“The main disadvantage of a system of symbolic logic is that it is a formal system divorced from intuition.” --- J. Barkley Rosser, Logic, (1978, , p. 10) .
“One advantage of a symbolic logic is that it can be made very precise, but an even greater advantage is that it can be changed to fit the circumstances.”
--- J. Barkley Rosser, Logic, (1978, , p. 522).
As promised at the beginning of this paper, we now arrive at the discussion Velupillai started his paper with, a consideration of the views of my late father, J. Barkley Rosser [Sr.], on these matters as best I understand them. based both on reading his work and remembering discussions with him (he died in 1989), with some of the more revealing discussions involving his close friend and long associate, Stephen C. Kleene as well 14. In this discussion we shall deal with the deeper foundational issues.. Velupillai suggests that Rosser was an ally of the constructivist position, and I have no disagreement whatsoever with his interpretation of the extension of the incompleteness theorem of Gödel made by Rosser in 1936  as supporting the “classic trade-off that great mathematicians consciously make between weakening/strengthening conclusions and weakening/strengthening hypotheses” (. p. 4). Furthermore, although as time proceeds it is for his earlier more pure, logic work that he is mostly remembered, Rosser was a serious applied mathematician later on, indeed always had this orientation to some degree as he had a masters degree in physics before he earned his Ph.D. in mathematics at Princeton under Alonzo Church.15 That background would manifest itself during World War II when he was perhaps the leading expert on rocket ballistics in the U.S.,16 and his first book was The Mathematical Theory of Rocket Flight  (still in print), which reflected his deep fascination with the exploration of space by humans,17 later shown by his deep involvement in the space program of the U.S.18 Furthermore, this deep pragmatism as reflected in the quotation above regarding the importance of being able to “be changed to fit the circumstances” suggests that he may have been open to any viewpoint as long as it could satisfy this ultimate criterion.
Without question he was a deep student of the controversies regarding the nature of the foundations of mathematics. His acute awareness of these debates is given by the following summary of them by him regarding the Axiom of Choice.
“X’s position: Any finite number of choices is permissible, but not an infinite number.
Y’s position: A denumerable number of choices is permissible, but not for any larger number.
Z’s position: Any number of choices is permissible.
Among actual mathematicians, perhaps a majority would applaud X’s position, certainly some would agree with Y, and others (who perhaps constitute a minority) agree wholeheartedly with Z. However, many mathematicians who would like to espouse the positions of X or Y find that this would leave them with no means of proof for certain theorems which they need in their research. Accordingly, they accept the position of Z, but with reluctance and a hope that someone will one day find proofs of their key theorems which do not involve an infinity of choices. Thus it comes about that we find papers written in which all the results of the paper depend on a theorem whose only known proof involves an infinity of choices; nevertheless, throughout the paper the author is careful to avoid the use of an infinity of choices.” --- Rosser, (Logic, 1978 , p. 491).
Velupillai and his allies in the program to constructivize the foundations of mathematical economics have argued precisely this last point at some length, that mathematical economist after mathematical economist has sought to prove the existence of general equilibrium and other crown jewels of economic theory, while assiduously avoiding mentioning the fact that their proofs are relying on unmentioned axioms that are not at all agreed upon universally. In this regard Rosser certainly had sympathy with these constructivist arguments, and his extensive research in numerical analysis and computer science more generally reinforces this perception.
However, we are now coming to the more difficult point of ascertaining his real position, and here I shall report my own personal observations of him. Something that makes things difficult is that while he often had strong views regarding things, he also tended to avoid specifically declaring his position, indeed enjoyed not doing so. To make matters worse, he would sometimes play the “devil’s advocate” and brilliantly argue for a position he disagreed with just to provoke his interlocuters.19 In the area of politics he strongly asserted the right of the citizen to secrecy of the ballot, and I never heard him actually state whom he had voted for in any election, although I generally had a good idea whom he favored.
There is a non-trivial relationship between these attitudes and the implications of the Gödel theorems with which he was so deeply associated. So within a system, both a statement and its opposite may be able to be proven. One response to this might well be intuitionism: overturning the law of the excluded middle can allow us to assert that both are true. However, another view, implied by the quote by Gödel at the end of Velupillai’s paper, is that there may be a definitely correct answer, a definite truth, but that it can only be known by moving to a “higher type,” that at a higher level of perspective the answer can be known. It was his hard pragmatism that inclines me to the view that he was not a post-modernist with regard to reality: he had a very hard appreciation of the very hard reality of the very hard facts that make themselves unavoidable to us, even if there are higher order things that we do not know or perceive whose reality or lack thereof remains a mystery.
So, while he never expressed his bottom line opinion on these matters, I have a sense of where he stood on some of them, especially compared to his old and close friend, the late Steve Kleene. I think that Kleene was more the constructivist, indeed even an intuitionist, whereas my father was in his heart of hearts a classical mathematician who accepted the law of the excluded middle, and who agreed with Hardy that mathematical reality truly exists outside of us. Let me deal with these issues in order.
