Yahadreas wrote:
Let us take a number system where the number 1 is written I and the number 2 is written II. In this number system, one shows the plus operator by placing two numbers next to each other. So, 1 + 1 will be written as II. In this number system, the equation 1 + 1 = 2 is shows as II = II. Is it not the case that II is contained in the concept II? So this would not be synthetic, correct? And if this is not synthetic, then how could the equivalent 1 + 1 = 2 be synthetic? It's the exact same logic, just with different visual representations.

Χριστόφορος wrote:
I often see this example used and I believe it was Kant's own example in defining what a synthetic statement is. it is said that the concept of "7 + 5" is not contained in the concept "12" and my question is why not? Isn't the proposition "7 + 5 = 12" simply a writing convention for displaying "1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = "1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1" in a more convenient and practical way? Isn't that what the number twelve represents fundamentally, the combination of twelve individual things?

Kant by way of plato.stanford.edu wrote:


In all judgments in which the relation of a subject to the predicate is thought . . . this relation is possible in two different ways. [Sic?] Either the predicate B belongs to the subject A something that is (covertly) contained in this concept A; or B lies entirely outside the concept A, though to be sure it stands in connection with it. In the first case, I call the judgment analytic, in the second synthetic.

http://plato.stanford.edu/entries/analytic-synthetic/#StaPri wrote:


One reason Kant may not have noticed the differences between his different characterizations of the analytic was that his conception of “logic” seems to have been confined to Aristotelian syllogistic, and so didn't include the full resources of modern logic, where the differences between the two characterizations become more glaring (see MacFarlane 2002). Indeed, he demarcates the category of the analytic chiefly in order to contrast it with what he regards as the more important category of the synthetic, which he famously thinks is not confined, as one might initially suppose, merely to the empirical. While some trivial a priori claims might be analytic, for Kant the seriously interesting ones were synthetic. He argues that even so elementary an example in arithmetic as “7+5=12,” is synthetic, since the concept of “12” is not contained in the concepts of “7,” “5,” or “+,”: appreciating the truth of the proposition would seem to require some kind of active synthesis of the mind uniting the different constituent thoughts. And so we arrive at the category of the “synthetic a priori,” whose very possibility became a major concern of his work. He tries to show that the activity of “synthesis” was the source of the important cases of a priori knowledge, not only in arithmetic, but also in geometry, the foundations of physics, ethics, and philosophy generally, a view that set the stage for much of the philosophical discussions of the subsequent century (see Coffa 1991t I).