, the intuitive ground is totally absent; the reason is that human beings have no intellectual intuition ( in spite of the fact that they have pure a priori intuitions).

  • For example, I have no intuition of myself as a permanent being. So I'm not entitled to say : "The I ( the thinking subject) is a substance".

  • Also, metaphysical concepts are fallacious, because they result from the fact that we endow with an objective/ ontological validity a principle of reason that is only a subjective necessity of our logical thought.

  • This principle is " for every conditionned thing that is given, the totality of its conditions must also be given". ( The main feature of reason is to look for conditions as says Kant in the beginning of the Transcendental Dialectics; Kant gives the example of the syllogism : "Socrates is mortal" . But why? Because he is a human being and all human beings are mortal. )

  • Endowing this logical principle with an objective ontological validity yields pseudo-concepts . These pseudo-concepts are metaphysical Ideas ( the Soul, the World, God).

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    While I may not agree with the wording at every point, I think the gist is correct. Giving the citations would strengthen the answer, though.
    – Philip Klöcking
    May 26, 2020 at 12:55
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    So the emphasis of Kant's project is that metaphysics is erroneous if it transgresses the boundary of "pure intuition" and not "pure concepts of understanding". This seems to render the importance and validity of concepts of understanding nullified. If these pure concepts of understanding have to be intuited using space and time, and therefore things like causality, cannot be applied without space and time, then why does Kant not agree with Hume that causality is a matter of habit and not transcendental? I am sure I am missing something. May 26, 2020 at 13:31
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    And I thing to further your argument, I would say 'exists' is something that we can only say when we apply the intuition of space there - hence all concepts of pure understanding are in themselves dependent on space and time. Is that a correct interpretation? May 26, 2020 at 18:24
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    Also, to add to this - can I say that since through the concept of space I can think of a unicorn in space and time - this is a legitimate assertion? Am i not using just pure concepts of space to posit a unicorn, similar to a triangle and sum of its angles? May 26, 2020 at 19:09
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    Thanks a lot. I mean, I would say then in this line of thought, even triangles need color of their borders or points of distinctions. Aren't they empirical too? But they are a priori according to Kant. If I were to agree that space and time are a priori then I would have to say that even unicorns are (they are something in space) just like a triangle. May 26, 2020 at 19:27

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