Laws Of Form

The *Laws of Form* (Spencer-Brown 1969) wikipedia , page did not enter the world of mathematical texts without presupposition. The problems to which the text responds, as well as some of the solutions presented, become accessible to a different and possibly deeper understanding when the *Laws of Form* are considered in the context of the history of mathematics and logic.

The *Laws of Form* are observed, in the perspective proposed here, as a late work of mathematical "counter-modernism", which revives in an idiosyncratic way justification-theoretical discussions of past days, and for this purpose chooses the logicism of the Frege/Russell program as a contrast foil of its own views; views which find a (mathematics-internal) prepared environment in the non-classical mathematics picture of constructivist intuitionism – also and especially as far as the mystical-spiritual moments of intuitionism are concerned.

At the same time, the *Laws of Form* carry out an aesthetic program of formalism that is almost antithetical to intuitionism (a "writing of form"), which takes up approaches from Ludwig Wittgenstein's Tractatus and develops them in a radicalizing way. By broadening the scope of attention in this way, the subsequent sketch of the *Laws of Form*'s major lines of thought may make clearer what is new about the *Laws of Form* in other ways – but also what may not.

Our contribution closes by taking up the initial question: To which problem does systems theory react with its (contingent, but precisely thereby: specifiable) take-up of figures of thought which it observes in the *Laws of Form*?

In the *Laws of Form*, George Spencer-Brown refers in many places to the relationship between logic and mathematics. He explicitly opposes a view according to which mathematics proceeds according to logical laws, and he names it as one of his central concerns "to separate what is known as the algebras of logic from the object of logic and to reconnect them with mathematics." (Spencer-Brown 1997: xxvi) It is not arithmetic that follows logic, but, according to Spencer-Brown, vice versa: logic is a derived, practical application of the arithmetic primordial laws of the generation of forms, their exchange and their order.

To whom does he say that? The idea that mathematics is applied logic cannot be called a relevant position of our time. In fact, Spencer-Brown addresses his objections to this logician reductionism to the mathematical past, namely to Bertrand Russell. The *Laws of Form* follows in the footsteps of another "cult book," Ludwig Wittgenstein's *Tractatus* (1998), in several respects, including that of working off Bertrand Russell. Nevertheless, the goal of Spencer-Brown's critique – the so-called "logicism" – is not a solo effort of Russell, but a co-production of several sources, the most important of which are the logical works of Gottlob Frege.

The title "Logicism" stands for a program of justification theory, which was pursued by Gottlob Frege as well as by Bertrand Russell and Alfred Whitehead and aimed at tracing back the whole of arithmetic (and mediated also the other fields of mathematics) to the (eternal, non-contingent, self-evident) logical basic laws.

Frege's program of reasoning owes itself to a double movement. On the one hand, through Frege's special invention of its own logical writing – the "Begriffsschrift" (1879) – logic gains medial autonomy from arithmetic and its symbolic writing (by being able to observe the writing medium of arithmetic operations as a form); on the other hand, this setting apart movement of logic is the precondition for logic's renewed turning to mathematics in order to justify it, as it were, from the outside. As will be explained below, it is precisely the change of media within logic initiated by Frege's Begriffsschrift that paves the way for Spencer-Brown's mathematical "writing of form."

While the notation of Boole's algebra of logic was still based on mathematical symbolic conventions, even a brief glance at a page in Frege's work Grundgesetze der Arithmetik (Frege 1998), for example, shows that an entirely new script was invented here, reminiscent of circuit diagrams, or a mixture of a route network diagram and a rare Asian script. To the practiced observer of demonstrations performed in conceptual writing, the mere figurative form is supposed to show what one would otherwise have to say meta-linguistically.

Why such a "conceptual writing"? Logic has always been confronted with the task of translating normal language expressions into expressions that were amenable to deductive processing (syllogistics, too, first had to translate facts into arrangements of categorically simple judgments). In the Tractatus, Ludwig Wittgenstein once again succinctly presents this basic problem of logic, which provided the motor for Gottlob Frege's development of his own "Begriffsschrift":

> In the colloquial language it happens very often that the same word designates in different ways – that is, belongs to different symbols – or that two words, which designate in different ways, are externally used in the same way in the sentence. Thus the word "is" appears as a copula, as an equal sign, and as an expression of existence; "exist" as an intransitive time word like "go"; "identical" as a property word; we speak of Something, but also of Something happening. [...] Thus the most fundamental confusions (of which the whole of philosophy is full) easily arise. (Wittgenstein 1998: 30 f., emphasis in the original).

A closer look at the "Begriffsschrift" would go beyond the scope of this essay, but one detail shall be briefly illuminated because it is able to establish references to Spencer-Brown's calculus writing. In Frege's Begriffsschrift, a judgment is always notated with the help of a sign composed of two strokes: A vertical stroke from which a horizontal one branches off (to the right). This sign expresses a simultaneity of two processes, more precisely: the assertions of a "that" and a "what": The vertical stroke, called "judgment stroke" by Frege, stands for the indication "a judgment has been made", the horizontal stroke, originally called "content stroke" by Frege, stands for the indication of the "ideas" (ibid.: 2) to which the judgment refers. We will encounter the combination of vertical stroke and horizontal branch again, in variation, in the calculus writing of the *Laws of Form*. Against the background of the aesthetic program we have observed, such analogies are not arbitrary trivialities of representation; rather, in our view, it is one of the offers of insight of symbolic-noting logic since Frege to give due attention to mediality, i.e., to the writtenness of calculi (cf. on this Bredekamp/Krämer 2003; Krämer 1991).

⇒ The “Justifiable” Mathematics

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digraph { Aesthetic Value Motive Intention Instruction Name Calling Recalling Law1 Indicate Motive Distinction [color=red penwidth=3] Containment Boundary Crossing Recrossing Law2 Content Sides In Out Aesthetic -> Value Value -> Name Name -> Indicate Name -> Calling Calling -> Law1 Recalling -> Law1 Indicate -> Value Value -> Motive Value -> Crossing Motive -> Distinction Motive -> Crossing Intention -> Crossing Instruction -> Crossing Distinction -> Boundary Distinction -> Containment Boundary -> Content Crossing -> Law2 Recrossing -> Law2 Crossing -> Boundary Content -> Sides Sides -> In Sides -> Out }