Ever Since Plato Philosophers
Have been impressed by the fact that there are universal propositions which we can apparently know to be true with absolute certainty, though we could not possibly observe all, or even a large proportion of the instances to which they apply. Such propositions, called "necessary propositions" by philosophers, are especially to be found in mathematics, the science which establishes its theorems with such infallible certainty that the expression "mathematical certainty" has become part of the everyday idiom. In Plato's dialogue Meno
, a slave boy is led by Socrates to "see" that a square whose side is a diagonal of a given square has exactly twice the area of the latter (the proposition in question is a special case of the "Pythagorean Theorum"). How can we know that it is so in every possible case of a square with inscribed diagonals? How can we know that not only the triangles we have drawn on the blackboard in order to "verify" by measurement that the internal angles add up to 180 degrees have an angle sum of 180 degrees but that all conceivable triangles have it? Philosophers who hold that experience is the only source of human knowledge ("empiricism") may, like John Stuart Mill, say that our conviction is nothing but a habit of association, built up by repeated observation that one property is conjoined with another, but a philosopher who holds such geometrical knowledge to be independent of experience, a priori
("rationalism"), retorts: If such were the case, why are we not convinced to an equal degree that all crows are black, or that all bodies have weight, or that the ground gets wet whenever it rains? It seems that we can conceive of exceptions to the latter propositions (even if we find it hard to believe that there ever will be any), in a way in which we cannot conceive of exceptions to the former propositions. If we found a square whose side is the diagonal of another foursided figure, yet whose area was not double that of the latter, we would conclude that the latter figure is not exactly square: indeed, we would abandon any previously entertained beliefs that were relevant except
the belief in the absolute validity of the theorum.
At any rate, this is the way Plato, Kant, Leibniz, and many other philosophers felt and feel about mathematical knowledge. In describing it as a priori
knowledge (following Kant's terminology), philosophers refer to its apparent independence of experience: we claim to know that two pebbles and two pebbles make four pebbles even on Mars, before having verified this by actual counting after a strenuous trip in a rocketship. Perhaps, we also believe that if there are crows on Mars they are black, yet as we would admit that this is only probable
on the basis of past experience: it is conveivable that on Mars or on some other planet there should be animals which are exactly like the animals we usually call "crows" except that their feathers are, say, red. In this sense our knowledge of the proposition that all crows are black is said to be empirical,
and traditionally empirical knowledge has been said to lack that certainty
which attaches to a priori
It is important to understand in exactly what sense a priori
knowledge is "independent of experience." No philosopher has ever denied that a child has to learn that two and two make four by learning to count, and that the latter process involves contact with concrete objects. But this only means that without sense-experience one cannot acquire the concepts
of number, in this case the concepts "two" and "four," that is, a child who has never learned to count, to associate different numerals with distinguishable objects, will not even understand what "two and two make four" means
. What the philosophers who believe in a priori
knowledge assert is only that once the concepts have been acquired, the proposition can be "seen" to be true by just thinking about it (by "the mere operation of thought," in Hume's phrase). Consider the statement "for any objects A, B, C: if A is bigger than B and B is bigger than C, then A is bigger than C." Obviously, a being without sense of sight and sense of touch would have no concept of the relation designated by the word "bigger," hence such a being would not even understand that statement. But in saying that its truth is independent of experience, philosophers only mean that anyone who understands
it will see that it is necessarily true, that it could not possibly be refuted by any observations at any time or place.
The philosopher not only makes the distinction between a priori
knowledge and empirical knowledge by reflecting on the appropriate methods of justifying our beliefs; he goes on to ask how a priori
knowledge is possible. Plato apparently was so perplexed by the fact that we can know universal propositions independently of experience that he had to invent a myth in order to account for it: the soul remembers visions it has enjoyed in a former disembodied life. Other philosophers, less poetical than Plato, tried to account for it in terms of a distinction between two kinds of entities, a distinction that played a vital role in Plato's philosophy: universals (Plato called them "forms"), and particulars. When we look at the blackboard we see particular triangles, but when we prove the Euclidean theorem about triangles we think of the universal triangularity,
i.e., that which all the particular triangles have in common and by virtue of which they are all triangles. Every particular triangle has a particular size, for example, but when we classify it as a triangle we abstract from this particular feature and focus attention of a property which it shares with similar figures; it is this common
property which philosophers call a universal.
Again, we can see particular cubical objects at different places at the same time, or at the same place at different times, but when we think about the nature of a cube (as when we say to ourselves "every cube must
have twleve edges") we think, in the terminology of those philosophers, about a universal that is identically present in all visible and tangible cubes. Whenever we classify a particular thing or event--in short, a "particular"--as being of such and such a kind, we consider it as an instance of some universal, or set of universals tied together by a single name, like "cow," "man," "table," "rain," "thunder." According to Locke's doctrine in the Essay Concerning Human Understanding
and Russell's in The Problems of Philosophy,
we can be certain that every particular which is an instance of universal A is also an instance of universal B, though we can never survey all past, present and future instances of these universals, if we "see" with our intellectual eye a certain relation between A and B, a relation which is sometimes called "necessary connection," sometimes "entailment." If we can see that squareness entails equilateralness, and that being a cube entails having twelve edges, then we can be sure in advance of sense-experience (think of the original meaning of "a priori"
in Latin: before!) that there are no squares that are not equilateral, nor cubes that do not have twelve edges.