KNOWLEDGE propositions Our knowledge of mathematics is a priori because mathematical truths are universal and necessary, whereas no generalization from experience can have those properties Yet many truths of arithmetic and geometry are synthetic, not analytic ‘That a straight line between two points is the shortest one is a synthetic proposition For my concept of straightness contains no notion of size, but only of quality’ (B, 16) Physics, too, contains synthetic a priori principles, such as the law of conservation of matter Finally, a genuine metaphysics is not possible unless we can have a priori knowledge of synthetic truths How such synthetic a priori judgements are possible is the principal problem for philosophy Its solution is to be found by reXection on the way that human knowledge arises from the combined operation of the senses and the understanding It is the senses that present us with objects; it is the understanding that makes objects thinkable Our senses determine the content of our experience; our understanding determines its structure To mark the contrast between content and structure, Kant uses the Aristotelian terms ‘matter’ and ‘form’ The matter of sensation would include what makes the diVerence between a splash of blue and a splash of green, or the sound of a violin and the sound of a trumpet If we isolate sensation from everything that really belongs to the understanding, we Wnd that there are two forms of pure sensory awareness, space and time: the common structure into which our perceptions are Wtted But in real life human beings never have purely sensory awareness For human knowledge, both senses and understanding are necessary: Neither of these faculties has a priority over the other Without the senses no object would be given to us, and without the understanding no object could be thought Thoughts without content are empty, awareness without concepts is blind The understanding is aware of nothing, the senses can think nothing Only through their union can knowledge arise (A, 51) In human experience any object of sense is also an object of thought: whatever is experienced is classiWed and codiWed; that is to say, it is brought by the understanding under one or more concepts In addition to the understanding, Kant tells us, human beings have a faculty of judgement The understanding is the power to form concepts, and the judgement is the power to apply them The operations of the understanding Wnd expression in individual words, while judgements are expressed in whole sentences A concept is nothing other than a power to 158