"Spencer-Brown's Laws of Form and Husserl's 'Strange World of the Purely Logical,'" paper given on August 4, 2022 at the Laws of Form Conference 2022 held at Liverpool University.

Abstract: I suggest some connections between Laws of Form and what Husserl called the "strange world of the purely logical." Specifically, I talk about: Husserl's background, especially as a mathematician, and his use of symbolic notation. Then, I compare his and Spencer-Brown's views on: the relationship between mathematics and logic, mentioning Boole; the fundamental particles from which numbers can be made; the structure of knowledge of the universe, especially Husserl's theory of manifolds; imaginary entities; Russell's theory of types. For lack of time, I only mention Spencer-Brown's and Russell's exchange on propositional functions and Husserl's theories about them.


Laws of Form and Husserl’s “Strange World of the Purely Logical”

I am here because Randy Dible asked me to suggest to you some connections between Edmund Husserl’s theories about what he called the “strange world of the purely logical,” about which I have written a lot, and Laws of Form, about which I know little.

Husserl was not really the lopsided philosopher commonly perceived by followers and foes alike to have spent most of his life cerebrating only about subjectivity. However, the evidence of his years-long work to lay bare and impart knowledge about the true and ultimate structure of objective reality became buried in the excitement generated by his extensive exploration of the realm of transcendental subjectivity, with which he was so smitten that he did not pursue his theories about pure logic as far as he could have. Indeed, in 1917, he wrote to Hermann Weyl of how, despite all the work he had devoted to formal logic, he had not followed it completely to the end, because it had had to be more important to him to develop his ideas about transcendental phenomenology. In 1930, he wrote to Georg Misch of how he had lost all interest in formal logic and all real ontology in the face of a systematic grounding of a theory of transcendental subjectivity.

Zealous followers followed suit. So, Husserl’s pioneering work in formal logic and ontology has not received the attention it deserves, even, or especially, from his most ardent followers. However, he left intact his austere scheme to come to clarity with respect to the central traits of reality and, since the 1980s, the Husserl Archives has been publishing his lecture courses on logic, theory of knowledge and science, two of which I have translated, which provide material needed to retrieve the map of the world of the purely logical he drew. So, lovers of laws of form can now experiment with his scarcely explored theories about what Spencer-Brown called “a reality independent of how the universe actually appears.” If, as he suggested, the initial exploration of the basic forms underlying knowledge is usually undertaken in the company of an experienced guide, then Husserl can be such a guide and his ideas what Spencer-Brown called “a key to a world beyond the compass of ordinary description.” (xxii)

Against this background, I want to talk about some connections I find between Husserl’s and Spencer-Brown’s theories. Given the time limit, I must aim for pithiness. I hope that not too much gets lost in the wash.

First, it is essential to know that Husserl was a mathematician by training, who for decades kept close company with the most outstanding and pioneering mathematicians of his time, namely Karl Weierstrass, Georg Cantor and David Hilbert and his circle. He was as knowledgeable, if not more so, about the latest developments in mathematics and logic as the makers and champions of modern logic and mathematics, say Frege, Russell and Quine, who were embraced by the Anglo-American philosophical establishment, which controlled those fields, and philosophy in general, in English-speaking countries, during the 20th century.

Second, Spencer-Brown stated that his “conscious intention” in writing Laws of Form had been “the elucidation of an indicative calculus” (xix). Husserl obviously could not have used that notation, but it is not well known that he did use symbolic notation. It figures in his lectures on the pure mathesis in his logic courses of 1902/03 and 1896 ‒ which contains a 23-page section on Boole ‒ and looks like that of Peirce and Ernst Schröder ‒ who did much to introduce Boole’s work on the continent. It has been widely used and is familiar and intelligible nowadays.

Third, for Spencer-Brown a “principal intention” of Laws of Form was “to separate what are known as algebras of logic from the subject of logic, and to re-align them with mathematics” (xiv). Accounts of the properties of Boolean algebras, he bemoaned, had revealed “nothing of any mathematical interest about their arithmetics.” And “nobody” had “made any sustained attempt to elucidate and to study the primary, non-numerical arithmetic” of Boolean algebra, so that when he had begun to see the need for this he had found himself “upon what was, mathematically speaking, untrodden ground” (xiv). Yet, he stressed, mathematics is a “powerful” way of “revealing our internal knowledge of the structure of the world, and only by the way associated with our common ability to reason and compute” (xv). He saw the relation of logic to mathematics as being “that of an applied science to its pure ground…” (83). The “subject matter of logic,” he wrote, “however symbolically treated is not, in as far as it confines itself to the ground of logic, a mathematical study. It becomes so only when we are able to perceive its ground as a part of a more general form, in a process without end. Its mathematical treatment is a treatment of the form in which our way of talking about our ordinary living experience can be seen to be cradled.” It was the laws of that form, rather than those of logic, that he wanted to record. (xx)

