D. Calvetti & E. Somersalo, Springer (2007), is a nice little book, with a remarkable good preface.
Here’s some quotes:
“The nature of mathematics is being exact, and its exactness is underlined by the formalism used by mathematicians to write it. This formalism, characterized by theorems and proofs, and syncopated with occasional lemmas, remarks and corollaries, is so deeply ingrained that mathematicians feel uncomfortable when the pattern is broken, to the point of giving the impression that the attitude of mathematicians towards the way mathematics should be written is almost moralistic. There is a definition often quoted, “A mathematician is a person who proves theorems”, and a similar, more alchemistic one, credited to Paul Erdös, but more likely going back to Alfréd Rényi, stating that “A mathematician is a machine that transforms coffee into theorems” (Footnote: That said, academic mathematics departments should invest on high quality coffee beans and decent coffee makers, in hope of better theorems. As Paul Turán, a third Hungarian mathematician, remarked, “weak coffee is fit only to produce lemmas”.)
Therefore is seems to be the form, not the content, that characterizes mathematics, similarly to what happens in any formal moralistic code wherein form takes precedence over content.“
On the reason why they do not use the formal framework of theorems and lemmas: “when obsessed with a formal language that is void of ambiguities, one loses the capability of expressing emotions exactly, while by liberating the language and making it vague, one creates space for the most exact of all expressions, poetry. Thus the exactness of expression is beyond the language. We feel the same way about mathematics.”
“Mathematics is a wonderful tool to express liberally such concepts as qualitative subjective beliefs, but by trying to formalize too strictly how to express them, we may end up creating beautiful mathematics that has a life of its own, in its own academic environment, but which is completely estranged to what we initially set forth.”
It cites Peter Lax on the occasion of his receiving the 2005 Abel Prize: “When a mathematician says he has solved the problem he means he knows the solution exits, that it’s unique, but very often not much more.”
“By following the principle of exclusion and subjective learned opinions, we effectively narrow down the probability distributions of the model parameters so that the model produced plausible results. This process is cumulative: when new information arrives, old information is not rejected, as is often the case in the infamous “model fitting by parameter tweaking”, but included as prior information. This mode of building models is not only Bayesian, but also Popperian in the wide sense: data is used to falsify hypotheses thus leading to the removal of impossible events or to assigning them as unlikely, rather than to verify hypotheses, which is in itself a dubious project. (Footnote: See A. Tarantola: Inverse problems, Popper and Bayes, Nature Physics 2, 492-492, (2006)) As the classic philosophic argument goes, producing one white swan, or , for that matter, three, does not prove the theory that all swans are white.”