Iain asked:

How is it possible to have a synthetic a priori statement? Rationalists seem to be suggesting that
2+2=4 is not analytic yet there seems to be no other way of describing it. I would say that writing
(2+2) is equivalent to writing 4, only in long hand. I would be grateful if you could explain.

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Although I do not want to defend Kant's view that there are synthetic a priori judgements, I think it is
easier to see what Kant had in mind if you do not consider so simple a proposition as 2 + 2 = 4. (After
all, although Kant would have disagreed, it might be that very simple arithmetical propositions like the
one you offered as an example is not synthetic a priori, but more complex ones are. Much may
depend on which examples you choose to think about.)

Suppose you consider a far more complex proposition in mathematics. For instance the Pythagorean
theorem that the square on the longest side of a right triangle is equal to the sum of the squares on
the other two sides. Now, do you think that writing the right hand side of this equation is simply short
hand for the left side? I think it would be difficult to maintain this unless you were already committed
to the view that all of mathematics is like the simple equation you gave as your example. And, if we
think about other (supposed) examples of synthetic a priories outside of mathematics, the view that
they too are "analytic" is even more implausible. Kant's prime example of a synthetic a priori was
"Every event must have a cause." It would be vastly implausible to think of this as an analytic
proposition (true "by definition") Indeed, to think that "event" could be "defined" as a something that is
caused, would be to confuse the terms "event" and "effect." That is, although "Every effect has a
cause" is analytic, since an effect is, by definition a caused event, it is not true that every event has a
cause is "true by definition."

So, what I am arguing is that even if it is difficult for you to see why 4 is not just "short hand" for 2 + 2,
it should not be difficult for you to see why it would be very implausible to argue the same thing about
propositions that have been said to be synthetic a priori.

Kenneth Stern

An analytic truth can be described as a truth which can be known by looking at the terms. Kant's
semantic example is "all bodies are extended" — it is part of the concept of a body that it is extended.
Synthetic truths on the other hand, such as "all bodies are heavy", are truths in which the concepts
are not analysable in terms of each other, such 2+2 = 4. To think of a body is to think of extension so
this is analytic, but to think of 2+2 is not to think of four, so this is synthetic, and it is also a priori
because you can come to know this regardless of empirical facts in the world.

You obviously think the above is simply false.

However, Gödel has shown that you cannot 'define' numbers on a purely axiomatic basis, as
most mathematicians had argued. Gödel's Incompleteness Theoremsshowed (oversimplifying
a very complex argument) that a number has properties not shared by its components. The number,
say, '12' has properties that '10' and '2' do not have, even when you add the properties of component
'10' to the properties of component '2'.

This supports Kant. If 10+2 has properties that 12 does not have they cannot be the same concept so
this is a synthetic truth known a priori.

Rachel Browne