How can it be, that a Wittgenstein scholar like Wilfrid Sellars, in his recent book Within the Space of Reasons (Harvard UP 2007),when he discusses Wittgenstein’s view in the Tractatus on predication, doesn’t hesitate to leave the entire debate in the philosophy of mathematics which was culminating in Wittgenstein’s time, completely without mention. Undoubtedly, the problem of predication in philosophical logics regards issues of quantification. So why does Sellars not address this debate which was so feverishly concerned with the “existential” or “ontological” status of socalled “imaginary” quantities, and with the theoretical and technical results of “complex” analysis – namely anything which involves electricity, electro-magnetism, and, eventually, the quantum level of things.
Wittgenstein’s views on predication, insisting that the configurations of objects are expressed by configurations of names, names being what Wittgenstein calls “complex signs”, was eminently concerned with this debate. I will support this view more extensively and discuss the arguments that make me reason so in a moment, before coming back to state my puzzled question about Sellars’ discussion. But first I would like to introduce the background to this debate which I mean here.
It can perhaps be said to have started, as an academic debate, 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.“ It is hard to see at first what this statement implies, and why Cayley holds it of such importance to devote his entire speech to it, and this with such a tone of gravity in his voice. Has not geometry, we are well justified to ask, at least since its analytic turn to the Cartesian Space of abstract representation lost its cosmologically ordered elementarity in favor of merely providing an imaginary plane for experimental science?
So what exactly is Cayley referring to with this imaginary space in geometry – what had happened?
The crucial sentence is the following specification Cayley gives: „I use in each case the word imaginary as including real.“ Both terms, imaginary and real, are meant in their number theoretical sense, but nevertheless, the issue Cayley wants to address is not one dedicatedly for mathematicians. Quite to the contrary, his concern is, as he continues: „This has not been, so far as I am aware, a subject of philosophical discussion or enquiry“.
The issue raised in this address concerns the grand question of whether and in what sense a notion of space is relying on experience and subjectivity. Yet the extraordinary take it presents, for philosophers, is that this question is raised out of the field of number theory. This is an unusual perspective. How can we, philosophically, conceive of space such that it features „as a locus in quo of imaginary points and figures“, or in other words: as the scene of the event of a peculiar kind of „elementarity“ where figures are articulated out of a numerical domain of which we must, somewhat paradoxically, trie to understand that it literally „includes the real“. By „including the real“ is meant that the numerical domain at stake is said to extend beyond the infinite number line of the real numbers. In their continuity, the domain of the real numbers comprehends all the positive and negative integers, zero, as well as all the rationals and the irrationals. It is indeed difficult to picture, mentally, what could be left out by the real numbers, but this is precisely the point of Cayley’s address. From the perspective of number theory, Cayley‘s question considers the possibility of a kind of intellectual intuition, and it considers that number theoretical questions of quantification may host something like forms of construction which might hive off such a notion of intuition out of the threatening swamps of unconditioned revelation in a mystical or theological sense.
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 a product of primes – just like the Fundamental Theorem of Arithmetics holds. This may seem like a fancy question for number-crunchers, not for intellectuals in general, but just consider that none of our electronically maintained infrastructures today would be working without those quantities. And yet, their usage is still today commonly put into rhetorical brackets which claim that only the „real“ part of these operations was of importance, philosophically, whereas the imaginary part is called „but a technical trick“ which we can apply when dealing with symbols. Contrary to this view, Cayley raised the question concerning the „nature“ of such tricks.
What was preoccupying Cayley, and many others in the second half of the 19th century, was the unsettling suspicion that we cannot exhaustively address reality by investigations following the Cartesian formula verum et factum convertuntur. With the emerging of 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 that could be settled but by a) concepts, or b) the constructivist stance working with algorithms and testing empirically. All sorts of symbolic quantities – not only zero, negatives, irrationals, infinitesimals, but also congruences and other genuinely algebraic numbers could be used as powerful tools to represent not only geometrical quantities anymore (as in Cartesian Analysis) but one level more abstract, the Integers which represent the geometrical quantities. Mathematics was exploring a newly developed capacity to render-present, symbolically and insofar intersubjectively, by pure acts of intellection and 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 negative Grössen in die 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 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 nearly all the different camps, from phenomenological schools to analytical ones, around the turn of the last century. It is often forgotten that Husserl, Whitehead and Russell all started out writing on this subject before the splitting into different vectors of valuing thought have emerged, i.e. before the publication of the Principia of the latter two, and before the phenomenological writings of the former. Let us give a brief diagrammatic 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: „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). The dazzling status of symbols in these newly emerging branches of Algebra, between concept and number, logics and mathematics, is so important because while the former is judging existential imports, or „content-conceived-and-formally- expressed“– 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. The dazzling status of symbols had been widely neglected because in the emerging fields of Symbolic Reasoning, to shortly summarize Whiteheads argument and motive for writing his Universal Algebra book, it cannot clearly be distinguished anymore whether a statement is to be treated as a mathematical statement or as a logical statement.
