Paul Isaac Bernays (17 October 1888 – 18 September 1977) was a Swiss mathematician, who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant and close collaborator of David Hilbert.
- Enlightened humanity has sought in rational deﬁniteness its liberating refuge from the dominating inﬂuence of the merely authoritative. At the present time, however, this has for a large part been lost to consciousness, and to many people scientific validity that has to be acknowledged appears as an oppressing authority.
- Bernays, Paul. "Comments on Ludwig Wittgenstein's Remarks on the foundations of mathematics." Ratio 2.1 (1959): 1-22.
- I shall now address you on the subject of the present situation in research in the foundations of mathematics. Since there remain open questions in this field, I am not in a position to paint a definitive picture of it for you. But it must be pointed out that the situation is not so critical as one could think from listening to those who speak of a foundational crisis. From certain points of view, this expression can be justified; but it could give rise to the opinion that mathematical science is shaken at its roots.
- Paul Bernays, Platonism in mathematics (1935) Lecture delivered June 18, 1934, in the cycle of Conferences internationales des Sciences mathematiques organized by the University of Geneva, in the series on Mathematical Logic.) Translation by: Charles Parsons
- The truth is that the mathematical sciences are growing in complete security and harmony. The ideas of Dedekind, Poincare, and Hilbert have been systematically developed with great success, without any con ict in the results. It is only from the philosophical point of view that objections have been raised. They bear on certain ways of reasoning peculiar to analysis and set theory. These modes of reasoning were first systematically applied in giving a rigorous form to the methods of the calculus. [According to them,] the objects of a theory are viewed as elements of a totality such that one can reason as follows: For each property expressible using the notions of the theory, it is [an] objectively determinate [fact] whether there is or there is not an element of the totality which possesses this property. Similarly, it follows from this point of view that either all the elements of a set possess a given property, or there is at least one element which does not possess it
- Paul Bernays, Platonism in mathematics (1935)
- Bernays's publications extend over the most diverse fields of mathematics … and are all marked by thoroughness and reliability … He is distinguished by a deep-seated love for science as well as a trustworthy character and nobility of thought, and is highly valued by everyone. In all matters concerning fundamental questions in mathematics, he is the most knowledgeable expert and, especially for me, the most valuable and productive colleague.
- P. Bernays has pointed out on several occasions that, since the consistency of a system cannot be proved using means of proof weaker than those of the system itself, it is necessary to go beyond the framework of what is, in Hilbert’s sense, finitary mathematics if one wants to prove the consistency of classical mathematics, or even that of classical number theory. Consequently, since finitary mathematics is defined as the mathematics in which evidence rests on what is intuitive, certain abstract notions are required for the proof of the consistency of number theory.... In the absence of a precise notion of what it means to be evident, either in the intuitive or in the abstract realm, we have no strict proof of Bernays’ assertion; practically speaking, however, there can be no doubt that it is correct...
- Kurt Gödel (1958, CW II, p. 241) as cited in: Feferman, Solomon. "Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on finitism, constructivity and Hilbert's program*." dialectica 62.2 (2008): 179-203.