The world is everything that is the case – I would like to take this famous line from Wittgenstein‘s Tractatus as a starting point. The world is not the totality of things, he says, but that of facts. I would like to consider, inversely, the world as the totality for whatever can be the case. After this inversion we can – a small deviation not withstanding – keep with Wittgensteins language game and call the totality of whatever can be the case the totality of artefacts. Artefacts capture and embody acts of concentration by intellect, not things that have happened or are given. What distinguishes them as artefacts from facts is that they conserve an act of concentration by condensating this act into manifest form. (1)

The crucial question, for considering the world as the totality of that which can be the case, regards what I would tentatively suggest to call the auxiliary structure needed for being able to say something about this world at all. My core interest in the following concerns the possibility of a philosophical grammar. I call it an auxiliary structure and not an infrastructure, because referring to it or behaving in it needs thought that considers. Such a grammar allows for conception, of course, but the act of conception it structures is ampliative. I would like to consider the possibility for a philosophical grammar in which conceiving is engendering-by-inference. Ampliative inferences are inferences capable of broadening a terms extension beyond the possibilities that were contained in the premises. Such thinking-as-conceiving involves an aspect of inception, of beginnings. My interest, in short, is to regard artefacts as articulations of such a philosophical grammar, in pursuit of an architectonics that were proper to the city today. Of course this article can be no more than an early step in this pursuit.

The brief sketch I would like to layout in this paper to support such an interest in artefacts as condensations of intellectuality departs from exploring a peculiar proximity between Ludwig Wittgenstein, Martin Heidegger and Gilles Deleuze. A proximity which may seem quite unlikely, at first sight, but for which I would like to argue along the following lines: all three share a common interest in the Kantian insight that reason conditions experience, but more importantly, they explore this insight in relation to acts of learning rather than objects of knowing. This distinction creates, despite all the obvious differences if not incompatibilities, a peculiar proximity among their thinking.

In their own individual ways, Wittgenstein, Heidegger and Deleuze have evoked the ancient sense of mathesis as an art of such conditioned learning. They have embraced the challenge that for learning, the conditions can never be sufficient nor clear and distinct. In such a sense of mathesis, to which I will refer to in the following as mathetical, learning is less concerned with issues of representation or recognition than with an act of appropriation and inhabitation of intellectual capacities and abilities. Learning in a mathetical sense involves a kind of privation which inverses the usual sense of the word: it involves a privation which engages in a relation of giving without depriving.

In the following I would like to extend on this aspect that for learning the conditions can never be sufficient nor clear and distinct. I will suggest to view artefacts in a broad sense – be it software, architecture, film, music, a piece of technology, suggestions for policies, tools for financing, business plans, recipes or theory books – as the manifest instances of acts of learning. I will regard artefacts as cases which are conceived and engendered by ampliative reasoning; this means, in full demand of that gesture, withstanding the temptation to subject thought to the comfort zone provided by artefacts if we assign them the a-conceptual status of incommensurable singularities, or the not-engendered one of generic stem cell like entities. (2)

Considered in their relation to the acts of learning which they manifest, rather than to their status as objects of know-how, artefacts are condensations from the outer space of intellectuality. They are aliens-from-within, if you like. Unlike pieces of art, they are popularized and in that sense de-capitalized acts of concentration. If it makes sense at all to say – with consideration – that they are, we might have to extend the conceptual leap from being to existence towards insistence, and claim that just like things are insofar as they are there (Dasein), artefacts are insofar as they are here. If being corresponds to things-as-they-are-named, and existence to things-in-their-thingness, we can say that insistence corresponds to the prespecific-genericity proper to things considered within the fantasmatics of what can be learnt mathetically.

1 Within the outer space of intellectuality

Wittgenstein had started to sketch out a philosophical grammar for addressing things-as- facts. A philosophical grammar for addressing things-as-artefacts considers things in their pre-specific genericity. It assumes they can be named, in this pre-specificity which they manifest, by mathetical instead of literal names. We can find such mathetical names, I would like to suggest, in algebraic polynomials.

