# Why not consider infinity and zero to be reciprocal

## Analysis in Hegel

### Summary

The second edition of Hegels *Science of logic* (1832) contains three remarks on the infinity of quantum, in which the various justifications of differential and integral calculus are dealt with. In so far as these three remarks go back to one in the first edition (1812), they alone show Hegel's ongoing preoccupation with this problem. He not only gains an overview of the various methods from the early days of analysis, but can also refer to the mathematical and philosophical problems that arise in each case. For Hegel, however, it is crucial that the essential objects of analysis such as infinity and limit are in themselves contradictory, so that they cannot be grasped analytically, but solely through the concept as a unity of opposites.

### Notes

- 1.
“Une théorie claire et précise de ce qu′on appelle Infini en Mathématique” was requested, see the note in [GW 7, 369]. The first number refers to the volume by Hegel,

*Collected Works*[19], the second number on the side. For the first book of*logic*[GW 11] the page numbers are given after the first edition (1812) [18] so that they can also be read. - 2.
With the title

*The objective logic*,*The doctrine of being*. The second book contains*The doctrine of essence*(Nuremberg 1813). - 3.
One result of this independence is an autonomous, abstract mathematics, which was practiced as pure mathematics in the 19th century, especially by the Berlin mathematical school, which was also inspired by Hegel in its way of thinking, see H. Boehme [2].

- 4.
See [1, 233a21f, 239b9f].

- 5.
[GW 11, 219]. See.

*Ethica*I, Prop. 8, Schol. 1 [33, Vol. 2, 7 f.]. - 6.
[GW 11, 75]. See.

*Epistole*L [33, Vol. 6, 210] - 7.
*Ethica*I, Prop. 15, Schol. [33, Vol. 2, 19]. - 8.
[33, Vol. 6, 12th letter, 51 f.] In the original:

*Infinitum actu negarunt*[32, Vol. 4, 59]. - 9.
[30, 524], 29th letter (old census),

*Van de Natuur van’t Oneindig*. It says, "de ruimte, tussen de twee kringen AB gestelt". The figure also appears in all editions, [31–33], but has been used since*Opera Posthuma*the figure described with "spatii duobus circulis AB, & CD, interpositi" [31, 59] cf.*Epistole*XII [32, vol. 4, 59]. But this is nonsensical because*A.*and*B.*are already two circles according to FIG. 3; d. H.*CD*was added by mistake in connection with the rest of the text, because there are called “de grootste ruimte AB, en de smallest CD” [30], or “maximum nempe AB, minimum verò CD” [31], so that the letters*A.*and*B.*first for circles and then for endpoints of lines, what for*C.*and*D.*should probably apply as well. - 10.
[32, Vol. 1, 198], see [33, Vol. 4, 71]. This figure also appears on the title page of the book in 1663, facsimile in [32, vol. 1, 125].

- 11.
This corresponds to the continuity equation for incompressible flows.

- 12.
[10, Pars II, XXXIII, p. 59], also in [11, 47].

- 13.
Here, too, Spinoza orients himself to Descartes, who first describes the movement of matter in a perfect circular ring, "so that no emptiness and no dilution or compression is necessary" [11, 46].

- 14.
[32, Vol. 1, 199], see [33, Vol. 4, 71]. This in turn results in FIG. 4, which accordingly emerged from FIG. 2. The remark in [GW 11, 428] that Hegel had identified these figures by mistake should therefore itself be an error.

- 15.
[1, 233b25], this definition by Aristotle would be contemporary for Spinoza.

- 16.
Accordingly, Spinoza sets in

*Ethics*presupposes that an infinite amount follows from movement and rest [33, Vol. 2, 34]. - 17.
It should be noted that Hegel elsewhere calculates an area using the

*Summation method*mentioned, whereby for him a direct connection with the infinite series is given [GW 21, 292]. In contrast to this, for Hegel the integral calculus is a calculus like the differential calculus and nothing other than its inverse [ibid.]. - 18.
According to the definition of Apollonios (

*homoimereis grammai*), d. H. Every two points on the line lie in parts that are congruent to one another [29, 105]. - 19.
A. Moretto in [26, 176]. He also considered the totality of the segments as that

*infinitum actu*Spinozas, with which he and Hegel had already anticipated G. Cantor's actual infinity. Apart from the fact that this totality does not correspond to Spinoza's argument, the real infinite is not to be confused with Cantor's transfinite sets, because these are determined and differentiated from one another, while for Spinoza there is only one*infinitum actu*gives that of substance. - 20.
*Messages on the doctrine of the transfinite*(1887)*,*in [5, 391]. - 21.
Letter to Dedekind (1899) in [5, 443].