For Husserl, the formalization of large tracts of mathematics in the 19th century had brought the parallels between its structures and those of logic to the fore, thus raising profound new questions about deep underlying connections between the two fields. This had unmasked close relationships between the propositions of logic and number statements and enabled logicians to develop a genuine logical calculus for calculating with propositions in the way mathematicians do with numbers, quantities and the like. Mathematizing logicians had done well to recognize the essential likeness between the formal mathematical disciplines and the disciplines of logical validity and to have raised them to a higher level of technical perfection by applying the same, completely appropriate, algebraic methods to them, thus expanding the scope of the exact mathematical disciplines.

He considered traditional syllogistic logic to be a piece of pure mathematics, namely, the pure mathematics of propositions and of predicates of possible subjects in general, where, he wrote, formalization “leads to a theory-form that can be understood as a special case of the formal genus-type ‘arithmetic.’” All the well-known algebraic propositions, ab = ba, the laws of association, distribution, hold,” he noted, “and the brilliant Boole saw that two closed domains of ordinary syllogistic logic could be dealt with as an arithmetic. So, by adopting the deductive theories of traditional syllogistic logic and gradually developing a mathesis of propositional, conceptual and relational meanings, 19th century mathematicians had but laid hold of a field that was their own.

Husserl taught that there was nothing extraordinary about the idea of calculating with concepts and propositions and a priori no reason why calculation should be limited to the arithmetical field, because the formal discipline of propositions in general and of concepts in general was a mathematical discipline of the same nature and used the same methods as familiar mathematical disciplines like arithmetic, which was the most marvelous tool devised for purposes of deduction, the science in which the deductive relations were analyzed most carefully. The fact that one could generalize, produce variations of formal arithmetic that led outside the quantitative domain without essentially altering formal arithmetic’s theoretical nature and calculational methods had shown him that there was more to the mathematical or formal sciences, or the mathematical method of calculation, than could be captured in purely quantitative analyses. For him, the essence of the mathematical did not lie in being quantitatively determinable, but in establishing a purely apodictic foundation of the truths of a field from apodictic principles, a matter of a rigorously scientific, a priori theory building from the bottom up and deriving the manifold of possible inferences from axiomatic foundations a priori in a rigorously deductive way requiring one and the same method everywhere, the algebraic method. For him, the essential thing in mathematics was not the objects, but its method which naturally flows into a purely symbolic technique.

Fourth, for Spencer-Brown, his “basic contribution to mathematics” had been “to discover the fundamental particles from which numbers and other, simpler, elements of mathematical systems can be made” (118). For Husserl, the concept of number was a paradigm of a purely logical concept, namely, a concept which is not limited to a special field of objects, but relates to objects in general in the most universal ways, which not only can and does figure in all the sciences, but is common and necessary to all sciences, because it belongs to what belongs to the ideal essence of science in general. No science is conceivable in which the number concepts cannot find an application. So, all purely mathematical concepts like unit, multiplicity, cardinal number, order, ordinal number, and manifold are purely logical, because they clearly relate in the most universal way to numbers in general and are only made possible out of the most universal concept of object.

Since each and every thing can be counted as one, to conceive the concept of number, or any arbitrarily defined number, we only need the concept of something in general. One is something in general. Each and every thing can be counted as one and out of the units all cardinal numbers built. Cardinal number is a specific differentiation of the concept of multiplicity which is the most universal logical concept combining objects in general. The first number in the number series is 2 As. From 2 As, we use definitions to form the new number 2 As and 1 A which we designate as 3 As, etc., then we obtain a series of the natural numbers, infinite in one direction.

