An Endless Succession of Mirrors

Especially interesting is the philosophical study of hitherto purely mathematical concepts and operations—with powers, roots, differentials, integrals, series—curves—and direct—functions. [source?]

‘The method of idealism is that of combinatorial experimentation.’ — F. Schlegel

IS ME EDIT WORKING

I. The Hindenburg School

It is well-established that both Novalis and Friedrich Schlegel took direct influence from an early form of combinatorics based around the school of Carl von Hindenburg. While the school’s rabid hyper-Leibnizian mission to ground the entirety of mathematics on the polynomial theorem failed, they pioneered the uses of tabular forms for combinatorial problems and certain of their results, particularly the work of Rothe, are used in computer science today. This, despite the amusingly grouchy dressing down Donald Knuth decides to give “Hindenburg’s hype” in The Art of Computer Programming.¹

The use of such permutation tables can clearly be seen in the model of the dialectic developed by Novalis in his Fichte-Studien, where the number of participant elements in a synthesis and therefore the number of combinatorial possibilities are expanded:

[Image from Fichte Studien]

He explicitly invokes “Hindenburg über das Infinitinomium” in his Allgemeine Brouillon, referring to “die Potenzierung des Infinitinoms”, the term Hindenburg used for the multinomial formula.²

II. Potenzierung

For Schlegel and Novalis, Potenzierung—exponentiation, or “raising to a higher power”—was the same thing as “transcendentalisation”: performing a critique on the thing in question. The two strongly associate this with dimensionality, so that the traditional Kantian search for conditions of possibility becomes analogous to finding an encompassing dimension. This structure bears similarities to the genitive formulation of Nishida Kitarō’s epistemic basho no basho, the “place of place”.³ Perhaps something of this is at work in the Sphäre/Muttersphäre conception of encompassing abstract space that we see in the Fichte Studies.⁴

Indeed, both Schlegel and Novalis’ writings abound with this use of the metagenitive or self-containment: transcendental philosophy is the “philosophy of philosophy”; Romantic poetry is the “poetry of poetry”. All of these have been subjected to critique by being raised to a higher dimension. According to their model, this is the same as philosophy³.

Of Romantic poetry, Schlegel proclaims in similar language that,

it can also—more than any other form—hover at the midpoint between the portrayed and the portrayer, free of all real and ideal self-interest, on the wings of poetic reflection, and can raise that reflection again and again to a higher power, can multiply it in an endless succession of mirrors.⁵

Elsewhere, he speaks of the ‘Potenzirung des Aufsatze, Analyse und Deduktion des eignen Verfahrens’ (KFSA 16, 49).

On the narrower question of what critique achieves, he writes: ‘critique in the narrower sense is the theory of the regular complete construction of the problem, e.g. of philosophy, and of philosophy as a science. It orders the data into necessary equations.’ [source?]

III. Romanticising the World

The philosophical stakes of Potenzierung are set out with startling directness by Novalis:

The world must be romanticised. This yields again its original meaning. Romanticising is nothing else than a qualitative potentisation. In this operation the lower self becomes identified with a better self. Just as we ourselves are a potential series of this kind. This operation is still entirely unknown. By giving the common a higher meaning, the everyday, a mysterious semblance, the known, the dignity of the unknown, the finite, the appearance of the infinite, I romanticise it—For what is higher, unknown, mystical, infinite, one uses the inverse operation—in this manner it becomes logarithmicised—It receives a common expression. Romantic philosophy. Lingua romana. Reciprocal raising and lowering.⁶

Here Novalis is referring to the result that, for example if 2³ = 8 then log₂(8) = 3, meaning that potentisation and logarithmicisation are reversible and provide a model for the ability to navigate up and down the scales of abstraction: “reciprocal raising and lowering”. One direction proceeds towards transcendental philosophy, or metaphysics as a science; the other towards the “common expression”, which would substantiate the democratic ambitions of the Romantic, who wanted to provide a bridge between scholarly philosophy and common parlance.

IV. The Encyclopaedia Project

The mathematical structure of romanticisation is not a local metaphor but the organising principle of Novalis’ encyclopaedic project. In entry 198 of the Allgemeine Brouillon, he writes that ‘science on the whole is generally the total function of the data and the facts—the n-th power of the binomial series of the data and the facts. Here combinatorial analysis would be necessary.’⁷ Series expansion techniques feature heavily throughout the encyclopaedia project, a clear indicator of the inheritance from von Hindenburg.⁸

He goes as far as to say that ‘science does not begin with an antinomy—binomy—but with an infinitinomy’. Clearly, he is extending the strict Kantian model of the antinomy—a pseudoproblem which drives philosophical reflection—by multiplying its terms indefinitely. The very basis of Novalis’ transcendental syntax of the Encyclopaedia is not just a set of virtually potent foundational paradoxes, but an infinite set of such paradoxes. [Develop: the false dichotomy, the expansion from binary to n-ary opposition.]

