"I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half proof = 0, and it is demanded for proof that every doubt becomes impossible." (Carl Friedrich Gauss, [letter to Friedrich Bessel] 1830)
"These sciences, Geometry, Theoretical Arithmetic and Algebra, have no principles besides definitions and axioms, and no process of proof but deduction; this process, however, assuming a most remarkable character; and exhibiting a combination of simplicity and complexity, of rigour and generality, quite unparalleled in other subjects." (William Whewell, "The Philosophy of the Inductive Sciences", 1840)
"Logic does not pretend to teach the surgeon what are the symptoms which indicate a violent death. This he must learn from his own experience and observation, or from that of others, his predecessors in his peculiar science. But logic sits in judgment on the sufficiency of that observation and experience to justify his rules, and on the sufficiency of his rules to justify his conduct. It does not give him proofs, but teaches him what makes them proofs, and how he is to judge of them." (John Stuart Mill, "A System of Logic, Ratiocinative and Inductive: Being a Connected View of the Principles of Evidence, and the Methods of Scientific Investigation", 1843)
"Logic does not pretend to teach the surgeon what are the symptoms which indicate a violent death. This he must learn from his own experience and observation, or from that of others, his predecessors in his peculiar science. But logic sits in judgment on the sufficiency of that observation and experience to justify his rules, and on the sufficiency of his rules to justify his conduct. It does not give him proofs, but teaches him what makes them proofs, and how he is to judge of them." (John Stuart Mill, "A System of Logic, Ratiocinative and Inductive: Being a Connected View of the Principles of Evidence, and the Methods of Scientific Investigation", 1843)
"But a thousand unconnected observations have no more value, as a demonstrative proof, than a single one. If we do not succeed in discovering causes by our researches, we have no right to create them by the imagination; we must not allow mere fancy to proceed beyond the bounds of our knowledge." (Justus von Liebig, "The Lancet", 1844)
"No step should ever be taken, and no statement should ever be made, without giving a legitimate reason therefor. A 'legitimate' reason means a reference to a definition, to an assumption, or to a proposition previously proved; it may also include reference to a statement in the hypothesis, or to a line arbitrarily drawn at the outset of the demonstration." (William L Schaaf, "Plane and solid geometry for home study", 1944)
"Whenever we discuss a geometric statement in order to prove it, we refer to previously proved statements for 'evidence'. In this process of referring back to earlier statements that have already been established by proof, it is clear that some few of the earliest statements must of necessity remain unproved - that is, they have to be taken for granted, since we must begin our chain of reasoning somewhere. It is not possible to prove every statement. Those statements which are thus taken for granted without proof are called assumptions. Hence in all logical reasoning, two considerations are of prime importance: (1) the meanings of the terms used, and (2) the basic assumptions made." (William L Schaaf, "Plane and solid geometry for home study", 1944)
"Dirichlet alone, not I, nor Cauchy, nor Gauss knows what a completely rigorous mathematical proof is. Rather we learn it first from him. When Gauss says that he has proved something, it is very clear; when Cauchy says it, one can wager as much pro as con; when Dirichlet says it, it is certain."
"The Higher Arithmetic presents us with an inexhaustible storehouse of interesting truths - of truths, too, which are not isolated but stand in the closest relation to one another, and between which, with each successive advance of the science, we continually discover new and sometimes wholly unexpected points of contact. A great part of the theories of Arithmetic derive an additional charm from the peculiarity that we easily arrive by induction at important propositions which have the stamp of simplicity upon them but the demonstration of which lies so deep as not to be discovered until after many fruitless efforts; and even then it is obtained by some tedious and artificial process while the simpler methods of proof long remain hidden from us." (Carl F Gauss, [introduction to Gotthold Eisenstein’s "Mathematische Abhandlungen"] 1847)
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