To questions as to how arithmetic came about and the foundations of arithmetic provided, Husserl answered that people analyzed the arithmetic propositions as they were first entertained by people. They found that certain relations were grounded in the concept of number. For instance, any two numbers are either equal or one is larger or smaller than the other. They found that certain combinations were grounded in the concept of number: addition, multiplication, subtraction, division, etc. Given with the elementary combinations were certain simple, directly intelligible laws that careful analysis traced back to a certain minimal number of laws no longer reducible to one another, which are a priori since they lie in the simple meaning of the concepts founding them. They are propositions about relations of ideas obtained by analysis of the universal concepts by digging more deeply into their meaning. The unending profusion of theories that arithmetic develops is fixed, enfolded in the axioms, each of which is a proposition that systematically unfolds from some side the meaning of cardinal number, itself the answer to the question “How many?”, or unfolds some of the ideas inseparably connected with it, following simple procedures. The field branches out into more and more theories and partial disciplines, new problems surface and are solved using the most rigorous methods.

Fifth, Spencer-Brown variously alluded to “investigations of the inner structure of our knowledge of the universe, as expressed in the mathematical sciences” (xviii), “our internal knowledge of the structure of the world” (xv), “mathematical form as an archetypal structure” (xvii), “the structure beyond ordinary experience in which all creation hangs together.” (xxii)

For Husserl’s part, he depicted the structure of the world of pure logic in a way which he considered to be a radical clarification of the relationship between formal mathematics and formal logic. He detected a natural order in formal logic consisting of three levels, on the first of which he placed the traditional Aristotelian logic of subject and predicate propositions and states of affairs. There numbers, for example, do not occur as independent objects about which something is predicated, but as form, thereby dependent, as when we say: ‘3 houses.’ Sets as objects do not occur there either, because in set theory, judgments are not made directly about elements, but about sets, whole totalities of elements. If we make such forms independent, new higher-order objects emerge. For him, traditional Aristotelian logic was but a small area of pure logic which had to be distinguished and segregated from the formal ontology of the broader sphere of pure logic, which included the mathematical disciplines and was immense in comparison.

In the disciplines of the two higher levels, it is a matter of investigating what is valid for higher-order object formations determined in purely formal terms, grounded in the essence of logical forms and dealing with objects in indeterminate, general ways. Husserl described the second level as an expanded, completely developed analytics in which one proceeds in a purely formal manner. One calculates, reasons deductively, with concepts and propositions. Signs and rules of calculation suffice because each procedure is purely logical. One manipulates signs, which acquire their meaning in the game through the rules of the game. One may proceed mechanically in this way and the result will prove accurate and justified. Numbers function entirely differently on this level, where statements about numbers in which numbers are the objects are found. As an example of such arithmetical propositions, he gave: “Any number can be added to any number.” Here, he located, the basic concepts of mathematics, the theory of cardinal numbers, of ordinals, set theory, mathematical physics, formal pure logic, pure geometry, geometry as a priori theory of space, the axioms of geometry as a theory of the essences of shapes, of spatial objects, but also the pure theory of meaning and being, a priori real ontology of any kind, ontology of nature, ontology of minds, natural scientific ontology, sciences of value, pure ethics, the logic of morality, the ontology of ethical personalities, axiology, pure esthetics, the logic of the ideal state or ideal world government as a system of cooperating ideal nation states, the ideal of a valuable existence, essence-propositions about objects insofar as they are objective truths and as truths have their place in a truth-system in general, etc.

On the third and highest level of formal logic Husserl placed his science of theory forms, his theory of manifolds or Mannigfaltigkeitslehre. It was to be a new method constituting a new kind of mathematics, the most universal of all, a technique for engaging in pure a priori analyses through an austere scheme of axiomatization, a matter of theorizing about possible fields of knowledge conceived of in a general, undetermined way and simply determined by the fact that the objects stand in certain relations, themselves subject to certain fundamental laws of such and such determined form, a science of deductive systems in general, a field of free, creative investigation made possible once form was emancipated from content. One is free to reason completely on the level of pure forms where the systems can vary in different ways. One finds ways of constructing an infinite number of forms of possible disciplines.

His manifolds were to be pure forms of possible theories which, like molds, are totally undetermined as to their content and not bound to any concrete interpretation, but to which thought must conform in order to be thought and known in a theoretical manner. Only a form is defined. It exists insofar as it is correctly defined, insofar as the axiom forms are ordered in such a way as to contain no formal contradictions, no violation of analytic principles. For example, one speaks of numbers in the formal sense, but one does not mean cardinal numbers, quantitative numbers, or anything of the kind, but anything for which formal axioms of the arithmetical prototype hold. If we drop the cardinal number meaning of the letters in the ordinary theory of cardinal numbers and substitute the thought of objects in general for which axioms of arithmetical forms are to hold, we no longer have arithmetic, but a purely logical class prototype of theory forms to which, besides innumerably many possible domains, the domain of cardinal numbers is also subject. For cardinal numbers, ab = ba holds. In constructing a manifold, though, one may just as well stipulate that ab ¹ ba.

