I'm new to the study of philosophy, feeling around slowly, finding my way. Am I misinterpreting, or did Hegel have contempt for mathematicians?

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    In general images are only really allowed when they're pertinent to the question (a diagram or chart or anything else containing information that is necessary to the question), generally that falls under the "this is not a blog" rule. – Not_Here Jul 23 '17 at 12:46
  • I think your impression is right. I would think the source of depreciating math for Hegel is the dialectical reasoning method that has the effect of denying the law of identity. That is, math operates under 'a=a,' but dialectical reasoning assumes that thesis contains anti-thesis within. – Nanhee Byrnes PhD Jul 24 '17 at 16:21
  • @NanheeByrnesPhD - Is it not the case that for Aristotle dialectical reasoning begins with a = a? – PeterJ Mar 30 at 13:03

Hegel considered mathematics as a subordinate science. Philosophy, as far as it shall be science, cannot borrow its methods from such a subordinate science as mathematics is.

Die Philosophie, indem sie Wissenschaft seyn soll, kann, wie ich anderwärts erinnert habe, hierzu ihre Methode nicht von einer untergeordneten Wissenschaft, wie die Mathematik ist, borgen, so wenig als es bei kategorischen Versicherungen innerer Anschauung bewenden lassen, oder sich des Raisonnements aus Gründen der äußern Reflexion bedienen. (Georg Wilhelm Friedrich Hegel: Wissenschaft der Logik - Kapitel 2)


From Hegel: Phänomenologie des Geistes, Vorrede, his contempt of mathematics should become crystal-clear:

Im mathematischen Erkennen ist die Einsicht ein für die Sache äußerliches Tun; es folgt daraus, daß die wahre Sache dadurch verändert wird. Das Mittel, Konstruktion und Beweis, enthält daher wohl wahre Sätze; aber ebensosehr muß gesagt werden, daß der Inhalt falsch ist. ... Die Evidenz dieses mangelhaften Erkennens, auf welche die Mathematik stolz ist und womit sie sich auch gegen die Philosophie brüstet, beruht allein auf der Armut ihres Zwecks und der Mangelhaftigkeit ihres Stoffs und ist darum von einer Art, die die Philosophie verschmähen muß. ... Das Wirkliche ist nicht ein Räumliches, wie es in der Mathematik betrachtet wird; mit solcher Unwirklichkeit, als die Dinge der Mathematik sind, gibt sich weder das konkrete sinnliche Anschauen noch die Philosophie ab. ... Eine Kritik jener Beweise würde ebenso merkwürdig als belehrend sein, um die Mathematik teils von diesem falschen Putze zu reinigen

Its contents is wrong, insufficent rekognition, poorness of its purpose and insufficiency of its stuff, ...

  • I am not sure that this isn't more a pun towards Kant and some of his remarks and methodological claims regarding mathematics, rather than a real sign of contempt for it. Good find, though! – Philip Klöcking Jul 23 '17 at 19:02
  • @Philip Klöcking: See the phrases from the Vorrede. – Heinrich Jul 24 '17 at 15:09

You are misinterpreting. As far as Hegel's biography goes, Hegel's relatives record mathematics was one of his favorites subjects, and as a theology student he would take long "sick leaves" to basically go home and study math and physics (and botany). He would have known more mathematics than essentially any other philosopher of his era.

There are a number of complications that make the naive answers others are rattling off unsuitable. One issue is what, in modern mathematics, Hegel would have called "mathematische Erkennen". Obviously Lawvere considers everything in modern category theory to correspond to the contents of the Wissenschaft Logik; so in that sense the core of what Hegel considered "philosophy" is currently taught in nearly every serious math department, but in nearly no philosophy department.

Most of modern number theory and topology would also, I believe, fall outside what Hegel is describing as "mathematics"; for the simple reason that they attempt to come up with a completely conceptual understanding of what numbers are from scratch (via Peano or similar) without assuming, at the beginning, an abstract system of numbers that behave like "everyday" quantities without referring to any specific count or measure. I suppose some touchy mathematicians might read things Hegel says about "math" and accuse him of failing to see what mathematicians were capable of - but Hegel wasn't denying you could come up with these pure foundations for mathematics (that's what was in the Wissenschaft Logik!), he was just sketching out a disciplinary division of labor between philosophy, abstract tools like mathematics and predicate logic, and the natural sciences that assumed his readers had some idea how the material was divided between philosophy, physics, and math in 1807.

Now, I have interpreted your question as one about the field of mathematics. If we are talking about people,

(a) There were some German philosophers who were floundering after they realized that it wasn't really easy to reduce Kant's philosophy to a deductive argument. Many of Hegel's cutting remarks about the difference between philosophy and syllogism/arithmetic/geometry are aimed at them, and yes Hegel had contempt for them. (They were not "mathematicians", however.)

(b) There was a tradition in German mathematics (still very much alive in the 1920s) that held that proofs were secondary to pure mathematical insight, that you can just visually see that, say, the Pythagorean theorem is true. In a sense Hegel respects that vision of mathematics (in that he sees geometrical intuition as genuinely powerful, within its proper domain) but he also was quite dismissive of the idea that these mathematical ideas couldn't have or didn't need any logical foundations. Here too his dismissiveness was mainly aimed at other philosophers (Schelling) who tried to make parallel claims about other fields, but some actual mathematicians definitely took it as an attack on their purely intuitive conception of math as well.

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    I like the answer, but I wonder if you could include references for the claims so that the reader would have somewhere to go for more information. This would also strengthen the answer. Welcome to this SE! – Frank Hubeny Oct 15 '18 at 9:41
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    @FrankHubeny you may have already found some of this: en.m.wikipedia.org/wiki/William_Lawvere ; ncatlab.org/nlab/show/… The second one may interest you too. I saw this last night. Even though it is over my head I found it very interesting too. – Gordon Oct 15 '18 at 15:12
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    @FrankHubeny also recently this at mathematics educators se. matheducators.stackexchange.com/questions/14596/… – Gordon Oct 15 '18 at 15:19
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    @Gordon Thank you! That does interest me. I'm going through the modal-logic tagged posts at the moment. – Frank Hubeny Oct 15 '18 at 16:40
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    Thank you to Gordon for adding additional links (and certainly feel free to leave them). I don't know exactly what references for claims you have in mind, Frank - links to WP and n-lab articles like what Gordon provided? Or page citations to Hegel's Gesammelte Werke? (or to biographies and major commentaries?) To be as direct as possible - I don't have time to annotate this in the style of an academic article and if I did, I wouldn't publish it in stack exchange. (I spent more time than I could really afford to spare writing it on a whim last night already.) – guest1806 Oct 15 '18 at 21:23

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