Regarding the law of the excluded middle, one piece of evidence, exhibited throughout his writings, is that he had no hesitation in using reductio ad absurdum arguments. I think that he viewed it as a matter of beauty implying truth, with a good proof by contradiction being the highest form of mathematical aestheticism. When I was 13, he sat me down one day and worked through the proof that there is no highest level of infinity. This is a diagonal proof by contradiction in which one assumes that one has found a set that possess the highest possible level of transfinite cardinality, and then one examines its power set, the set of all its subsets, and shows that one cannot establish a one-to-one correspondence between this set and the one that was assumed to be at the highest level of infinity, with the failure of the one-to-one correspondence amounting to there always being at least one member of the power set that is left over not being associated with a member of the original set, just as in Cantor’s proof that the real number continuum is at a higher level of infinity than the denumerable natural numbers.
In this he contrasted with his friend Kleene. At a certain point I became fascinated by arguments regarding Zen koans and the possible reality of apparent contradictions holding simultaneously. When I asked my father about this he told me to ask Steve Kleene about it. When I did, Kleene told me about intuitionism, the first time I heard about it. Now, while my father makes frequent references to “intuition” in his works, he never used the term “intuitionism” that I am aware of. I think that in the end he simply did not buy it, although I suspect that it also involved the view of Keats, and that it was fundamentally an aesthetic judgment: he found a good proof by contradiction to be supremely elegant and beautiful, and intuitionism rules them out.
The question reappears implicitly in his 1969 book, Simplified Independence Proofs . There he redoes the forcing proof by Paul Cohen  of the independence of the continuum hypothesis into Boolean algebraic terms. As noted earlier, like fuzzy logic, Boolean algebra admits of truth values that are not just one or zero. This would follow the approach developed by Rosser and Turquette  in their work on multiple-valued logic. This looks on the surface like it could be consistent with the possible intuitionistic assertion that A may be both true and not true at the same time. But this is not the case in the original formulations. The Boolean formulation can sometimes be interpreted as asserting probabilistic truth, as in the many-valued logic.20 Thus the statement that “there is a 30 percent chance of rain today” can be viewed as an Aristotelian assertion: it may be definitely true or false (although it may not be easy to establish if it is or not, as the appearance of rain today does not disprove it). Booleanism and Aristotelianism are perfectly consistent, if considered within the penumbra of many-valued logic.21
Another piece of this is the matter of whether or not one “invents” or “discovers” theorems. Are theorems “constructed” (invented), or are they expositions about a real mathematical reality outside of us that the mathematician discovers? I can report that my father always used the latter terminology: for him, for all his willingness to consider many different possible systems of logic and mathematics, theorems were “discovered.” They are sitting out there somewhere for the brilliant mathematician to find them. Thus, while he never expressed a per se opinion on the matter, I think that he ultimately agreed with Hardy that mathematical reality is a real Platonic reality.22
Yet another piece of evidence is that he was quite an admirer of nonstandard analysis, supporting the tenuring of Robinson’s strong ally, H. Jerome Keisler, in the mathematics department at the University of Wisconsin-Madison (Kleene also was involved as well, initially hiring Keisler when he was department chairman,). In the second edition of his Logic for Mathematicians (which was simply entitled, Logic for the second edition), he added several appendices . The final one (D) is about nonstandard analysis and is quite sympathetic to its arguments. In the next to last paragraph of the entire book (, p. 560), he cites its ability to provide an easy solution for the Dirac delta function, and in the final paragraph he approvingly cites Keisler’s book, Elementary Calculus  which Errett Bishop had strongly criticized . Of course, as noted above, Hellman and others see nonstandard analysis as ultimately consistent with a constructivist approach,23 and Abraham Robinson himself fell back on a pragmatic approach, ultimately arguing that the ontological status of infinitesimals was meaningless, even as they might be useful, if only for understanding the history of mathematics.
However, in the end, I think that for my father it all came together. I believe that he agreed with “Z” above from his Logic, but like Hellman he saw a consistency between the pragmatic and computable aspect of constructivism in its broader perspective with there being a real existence of a higher mathematical reality that includes infinitesimals and transfinite cardinals, even inaccessible ones. More than once I heard him repudiate Hardy’s clam that “real” mathematics was ultimately “useless,” noting that theorems that he proved when he was young and thought to be strictly part of totally pure mathematics, turned out later to have practical applications. Thus, I think he believed in the ultimate unification of the Platonic ideal with the Aristotelian material reality somehow. If there was a deity for him, that being was somehow connected with that level of inaccessible transinfinity that he seemed to be trying to reveal to me without saying it when he showed me the proof that there is no highest level of infinity, when I reached the age that a young man is supposed to decide his religious identity, but which he may have seen as also manifesting itself somehow in the most concrete reality that we live in.