Also Russell has 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, and the diverse „metageometries“, as he calls them, that had emerged on this basis. 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 that is absolutely free from any psychological or empirical uncertainty. On the other side, 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 – ten years before Russell and 11 years before Whitehead), and much later in his career he returned to this issue and devoted a study on The Origin of Geometry (1936).
Even after the advances in propositional calculi, in formalizing propositions as function and classes as sets, after the introduction of the notion of the „incomplete symbol“ by Russell and Whitehead as a kind of a bridging principle across levels of ramification and between types, even after Wittgensteins Tractarian semiotization of the incomplete symbol notion into a sign-symbol dynamics which gives primacy to on the rôle of learning instead of representing a piece of knowledge, even after Heidegger‘s Sein und Zeit 1927, the algebraic symbolization which had broken free within mathematics and logics could not in any way be regarded as settled. [1]
In The Origin of Geometry Heidegger‘s teacher Husserl asked about the premises for the commonly accepted practice according to which every geometrical figure can be defined algebraically, by an equation from which – so the problematic practice – „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 algebraic-analytical methods calculate directly with what was then called „pure quantities“, quantities which supposedly do not need the assumption of intuition for working out their proofs rigorously. Whitehead had also commented on this generalization of the quantity notion: „The introduction of the complex quantity of ordinary algebra, 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.
It is important to realize that there were, at the time, two competitive vectors emerging. One which claimed a possible mathematization of all of logics, on the assumed basis that all inferential necessities ought to be dissolved into a probabilistic framework where we can at best, as we know it from physics, look for regularities as natural laws by a combination of mathematical and empirical inquiry. Of such characteristics are the Laws of Thought which George Boole had in mind when he developed the kind of symbolic algebra in 1854, which, in combination with IT, brings us the powerful and pragmatic practical operability with formal languages as we know it today. In theoretical terms, however, the competing vector has been much more successful sofar, namely the logification of mathematics in the tradition of Gottlob Frege up to Carnap, but affecting also Heidegger and featuring Alain Badiou perhaps as its latest keeper of that Grail.
The secret danger that is being kept or contained, in this grail, is the susceptibility of the argument which related calculability – and especially, today, computation – with necessity to the adventures of abstraction. This argument had always been a philosophical-political argument, and it is such also today. Only, it had seemed for a short time span as if it could be politically sentenced into a proper place, if only it were guarded safely enough within stacks and stacks of complicated and formalized generalizations from the pleasures and fertilities of abstract thought and learning.
Whitehead, in any case, was well aware of this susceptibility of the necessity argument when he explained the confusion around the status of abstract algebra‘s symbols, which works with mathematical definitions that take the form of propositions, i.e. they embody an act of inference, a subject-predicate relation – with the crucial difference, however, that different from logical defintions, mathematical concepts are never concerned with a transitive, in the sense of extensive, dimension of their definitions. This is what makes up their abstract symbolicness: neither form nor content, in any direct sense, they are something like the encoding of a form of structure for an unknown quantity. As such, as an encoding of a form of structure – notice the indefinite articles in both cases, an encoding, and a form of structure – algebraic quantity-expressions are „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), writes Whitehead and distinguishes such conventional mathematical propositions from logical ones by pointing out that the former make no existential imports what so ever. The troubling question hence, regarding the symbolicalness of those symbols is, that those propositions take the same form as logical propositions, and criteria for distinguishing them both is what is at stake. One, after all, is expected to regulate our statements about the world, while the other is not commonly ascribed a direct regulatory rôle in reasoning. Without clarifying this, any algebraic take on geometrical and also mechanical, technical questions would rely on nothing but analogy, on the assumed affinity of those abstractly created entities with the properties of real existing things.[3] In effect, this means that the link between calculability and necessity were broken.
Let us at this point interrupt and quickly detour briefly to Sellars discussion. He starts out by the following quotes from the Tractatus 3.1432:
‘We must not say: The complex sign “aRb” says “a stands in the relation R to b”, but we must say, ‘That “a” stands in a certain relation to “b” says that aRb.’
Sellars discussion centers around the Russellian interest of how Wittgenstein’s “complex signs” might be understood as “facts”. Yet this interest is on a level not abstract enough to address the issue at stake, as Wittgenstein was not speaking in the language of the Principia, nor does it seem maintanable to me that Wittgenstein had subjected his own thought in the Tractatus to the scope and interest of the Principia, and the ideas about types and ramifications put forward therein). Much rather, I would like to think, has Wittgenstein put is own thought in relation to the very problem that the Principa as well tried to tackle with: namely how to address this symbolicness of symbols introduced into intersubjective reasoning by abstract algebra. It is clear that Russell himself held a clearly conservative stance on the challenges by symbolic algebra, as his Dissertation made very clear. It also seems rather straight forward that Russell – different than Whitehead – has tried to “fix”, as he might call it, the strictly realist solution put forward in the Principia, for the powerful new ways of dealing with abstract symbols, in their algebraic ideality, in logics and mathematics. Clearly, it seems to me, Wittgenstein did not think this a satisfying stance (consider that he always thought Russell does not understand his views in the Tractatus properly) and seeked his own ways. Ok, let go back to our larger contextualization of the problem.
