I'm reading The Principles of Non-Philosophy and re-reading Philosophy and Non-Philosophy by François Laruelle and I keep coming across this description of the philosophical decision described as a "fractional matrix in 2/3rd terms." For example he states in the first work:

The most universal invariant trait of philosophy is a fractional matrix in 2/3 terms: it gives itself an interiority and an exteriority, an immanence and a transcendence simultaneously, in a synthetic or hierarchal structure, the one overcoming the other in turn. This matrix of “Philosophical Decision” can be read as the identity of a double relation of philosophy to itself. First, an identity of 2/3 (insofar as the third term, synthesis, is immanent to the dyad, philosophy being in need of itself). Second, a 3/2 identity (insofar as the term of synthesis is transcendent to the dyad, philosophy being in excess of itself). Through this structure, philosophy claims to determine itself beyond all its empirical determinations which it only calculates in order to prescribe it in an auto-position in which it is titular, an auto-comprehension or auto-legislation, auto-naming, etc. In this formal trait the circularity of philosophical argumentation takes root along with its procedures of auto-validation.

Ross Wolf (see http://thecharnelhouse.org/2008/07/03/thoughts-on-francois-laruelle%E2%80%99s-preface-and-introduction-to-principles-of-non-philosophy-as-translated-by-fractal-ontology%E2%80%99s-taylor-adkins/) states that it is a reference to Hegel's "henology" and describes it in the Fichtean terms of thesis-antithesis-synthesis: "The “3” side invariably refers to the transcendentally exterior “synthesis,” while the “2” refers to the immanently interior dualism of “thesis” and “antithesis” (to use crude Fichtean terms)." However, I still want to know that the significance of Laruelle's use of a fractional matrix is. Can anyone conceptualize this or define a "fractional matrix" in either layman's terms or "philo-speak"?

  • I don't suppose its a matrix of fractions; or a fraction of matrices. Sep 1, 2014 at 14:05
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    I'm guessing this means to reduce philosophy to a bunch of interrelated 'proportions' scattered all over the place, to prove it leads to reusage of the same content, or at least maintains the same overall content value, and it uses this content to promote its own ends.
    – dwn
    Jan 23, 2015 at 4:14
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    You may find some clarification in this Differend Komplex blog post that references Kolozova's recent book on Laruelle (which might also be helpful here.)
    – Joseph Weissman
    Dec 30, 2015 at 16:51

1 Answer 1


I don't consider the following to be a full answer in the terms given; but it may help, or you may know this already. One point to carry in mind is that Laruelle is reifying the relationships implicit in mathematics on a philosophical manner, and hence shouldn't be taken literally as mathematical concepts thought mathematically.

It looks like Hegels solution to the problematic set out in Aristotles Physics, on the number of principles (arche) of nature: a contrary acting on another; and A struggles to say whether this is two, three, finite or infinite - saying that whilst all philosophers have agreed that contraries are implicated in nature, there is no consensus on their number.

Hegel turns this into one; which is immanently a dyad, which it immediately and overcomes - hence an 'identity'.

When the overcoming is transcendental, Laruelle is writing this as 2/3; and when immanent 3/2 - which is a neat allegory in mathematical terms to summarise the content of the above - recalling that a fraction is one number, and that inverting fractions is a useful thing to do with fractions and that transcendence and immanence have opposite or inverse senses.

I have no suggestions to why Laruelle thinks of this as a 'matrix', unless he is conceptualising the unfolding of contraries as such.

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