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Just thought I’d make this thread to place the work on & conversation about Hegel and mathematics in the broader context of the relationship between mathematics and German philosophy around the time of Hegel.
I haven’t explored Fichte yet, but, Hegel’s Wissenschaft der Logik isn’t the only German Idealist work that admits a mathematical interpretation. Novalis saw Fichte’s Wissenschaftslehre as the “mathesis of mathesis” and David W. Wood (a philosopher (with an undergrad degree in mathematics) specializing in Fichte and Novalis) has an interesting looking book here on Fichte’s Wissenschaftslehre with regards to the mathesis of the mind, the parallels between philosophy and geometry, platonism, and an “”ursprüngliche” or original geometry - that is to say, a synthetic and constructivistic conception grounded in ideal archetypal elements that are grasped through geometrical or intelligible intuition”:
Novalis also has some really interesting comments on mathematics, such as these:
“In the end, mathematics is only common, simple philosophy, and philoso- phy, is higher mathematics in general. In particular, higher mathematics connects common mathematics with the system of mathematics, while the latter borders on the philosophy of mathemat- ics—or philosophical mathematics, just as systematic science is generally always the precursor and boundary of a higher degree of science—of the philosophical degree.222 (Degrees of scientific character. The highest degree of scientific character would be termed philosophy). The philosophical degree again divides into 3 parts and immediately—passes over into the higher series, or into the higher de- gree of the philosophy of philosophy, and so on. (Just as the man of nature passes over into the common and complex human being, so too pure science into the common and higher. Higher science is the transition to a system, just as the scholar, or complex man is the transition to the systematist).”
“(Has philosophy originated from the contemplation of mathematics?) Philosophy is universal—or higher mathematics—the animating principle of mathematics—poetical mathematics—Or the substance, if mathematics is the form. Mathematics is merely objective philosophy—formal philosophy—and so- called philosophy—is merely subjective philosophy or mathematics—real philoso- phy. By combining them in a manner analogous to that of the combination of chemistry and mechanics—there arises substantial—synthetic—philosophy—or mathematics, or physics. Contrasted with philosophy, physics is mathematics— while contrasted with mathematics—it is philosophy”
“All the universal sciences—e.g. physics and mathematics, etc., really resemble philosophy in one respect—they are Proteusses—universal substances—indications etc.”
“The mathematical method is the essence of mathematics. Whoever fully understands this method, is a mathematician. As the scientific method in general it is extremely interesting, and perhaps supplies us with the most accurate model for the classification of knowledge, or for the faculty of experience. Axioms and postulates denote the theoretical (a.) and practical (b.) cognitive faculty as such. Problems denote the desire. Solution and proof, the analytic (ad a.) and synthetic (ad b.) ability. Explanations and corollaries also have their significance. This reveals that our desire for knowledge is the intelligence’s desire for life, a play of intellectual forces.”
“Mathematics is genuine science—because it contains created knowledge—the products of its own spiritual activity—and because it methodically inspires. It is art, because it has fashioned inspired procedures into rules—because it teaches one to be a genius—and because it replaces Nature with reason. Higher mathematics is concerned with the spirit of quantities—with their political principle—with the world of quantities”
“All sciences should become mathematics. Up to now, mathematics has merely been the first and simplest expression or revelation of true scientific spirit.”
“The external is the common. The internal, is the particular./ The inte- gration is much more difficult than the differentiation. In relation to physics and philosophy./ The science that joins and puts both into contact with one another—that instructs in deriving the particular from the common, and the inverse, as well as with the external and internal aspects—this science is the connecting—and higher science. If the first is quantative mathematics, and the second qualitative mathe- matics, then the third is relative mathematics—which appear in four systems of elements and in a single universal system.”
“The highest life is mathematics.”
“Pure mathematics is religion.”
“Pure mathematics is the intuition of the intellect as a universe. Genuine mathematics is the actual element of the magician … In music it appears formally, as revelation - as creative idealism”
Also, Schlegel had this to say about mathematics: “Mathematics is, as it were, sensual logic. It relates to philosophy as the material arts, music and sculpture, relate to poetry.”
Hi trent,
this is all interesting. If you want to stick with the spirit of the Lab, then you should record all that nice information in an Lab page!
Maybe at Novalis or maybe at objective idealism, or maybe at philosophy of mathematics. Or maybe you’ll create a new page with a title of your choice.
Then we could eventually interlink with related material that we have, etc.
@trent - it would be a good idea to drop a note pointing to what you created or added. Then people who don’t watch the latest changes list can see your work :-)
Added everything except for the link to the wikipedia entry on the novalis page.
Thanks, trent.
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