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The phrase a priori is a Latin term which literally means before (the fact). When used in reference to knowledge questions, it means a type of knowledge which is derived without experience or observation. Many consider mathematical truths to be a priori, because they are true regardless of experiment or observation. For example:

2 + 2 = 4

The above is a statement which can be known a priori.

When used in reference to arguments, it means an argument which argues solely from general principles and through logical inferences.

The term a posteriori literally mean after (the fact). When used in reference to knowledge questions, it means a type of knowledge which is derived through experience or observation. Today, the term empirical has generally replaced this. Many empiricists, like Locke and Hume, have argued that all knowledge is essentially a posteriori and that a priori knowledge simply isn't possible.

The distinction between a priori and a posteriori is closely related to the distinctions between analytic / synthetic and necessary / contingent.


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--Last edited by saucer on 2007-01-14 11:47:06 --

The external world is purely a chance matter, if anything can happen in the broadest possible sense, if the sun may not rise tomorrow, if, as a matter of fact, there may be no sun, or only a sputnik, when tomorrow comes, if there may be no tomorrow, etc., can there be any assured statement at all about apples? Why, for instance, don't apples disappear and appear randomly while we are counting them?

If, on the other hand, the external world has some degree of regularity mixed in with its chance elements, why expect that regularity to coincide, in even the remotest way, with the a priori mathematical expectations of human minds? Such questions can be multiplied without limit. Once one has made the Cartesian separation of mind and matter, of a priori and a posteriori, one can never get them back together again.

A strict a priori view is also open to more practical objections. If mathematics is known a priori, why have paradoxes sometimes arisen within mathematics? In the past actual contradictions have sometimes arisen from starting assumptions that appeared to be a priori true. The contradictions obtainable from conditionally convergent sums and integrals, and Russell's paradox concerning the set of all sets that do not contain themselves, are cases of this kind. Mathematical theory has also produced various counter-intuitive results, like the space-filling curves of Peano and everywhere-continuous nowhere-differentiable functions.

The paradoxes seem less threatening today, partly because mathematicians adopt a more conventionalist attitude toward them. Whatever axioms may be "convenient" are adopted, and the results are simply conventions of language. (See below on this.) Partly the paradoxes have been disposed of by modifying the axioms (to avoid contradictions) or modifying one's intuitions (to square with the theories). Nevertheless, the history of the paradoxes illustrates that supposedly a priori mathematical convictions are not always reliable.

It is understandable that these difficulties on the a priori side have led people to cast about for an a posteriori solution. In this case one emphasizes the supposed inductive character of mathematical knowledge. Mathematical knowledge is viewed as a generalization from experience in the world. One comes to believe that 2+2 = 4 from repeated experience of two objects plus two objects making four objects.

So far so good. But no one has repeated experience of 2,123,955 objects plus 644,101 objects making 2,768,056 objects. So why do people believe that 2,123,955+644,101=2,768,056? The consistent reply would be, "People generalize on the basis of their previous experience with small numbers." Unfortunately, in the word "generalize" are concealed all the problems that we began with. We may ask, "Why does a person "generalize" in one way rather than another?" Why, after observing that 3+2 = 5, 4+2 = 6, . . . , 12+2 = 14, does a person conclude (generalize?) that 13+2 = 15 rather than 13+2 = 16 or even 13+2 = 13? Doubtless this seems natural to any educated person nowadays, partly because he was taught such things. But how did he come to know it at the beginning?

In terms of a consistently a posteriori viewpoint, the answer can only be that a person generalizes this way because of previous experience with other generalizations. He has had experience before with detecting patterns. In other words, the step to 13+2 = 15 is based on generalization from previous experience, previous experience of other generalizations. But why has he generalized in this particular way from those other generalizations? Because he has generalized from previous experience of generalizing from previous generalizations. Etc. Apparently, one can escape this regress only by saying at this point, "Because that's the way the human mind operates." And then one is confronted with an a priori knowledge, or at least a priori heuristics, that is, ways of arriving at knowledge.

The a posteriori solution is also open to more practical, prosaic objections. What about the constantly growing quantity of abstract, non-visualizable mathematical entities? To claim that transfinite numbers, topological spaces, and abstract algebras are somehow impressed on us from sense experience takes some stretch of the imagination.

A third attempted solution to the problem of mathematical knowledge deserves mention, if only because of its wide-spread popularity among mathematicians themselves. This is the view that mathematics is, in some sense, a mere convention of our language, and thus not "knowledge" at all. 2+2 = 4 because we have agreed in our language to use words "two" and "four" in just that way.6 Or, to put it another way, in saying "2+2 = 4” we are just saying “A is A” in a roundabout way (A. J. Ayer).7 Or, "2+2 = 4 because it follows from our (conventionally determined) postulates" (formalists).

All these "conventionalist" answers are really so many variations of the a priori solution, inasmuch as one can still ask the same unanswerable questions about why mathematics should prove so useful in dealing with the external world. If it is pure convention, why should this be? Or if one says that the conventions are chosen because they are useful, one moves into the a posteriori camp, where he is confronted with the same unanswerable questions about the role of generalization.  

--Last edited by very very small tic on 2007-02-04 15:05:54 --