Th e key to Kant’s philosophy is in the way he connected a priori judgments to a posteriori judgments. Scholars say this is how Kant put “philosophy back in the saddle.” In a stroke of genius, Kant combined the two judgments into a third judgment that he called synthetic a priori. What seemed like a contradiction actually worked. Kant explained that we make synthetic a priori judgments in mathematics, science, and ethics. His argument for mathematical judgments as synthetic is as follows:
All mathematical judgments, without exception, are synthetic. . . .
We might, indeed, at fi rst suppose that the propo-sition 7 + 5 = 12 is a merely analytic propopropo-sition, and follows by the principle of contradiction from the con-cept of a sum of 7 and 5. But if we look more closely we fi nd that the concept of a sum of 7 and 5 contains nothing save the union of the two numbers into one, and in this no thought is being taken as to what that single number may be which combines both. Th e con-cept of 12 is by no means already thought in merely thinking this union of 7 and 5; and I may analyze my concept of such a possible sum as long as I please, still I
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shall never fi nd the 12 in it. We have to go outside these concepts, and call in the aid of the intuition [synthetic]
which corresponds to one of them, or fi ve fi ngers, for instance . . . adding to the concept of 7, unit by unit, the fi ve given in intuition. For starting with the number 7, and for the concept of 5 calling in the aid of the fi ngers of my hand as intuition, I now add one by one to the number 7 the units which I previously took together to form the number 5, and with the aid of that fi gure see the number 12 come into being. Th at 5 should be added to 7, I have indeed already thought in the num-ber 12. Arithmetical propositions are therefore always synthetic. Th is is still more evident if we take larger numbers. For it is then obvious that, however we might turn and twist our concepts, we could never, by the mere analysis of them, and without the aid of intuition, discover what is the sum. 25
Critical Philosophy
Kant lived in this house in Königsberg, East Prussia. His mother, Anna, took young Immanuel for walks in the nearby meadows and fi elds, teaching the curious child about the seasons, plants, and animals.
In mathematics, the judgment that 7 plus 5 equals 12 is a priori. It is always true. Seven plus 5 has to equal 12. At the same time, this judgment is synthetic because we cannot get the number 12 merely by analyzing the numbers 5 and 7. Th is is where experience comes into play: to make the synthesis of the concepts 7, 5, and plus. Th e plus sign (+) has diff erent meanings when used in diff erent circumstances. On top of a church, the plus sign could signify a cross, or tipped to the side, the plus sign could mean “railroad crossing.” Th is also is true for the equals (=) sign. If the lines of the sign were extended, the symbol could look like a road, parallel lines, or a railroad track.
Th is means that we must fi rst learn by experience the cir-cumstances under which we use these signs. For example, when we see the plus (+) sign in mathematics, we have learned that that symbol means to add. Th at is synthetic. We know that 7 plus 5 always equals 12. Th at is a priori. We then have synthetic a priori, and that, for Kant, is how we get knowledge.
Another example Kant gave of synthetic a priori knowledge was using the statement, “Th e straight line between two points is the shortest.” Th e statement is a priori because it is always true. Yet, the idea “straight” does not contain the idea “shortest.”
Th us, “shortest” is synthetic, depending on the situation. For example, to say that John is the shortest boy in his history class has nothing to do with a straight line.
Just as little is any fundamental proposition of pure geometry analytic. Th at the straight line between two points is the shortest, is a synthetic proposition. For my concept of straight contains nothing of quantity, but only of quality. Th e concept of the shortest is wholly an addition, and cannot be derived, through any process of analysis, from the concept of the straight line. Intuition
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[synthetic], therefore, must here be called in; only by its aid is the synthesis possible. 26
Kant used this method to show that Hume was mistaken when he said there is no necessary connection between cause and eff ect. Without the law of cause and eff ect, this world would have no order. We would never know what to expect. If you threw a ball, you would never know if it would bounce, disap-pear in the clouds, or grow green feathers. What Kant calls into play is both a priori reason and experience. To Kant, it is syn-thetic a priori, his term for necessary connection. We know the law of gravity will bring the thrown ball back down to Earth.
Th us, we know the law of cause and eff ect to be true.