The etymological meaning of polynomials is having many family names. Polynomials name heterogenous things, hybrids that comprehend aspects of many generic lineages. Polynomials name things that have no natural belonging – if natural belonging means that the identity expressed belongs to exactly one genus or genre.

From a pragmatic viewpoint, we can characterize the mathetical context of polynomials today as follows. For all sciences working with methods of probabilistics, factoring polynomials is as ordinary, as elementary and capability-dependent a practice as composing more or less well-formed, more or less well-reasoned arguments in sentences is. Polynomials feature in systems of differential equations, and they are especially useful when describing processes which do not unfold uniformly and steadily in space and time. Polynomials allow to map, probabilistically, they are the building blocks of all sciences involving computation and electronic technology today.

Such a grammar is a formulaic grammar, and the identities it names are both generic and pre-specific. Hence they are – in a fully determinate yet never exhaustively determinable sense – evoked. They are literally called out, summoned, roused, out of the outer space of intellectuality we all engage with when we learn. These pre-specific identities are only insofar as they are articulated, a bit like daimonions, manifest voices, like those we encounter in erotic recognition. Viewed in their prespecificity, these identities were more adequately called erogenic than generic; they vibrate of erogenous affectivity, these identities are the pure skinning and membraning of the restless intermingling of abundant responsitivity, which is proper to things when dissolved into the symbolicalness of polynomials. Without the evocation of grammatical articulation, the polynomial space of predication comprehends a mere happening of subtle violence.

The attractive promise is that such a grammar may provide us with the ability for systematic and structural thought in domains where reason is not only insufficient but also abundant. In other words: whenever we refer to the probable, the topical, the urban. The space of polynomial predication comprehends virtually anything that can be thought rigorously. As such, it exhausts neither thought nor the outer space of intellectuality – it relies on input or investment that cannot be deduced or induced from within it; but it does provide the virtual consistency of how we can consider this where where all the artefacts that ever were, are or will be here, as manifestations of acts of intellectual concentration, are being conceived. The outer space of intellectuality is vaster than we can imagine or overview, in any one moment or in any summation of moments. But we can be positive that it is not infinite. It is abundant yet not unlimited, because it is only insofar as it is inspired by the thinking of people, finite beings.

The modality of its consistency is virtually real – not actual, and not even necessarily actually possible. It consists in the purity proper to differences of which the parts are potentially set in relation to anything at all, within the boundaries provided by the space of polynomial predication as it is dramatized. Within the outer space of intellectuality, the differences are not yet actually related to a code that joins them; they are not rendered present, they are not symbolized. Or in other words, the erogenous skinning, membraning of this differential space is not yet responding to something. It is not yet organized by the reciprocal liminalization provided by a formulaic identity-relation, i.e. by an algebraic equation. As such, the outer space of intellectuality is the springing origin of the dignity of thinking. Its fertility engenders. It is an erotic, a cultivated fertility, not that of natural reproduction qua multiplication. Thought that strives for articulating the contradictory consistency of this space – it‘s unjoined differences-in-relation – involves the genesis of its actualizations. Thought that claims authority within the outer space of intellectuality is involved in a static genesis, through its acts of intellection. It does nor revolve around a center. Rather it involves condensation, a skinning, a membraning, falling off from its rotating acts of concentration. The products of this fertility are subtle and vulnerable, to the point of their virtual non-existence – if the grammar that expresses these condensations is incapable of receiving them as cases.

