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Ivor Grattan-Guinness found the effectiveness in question eminently reasonable and explicable in terms of concepts such as analogy, generalization, and metaphor. He emphasizes that Wigner largely ignores "the effectiveness of the natural sciences in mathematics, in that much mathematics has been motivated by interpretations in the sciences".
In mathematics, an extraneous solution (or spurious solution) is one which emerges from the process of solving a problem but is not a valid solution to it. [1] A missing solution is a valid one which is lost during the solution process.
The word proof derives from the Latin probare 'to test'; related words include English probe, probation, and probability, as well as Spanish probar 'to taste' (sometimes 'to touch' or 'to test'), [5] Italian provare 'to try', and German probieren 'to try'.
The seemingly "simple" elementary brain-teaser asks one student "Reasonableness: Marty ate 4/6 of his pizza and Luis ate 5/6 of his pizza. Marty ate more pizza than Luis.
Roberto Poli of McGill University delivered a number of lectures entitled The unreasonable ineffectiveness of mathematics in cognitive sciences in 1999. The abstract is: My argument is that it is possible to gain better understanding of the "unreasonable effectiveness" of mathematics in study of the physical world only when we have understood the equally "unreasonable ineffectiveness" of ...
The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...
Foundations of mathematics are the logical and mathematical framework that allows the development of mathematics without generating self-contradictory theories, and, in particular, to have reliable concepts of theorems, proofs, algorithms, etc.
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