On Conjecture (1900-1949)

"It is best to prove things by actual experiment; then you know; whereas if you depend on guessing and supposing and conjecturing, you will never get educated."  (Samuel L Clemens [Mark Twain], "Eve’s Diary", 1906)

"The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules.[...] One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question that can at all be formally expressed in these systems. It will be shown below that this is not the case, that on the contrary there are in the two systems mentioned relatively simple problems in the theory of integers that cannot be decided on the basis of the axioms." (Kurt Gödel, "On Formally Undecidable Propositions of Principia Mathematica and Related Systems", 1931)

"An inference, if it is to have scientific value, must constitute a prediction concerning future data. If the inference is to be made purely with the help of the distribution theory of statistics, the experiments that constitute evidence for the inference must arise from a state of statistical control; until that state is reached, there is no universe, normal or otherwise, and the statistician’s calculations by themselves are an illusion if not a delusion. The fact is that when distribution theory is not applicable for lack of control, any inference, statistical or otherwise, is little better than a conjecture. The state of statistical control is therefore the goal of all experimentation." (William E Deming, "Statistical Method from the Viewpoint of Quality Control", 1939)

"To say that mathematics in general has been reduced to logic hints at some new firming up of mathematics at its foundations. This is misleading. Set theory is less settled and more conjectural than the classical mathematical superstructure than can be founded upon it." (Willard Van Orman Quine, "Elementary Logic", 1941)

"[…] analogy [is] an important source of conjectures. In mathematics, as in the natural and physical sciences, discovery often starts from observation, analogy, and induction. These means, tastefully used in framing a plausible heuristic argument, appeal particularly to the physicist and the engineer." (George Pólya, "How to solve it", 1945) 

"Inference by analogy appears to be the most common kind of conclusion, and it is possibly the most essential kind. It yields more or less plausible conjectures which may or may not be confirmed by experience and stricter reasoning." (George Pólya, "How to Solve It", 1945)