In this, thought in such domains of abundant yet insufficient reason is different from the dynamics of dialectial inferential movement. Dialectical inference negates the process of condensation, its mediality and simulacras, and instead of affirming this process it strives for the annihilation of contradictions through normalization. Mathetical inference, on the other hand, strives to articulate the contradictory in myriads of ways. It strives to articulate polynomial names as quantities, not names as generas or singularities. Polynomial names involve diverse ranges of powers, articulating them into formulaic systems of equations means mediating between the involved ranges and orders of powers. Mathetical inference can dissolve the violence involved thereby through a posology of pharmaceutical doping. Such a posology, in the proper sense of logos, does not need to be invented anew, I would like to suggest with the lines of argument presented here. Rather, it can be found in and extracted from algebraic structuralism which attends to the world as the totality of artefacts. Algebraic structuralism is a categorical structuralism, (5) and it looses the dogmatism that usually goes along with categorical thinking if we relate it to domains of abundant and insufficient reason. (6)

Heidegger has paid attention to the mathetical alternative to the notion literacy. He referred to it with cautious consideration as the mathematical, and meant by it, in an open sense, that which can be learnt. In Die Frage nach dem Ding (1935/36)7, Heidegger comprehends genuinely philosophical thought as thought revolving around the notion of the thing. The mathematical is concerned with things, he says, insofar as we can learn about things. Not simply how to use them, name them, or master them, but rather how we can learn about things in their thingness. About bodies as bodiliness, plants as plantness, etc. With these abstract terms Heidegger does not refer to an idea of a thing, but to a certain kind of intellectual experience of an object as a thing in a certain appearance. This experience is conditioned by a sort of intellectual intuition, yet it only concerns the possible awareness of our interiority. Such experience is not real, for Heidegger, it is purely projective.8 He introduces the mathematical as that which at one and the same time gives things to us, and allows us to learn about them: „The mathematical, this is what we intrinsically already know about things, what we do not have to extract or abstract from things but what we, in a certain way, bring along ourselves“. (9)

Learning, he continues, is a giving to oneself what one already has. It contains an element of a-substantiality which for Heidegger is, in this certain mathematical way, strictly personal.

3 There is a naturalness proper to reasoning

Different from Heidegger, Gilles Deleuze has suggested to consider a possible generalization of this peculiar element of a-substantiality that is involved in mathetical learning.10 He conceives of purity not as an attribute, but as an elementarity, as a transcendental quasi-naturalness proper to reasoning. This elementarity, for Deleuze, is conditioned by three inseparable principles: that of pure quantitability, complemented by those of pure qualitability and pure potentiality. Raising these terms, the quantitative, the qualitative, and the potential, to the level of –abilities is crucial for understanding Deleuze. He calls them principles, but – by raising them to the level of abilities – he manages to call them so in such a way that they do not presuppose anything given. These principles do not allow us to recognize, imagine or picture ideas by thought. Rather, ideas bath in this pureness as a natural elementarity, and this pureness grants that thought is natural in a different way from assuming its Good Nature. Within such a setting, ideas need to be indexed before they can be treated analytically or synthetically. They need to be actively coded. Thought can engender thinking, within this elementarity of pureness, because thought is mobilized by what Deleuze in a later work calls the surplus value of code. (11) Deleuze conceives of ideas as the differentials of thought, and thinking for him involves determining – reciprocally – the differential relations contained within them. (12) Thus, Deleuze presupposes a naturalness for reasoning which precedes the assignability of truth or falseness to any act of thought in particular. This naturalness itself provides conditions, yet neither sufficiency nor well- foundedness for emerging thoughts.

Considered together, Deleuze‘s ideas and this elementarity of pureness make up for considering reason not from the point of view of its conditioning, but from the point of view of inception or genesis, as Deleuze called it. Precisely because he assumes a naturalness to reason, Deleuze can hold, in a mathetical sense, that reasoning depends upon learning. (13)

4  Considering the symbolicalness of symbols

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 – one that is not grounded in geometry nor in arithmetics nor in language – would be profoundly mislead; they both held firm – albeit in different versions – that symbols need to regard necessary facts. (26)

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. (27)

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. (28) 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. (29)