He suggested this meaning for the theory of non-Euclidean manifolds: “Let there be a domain in which the objects are subject to certain forms of relation and connection, for which axioms of such and such a form are valid, then for a domain formally constituted in this way, a mathematics of such and such a form would be valid, there would then result propositions of such and such a form, proofs, theories of such and such a form. There is no domain. There are no actually given concepts, connections, relations and axioms. One simply says, if one had a domain, and if axioms of such and such a form obtained for it” (Husserl 1906/07, §19c-d).

Husserl believed that it was up to mathematicians, as the only competent engineers of deductive structures, to construct such theories and theoretical disciplines, while it was up to philosophers to engage in complementary reflections on the essence and meaning of the basic concepts and laws. He believed that the idea of a theory of forms of meanings as a discipline prior to the disciplines of logical validity was still completely beyond the ken of mathematizing logicians.

Sixth, Spencer-Brown stressed that by failing to make use of the crucially important fact that a calculus and its interpretation are distinct, one cuts oneself off from readily available forms of simplification, one of them, frequent in mathematics, being the underlying use of a construction that is devoid of interpretation in the particular application, but can be used to shorten the way to an answer in it (91). As an example, he gave the √ -1. He said that “perhaps the most significant thing” about his calculus “from the mathematical angle” was that it enabled one to use complex values in the algebra of logic, which are “the analogs, in ordinary algebra, to complex numbers” where they “are accepted as a matter of course,” the “more advanced techniques” being “impossible without them.” Yet in Boolean algebra, and thus “in all our reasoning processes, we disallow them.” He had seen how their Boolean counterparts worked perfectly well in practical engineering but, like the first mathematicians to use ‘square roots of negative numbers,’ had felt guilty about using them because he “could see no plausible way of giving them respectable academic meaning.” But, he was “quite sure there was a perfectly good theory that would support them,” if only he could think of it (x). So, he extended the concept of imaginaries to Boolean algebras, so that a valid argument might contain four classes of argument: true, false, meaningless and imaginary, something he saw as having profound implications for mathematics, logic, philosophy and even physics (xi). That imaginary values could be used to reason towards a real and certain answer, combined with the fact that they were not being used so in mathematical reasoning and with the fact that certain equations plainly could not be solved without using them, he reasoned, “meant that there must be mathematical statements (whose truth or untruth is in fact perfectly decidable) that cannot be decided by the methods of reasoning to which we have hitherto restricted ourselves.” If “we confine our reasoning to an interpretation of Boolean equations of the first degree only, we should expect to find theorems that will always defy decision, and the fact that we do seem to find such theorems in common arithmetic may serve, here, as a practical confirmation of this obvious prediction.” (81)

For his part, Husserl maintained that his chief purpose in developing his theory of manifolds had been to find a theoretical solution to the thitherto unclarified problem of imaginaries, to how in the realm of numbers, impossible, non-existent, meaningless concepts could be dealt with as real ones. In the early 1890s, he wrote to Carl Stumpf of how, in trying to understand how operating with contradictory concepts could lead to correct theorems, he had found that for imaginary numbers like √2 and √-1, it was not a matter of the possibility or impossibility of concepts. Through the calculation itself and its rules as defined for such numbers, the impossible fell away and a genuine equation remained. One could calculate again using the same signs, but referring to valid concepts, and the result was again correct. The calculation remained correct if it followed the rules even if one wrongly imagined that what was contradictory existed or held the most absurd theories about the content of the corresponding concepts of number. So, this must be a result of the signs and their rules.

