What does Kant mean when he says that all Mathematical judgments are synthetic?

Answer 1

Basically, synthetic judgments, like those found in math, can't be proven based on themselves, they must have an outside principle to justify them.

Analytical: The number 4 is what it is. You can know 4 all on its own.

Synthetic: What is the relationship of 2 and 4? Well, you need to bring in a math function to get there. 2 plus 2 brings you to 4.

Found the following online at the site below. There's a lot more after this that can help clear this up for you.

"But Kant also made a less familiar distinction between analytic and synthetic judgments, according to the information conveyed as their content. Analytic judgments are those whose predicates are wholly contained in their subjects; since they add nothing to our concept of the subject, such judgments are purely explicative and can be deduced from the principle of non-contradiction. Synthetic judgments, on the other hand, are those whose predicates are wholly distinct from their subjects, to which they must be shown to relate because of some real connection external to the concepts themselves. Hence, synthetic judgments are genuinely informative but require justification by reference to some outside principle."

Answer 2

Kant lived in the 1700's and the knowledge of mathematics was not as honed as today. He made minor flaws in his philosophy of mathematics in his book, Critique of Pure Reason. This is a good website to read up on it.--http://www.spaceandmotion.com/Philosophy-Immanuel-Kant-Philosopher.htm

Answer 3

Math can only be proven or disproven by other Mathematical formulas. Mathematics is a construct of the human intellect and is not found in nature.

That doesn't mean that it can't help to explain nature or that it isn't useful (very useful in certain situations) but in the end it is not real. You can't touch math, you can't see math, you can't taste math.

You can't experience Math in any way except within the cozy confines of our clever yet fallable human brains. Thats why Math purists consider Math to be an art and not just a tool of Science. (you can get Math degrees from the faculty of Arts from most Universities)

Answer 4

I think he meant that math is based on axioms, and axioms cannot be proven or disproven, we just accept them as evident. Other than this, Kant is obscure even to native German speakers.
The novel "Zen and the Art of Motorcycle Maintenance" helped me to understand Kant in a roundabout way.