(5)  There is currently, and somewhat symptomatically, a great debate emerging within philosophy interested in metaphysics and the foundational issues of mathematics. It concerns the relation between set theory and category theory. Cf. for a condensed and rich introduction to many of the issues involved: Mary Tiles‘ review of Penelope Maddy‘s book Defending the Axioms: On the Philosophical Foundations of Set Theory, Oxford University Press, 2011, accessible online at Notre Dames Philosophic Journal, Sept.13 2012: http:// ndpr.nd.edu/news/25941-defending-the-axioms-on-the-philosophical-foundations-of-set-theory/ (accessed: July 15th 2012). Also, with many further references, the article on Category Theory at Stanford Encyclopedia: Marquis, Jean-Pierre, “Category Theory”, The Stanford Encyclopedia of Philosophy (Spring 2011 Edition), Edward N. Zalta (ed.), http://plato.stanford.edu/archives/spr2011/entries/category-theory/ (accessed: July 15th 2012).

(6)  If we conceive of such genesis in terms of a grammar, we can avoid subscribing to what Hegel had wisely stigmatized as „the bad infinity“ – which would mean ascribing artefacts the status of an absolute self- or auto-conditioning. Though infinitary in regard to its engagement with an element of the probable, such a grammatical structuralism is nevertheless finitary in the actualizations it conditions. We can express that and only that which the grammatical structure allows to be the case. The powerful move of articulating the polynomials as quantities, symbolically, instead of names as generas or individuals, lies in its capacity to deal with what linguists call „grammaticalization“, the becoming of grammar. Cf. for the relevance of grammaticalization and IT technology especially Bernhart Stiegler: „Nanomutations, hypomnemata and grammatisation“, online at: http://arsindustrialis.org/node/2937 (accessed July 15th 2012). Cf. also footnote [13].

(7)  Heidegger, Martin: Die Frage nach dem Ding. Zu Kants Lehre von den transzendentalen Grundsätzen, in: Martin Heidegger, Gesamtausgabe, 2. Abteilung: Vorlesungen 1923-1944, Band 41, Frankfurt am Main, Vittorio Klostermann 1984.

(8)  It is purely projective in a specific way, which Heidegger clarifies as follows: „Where the casting of the mathematical design is ventured, the pitcher of this cast places herself on a ground which comes to be itself projected itself only in the design ventured. There is not only a liberation proper to mathematical design, but also a new kind of experience and designing of freedom itself, i.e. of the actively resumed attachment. Within a mathematical design, an attachment to the conditions claimed is taking place.“ My own translation, cf. the original German: „Wo der Wurf des mathematischen Entwurfs gewagt wird, stellt sich der Werfer dieses Wurfes auf einen Boden, der allererst im Entwurf erworfen wird. Im mathematischen Entwurf liegt nicht nur eine Befreiung, sonder zugleich eine neue Erfahrung und Gestaltung der Freiheit selbst, d.h. der selbstübernommenen Bindung. Im mathematischen Entwurf vollzieht sich die Bindung an die in ihm selbst geforderten Grundsätze. Befreiung zu einer neuen Freiheit.“ Heidegger (1984), p. 97.

(9)  Heidegger 1984: „Das Mathematische, das ist jenes an den Dingen was wir eigentlich schon kennen, was wir demnach nicht erst aus den Dingen herholen, sondern in gewisser Weise selbst schon mitbringen“, p. 74.

(10) cf Deleuze, Difference and Repetition [1968], transl. by Paul Patton, Columbia University Press 1994. In what follows I refer mainly to the chapter entitled „Ideas and the Synthesis of Difference“, p. 168-221.

(11)  Gilles Deleuze and Félix Guattari, Mille Plateaux [1980], transl. by Brian Massumi, University of Minnesota Press 1993; cf. especially „The Geology of Morals“, pp. 53ff.