- 22.
According to Aristotle (and Spinoza see above) a continuous line does not consist of points. [1, 231a25].

- 23.
Cf. the remarks by W. Bonsiepen [3].

- 24.
[GW 11, 229]; see.

*Principia book*II, chap. II, Lemma II [27, 256]. - 25.
Hegel used not only Hauff's introduction, but also his almost 60-page appendix on the history of calculus, after which he dealt with the methods of Kepler, Cavalieri, Fermat, Barrow, Newton and Leibniz.

- 26.
D’Alembert, Jean le Rond: Mélanges de literature, d’histoire, et de philosophie. Amsterdam (1767); quoted from [4, 248].

- 27.
D’Alembert and Diderot: Encyclopädia, Vol. 9 (1765),

*Limit;*quoted from [16, 91]. - 28.
GW 7, note p. 369.

- 29.
See [25, 18]. In the 2nd edition of the

*logic*Hegel deletes the name of L'Huilier at this point [GW 21, 258], but in contrast to M. Wolff, no reference to Cauchy can be derived from this [35, 215], because the method of limits was also used by other authors , e.g. B. Carnot and Lacroix, it is therefore sufficient to assume that Hegel no longer wanted to specifically identify the method by a name. - 30.
Cf. [25, 20], "de thunder par là à la solution de cesproblemèmes toute la rigeur des demonstrations des anciens."

- 31.
GW [21], footnote p. 262; see [25, 341]

- 32.
Hegel refers to Lagrange, II.P., II.Chap., [25, 190 f.]. See A. Klaucke [22].

- 33.
*Encyclopedia*, 2nd edition, Berlin 1827, § 270 [GW 20, 268]. - 34.
*Principia,*Book I, chap. 2, Theorem IV, Cor. 1 [27, 64]. - 35.
Stekeler-Weithofer refers to the strange reception "that Hegel, who radically criticizes all hypostasis of our concepts, including those based on the theory of consciousness, is himself generally understood as an a priori metaphysician" [34, 216] 1801 to refer: "

*Philosophical discussions about the planetary orbits*“, In which Hegel polemicises against the Titius-Bode series and sets up a separate series for the distances between the 7 planets known up to then. However, he does not claim that there could be no more planets, which was assumed by contemporaries after the discovery of the planetoid Ceres. See the comment by W. Neuser in [17]. - 36.
See.

*Cours d'analysis*, Introduction [7, ij] "Quant aux méthodes, j‘ ai cherché à leur donner la rigeur qu ’on exige en géométrie, de manière à ne jamais recourir aux raisons tirées de la généralité de l‘ algèbre. " - 37.
In [12] Dirksen does not mention the mean value theorem in the résumé [8. Leçon; 8, 44] nor the intermediate value theorem in his review of

*A.L. Cauchy's textbook on algebraic analysis. Translated from the French by C. L. B. Huzler*[13], see [7, Note III]. - 38.
M. Wolff sees this as a criticism of Hegel's Cauchy's theory of limits [35, 231], but A. Moretto [26, 187 f.] And A. Klaucke [22, 143 f.] Already refer to an identical text by Lacroix as Reference, see [24, 189 f.].

- 39.
In it he writes about the differential quotient: Le rapport lui-même pourra converger vers une limite,…, qui sera la dernière raison des differences infiniment petites \ (\ Delta y \) et \ (\ Delta x \). Le rapport \ (\ frac {dy} {dx} \) coïncide avec la dernière raison des quantités infiniment petites \ (\ Delta y \) et \ (\ Delta x \) [9, 288].

- 40.
With the title

*The subjective logic or the doctrine of the concept.* - 41.
“The shining example of the synthetic method is that

*geometric*Science "[GW 12, 226]. With this, Hegel stands out critically from Kant, for whom the sum \ (5 + 7 = 12 \) is already a synthetic proposition [20, B 49], while Hegel sees this as a task that has to be solved analytically by the Members of the sum are added together [GW 12, 206].

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### Author information

### Affiliations

Department of Mathematics / Computer Science, University of Bremen, P.O. Box 330440, 28334, Bremen, Germany

Harald Boehme

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Correspondence to Harald Boehme.

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### Cite this article

Boehme, H. Analysis in Hegel. *Math semester***61, **159-181 (2014). https://doi.org/10.1007/s00591-014-0136-2

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- Hegel
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- Differential calculus
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