The Elements has been considered as offering models of scrupulously rigorous demonstrations. It is certainly true that in point of rigour it compares favourably with its modern rivals; but when examined in the light of strict mathematical logic, it has been pronounced by C.S. Peirce to be "riddled with fallacies." The results are correct only because the writer's experience keeps him on his guard.
Florian CajoriRelated topics
correct experience guard light point rigour keepsRelated quotes
The true function of logic ... as applied to matters of experience ... is analytic rather than constructive; taken a priori, it shows the possibility of hitherto unsuspected alternatives more often than the impossibility of alternatives which seemed prima facie possible. Thus, while it liberates imagination as to what the world may be, it refuses to legislate as to what the world is.
Bertrand Russell
One could say that what differentiates ancient from modern philosophy is the fact that, in ancient philosophy, it was not only Chrysippus or Epicurus who, just because they had developed a philosophical discourse, were considered philosophers. Rather, every person who lived according to the precepts of Chrysippus or Epicurus was every bit as much a philosopher as they.
Pierre Hadot
I would take school instruction out of the hands of the old order of decrepit, stammering, journeymen-teachers as well as from the new weak ones, who are generally no better for popular instruction, and entrust it to the undivided powers of Nature herself, to the light that God kindles and ever keeps alive in the hearts of fathers and mothers, to the interest of parents who desire that their children should grow up in favour with God and man.
Johann Heinrich Pestalozzi
I am surprised to see that today everything that does not amount to surrender pure and simple to generalized capitalism, let us call it thus, is considered to be archaic or old-fashioned, as though in a way there existed no other definition of what it means to be modern than, quite simply, to be at all times caught in the dominant forms of the moment.
Alain Badiou
That this subject [of imaginary magnitudes] has hitherto been considered from the wrong point of view and surrounded by a mysterious obscurity, is to be attributed largely to an ill-adapted notation. If for instance, +1, -1, √-1 had been called direct, inverse, and lateral units, instead of positive, negative, and imaginary (or even impossible) such an obscurity would have been out of question.
Carl Friedrich Gauss