He saw his theory of manifolds as the key to the only possible solution to that problem. Understanding the nature of theory forms had shown him how reference to impossible objects could be justified. It was formal constraints banning meaningless expressions, meaningless imaginary concepts, reference to non-existent and impossible objects that was restricting theoretical, deductive work, but resorting to the infinity of pure forms and transformations of forms freed one from such conditions and explained why having used imaginaries, what is meaningless, must lead, not to meaningless, but to true results. There are no negative numbers in the arithmetic of cardinal numbers because the meaning of the axioms is so restrictive as to make subtracting 4 from 3 nonsense. Irrational numbers, √ -1 are meaningless there, and so on. We cannot arbitrarily expand the concept of cardinal number, but we can abandon it and define a new, pure formal concept of positive whole number with the formal system of definitions and operations valid for cardinal numbers. And, this formal concept of positive numbers can be expanded by new definitions while remaining free of contradiction. One can operate freely within a manifold with imaginary concepts and be sure that what one deduces is correct when the axiom system completely and unequivocally determines all the configurations possible in a domain through a purely analytical procedure. It is the completeness of the axiom system that gives one the right to operate freely. A domain is complete when each grammatically constructed proposition solely using the language of the domain is, from the outset, determined to be true or false in virtue of the axioms.

Seventh, “To make their mathematical logic conform with the logic of Aristotle,” Spencer-Brown thought, Russell and Whitehead “introduced the theory of types, an “arbitrary principle… that effectively forbade all arguments that would have to be represented by equations of degree higher than unity.” They mistakenly introduced it “expressly to disallow complex values,” so that “in this field, the most advanced techniques, though not impossible, simply did not yet exist” and one was still constrained in one’s reasoning processes to proceed as in Aristotle’s time. However, all that had to be done, he believed, was to show that the self-referential paradoxes discarded with the theory were no worse than similar self-referential paradoxes considered acceptable in the ordinary theory of equations and thus resolvable by introducing imaginaries. It was possible to solve equations of higher degree that the theory of types had excluded from ordinary logic and that had been undertaken. He said that when, in 1967, he showed Russell the proof that his theory was unnecessary, he abandoned his belief in it and admitted that it was “the most arbitrary thing” he and Whitehead had ever had to do and was glad to see the matter resolved. However, though Russell and Spencer-Brown agreed that a mathematical proof with a logical argument requiring an algebraic equation of higher degree was possible, neither of them could then imagine what it might look like. (x-xi, xviii, 198)

In any case, Russell had recognized decades earlier that the theory of types could not be “the key to the whole mystery” of the onerous contradiction he had worked so hard to evade. He saw that deeper problems caused it to break out afresh and “further subtleties… needed to solve them.” After all, the theory was but an ad hoc attempt to evade the contradictions derivable in Frege’s system by restoring the formal structure that he programmed his system to break. Frege had placed the need to lay hold of self-subsistent, independent, objects at the heart of his system in a way that forced him to devise a law to let him pass from a concept to its extension, something which he believed was “forbidden by the basic difference between first and second level relations.” Yet, he temporarily convinced himself that he might assume “an unprovable law” to legitimate the illicit transformation.

Russell said that the contradiction had taught him that classes could not be independent entities in the same sense in which things are things, that if a word or phrase that is devoid of meaning is wrongly assumed to have an independent meaning, false abstractions, pseudo-objects, paradoxes and contradictions were apt to result. Along with his theory of definite descriptions, no-classes theory, axiom of reducibility, the theory of types was one of his various attempts to get rid of the contradiction-generating imaginary objects that Frege’s system generates. However, he believed that “without a single object to represent an extension mathematics crumbles,” ‒ as if mathematics would crumble because some thinkers had devised some bad theories.

In comparison, for Husserl, as seen, sets and classes were not independent entities occurring in traditional Aristotelian logic, but higher-order object formations figuring on the second level of formal logic, because in them, judgments are not made directly about elements, but about sets, whole totalities of elements. In fact, Frege, Russell and Husserl all concluded that the essential differences between dependent and independent meanings were of the highest importance, inviolable and “founded deep in the nature of things,” so that antinomies, contradictions, paradoxes, fallacies, nonsense, confusion, absurdity, mysteries inevitably result were they are not respected. Specifically, insidious problems with pseudo-objects, inference, existential generalization, type ambiguities, substitutivity of identity, semantical paradoxes, namely much of what analytical philosophers have been battling since Frege’s time, inevitably creep into reasoning. Frege’s logic in fact led to what he himself called a thicket of contradictions, what Russell called a bewildering maze, to Quine’s fragmented world of rabbit parts, stages, and fusions, river stages and kinship, person stages where the ontologies of physical and mathematical objects are but myths relative to an epistemological view, not to mention the deep confusion that, for a while, during the 1980s, analytic philosophers admitted to experiencing.

As an eighth connection, I might propose Spencer-Brown’s and Russell’s exchange regarding functions and Husserl’s theories about the same, but my time is up.