What is the relationship between mathematics and literature
What is the relationship between evidence in mathematics and observations in physics?
The linked 1993 paper contains a serious critic of the theoretical physics practices of the time:
This unreliability is certainly a problem in theoretical physics, where primary literature often becomes so irrelevant that it is abandoned on a large scale. IM Singer compared physics literature to a blackboard that has to be erased regularly. Physicists traditionally benefit much less from the historical background of a problem and are less inclined to search the literature. The citation half-life of physics papers is much shorter than in mathematics.
A similar critic applies to the practice of publishing research announcements:
Some areas of the Russian mathematics school have extensive theoretical working traditions, usually carried out through advance research announcements. We will mention only two of the numerous possible examples. [...] 1954 Kolmogorov announced that [...] evidence for the analytical case was [...] provided by Arnold in 1959 and by Moser in 1962 for the smooth case.
[...] In 1973 the respected mathematicians Dobrushin and Minios published an announcement of this result. Two years later, when the Russians had given no hint of evidence, Glimm, Jaffe, and Spencer resumed their work on the problem, eventually giving two different pieces of evidence. A few years later, Dobrushin and Minios issued a retraction of their original announcement.
The paper points to the devastating consequences that similar practices in mathematics had in the past and tries to suggest solutions on how to avoid these consequences and enable theoretical work:
Theoretical work should be explicitly recognized as theoretical and incomplete. In particular, much of the bottom line credit must be reserved for the rigorous work that confirms it.
Only one solution is proposed for research announcements (with authors' opinions being disclosed):
Research announcements should not be published except as summaries of full versions that have been accepted for publication. Citations of unpublished work should clearly distinguish between announcements and full preprints.
I couldn't access the 2015 Hirzebruch lecture in Bonn. It would certainly be interesting to see if Jaffe believes the problematic practices persist or if he is more likely to report the success of his proposed solutions.
While this may not be the answer the questioner was looking for, the question contains a link to the 13-page opinion piece and a link to an unavailable lecture. It should be noted, therefore, that the submission contains strong statements that overshadow possible discussions (even if they may be true) of details of the proposed analogies used in the submission.
Jaffe justifies saying "experimental math" to find evidence for theorems about:
A relevant observation is that most theoretical physicists have a great deal of respect for their experimental counterparts. The relationship between physics and mathematics would be much easier if physicists recognized mathematicians as "intellectual experimenters" rather than scornfully viewing them as uselessly compulsive theorists. The typical attitude of physicists towards mathematics is illustrated by a passage from a book by PW Anderson: "We are talking about theoretical physics here, and so of course mathematical rigor is irrelevant and impossible."
This makes it clear that Jaffe is talking about sociological phenomena here. Since no comparable phenomena exist (or existed) within the mathematical community itself, there was no need to express this kind of feeling through type theorists like Martin-Löf. The close relationship between verifiability, falsifiability and expressiveness on the one hand and experimental observation in physics, accuracy and proof in mathematics, and the absence of metaphysical speculations in philosophy has been clearly expressed by proponents of logical positivism.
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