Fidelity in Mathematical Discourse: Is One and One Really Two?. Davis, P. J. 79(3):252–263.
Fidelity in Mathematical Discourse: Is One and One Really Two? [link]Paper  doi  abstract   bibtex   
[Excerpt] "I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable." [\n] BERTRAND RUSSELL, Portraits from Memory. [\n] [...] [Conclusions] Symbols and operations do not have a precise meaning, but only a probabilistic meaning. [\n] A derivation of a theorem or a verification of a proof has only probabilistic validity. It makes no difference whether the instrument of derivation or verification is man or a machine. The probabilities may vary, but are roughly of the same order of magnitude when compared with cosmic probabilities. [\n] Mathematics has some of the aspects of an experimental science. We are saved from chaos by the stability of the universe which implies the repeatability of ex- periments and the self-correcting features of usage. [\n] Mathematics has been Platonic for years. Does this rob it of a certain freedom and vitality which might be obtained by openly recognizing its probabilistic nature? [\n] It is possible that a new type of mathematics might develop in which the "derivations" or the "processes" are so enormously long that the probabilistic nature of the result will be an integral feature of the subject. [\n] [...]
@article{davisFidelityMathematicalDiscourse1972,
  title = {Fidelity in Mathematical Discourse: Is One and One Really Two?},
  author = {Davis, Philip J.},
  date = {1972-03},
  journaltitle = {The American Mathematical Monthly},
  volume = {79},
  pages = {252--263},
  issn = {0002-9890},
  doi = {10.2307/2316620},
  url = {https://doi.org/10.2307/2316620},
  abstract = {[Excerpt]

"I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable." [\textbackslash n] BERTRAND RUSSELL, Portraits from Memory.

[\textbackslash n] [...]

[Conclusions] Symbols and operations do not have a precise meaning, but only a probabilistic meaning. 

[\textbackslash n] A derivation of a theorem or a verification of a proof has only probabilistic validity. It makes no difference whether the instrument of derivation or verification is man or a machine. The probabilities may vary, but are roughly of the same order of magnitude when compared with cosmic probabilities.

[\textbackslash n] Mathematics has some of the aspects of an experimental science. We are saved from chaos by the stability of the universe which implies the repeatability of ex- periments and the self-correcting features of usage.

[\textbackslash n] Mathematics has been Platonic for years. Does this rob it of a certain freedom and vitality which might be obtained by openly recognizing its probabilistic nature?

[\textbackslash n] It is possible that a new type of mathematics might develop in which the "derivations" or the "processes" are so enormously long that the probabilistic nature of the result will be an integral feature of the subject.

[\textbackslash n] [...]},
  keywords = {*imported-from-citeulike-INRMM,~INRMM-MiD:c-7205687,computational-science,epistemology,mathematical-reasoning,mathematics,semantics,software-errors,software-uncertainty},
  number = {3}
}
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