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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.

Cotton Headed Ninnymuggins
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Rajan Aggarwal
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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 '20 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? – Rajan Aggarwal May 25 '20 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 '20 at 18:31
  • The direct sources to answer the part on *meta*physics 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 '20 at 20:09
  • When you define "due boundaries" as "possible experience", do you mean to say that my proposition about the triangle is legitimate since I CAN possibly experience it (since it conforms to the a priori concept of space), but for metaphysical objects like Soul or Free Will, I cannot do that. Or is it the other way around? I need to see a triangle first to deduce anything at all (that would not sound right to me because understanding is presumed in experience, and using understanding alone I came to the above conclusion about the sum of angles). Or do you mean I need to empirically test it? – Rajan Aggarwal May 25 '20 at 21:44
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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 '20 at 04:03
  • As @Conifold said. The judgement about triangles is a priori exactly because you do not have to **see** one in order to know its properties, you can *construct* (or *deduce*) from the concept of a triangle (in euclidean planes, that is) that the sum of the inner angles has to be 180 degrees. You cannot experience all mathematical objects, eg. a perfect circle. There is no orthogonal relation in nature. But concepts like *dancing* and *needle* are empirical, while *angels* are not. So it is moot to even bother bringing them together in synthetic judgements **about angels**, of whom we know not. – Philip Klöcking May 26 '20 at 07:44
  • Thanks. So basically, if it is consistent with my a priori (pure) intuition, it is a legitimate inquiry. But Kant is claiming that metaphysics faults here, which is surprising since it utilises a priori concepts of understanding (like causality), which for me is analogous to pure intuition. I guess on a general level I am just not convinced about how pure intuition is distinct from pure concepts of understanding and why there's a precedence. – Rajan Aggarwal May 26 '20 at 13:06

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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 '20 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. – Rajan Aggarwal May 26 '20 at 13:31
  • As to causality, there seems to be 2 options: either causality is a feature of reality in itself ( old dogmatic metaphysics) or causality reduces to a mental habit that we "project" onto reality ( Hume). Kant finds a way to escape this dilemma : caisality is an objective feature of reality ( contra Hume) but not of reality " in itself", only of phenomenal reality, that is, a feature belonging to the objects of experience. –  May 26 '20 at 15:01
  • Kant cannot satisfy himself with the inductive view of empirical science .According to him, knowledge requires certainty, and certainty requires necessity and universality. In turn, necessity and universality are impossible without the *a priori*. So even empirical science needs *a priori* principles. But this principles are only objective as long as they function as a grounding of the possibility of experience, that is, as long as they are used immanently. When one wants to use them in a transcendent way ( in metaphysics) they loose all " sense and meaning" . –  May 26 '20 at 15:19
  • "Causality is an objective feature of phenomenal reality". Why does he say that. Before I made this question up I was of the opinion that it is an objective feature because without causality we do not experience things (it's a priori) just like space and time. It's a necessary condition for experience, much like space and time. And if you can abuse (for lack of a better word) space and time and create math, why can't you do the same with causality. This basically is the crux of my confusion. – Rajan Aggarwal May 26 '20 at 16:26
  • The reason you give in order to account for the objective reality of causality is perfectly ok in Kantian terms. –  May 26 '20 at 16:28
  • I fail to understand what you are refering to by " abusing" space, time or causality. –  May 26 '20 at 16:29
  • Abusing space, time and causality here means thinking/conceiving of things without experience and using space, time and causality alone to justify some conclusions. Now in geometry, in this sense, space is being "abused" (sorry for the lack of vocabulary). However, when without any experience, we postulate things with causality alone, then he interjects saying it should be possible to think of them in terms of space and time, and not causality alone. This is something I struggle with. Why am I allowed to think in space terms ONLY and not causality terms ONLY. – Rajan Aggarwal May 26 '20 at 16:51
  • Basically what is the primary distinction between causality (concepts of understanding) and space (pure intuition) - and why is the latter necessary for concluding anything based on causality alone but it doesn't work that way vice versa. For example I can conceive of a still object (only space is used here), but not of causality outside the realm of pure intuition. Why is that? – Rajan Aggarwal May 26 '20 at 16:54
  • When reduced to its purely intellectual contect, "causality" boils down to the " if...then " logical relation. In order to mean anything substantial, causality needs to be fullfilled with the temporal relation of succession : " everything that happens ( begins to exists) requires something that comes before it, according to a rule" . So you need time to make causality meaningful and useful epistemically. –  May 26 '20 at 17:07
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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? – Rajan Aggarwal May 26 '20 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? – Rajan Aggarwal May 26 '20 at 19:09
  • Maybe this could be the theme of another question you could ask on MSE. –  May 26 '20 at 19:13
  • Note that the concept of a unicorn is not *a priori* and hat you cannot construct such a concept in the pure intuition on space. You need to use empirical data ; for example, the unicorn you imagine has a color. –  May 26 '20 at 19:17
  • If ever you have the possibility to find a copy of this book, it is an excellent one : Paul Guyer, Kant , Routledge ( Routledge Philosophers Collection) –  May 26 '20 at 19:24
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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. – Rajan Aggarwal May 26 '20 at 19:27