I imagine that we are going to talk about the apriori-aposteriori and synthetic-analytic distinctions in due course, but since we didn’t get to them in our last class, I thought I would put in an early request. I was also hoping you could touch on an objection to the distinction from logical positivists (in particular A.J. Ayer whom I believe is not exactly known as a “careful” Kant scholar), who claim that 7+5=12 is apriori, but not synthetic, and the idea that mathematics in general is nothing more than an array of analytic truths or tautologies. I’m not sure I agree with the objection, but I am rather uncertain what exactly it is that makes 7+5=12 synthetic.

## Why is 7+5=12 a synthetic proposition?

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February 1, 2010 at 9:22 pm |

Leibniz held that arithmetical truths (such as 7+5=12) were analytic, which is to say that they could be reduced to logical definitions and truths, or could be derived from the analysis of concepts alone. The logicist program of the early twentieth century (attempted by Russell, Whitehead, and others) tried to show that Leibniz was right about this. But Kant held that arithmetical truths are not analytic, but synthetic. His most famous discussion appears at B16 of the CPR (see text below). But briefly: the key to understanding Kant on this point is to see that a synthetic truth is shown to be synthetic insofar as any demonstration of its truth requires recourse to intuition (that is, it requires recourse to what is *given*, either in time or space); that is to say, a synthetic truth is shown to be synthetic insofar as any demonstration of its truth requires recourse to some construction in time or space. Thus to know that the shortest distance between two points is a straight line, I need to construct a line (imaginatively) in space, and in order to know that 7 + 5 = 12, I need to construct (imaginatively) the successive adding (and thus the temporal adding) of units which make up 7 and 5 and 12.

Kant’s text at B16, of the CPR:

“We might, indeed at first suppose that the proposition 7 + 5 = 12 is

a merely analytical proposition, following (according to the principle

of contradiction) from the conception of a sum of seven and five.

But if we regard it more narrowly, we find that our conception of

the sum of seven and five contains nothing more than the uniting of

both sums into one, whereby it cannot at all be cogitated what this

single number is which embraces both. The conception of twelve is by

no means obtained by merely cogitating the union of seven and five;

and we may analyse our conception of such a possible sum as long as

we will, still we shall never discover in it the notion of twelve.

We must go beyond these conceptions, and have recourse to an intuition

which corresponds to one of the two–our five fingers, for example,

or like Segner in his Arithmetic five points, and so by degrees, add

the units contained in the five given in the intuition, to the

conception of seven. For I first take the number 7, and, for the

conception of 5 calling in the aid of the fingers of my hand as

objects of intuition, I add the units, which I before took together

to make up the number 5, gradually now by means of the material image

my hand, to the number 7, and by this process, I at length see the

number 12 arise. That 7 should be added to 5, I have certainly

cogitated in my conception of a sum = 7 + 5, but not that this sum

was equal to 12. Arithmetical propositions are therefore always

synthetical, of which we may become more clearly convinced by trying

large numbers. For it will thus become quite evident that, turn and

twist our conceptions as we may, it is impossible, without having

recourse to intuition, to arrive at the sum total or product by

means of the mere analysis of our conceptions. Just as little is any

principle of pure geometry analytical. “A straight line between two

points is the shortest,” is a synthetical proposition. For my

conception of straight contains no notion of quantity, but is merely

qualitative. The conception of the shortest is therefore fore wholly

an addition, and by no analysis can it be extracted from our

conception of a straight line. Intuition must therefore here lend

its aid, by means of which, and thus only, our synthesis is possible.”