(12)  I elaborated on this Deleuzean notion of ideas as the differentials of thought in Vera Bühlmann, inhabiting media. Annäherungen an Herkünfte und Topoi medialer Architektonik, PhD University of Basle 2009 / 2011, published online: http://www.edoc.unibas.ch/1354/2/Promotion_Bühlmann_Juli2011.pdf (accessed July 15th 2012). Cf. especially the subchapters comprehended in the section “Funktion, Sinn und Form”: pp. 120-189.

(13)  Deleuze sais for example: „Just as the right angle and the circle are duplicated by ruler and compass, so each dialectical problem is duplicated by a symbolic field in which it is expressed.“ The capacity of such symbolic fields considered as tools he defines as follows: „Instead of seeking to find out by trial and error whether a given equation is solvable in general, we must determine the conditions of the problem which progressively specify the fields of solvability in such a way that ‘the statement contains the seeds of the solution’. This is a radical reversal in the problem-solution relation, a more considerable revolution than the Copernican.“ Deleuze [1968], pp. 179/180.

(14)  In his Algebra of Logic George Boole has introduced three laws according to which logical identity relations can be established algebraically: the distributive law, the commutative law, and the idempotency law (which Boole also called the index law). All three regulate the establishment of symmetry relations within not fully determined givenness (hence Boole‘s crucial involvement of probabilistics into logics). It is crucial to understand the abstract move involved in considering an algebra of logics, as this is directly inverse to the consideration of logical foundations for mathematics (i.e. Frege‘s and Russell‘s Logicism). While the latter seeks in logics a formal representation of abstract identities, the latter seeks in mathematics the means for learning to think abstract identities in ever more differentiated manners. For algebraists, abstract identity is treated as a difference relation, while for logicists abstract identity is treated as the representation of a unified relation. While logicists seek, ultimately, to determine a unified universe (of discourse), and proceed by seeking to analyze elementary sets or atomic units from which to build up by generalization, algebraists like Boole and Dedekind proceed inversly. They do not begin with assuming ultimately fully analyzable elementary units to work with. They begin with assuming a never exhaustively representable abstract unity as the „universe of thinkable objects“. Hence the former approaches are referred to as „finitary approaches“, and the algebraic approaches as „infinitary approaches“. By this, algebraists can experimentally test logical structures. Thereby, the values of mathematical rigor and exactness are no less important for algebraists than for logisists, yet they are seeked for in the exact definition of concepts to be experimentally applied within the infinite universe of the thinkable. Logicists, on the other hand, seek exactness for the definition of the whole universe of discourse, in order to legitimate the application of one defined concept as opposed to another definition of that concept – this features most prominently in Frege‘s „metaphysics“ of Three Empires (the empire of physical/empirical „objects“, that of psychological „objects“, and that of logical „objects“), as well as in Popper‘s many world theory. I would strongly like to suggest, without being able to go into details here, that Booles laws of thought are misunderstood profoundly, if considered as an early version along the same lines as sets of axioms are proposed in the foundational discourse on mathematics. Boole‘s laws of thought are less like logical axioms than like experimental laws. They are like postulated laws of nature that allow to regulate and orientate experiments, within the „nature“ of the intellectual Universe of Thought (rather than that of the Physical Universe). Obviously, some of the most diversely and all too often overheatedly disputed issues are at stake here. Introductory articles on George Boole and on his Algebra of Logics that are rather uncorrupted by taking a specific stance on the foundational question in philosophy of mathematics can be found at Stanford Encyclopedia: Burris, Stanley, “George Boole”, The Stanford Encyclopedia of Philosophy (Summer 2010 Edition), Edward N. Zalta (ed.), http://plato.stanford.edu/archives/sum2010/ entries/boole/; and Burris, Stanley, “The Algebra of Logic Tradition”, The Stanford Encyclopedia of Philosophy (Summer 2009 Edition), Edward N. Zalta (ed.), http://plato.stanford.edu/archives/sum2009/ entries/algebra-logic-tradition/ (accessed July 15th 2012). Cf. also footnote [6].

