nLab Attempt to Introduce the Concept of Negative Quantities into Philosophy






The basis of it all

 Set theory

set theory

Foundational axioms

foundational axioms

Removing axioms



“Versuch, den Begriff der negativen Grössen in die Weltweisheit einzuführen”“ (1763) is a short article of Immanuel Kant concerning the philosophical foundations of the integers, in particular the negative integers.


Diese kleine Schrift ist eine der tiefsinnigsten und lichtvollsten, die nicht bloß Kant geschrieben, sondern welche die philosophische Literatur überhaupt aufzuweisen hat. Man tut Kant nicht unrecht, wenn man behauptet, daß sie ihm wie ein Meteor entschlüpft und selbst nicht wieder zu Gesicht gekommen ist.1 Karl Rosenkranz (1840, p.118)

Starting with a discussion of the meaning of the minus sign in negative numbers and as a designator of the subtraction operation, Kant sets out to distinguish between (predicate) negation in logic and substraction in arithmetic. He is thereby led to argue for a sharp distinction between logical and real opposition: whereas the former leads to logical contradictions, the latter does not.

Since he conceives logical contradictions as the simultaneous predication of a predicate pp and its negative ¬p\neg p to the same subject SS, a contradiction destroys the possibility of the thing represented by the subject.

In contrast, in a real opposition, e.g. two forces acting in opposite directions on the same point mass, merely the effects of the opposites are cancelled without affecting their reality, i.e. the cause is still effective but its effect are covered by the effects of its opposite.

As result, a gulf opens between the realm of logic with its analytic modes of reasoning and the realm of reality throwing into crisis the rationalist thought of the Leibniz-Wolffian school that maintained that in principle all truth was analytical and that denied any difference in principle between empirical and logical truths.

Hence the distinction between logical and real oppositions gave Kant strong incentive to reconceptualize the relation between logical and empirical propositions finally leading to the distinction between the formal and transcendental logic in his Critique of Pure Reason of 1781.

On more general grounds, Kant also recalibrates in the article the relation between philosophy and mathematics, urging the former to modesty e.g. when it comes to deny the existence of infinitesimals on metaphysical grounds.


The concept of real opposition preserving (the reality of) the opposites positively in its result proved to be decisive not only for Kantian but also for the post Kantian philosophy since, as already pointed out by K. Rosenkranz surrounding the discussions around the Hegelian logic in the 1840-50s, it can be viewed as a germ of Hegel’s concept of contradiction and sublation in the ‘Wissenschaft der Logik’. In fact, the section on ‘contradiction’ in the second volume even has a discussion of arithmetic and negative numbers (cf. Wolff 2010).


  • M. Giovanelli, Trendelenburg and the Concept of Negation in Post-Kantian Philosophy, to appear in Munk (ed.), Proceedings of the Amsterdam 2010 Colloquium: Natur des Denkens und das Denken der Natur: Spinoza, Trendelenburg und H. Cohen. (draft)

  • Abraham Gotthelf Kästner, Anfangsgründe der Arithmetik, Algebra, Geometrie, ebenen und sphärischen Trigonometrie, und Perspectiv, Göttingen 1758. (gdz)

  • Srećko Kovač, In what sense is Kantian principle of contradiction non-classical?, Logic and Logical Philosophy 17 (2008) pp.251-274. (link)

  • Karl Rosenkranz, Geschichte der Kant’schen Schule, Akademie-Verlag Berlin 1987[1840].

  • Michael Wolff, Der Begriff des Widerspruchs - Eine Studie zur Dialektik Kants und Hegels, Frankfurt UP ²2010.

  1. ‘This small text is one of the deepest and most lightfull not only in the writings of Kant but in the whole philosophical literature. One does no injustice to Kant if one says that it snapped out of him like a meteor and was never seen again by him.’

Last revised on December 21, 2019 at 20:36:04. See the history of this page for a list of all contributions to it.