For Husserl with his strong methodological sensitivity, hence, an interest in the status of those signs as a genuinely symbolic one was profoundly misleaded. For him it was clear that every logical practice centers around the distinction between necessity and contingency, all reasonable dealings in the form of conceptualization and propositions, in short: all acts of inference need to be grounded on some intuitive facts.[3] Algebraic-analytical methods, so Husserl held, are not actually working with concepts and propositional forms, they are merely making use of what he calls Hilfsbegriffe, auxiliary concepts, like that of the imaginary, the irrational, the continuous, or the differential and the integral. All the mathematical concepts which involve them – hence any definition which makes reference to one of these auxiliary concepts – must be made dependent on elementary arithmetics, i.e. on a logical definition of numbers. „All the complicated and artificial constructions, which are also called numbers, the fractions and the irrationals, the negative and the complex numbers, have their origin and permanence in the elementary concept of numbers and the relations defining those; along with the latter, also the former would fall, yes, mathematics at large would dissolve. Every philosophy of mathematics ought to start out with an analysis of the concept of number“ (p.8). [4]
The troubling question can be put like this: how can we conceive of the symbolicalness of symbols in Universal Algebra? For Whitehead it was an open question. For Russell, just as for Husserl, it was clear that assuming for symbols a status of their own (i.e. a status as quantities!) – one that is not grounded in geometry nor in arithmetics nor in language – would be profoundly misleaded; they both held firm – albeit in different versions – that symbols need to regard necessary facts.
Yet with algebraic expressions, there is an objectivity proper to symbolic encodings that allows the encoded to be referred to and represented in purely general terms. This generality is not gained by strictly deductive reasoning, and it nevertheless does not depend upon psychological subjective experience. Conceiving of a genuine symbolicalness of symbols means tackling with the primacy of abstract algebra as the means for formulating symbolic constitutions. These constitutions provide the structures for what can be expressed as the cases of this peculiar algebraic generality. Strictly speaking, the fundamental theorem of algebra leaves the general applicability of arithmetics problematic. If algebra is granted a universal status, applying arithmetics turns into a practice of engendering solutions as cases, i.e. of calculating solutions which are not, strictly speaking, necessary solutions. For the majority of philosophers, an affirmation of this would be a straight forward capitulation of enlightenment philosophy at large, because it means that the strong link between calculability and necessity were broken, and along with that, the distinction between philosophy as metaphysics and philosophy capable of critique.
Yet if Algebra‘s universal status is considered as complementing a probabilistic element, into which the formula – i.e. the algebraic identity-as-relation-to-be-established – is seeked to be integrated, all that the fundamental theorem of algebra asks for, philosophically, is to ascribe a different modality to the abstract objects of mathematics and logics than that of necessity or contingency.
I read Deleuze‘s concept of the virtual in these terms, as the modality for the experienciability of things which are not merely possible but real. Virtually real means in principle fully determinate yet never actually exhaustively determinable. We can consider the virtual as the modality of the things engendered by abstract thought. The symbolicness of symbols encodes forms of structure for determining unknown quantities, and is itself neither form nor content. Such algebraic quantity-expressions can be considered „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), writes Whitehead about such vectors of imaginary verticality.
But this is for another post.
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[1] The original titel hence was The Origin of Geometry as intentional-historical Problem, it appeared in the same year as Heidegger‘s reflections on the mathematical from which we started out. This late book of Husserl served as a starting point for Derrida‘s endeavor to formulate a fully generalized logic, a kind of a logified logic, one crafted not by mathematization and symbolization but by by welding together grammar and logic into a non-negotiable bunker, barring existential import, in short: quantification, by sentential judgement (not conceptual judgement!) into absentia from legitimate discourse altogether. For Derrida, this is the necessary condition for treasuring metaphysical indeterminacy against the threats of positivism. If I had time to expand on this, I would have liked to point out an almost tragic complicity between the two stances – grammatology and positivism. Yet with regard to my argument here, which aims to plot out the main pillars for considering the articulate-ability of quantities, such an excursion is of minor interest and would be nothing but commentary.
[2] with that the generality of their expressed identity (which would be: the determined unknown quantity) is genuinely heterogenous, as the name polynomials, many-nominal, directly points out. Unknown quantities, expressed as a determined identity in algebraic form, take their „origin“ in a symmetry break. Of such a heterogenous character is the generalization of quantity which Symbolic Reasoning was practicing: it became according to the indexes which the specific code allows for.
[3] These must be accounted for by other means than those provided by the spatiotemporal a priori which Kant had established for legitimating the status of inferences as judgements .
[4] „Alle die complicirteren und künstlicheren Bildungen, die man gleichfalls Zahlen nennt, die gebrochenen und irrationalen, die negativen und complexen Zahlen, haben ihren Ursprung und Anhalt in den elementaren Zahlbegriffen und den sie verbindenden Relationen; mit den letzteren Begriffen fielen auch die ersteren, ja fiele die gesammte Mathematik fort. Mit der Analyse des Zahlbegriffes muss daher jede Philosophie der Mathematik beginnen.“ (p.8)
monas oikos nomos 