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There is a common difficulty in all interpretations of Kant's philosophy of mathematics, including my texts. To answer the question: What is an intuition? Even the researches that are related completely with the term 'intuition' start without a thorough definition of the term. So now, for a while, I have been trying to formulate in one sentence what an intuition is. First of all intuition Kant uses is a technical term and does not have the same meaning with the one that is used in English, namely "(knowledge from) an ability to understand or know something immediately without needing to think about it, learn it or discover it by using reason" (Cambridge online dictionary). Kant uses Anschauung in German which means view or opinion, but the term is translated to English as intuition since Kant interchangeably uses Latin word 'intuitus' (cf. Potter, Reason's Nearest Kin, p. 21). I know I still did not give a definition of it. And it will still take some more explanations to give a proper description or definition.
First, let us look how Kant uses the term 'perception'. When representation is accompanied by consciousness it is perception (Critique, B376, Bennett Translation) and when perception is a state of person it is sensation and when it is as perception of something it is cognition (ibid). Finally when a cognition is related directly/ immediately to an individual object, it is intuition and when it is related indirectly and with general properties of several objects it is a concept (ibid). This is the warm up for the epistemic stuff although we will not use the terms 'perception' and 'cognition' again. I just thought this paragraph gives a nice categorization in level base, and helps us to understand that concept and intuition are both representations of objects and are at the same level.
We are almost there... But let's understand a bit throughly what sensibility (Sinnlichkeit), the mere forms of sensibility and sensation (Empfindung) are. Sensibility is "the name of the capacity for acquiring representations that reflect how we are affected by objects. So objects are given to us by means of sensibility, and that's our only way of getting intuitions" (Critique, B33, Bennett Translation) The mere forms of sensibility are present a priori in the human mind and these are a priori intuitions. These appear in the mind "even when there is no actual object of the senses and sensation" (ibid, B35). And "when an object affect us, it's effect on our capacity for representation is sensation" (ibid, B34). Bennett notes in his remark to B35 that "sensation refers to the detailed content of what the senses dish up and the senses refers to every aspect of our capacity for passively receiving data". I haven't decided whether the difference between sensation and sensibility is important, but it is crucial to understand the difference between sensibility and the mere forms of sensibility since this is how Kant distinguishes empirical and a priori intuitions.
Now one more step: "If a representation contains sensation (which presupposes the actual presence of the object), it counts as empirical; if no sensation is mixed into it, the representation is pure. [Recall that ‘representation’ = ‘intuition or concept’.] We can call sensation the ‘matter’ of sensible knowledge; and what is left when that is removed is the ‘form’. Thus pure intuition contains merely the form under which something is intuited..." (Critique, B75, Bennett Translation)
Examples? Consider the representation "body", remove from it everything that belongs to the understanding: substance, force divisibility... (we eliminated the concept part), remove from it everything that belongs to sensations: impenetrability, hardness, color... what is remaining: extension and shape (cf. ibid B 35) This is pure intuition. But, note that "[it] is just a fact about our nature that our intuition can never be other than sensible, all there is to it is our being affected by objects in a certain way." (ibid B75)
So what is an intuition? An intuition is a singular and immediate representation of an object, when the object is given to us by sensibility. Why Kant does not call it merely an object than? Because objects are out there, they act on the senses and the reason thinks about them, but what's an object in the human mind? It can only be represented and it is either a concept or an intuition, depending on the faculty (sensibility/ reason -more correct term would be Understanding-) that deals with it.
Then what is a pure or a priori intuition? A priori intuition is the mere forms of sensibility. You cannot say the form of intuition is a priori intuition since intuition comes represented through sensibility whereas a priori intuition resides a priori in human mind. And what can be mere forms of sensibility? These are primarily space and time, and then comes the a priori intuitions such as extension, shapes... Kant at some point believed that numbers were a priori intuitions too (cf. Kant, Inaugeral Dissertation, §23, referred by Potter, Reason's Nearest Kin, p. 42) but let's not get into the number talk here since it's a whole another story, which hopefully I will deal with later.
Review of Hintikka's text? Coming up after a bit more clarifications...