A resurfacing of the debate about amphiboly and concepts, and the relation of this theme to number theory, took place in the 19th century and can perhaps best be associated with an address by Arthur Cayley, a British algebraist working on variational calculus and invariance-theory, to the the British Academy for the Advancement of Science in London in 1883. There is a notion, he told his fellow intellectuals, which is „really the fundamental one (and I cannot too strongly emphasize the assertion) underlying and pervading the whole imaginary space in geometry“ – and this notion, he continued, had not yet been a subject for philosophical discussion or enquiry. It is hard to see at first what this statement implies. Had not geometry, at least since its analytic turn to the Cartesian Space of abstract representation, long lost its attunement with a cosmological order in favor of merely providing an projective plane for experimental science? Have experiments, especially if based on mathematics, not always been “imaginary” ?
The imaginary numerical domain Cayley is referring to is that of the Complex Numbers, and what this domain allows – as we could perhaps put it – is operations on real infinities. The crucial point about them is that their conditioning cannot be thought of as natural, if we understand natural by its more conventional notion, namely that the quantities describing it (the integers) need to be factorizable in a unique and necessary way as the product of primes – just like the Fundamental Theorem of Arithmetics holds.
What is at stake with the imaginary numbers is a conflict over primacy, in philosophical terms, between the Fundamental Theorem of Arithmetics and that of Algebra. This conflict literally involved how to think about ‘roots’. While this is meant in its thoroughly technical, mathematical sense, it seems clear that in its philosophical relevance, reducing the question at stake to technicalities doesn’t help to relax the virulency of the problem at stake – the problem of roots being, after all, the most powerful ideological instrument in a political climate of emerging nation states, their striving for hegemony, and the emerging discussion around scientific notions of identity, race, legitimacy arguably stripped from all ‘pre-modern’, ‘pre-critical’ theological or metaphysical ‘ballast’.
Thus, it is unsurprising that this so-called “complex” analysis, which introduced the necessity in philosophical logics to complexify their notion of quantification with a non-absolute, ungrounded, purely relational aspect of quantifiability, provoked a problem of predication. Philosophical logics which became almost feverishly concerned with the “existential” or “ontological” status of so-called “imaginary” quantities.
What was preoccupying Cayley, and many others in the second half of the 19th century, was that with the emerging new approaches linking algebra and arithmetics, the status of numbers had grown problematical in a new way. While on the one hand, the applicability of arithmetics could be extended to completely new domains, these extensions grew ever more non-intuitive and eventually involved purely symbolical considerations. As such, they could be settled, but the settlement of them raised the role of method into a problematical status. Mathematics was exploring a newly developed capacity to render-present by pure acts of intellection and to an incredibly large degree independent of experiments involving any kind of direct measuring.
Let us extend more on the particular context in which Cayley‘s address can be located from today‘s retrospective. The full passage of what we have cited goes as follows:
“In arithmetic and algebra, or say in analysis, the numbers or magnitudes which we represent by symbols are in the first instance ordinary (that is, positive) numbers or magnitudes. We have also in analysis and in analytical geometry negative magnitudes; there has been in regard to these plenty of philosophical discussion, and I might refer to Kant’s paper “Ueber die negativen Grössen in der Weltweisheit (1763)”, but the notion of a negative magnitude has become quite a familiar one, and has extended itself into common phraseology. I may remark that it is used in a very refined manner in bookkeeping by double entry. But it is far otherwise with the notion which is really the fundamental one (and I cannot too strongly emphasize the assertion) underlying and pervading the whole imaginary space (or space as a locus in quo of imaginary points and figures) in geometry: I use in each case the word imaginary as including real. This has not been, so far as I am aware, a subject of philosophical discussion or enquiry. As regards the older metaphysical writers this would be quite accounted of by saying that they knew nothing, and were not bound to know anything, about it; but at present, and, considering the prominent position which the notion occupies – say even that the conclusion were that the notion belongs to mere technical mathematics, or has reference to nonentities in regard to which no science is possible, still it seems to me that (as a subject of philosophical discussion) the notion ought not to be thus ignored; it should at least be shown that there is a right to ignore it. ” (p. 784)
Indeed, Cayley’s appellation to philosophers for attending to the imaginary units was not without impact. When asking ourselves what the relevance of all of this might be to us today, it is important to be aware that the number theoretic take on the problem of space and experience has transversed, around the turn of the last century, nearly all the different camps, from phenomenological schools to analytical ones. It is often forgotten that Husserl, Whitehead and Russell all started out writing on this subject before the splitting up into different vectors of valuing method in thought have emerged, i.e. before the publication of the Principia by Russell and Whitehead, and before the phenomenological writings of Husserl. Let me give a brief indexical sketch through the larger context around the turn of the last century.
