How is 'Pure Intuition' possible according to Kant?

Question

One of the key passages is from Prolegomena to Any Future Metaphysics (1783). According to Kant pure intution is the means to obtain mathematical theorems as synthetic a priori propositions. This possibility

must be grounded in some pure intuition or other, in which it can present, or, as one calls it, construct all of its concepts in concreto yet a priori. (§7)

I do not see what the content of this type of intuition can be. Because Kants excludes the presence of any object as content of pure intuition. Instead he says:

There is therefore only one way possible for my intuition to precede the actuality of the object and occur as an a priori cognition, namely if it contains nothing else except the form of sensibility, which in me as subject precedes all actual impressions through which I am affected by objects. For I can know a priori that the objects of the senses can be intuited only in accordance with this form of sensibility. From this it follows: that propositions which relate merely to this form of sensory intuition will be possible and valid for objects of the senses; also, conversely, that intuitions which are possible a priori can never relate to things other than objects of our senses. (§8)

1. How is intuition possible without the presence of any object of intution? Of course I can make a mathematical construction only in my mind. But even then I imagine and remember some objects in question.

2. How can mathematical propositions be verifed by the means of pure intution?

3. Which mathematical propositions are obtained by pure intuition, do we have some examples?

A similar question has been asked What is "intuition" for Kant?

Answer 1

Kant writes:

There is therefore only one way possible for my intuition to precede the actuality of the object and occur as an a priori cognition, namely if it contains nothing else except the form of sensibility, which in me as subject precedes all actual impressions through which I am affected by objects.

I think this says a lot. Intuition, according to Kant, contains nothing else except the form of sensibility. This form precedes all actual impressions made by objects (or through which you are affected by objects). So pure intuition contains no objects. Rather, it contains the form of sensibility.

For I can know a priori that the objects of the senses can be intuited only in accordance with this form of sensibility.

Again, this form of sensibility precedes intuition. The a priori form of sensibility is the base of the pure intuition about objects. The objects are not part of the form of sensibility, but the intuited objects are intuited in accordance with the a priori forms of sensibility. So the a priori forms of sensibility are, in a sense, the objects of pure intuition. It are these a priori forms that facilitate pure intuition.

From this it follows: that propositions which relate merely to this form of sensory intuition will be possible and valid for objects of the senses;

So propositions regarding the a priori forms of sensibility (sensory intuitions, pure intuitions) are possible and valid for objects that impress themselves on us via the senses. Pure intuition uses the pure a priori sensory forms, the forms of sensibility, to intuit the objects that impress themselves via the senses. The objects "click" or "fall into" the a priori forms of the sensory sensibility. This is the pure intuition.

also, conversely, that intuitions which are possible a priori can never relate to things other than objects of our senses.

And let it be clear. A priori pure intuition, making use of the a priori forms of the sensibility, or propositions derived from these, only relates to the objects that impress themselves via the senses.

A tight construction. I'm not sure, but I think the forms of the sensibility on which the intuitions of objects rely, are the forms of mathematics. The propositions about these a priori forms are the basis of pure intuition about objects presenting themselves through the senses. Pure intuition is using the pure, a priori form of math.

Answer 2

1. How is intuition possible without the presence of any object of intuition ...

This is exactly the same question he asks in section eight. I'd advise you to read it more closely and ponder what he has to say. He says:

But with this step the difficulty seems to grow rather than diminish. For now the question runs: How is it possible to intuit somethimg a priori? An intuition is a representation of the sort which wpuld depend immediately on the presence of an object. It therefore seems impossible originally to intuit a priori, since then the intuition would have to occur without an object being present, either previously or now ... [thus] how can the intuition of an object precede that of an object itself?

Ge answers that the only possibility is that when the object is nothing other than the 'form of the sensibility'.

2. How can mathematical propositions be verified by pure intuition?

Kant states earlier that "mathematical judgements are one and all synthetic. This proposition seems to have escaped the observations of analysts of human reason up to the present, and indeed to be directly opposed to their conjectures, although it is incontrovertibly certain amd very important in it's consequences."

He is correct in this assessment as Gauss was inspired by the synthetic description of geometry to discover non-Euclidean geometry. He also states "that first of all it must be observed that properly mathematical statements are a priori and not empirical judgements because they carry necessity with them which cannot be taken from experience." Thus they are synthetic a priori concepts. They are "verified" through "going beyond the concept to that which is contained in the intuition corresponding to it."

3. Which mathematical propositions are obtained by pure intuition.

In section two, he mentions 7 + 5 = 12 and also the straight line between any two points is the shortest.

Answer 3

Note: I will be citing the B-edition, as trans. by Meiklejohn.

Space and time are not only forms of intuition, but formal intuitions, and these descriptions are subtly different. Intuition in general maps to reference/extension where discursion maps to sense/intension; intellectual intuition is the ability to refer 'merely' by 'trying' to refer, without having to 'land on a target' when exercising this ability. (Hence intellectual intuition crystallizes omniscience and omnipotence together, resolving in Kant to a divine faculty.) So as a form of intuition, space is a form of referring to externally different things (numerical differentiation).

But space, as a formal intuition, is then also a "pure object":

But apart from this relation, a priori synthetical propositions are absolutely impossible, because they have no third term, that is, no pure object, in which the synthetical unity can exhibit the objective reality of its conceptions.

That is, we can refer to our ability to refer.

Apriority, in Kant, can be effectively reduced to epistemic proactivity. Counting might seem like looking, but whereas your eyes happen to see things, your mind does not "happen" to count them, but you proactively count things (when you count them). (This point is obscure, esp. respecting the official definition of apriority at the start of the first Critique, but Kant makes it in the Groundwork (I think) when he compares the spontaneity of the understanding to that of reason.) So a priori intuition is proactive reference (indistinguishable from intellectual intuition in the case of a being with no passive faculty of consciousness).

Questions of more detailed applications of these principles, to mathematical questions, are beyond me at the moment, so I will just leave my answer at the first subquestion from the OP. My closing remark: space is the form of the external sense inasmuch as "external" means "different from," again going back to numerical differentiation. Time is the form of the inner sense, but this does not mean the form of my own inner self alone, but rather the overall principle of numerical identity over time. (This is how Kant traces together notions of permanent substance, skepticism about the external world, the integrity of spatial and temporal reference points, etc.)