Scott Douglas Jacobsen: Of all possible mathematical knowledge, what do we really know? You were a distinguished professor in the past. We have written a text on this.
Herb Silverman: Here is what we know about mathematics. Mathematicians start with axioms (assumptions) and see what conclusion may logically be deduced (proved) from these axioms. The nineteenth century mathematician Leopold Kronecker once said, “God created the integers, all else is the work of man.” I interpret this statement to be more about the axiomatic approach than about theology. Mathematicians often begin with axioms that seem “self-evident,” because they are more likely to lead to real-world truths, including scientific discoveries and accurate predictions of physical phenomena. But if at least one axiom is false, then the conclusion may not be scientifically applicable. Unlike with applied mathematicians, theoretical mathematicians are not so concerned with whether their axioms are true. Axioms in some branches are contradictory to axioms in others. In non-Euclidean geometry, we replace Euclid’s parallel axiom with a different axiom. The axioms in Euclidean geometry have led to discoveries on planet Earth; results from the axioms in non-Euclidean geometry were applied many years later by Einstein for his general theory of relativity, when he showed we live in a non-Euclidean four-dimensional universe, consisting of three-dimensional space and one-dimensional time.