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Kant, Prolegomena to any Future Metaphysic

If you disagree i shant argue i shall merely make

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Unformatted text preview: e earlier thinkers saw that all the inferences of mathematicians proceed according to the law of contradiction, and wrongly slipped into thinking that mathematical truths were known from the law of contradiction. This was a great mistake. The law of contradiction can lead one to a synthetic proposition, but only from another synthetic proposition. (Still, it must be borne in mind that mathematical propositions are always a priori judgments, not empirical ones. They carry necessity with them, and that cannot be learned about from experience. If you disagree, I shan’t argue; I shall merely make this claim about the propositions of pure - that is, non-empirical - mathematics!) 10 One might think that the proposition 7 + 5 = 12 is analytic, and that it follows according to the law of contradiction from the concept of the sum of 7 and 5. But if we look more closely, we find that the concept of the sum of 7 and 5 contains only the uniting of 7 and 5 into a single number; and in thinking this we do not have the least thought of what this single number is in which the two are combined. I can analyse my concept of the uniting of seven and five as long as I please - I shall never find 12 in it. I have to go outside these concepts and - with the help of an intuition that corresponds to one of them (my five fingers for instance) - add the 5 given in intuition to the concept of 7, adding them one by one. Thus we really amplify our concept by this proposition 7 + 5 = 12, and add to the first concept a new one that was not thought in it. That is to say, arithmetical propositions are always synthetic. It will be easier to grasp this if we take larger numbers. It is obvious that however we might turn and twist our concept of the sum of 38976 and 45204 we could never find 84180 in it through mere analysis, without the help of intuition. [Kant’s use of the term ‘intuition’ needs to be explained; the explanation will occupy five paragraphs.Traditionally, the word has had two meanings. ŸIn one it contrasts with ‘demonstration’ - you know something intuitively if it is immediately self-evident to you, whereas demonstrative knowledge involves a series of deductive steps. ŸIn the other meaning - which alone is relevant to Kant - our faculty of ‘intuition’ is our ability to be mentally confronted by individual things, to have in our minds representations of the things and not merely of certain features or properties of them. Kant uses ‘intuition’ to stand not just for the faculty but also for the mental representations that it involves. Thus, for example, when you see the Lincoln Memorial you have an intuition of it, and this is an exercise of your faculty of intuition. That intuition is a ‘sensible’ one, meaning that you get it through your senses. It stands in contrast with a concept of the Lincoln Memorial such as the concept or abstract thought of a large white memorial to a great American statesman. Having in your mind a (conceptual) representation of a large white memorial etc. is quite different from having in your mind an (intuitive) representation of the Lincoln Memorial, that one particular individual object. [Now, Kant holds that we are also capable of having in our minds intuitions that do not come from the senses; he calls them ‘pure’ or ‘a priori’ intuitions. When in the previous paragraph he speaks of the intuition of my five fingers, that is a Ÿsensible intuition: I feel or look at the fingers. But he believes - as we shall see in section 7 - that pure mathematics involves Ÿpure intuitions: for example, a geometer works out the properties of circles not by merely taking the abstract concept circle and analysing it, but by somehow giving himself a pure intuition of a circle, and working out the properties of all circles from that. This is something like imagining-a-circle, but it is not ordinary imagination, which is copied from sense experience. [The basic idea is something like this: Every time you see or feel something circular, various aspects of your mental state are contributed by the sensations that come from outside you, and others are contributed by your understanding, i.e. the concept-using faculty. If all of that were somehow stripped off, what would be left is a very thin, abstract 11 intuition of the circular thing just as a circle. That is, nothing would be left of it but its purely spatial or geometrical properties; they will be the same for every circular thing; so the stripped down intuition will be the same in each case. That stripped down intuition is what Kant calls a ‘pure intuition’ of a circle. According to him, this is not contributed by sensation from outside you; rather, it is conferred on your mental state by your own mind, specifically by your own faculty of sensible intuition. You are so built, he thinks, that you have to experience the world outside yourself as spatial, not because the outer world is spatial but because you impose spatiality on the intuitions you have of it. Kant puts this, sometimes, by saying that what is represented in a pure intuition is the form of your sensibility or of your sensible i...
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