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Bishop Berkeley, prompted by the irreligion of Newton and his sponsor Edmund Halley, attacked certain illogicalities, notably circular argument, in Newton's presentation of the differential calculus; his criticisms exercised mathematicians for over a century. The following quotation is from The Analyst:
It must, indeed, be acknowledged that he used fluxions, like the scaffold of a building, as things to be laid aside or got rid of as soon as finite lines were found proportional to them. But then these finite exponents are found by the help of fluxions. Whatever therefore is got by such exponents and proportions is to be ascribed to fluxions: which must therefore be previously understood. And what are these fluxions? The velocities of evanescent increments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities? [Berkeley, 1734, p. 130]
Newton's formulation of the differential calculus is a transformation in K. “The ghosts of departed quantities” expresses the negative of the column 1 dimension of his formulation. The transformation in K is effected by discarding the ‘scaffolding’ of fluxions, “the ghosts of departed quantities”. The discarding of the scaffolding may be regarded as a step to achieve finite lines “proportional to them”, a category H3 formulation; or, “the finite lines … proportional to them” may be regarded as an F3 formulation used as a column 2 formulation to prevent emergence of the “ghosts of departed quantities” and the psychological turbulence that such an emergence would precipitate; Newton did have what we would today regard as a psychotic breakdown in which, in his own words, he lost “the former consistency of
his mind“ and from which he emerged, according to J. M. Keynes, ”slightly ‘gaga‘“. Keynes's paper, which was read by his brother at the Centenary Celebrations held in July 1946, contains material which wi