Is all of mathematics based on logic?

Generally speaking, mathematical logic investigates the nature and limitations of logical systems. For example, consider the question: "can every true statement be proved?" This is a problem about logic. When we study geometry, algebra and other branches of mathematics we are interested in using logical thinking, but the purpose is to understand something about shapes or symmetries.

Are there known results in mathematics that are not based on logic?

Well, there's always Ramanujan's approximation of $\pi$ as $\sqrt[4]{\dfrac{2143}{22}}$ , based on a dream which he had one night about a certain Hindu goddess that his family worshipped. :-)