The idea of a “computer program” originates from work of Kurt Gödel (1931) and Alan Türing (1936-7) on foundations of mathematics, and it is the main ingredient for the proof of the so called Undecidability theorem. This astonishing result establishes that we will never have a universal computer program capable of answering any well-posed question on numbers. This means that at any time T of history there will be problems in arithmetics that are solvable but cannot be solved with all the methods (algorithms) available at time T. The mathematical knowledge human minds will ever reach, will always be incomplete.
This result has important philosophical implications (see here: https://arxiv.org/abs/0809.3691)
Mental creativity and the principle of non-contradiction lead us to mathematical evidence, in a similar way as experimental creativity and observation lead us to physical evidence. Mathematics is the knowledge about the modes and forms of mental operations with numbers.
Paraphrasing the philosopher Moses Mendelssohn one can state: Reality is intelligible or thinkable (for us) because it itself is thought (by God).
Mathematics is the way God shapes our ordinary world in order we can calculate and predict it, and live in comfortably.
Gödel’s theorem is interpreted by Stephen Hawking in this way: “There is not an ultimate theory that can be formulated as a finite number of principles.” (http://www.hawking.org.uk/godel-and-the-end-of-physics.html)
Both physical and virtual reality are real to the extent that they are underpinned by an omniscient mind.
- 2014: On the Axiom of Choice
Flora Dellini, Marco Natale and Francesco Urso, Slideshare.
A brief description of the axiom of choice, with some philosophical considerations.
Alfred Driessen. In: Eeva Martikainen (Ed.) Human approaches to the universe, Luther-Agricola-Society, Helsinki, pp. 66-74
In this contribution an attempt is made to analyze an important mathematical discovery, the theorem of Gödel, and to explore the possible impact on the consistency of metaphysical systems. It is shown that mathematics is a pointer to a reality that is not exclusively subjected to physical laws. As the Gödel theorem deals with pure mathematics, the philosopher as such can not decide on the rightness of this theorem. What he, instead can do, is evaluating the general acceptance of this mathematical finding and reflect on the consistency between consequences of the mathematical theorem with consequences of his metaphysical view.
The findings of three mathematicians are involved in the argumentation: first Gödel himself, then the further elaboration by Turing and finally the consequences for the human mind as worked out by Penrose. As a result one is encouraged to distinguish two different types of intellectual activity in mathematics, which both can be carried out by humans. The astonishing thing is not the distinction between a formalized, logic approach on the one side and intuition, mathematical insight and meaning on the other. Philosophically challenging, however, is the claim that principally only one of these intellectual activities can be carried out by objects exclusively bound to the laws of physical reality.
Alfred Driessen and Antoine Suarez, Editors. Kluwer Academic Publishers, Dordrecht, The Netherlands.
This book offers a series of contributions written by scientists interested in a philosophical reflection on recent advances of science. The reader will find generally understandable presentations of recent results from mathematics, like the theorems of Gödel and Turing, and physics, mostly related to EPR “Gedanken” experiments and Bell’s theorem. In the case of physics, special attention is directed to old and new experiments supporting a nonlocal approach. Especially worth mentioning is the until now unedited contribution of the late John Bell on Bell’s theorem held on 22 January 1990 in a Seminar at CERN.
Profound scientific theorems in modern mathematics and physics shed new light on two fundamental questions often only implicitly dealt with: is mathematical truth a purely man-made construction and is the physical reality behind the phenomena at least in principle always observable? The answers to both questions are closely related to the possible existence of an omniscient and omnipotent being. In this sense mathematical undecidability and quantum nonlocality are proposed as a possible road to metaphysical principles and eventually to God.