Indeed, In the ‘Freiburg Natural Scientific Studies’ he imputes this combinatorial art to nature as such:

Nature incessantly adds, subtracts, multiplies, raises to a higher power etc. The applied mathematical sciences show us Nature as a mathematician. Physics is real mathematics.⁹

V. Schlegel’s Fragments

Schlegel’s notebooks and fragments develop the same operation across a range of registers. Its most compressed expression is the self-referential structure of irony:

Socratic irony is reciprocal parody, potentiated parody.¹⁰

The fragment on critique makes explicit the mathematical grammar:

622. Critique = π²/0. All critique is potentiated. All abstraction, but especially the practical kind, is critique. Critique only exists where the Absolute and Empirical are synthesized, not where one of the two is isolated and potentiated; but potentiation itself is already an approximation to critique.¹¹

Note that for Schlegel, /0 designates the Absolute—so that k/0 means “absolute critique”, and the formula captures critique as the ratio of the infinitely repeated (π²) to the Absolute. On the operation itself:

910. Potentiation is a modern operation; combination of the individual with itself.—Beginning, closing, contracting operations for the systematic style, likewise cyclicising ones.¹²

This typology maps suggestively onto the language of graph theory: the potentiated entity as the self-loop (an edge connecting a vertex to itself), the source node, the sink node, graph contraction, and cycles. His systematic style does appear to be a consideration of linkages between concepts.

Schlegel’s concept of Witz—wit as the combinatorial operator of thinking—belongs to the same field. In a celebrated passage, he writes:

If wit in all its manifestations is the principle and the organ of universal philosophy, and if all philosophy is nothing but the spirit of universality, the science of all the eternally uniting and dividing sciences, a logical chemistry: then the value and importance of that absolute, enthusiastic, thoroughly material wit is infinite… The best ones are échappées de vue into the infinite. Leibniz’s whole philosophy consists of a few fragments and projects that are witty in this sense.¹³

VI. Kant and the Ars Combinatoria

Early in his career, Kant rejected the possibility of representing the relationships between concepts using mathematical notation—a position that would appear to chime with his broader rejection of the mathematisation of epistemological method. By the time he arrived at the Critique of Pure Reason, however, his table of the categories, which constituted a dissection of the faculty of understanding itself, had begun to bear some resemblance to the early combinatorial systems we have been discussing. In a letter, Kant himself urged the mathematician Johann Schultz to formalise the table using the ars combinatoria:

These and the other, partly mentioned properties of the table of concepts of understanding seem to me to contain material for a perhaps important discovery, which I cannot pursue and which is reserved for a mathematical mind like yours, to bring an artem characteristicam combinatoriam into practice with it, which, if it is possible anywhere, would have to work particularly well with the same elementary concepts…

This attempt to solicit an in-house mathematician to support the critical project apparently bore no fruit. Jacob Sigismund Beck, who would go on to compile a three-volume Explanatory Extract from the Critical Writings (1793–96) in an attempt to popularise the critical project, also drew comparisons between it and the work of von Hindenburg:

A thought of Mr. Hindenburg, which you had the kindness to communicate to me, is indeed very flattering to me as far as confidence is concerned, but exceeds my mathematical knowledge far too much for me to even attempt to apply the combinatorial method to philosophy.

As with the table of categories itself, this formalisation would extend to the derivative concepts (still pure) called predicables (i.e. under causality go force, action, passion). Kant suggests that this formalisation project could ‘completely paint the family tree of pure understanding’.

The existence of these letters really puts the lie to an interpretation held to this day according to which Kant’s refutation of certain key Leibnizian gestures would extend to the symbolic formalisation of philosophical concepts and their composition. The fault-line can be seen in the enduring dogmatisms of the positivist Fregean inheritance and the “phenomenological” school of transcendental philosophy who seem to have ignored the B draft of the Critique of Pure Reason, the Metaphysical Foundations of Natural Science and the Opus Posthumum. Indeed, it seems as though Kant actively sought out a mathematician with adequate philosophical understanding to undertake the task of mathematising the concepts of the understanding.

Further Threads

• Schlegel on the Wechselerweis — the alternating proof
• Hölderlin and the Wechsel der Töne
• Schelling on dimensionality in the Ages of the World
• Weyl on group theory and transcendental critique
• Vuillemin on the group
• Timmermans
• Rabouin, Mathesis Universalis
• Catren on categories and phenoumenon
• Neubauer, Von Symbolismus
• Baroque combinatorial poetry

‘The so-called unity of the group is a function of nothingness. Zero becomes the unity of everything.’ — Nishida Kitarō, Diaries (upon revisiting Parmenides, Cusanus and Duns Scotus)

「我心深き底あり喜びも憂の波もとどかじと思ふ」