Whitehead had published his A Treatise on Universal Algebra in 1898, in order to present „a comparative study of the various Systems of Symbolic Reasoning“ that had been allied to ordinary Algebra since mid 19th century. Those Systems of Symbolic Reasoning, as Whitehead calls them, had been looked upon „with some suspicion“ by mathematicians and logicians alike – as he puts it: „Symbolic Logic has been disowned by many logicians on the plea that its interest is mathematical, and by many mathematicians on the plea that its interest is logical“ (p. vi). To shortly summarize Whiteheads argument and motive for writing his Universal Algebra book, we can say that the dazzling status of symbols at the time (and still today!) had been widely neglected because in the emerging fields of Symbolic Reasoning, it cannot clearly be distinguished anymore whether a statement is to be treated as mathematically or logically. The established methodical relations and notions of concept and number, logics and mathematics have been thwarted by this new role of symbols. The conceptual or logical way of symbolic reasoning is concerned with judgements involving existential imports – what we commonly distinguish today as predicate logic from propositional calculus. The latter, strictly mathematical way of symbolic reasoning, is held to be concerned with conventional definitions only, without existential import.
Also Russell had published, prior to his work in the Principia, on these confusions. He has given – one of the most informative accounts of what was going on throughout the 19th century in terms of an Algebraization of Geometry. In his dissertation An Essay on the Foundations of Geometry (1897) Russell is concerned primarily with reconsidering the notion of the Kantian a priori, and its distinction between „the necessary and the merely assertorical“ in such a way that a notion of knowledge may be maintained which is absolutely free from any psychological or empirical uncertainty. Roughly 10 years earlier, Husserl had completed his Habilitation with a study on the Notion of Number in terms of Psychological Analysis (Zum Begriff der Zahl, psychologische Analysen, 1887 – against which Russell’s stance was surely directed, among others.
Here, Husserl asked about the premisses for the commonly accepted practice according to which every geometrical figure can be defined algebraically: „it shall be possible to infer directly from algebraic relations to geometrical ones, without any chance for habits of thought to impose themselves with the necessity of intuitive facts“, as he put it. The crucial wording is that of ‘intuitive fact’ – for Leibniz and his interest in intellectual intuition, this would have been a severely fallacious formulation. We owe the link between facticity and intuition to Kant – this, indeed, is what he thought to gain by banning amphiboly from his notion of reflection. Kant knew well that leaving the role of his forms of intuition – Euclidean space and Newtonian time – to arithmetics would re-introduce the Leibnizian specter into his system. As far as I know, he did consider this at one point, but quickly dispensed it again. This is, however, exactly what Frege had set out to develop. Frege did not addressed the problem of the amphibolic life of quantities within symbolic formulas as a problem. Instead he declared, undoubtedly himself in full awareness of the problem, the natural numbers as ‘ideal positivity’ from which everything else can be inferred.
Whitehead had commented much more cautiously on the problematic algebraic generalization of the quantity notion: „The introduction of the complex quantity of ordinary algebra” he wrote, “an entity which is evidently based upon conventional definitions, gave rise to the wider mathematical science of today. The realization of wider conceptions has been retarded by the habit of mathematicians, eminently useful and indeed necessary for its own purposes, of extending all names to apply to new ideas as they arise. Thus the name of quantity was transferred from the quantity, strictly so called, to the generalized entity of ordinary algebra, created by conventional definition, which only includes quantity (in the strict sense) as a special case“ (p. vii/viii). Conventional definitions, for Whitehead, refer to mathematical definitions. What is at stake is the relation of calculability – and especially, today, computation – to the necessity of the solutions thus produced.
Whitehead had tried to clarify the confusion around the status of abstract algebra‘s symbols, and insisted on the problematicity involved in the distinction between logics and mathematics. To him, the abstract symbolicness of algebraic quantities was „pure“ in a quasi- Kantian sense, they make reference to no specific magnitudes at all, and work only with conceptual definitions. „It sets before the mind by an act of imagination a set of things with fully defined self-consistent types of relations“ (p. vii), he writes. He saw very clearly that without attending dedicatedly to this obscure ‘symbolicness’ proper to the Symbols used in Algebra, the distinction between mathematics and logics gets confused. Either way of subsuming (and hence conflating) one over the other – as an act of hygienic purification of this obscure ‘symbolicness’ proper to the Symbols in Algebra – comes at the cost of instigating a move of totalization and enclosure: logics under mathematics leaves us with the necessity of contingency, while mathematics under logics leaves us with contingency of necessity.
What I would like to suggest is that the symbolic nature of polynominality occupies an unsought and hypothetical vantage point in the political and economical context for all of 20th century philosophy – indeed, it seems to lie at the heart of the development of a social theory.
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