"Allow us now a hypothesis that is arbitrary but not self-contradictory. One might encounter instances where using a function without a derivative would be simpler than using one that can be differentiated. When this happens, the mathematical study of irregular continua will prove its practical value." (Jean-Baptiste Perrin, 1906)
"Consider, for instance, one of the white flakes that are obtained by salting a solution of soap. At a distance its contour may appear sharply defined, but as we draw nearer its sharpness disappears. The eye can no longer draw a tangent at any point. A line that at first sight would seem to be satisfactory appears on close scrutiny to be perpendicular or oblique. The use of a magnifying glass or microscope leaves us just as uncertain, for fresh irregularities appear every time we increase the magnification, and we never succeed in getting a sharp, smooth impression, as given, for example, by a steel ball. So, if we accept the latter as illustrating the classical form of continuity, our flake could just as logically suggest the more general notion of a continuous function without a derivative." (Jean-Baptiste Perrin, 1906)
"If, to go further, we [...] attribute to matter the infinitely granular structure that is in the spirit of atomic theory, our power to apply to reality the rigorous mathematical concept of continuity will greatly decrease." (Jean-Baptiste Perrin, 1906)
"It must be borne in mind that, although closer observation of any object generally leads to the discovery of a highly irregular structure, we often can with advantage approximate its properties by continuous functions. Although wood may be indefinitely porous, it is useful to speak of a beam that has been sawed and planed as having a finite area. In other words, at certain scales and for certain methods of investigation, many phenomena may be represented by regular continuous functions, somewhat in the same way that a sheet of tinfoil may be wrapped round a sponge without following accurately the latter's complicated contour." (Jean-Baptiste Perrin, 1906)
"Mathematicians, however, are well aware that it is childish to try to show by drawing curves that every continuous function has a derivative. Though differentiable functions are the simplest and the easiest to deal with, they are exceptional. Using geometrical language, curves that have no tangents are the rule, and regular curves, such as the circle, are interesting but quite special." (Jean-Baptiste Perrin, 1906)
"The fundamental laws of chemistry which are well known to you and which are laws of discontinuity (discontinuity between chemical species, and discontinuous variation according to the 'multiple proportions' in the composition of species made from the same simple bodies) then become immediately clear: they are imposed solely by the condition that the molecule constituting a compound contains a necessarily whole number of atoms of each of the simple bodies combined in this compound." (Jean-Baptiste Perrin, "Discontinuous Structure of Matter", [Nobel lecture] 1926)
"We are, finally, forced to think that each grain only follows the portion of liquid surrounding it, in the same way that an indicating buoy indicates and analyses the movement all the better if it is smaller: a float follows the movement of the sea more faithfully than a battleship. We obtain from this an essential property of what is called a liquid in equilibrium: its repose is only an illusion due to the imperfection of our senses, and what we call equilibrium is a certain well-defined permanent system of a perfectly irregular agitation. This is an experimental fact in which no hypothesis plays any part." ("Discontinuous Structure of Matter", [Nobel lecture] 1926)
"It is thus that statistics reveals more and more the inconstancy and the irregularity of much social phenomena, when in lieu of applying it to a great nation altogether, one descends to a province, a town, a village." (Jean-Baptiste Perrin)
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