Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken th...e basis on which I intended to build my arithmetic.... It is all the more serious since, with the loss of my rule V, not only the foundations of my arithmetic, but also the sole possible foundations of arithmetic seem to vanish.LESSATTRIBUTION DETAIL »
It is natural, now, to think of there being connected with a sign (name, combination of words, letter) besides that to which the s...ign refers, which may be called the reference of the sign, also what I should like to call the sense of the sign, wherein the mode of presentation is contained. In our example, accordingly, [an example in which lines a, b, and c all intersect at a single point] the reference of the expressions 'the point of intersection of a and b' and 'the point of intersection of b and c' would be the same, but not their senses. The reference of 'evening star' would be the same as that of 'morning star,' but not the sense.LESSATTRIBUTION DETAIL »
I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequentl...y a priori. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction.LESSATTRIBUTION DETAIL »