The contrast between a priori and empirical knowlege

Suppose someone knows that ravens are black, that Caesar was born before Caligula, that hydrogen molecules always consist of two atoms, or that there will be a gale tomorrow. These are clear-cut examples of what philosophers have meant by empirical knowledge. Each of these pieces of knowledge is based on experience in this sense: to know any one of these facts a person must not only understand what is meant but must also possess evidence drawn from sense experience - that is, evidence regarding what has been seen or heard or felt or smelled or tasted. In order to know that ravens are black, I must not only understand what this means, I must also have seen ravens, or seen feathers that they have left behind, or heard reports from observers that they have seen such things, or something of the sort. Of course, even without evidence, a person could 'believe' that ravens are black, that Caesar was born before Caligula, that hydrogen molecules contain two atoms, or that there will be a gale tomorrow. But belief, even if true, is not knowledge when it lacks justification. The point is that only sensory observations can provide the kind of justification needed to entitle a person to say that he knows facts like these. If I do not have any sort of observational evidence relative to ravens, then it is certainly false to say that I know them to be black. To claim to know this kind of thing without knowing it on the basis of evidence gained from sense experience would be self-contradictory.

Summing up, we may define empirical knowledge as knowledge that requires justification from experience.

There are other examples of knowledge, however, which do not depend upon experience in this way. Suppose someone knows that ravens are birds, that Caesar either was born before Caligula or was not born before Caligula, that hydrogen molecules are molecules, or that there will be a storm tomorrow if there is a gale. These are clear-cut examples of what philosophers have meant by a priori knowledge. A person does have to have observed ravens directly or indirectly in order to be entitled to say that all ravens are birds; he does not have to have looked into Roman history to know that Caesar either was or was not born before Caligula; he need not have watched physicists' experiments with hydrogen in order to know that hydrogen molecules are molecules; nor need he have seen tomorrow's weather map to know that there will be a storm if there is a gale. In these cases the only experience that is required is whatever experience may be needed in order to enable him to understand the words in which the knowledge is expressed: no sense experience beyond this is necessary to justify his claim that he knows. Summing up, we may define a priori knowledge as knowledge that does not need to be justified by experience.

This distinction between a priori and empirical knowledge is of philosophical importance both because of the clarification it effects and because of the problems it raises. It helps us to see that subjects such as physics, biology, and history, all of which are primarily concerned with matters of empirical knowledge, must therefore rely upon observations in order to establish their conclusions. A subject like logic, in contrast, is concerned only with a priori knowledge (logic seeks a priori knowledge of the rules governing the validity of arguments) and therefore need not rely upon observations in reaching its conclusions. The question then arises, is mathematics like physics, or is it like logic, in this respect? Or is it partly like the one and partly like the other? And one very general philosophical problem that the distinction raises is the problem of how we attain a priori knowledge: is it through some special insight into reality, or through insight into our own minds, through understanding of language, or what? If mathematical knowledge is a priori, not justified by experience, then what is it based on?

- Stephen Barker, Philosophy of Mathematics, Prentice Hall, 1964.