Gödel Incompleteness Theorem Essay

1845 Words4 Pages

What is the significance of Gödel’s Incompleteness Theorems for the philosophy of Mathematics?
Gödel’s incompleteness theorems, established in the first half of the twentieth century, have transformed the way many mathematicians, philosophers and even computer scientists have thought about mathematics. Although throughout the entirety of his work, he is neither concise nor always clear; it is obvious Gödel's theorems unearth a series of restrictions of an axiomatic and mechanical view of mathematics. He is understood to claim that any complete axiomatic system cannot be consistent. His theorems changed the understanding of various fields of philosophy, particularly to the philosophy of mathematics; they pose prima facie problems for Hilbert's program and directly to logic, to intuitionism and also invites controversial comparisons between the scope of mathematics and the human mind. The extent of the first will be the focus of this essay. I will discuss the efforts of Gödel to unveil a new era of mathematics, in doing so he successfully discovered a flaw in mathematicians reasoning, but whilst his theorems were non-the-less significant, a physical change in mathematics has not been dramatic; the theorems did not over-rule the astounding perfection mathematics has already established.
Before we consider such significance, I feel it’s important to confirm a mutual understanding of the theorems and its foundation. Considered to be one of the greatest philosophical mathematicians of a generation David Hilbert published his pursuit of an ideology for a systematic basis for arithmetic; ‘turning every mathematical proposition into a formula…thus recasting mathematical definitions and inferences in such a way that they are unshakeable and...

... middle of paper ...

...ems and to arithmetic. A mathematician says 2+2=4, and Gödel says prove it. Some things require no more proof than the thought of its negation. Some non-axiomatic truths do not require proof; Gödel’s threat is in instance rather than reality.
Lecture
David Hilbert; The Foundations of Mathematics, The modern development of the foundations of mathematics in the light of philosophy 1927
Journal
Kurt Gödel, Collected Works, Volume III (1961) publ. Oxford University Press, 1981
Essay
Michael Detlefsen, Hilbert’s Program: An Essay on Mathematical Instrumentalism, Kluwer Academic Publishers 1986
Journal
Daniel Isaacson, Arithmetical truth and hidden higher-order concepts in W.D. Hart, ed., The Philosophy of Mathematics (OUP, 1996)
Journal
David Hilbert, ‘On the infinite’, in Benacerraf and Putnam, eds., Philosophy of Mathematics: Selected Readings, 2nd ed. (CUP, 1983)

More about Gödel Incompleteness Theorem Essay

Open Document