"It is common sense to take a method and try it. If it fails, admit it frankly and try another. But above all, try something." (Franklin D Roosevelt, Address at Oglethorpe University, 1932)
"The more difficult and novel the problem, the greater is likely to be the amount of trial and error required to find a solution. At the same time, the trial and error is not completely random or blind; it is, in fact, rather highly selective. The new expressions that are obtained by transforming given ones are examined to see whether they represent progress toward the goal. Indications of progress spur further search in the same direction; lack of progress signals the abandonment of a line of search. Problem solving requires selective trial and error." (Herbert A Simon, "The Sciences of the Artificial". 1969)
"We know the laws of trial and error, of large numbers and probabilities. We know that these laws are part of the mathematical and mechanical fabric of the universe, and that they are also at play in biological processes. But, in the name of the experimental method and out of our poor knowledge, are we really entitled to claim that everything happens by chance, to the exclusion of all other possibilities?" (Albert Claude, [Nobel Prize Lecture], 1974)
"Mathematics is not a deductive science - that’s a cliché. When you try to prove a theorem, you don’t just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork." (Paul R Halmos, "I Want to Be A Mathematician", 1985)
"Heuristics are rules of thumb that help constrain the problem in certain ways (in other words they help you to avoid falling back on blind trial and error), but they don't guarantee that you will find a solution. Heuristics are often contrasted with algorithms that will guarantee that you find a solution - it may take forever, but if the problem is algorithmic you will get there. However, heuristics are also algorithms." (S Ian Robertson, "Problem Solving", 2001)
“Another feature of Bourbaki is that it rejects intuition of any kind. Bourbaki books tend not to contain explanations, examples, or heuristics. One of the main messages of the present book is that we record mathematics for posterity in a strictly rigorous, axiomatic fashion. This is the mathematician’s version of the reproducible experiment with control used by physicists and biologists and chemists. But we learn mathematics, we discover mathematics, we create mathematics using intuition and trial and error. We draw pictures. Certainly, we try things and twist things around and bend things to try to make them work. Unfortunately, Bourbaki does not teach any part of this latter process.” (Steven G Krantz, “The Proof is in the Pudding: The Changing Nature of Mathematical Proof”, 2010)
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