Abstract
English translation of Moritz Pasch,“Der Begriff des Differentials,” Annalen der Philosophie und philosophischen Kritik 4 (1924), pp. 161–187. Hans Vaihinger (co-founder of the Annalen der Philosophie) argued that various mathematical theories cannot be true because close inspection shows them to be absurd. Since Vaihinger recognized the utility of these theories, he confronted the problem of explaining how absurd theories can be useful. Pasch insists there is no need for such an explanation because the theories Vaihinger found absurd are, in fact, logically impeccable. To support this claim, Pasch offers a quick overview of the foundations of differential calculus and a more detailed analysis of Fermat’s rule for maxima and minima.
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- 1.
See [3], p. 305.
- 2.
See the explanation in §13 of [2].
- 3.
[We have
$$\forall x\,\forall\varepsilon>0\,\exists\delta>0\,\forall x_1(|x_1-x|<\delta\longrightarrow 5\cdot|x_1-x|\cdot|2x+x_1|<\varepsilon).$$For example, if \(\varepsilon\leq1\leq x\), we can let
$$\delta=\frac{\varepsilon}{30x}.$$Pasch has shown how to construct counter-examples to the stronger condition that
$$\forall\varepsilon>0\,\exists\delta>0\,\forall x,x_1(|x_1-x|<\delta\longrightarrow5\cdot|x_1-x|\cdot|2x+x_1|<\varepsilon).$$By way of contrast, Pasch earlier noted that
$$\forall\varepsilon>0\,\exists\delta>0\,\forall t,t_1(|t_1-t|<\delta\longrightarrow|a|\cdot|t_1-t|<\varepsilon)$$since we can let
$$\delta=\frac{\varepsilon}{|a|}.\big]$$ - 4.
Here ∞ is neither positive nor negative. It is the “indefinite infinity.” So, here, \(-\infty=\infty\); and \(\infty+\infty\) is just as illegitimate as \(\infty-\infty\).
- 5.
[Suppose \(\varepsilon =0.0001\) and \(x=1\). Then any positive number less than 0.000006 is “sufficiently small” and, hence, is a permissible value of dx. So any positive number less than 0.00009, is a possible value of \(d_xy\;(=15dx)\). On the other hand, if \(x=0\), the only possible value of d xy is 0.]
- 6.
So Vaihinger is quite wrong to insist that, in his notation, I=II “is only possible if \(e=0\).”
- 7.
1, p. 66.
- 8.
[Pasch is referring to two figures not reproduced here. The first depicts a curve whose slope is negative everywhere except for an inflection point M where it is 0. The second depicts a curve whose slope is negative everywhere. On this second curve, M is an arbitrary point. In both figures, M has coordinates \((x,f(x))\), while Q has coordinates \((x,f(x+e)\).]
References
Fermat, Pierre de. 1861. Varia Opera Mathematica. Berlin: Springer.
Pasch, Moritz. 1882. Introduction to differential and integral calculus. Leipzig: B.G. Teubner.
Pasch, Moritz. 1915. Bounds and limits. Monatshefte für Mathematik und Physik 26:303–308.
Vaihinger, Hans. 1918. The philosophy of ‘As If’, 3rd edn. Berlin: Reuther & Reichard.
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Pollard, S. (2010). The Concept of the Differential. In: Pollard, S. (eds) Essays on the Foundations of Mathematics by Moritz Pasch. The Western Ontario Series in Philosophy of Science, vol 83. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9416-2_10
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