As I mentioned earlier, Kant argued that there are two kinds of judgment which he called analytic and synthetic. Analytic judgment is just an analysis of concepts. I can say with supreme confidence that if there's a storm then it's raining. I don't have to know what the weather conditions are; the sentence
1. If there's a storm then it's raining.
is guaranteed to be true. I know that it's true just by analyzing what it means. Part of what I mean by "there's a storm" is that it's raining (please don't tell me about snow storms. There isn't a simple relationship between words and concepts). My judgment on sentence 1 is an analytic judgment.
By contrast, consider
2. If there's a storm then I can't go for a walk.
Sentence 2 requires some knowledge of storms, walking, and physical discomfort. There is nothing about "there's a storm" that means that I can't go for a walk. This is a synthetic judgment because I have to synthesize new knowledge. By contrast, the judgment of 1 is not really knew knowledge, I'm just affirming what was already known.
There is another difference between 1 and 2. Sentence 1 is judged a priori, prior to any experience. I don't have to learn from experience what my concepts entail, it's built into the concepts. The knowledge is timeless. By contrast, sentence 2 involves a posteriori knowledge, knowledge that I learned from experience. I had to have some experience of walking in a storm, or at least of being cold and wet, to know that I don't like the experience.
So there are two kinds of judgment, one kind is simply what we know about our own thoughts. It is analytic and a priori. Since it doesn't rely on experience, it is impossible that any experience could contradict an analytic judgment. Analytic knowledge is certain but trivial.
The other kind of judgment is based on experience. It synthesizes different bits of knowledge to get something entirely new. It is what we know about nature (philosopher-speak for the physical world). This knowledge is synthetic and a posteriori. Because it is a posteriori, it is contingent on our experience. It is uncertain but it is nontrivial.
So, what kind of knowledge is mathematics? It isn't analytic. The concept of being the number 7 doesn't entail the concept of being prime. It isn't as though my understanding of 7 is such that I can't imagine 7 being divisible by any other number. 7 is just one more than 6. To know that it isn't divisible by any other number I have to go over all the candidates in my head and see if they divide it. I have to take multiple facts and join them together to arrive at the conclusion that 7 is prime. This makes the knowledge synthetic.
But mathematics isn't a posteriori. I don't have to take seven pebbles and try to divide them up into equal groups to see if it is possible. I don't have to try it with teacups and stars and other physical objects to see what sort of objects it applies to. I can figure it out in my head, and the knowledge is universal and certain. So mathematic judgments are a priori.
Kant concluded that mathematics is both synthetic and a priori. It is nontrivial knowledge synthesized from other facts and yet it is certain and non-contingent. This bothered a lot of people.
Think what this implies. It seems obvious that all our knowledge of nature must be a posteriori because it is all contingent. I didn't know that sugar was sweet until I tasted it. I didn't know that fire burns until I touched it. And because of this, I don't really know that fire will burn the next time I touch it.
Experience is contingent. It depends on the laws of nature which might just as well have been different. At least, there is nothing logically contradictory with the possibility that fire will no longer burn tomorrow. This is the very definition of the physical world isn't it? It is that part of my experience which can only be known through experience.
But now I take four saucers out of the cupboard and take them over to the table. Then I go back and get four teacups and put them on the saucers, and they match. Every teacup has exactly one saucer and every saucer has exactly one teacup.
How could I have known that? I didn't experiment. I may never have set the table before in my life, yet I can still figure this out. I predicted an experience, not on the basis of prior experience, but on the basis of a priori judgment. It's almost magical, when you think about it. How can I know that four physical objects will always match four other physical objects?
I know of three solutions to this problem. You can argue that mathematics is actually analytic knowledge after all. That was the logicist solution. You can argue that mathematics is actually a posteriori knowledge after all. That's the most popular solution today. Or, you can argue that although mathematics is really synthetic and really a priori it is not really knowledge of the physical world. That was Kant's solution.
OK, I think I'm finally done with the background. In the next philosophy post I may actually make a point.