Another position was endorsed by David Hilbert: finite mathematical objects are concrete objects, infinite mathematical objects are ideal objects, and accepting ideal mathematical objects does not cause a problem regarding finite mathematical objects. More formally, Hilbert believed that it is possible to show that any theorem about finite mathematical objects that can be obtained using ideal infinite objects can be also obtained without them. Therefore allowing infinite mathematical objects would not cause a problem regarding finite objects. This led to Hilbert's program of proving consistency of set theory using finitistic means as this would imply that adding ideal mathematical objects is conservative over the finitistic part. Hilbert's views are also associated with formalist philosophy of mathematics. Hilbert's goal of proving the consistency of set theory or even arithmetic through finitistic means turned out to be an impossible task due to Kurt Godel's incompleteness theorems. However, by Harvey Friedman's grand conjecture most mathematical results should be provable using finitistic means.