(16)  Deleuze (1968), p. 171

(17) cf Deleuze (1968), pp. 216ff. Also his article „The Method of Dramatization“, in Desert Islands and other texts, 1953–1974, semiotexte 1994, pp. 94-116.

(18)  cf. Heidegger (1984), „This concept of pure understanding, quantity, is nothing else but the synthesis which empowers appearances to appear as specific figurations in space. Hence, all appearances, insofar as they are intuited, are quantities, extensive quantities (space). It is the same precondition which allows the encounter of that which encounters, which brings what counters into constellation. Proof is a going in circles [ein Kreisgang]. If we see through and enact this going in circles [diesen Kreisgang], we may receive, as knowledge, the pivot around which everything circles“. My own translation, cf. the original German: „Dieser reine Verstandesbegriff Quantität ist aber nichts anderes als jene Synthesis, kraft deren Erscheinungen als bestimmte Raumgestalten erscheinen können. Also sind alle Erscheinungen als Anschauungen Quantitäten, und zwar extensive (Raum). Es ist dieselbe Bedingung, die das Begegnende begegnen lässt und die es als Gegen zum Stehen bringt. Der Beweis ist ein Kreisgang. Wenn wir diesen Kreisgang als solchen durchschauen und vollziehen, gehen, bekommen wir eigentlich zu wissen, worum sich alles „dreht“.“ p. 252.

(19)  Deleuze, with this characterization of elementary pureness of reason, in which thought dramatizes ideas through involving them into spatio-temporal dynamism by specifying those dynamisms, opens up sights onto a philosophical domain that is conditioned by abundant and insufficient reason. I would like to suggest that this is the domain of symbolic algebra, and that the cases engendered by it – the artefacts – make up all the spatio-temporal dynamisms conceivable. Every single artefact embodies a multitude of acts of concentration which was necessary for dramatizing an idea. It is in this sense that they conserve intellectual energy. What for Heidegger is „the power of synthesis which brings appearances into extensive form“ (cf. footnote 18), the dynamisms conceived through their extension as artefacts make this „power of synthesis“ accessible in terms of intellectual energy: artefacts allow us to apply, store and construct with symbolically encapsulated acts of Heidegger‘s „power of synthesis“. Intellectual energy can be conceived as encapsulatable, storable, transmittable, and even transformable in artefacts of any kind, i.e. of any symbolic constitution. The power of synthesis thereby acquires a qualifiable quantification. For the energy conserved, by such acts of concentration articulated into the manifest form of artefacts, depends upon our ability to understand it. If we naturalize those artefacts, as autolog or self-sufficient objects, without esteem for the acts of concentration they encapsulate, they don‘t conserve but consume energy, through annihilation of their mediality. Because then, there are always too many artifacts, and too many which are not optimally configured, or not to the right optimization configured, etc. The wealth of artefacts then inevitable appears like a waste of resources, intellectual as well as material resources. Cf. footnote [2] on Koolhaas‘ generalization of the concept of paradise in what he calls The Generic City and Junkspace – without understanding and esteem, artefacts must be seen, within the context of Western history at least, as establishing a realm of pure guilt.

(20)  Cayley, Arthur (1996) [1883],“Presidential address to the British Association“, in Ewald, William, From Kant to Hilbert: a source book in the foundations of mathematics. Vol. I, II, Oxford Science Publications, The Clarendon Press Oxford University Press, pp. 542–573, reprinted in collected mathematical papers volume 11.

(21)  Cayley‘s is not a solitary voice at that time. Within the last decades of the 19th century, Gustav Lejeune Dirichlet, Ernst Kummer, Leopold Kronecker and Karl Weierstrass all wrote on the theory of numbers involving algebraic quantities; Husserl published his Habilitationsschrift entitled Über den Begriff der Zahl (1987); Dedekind wrote Über die Theorie der ganzen algebraischen Zahlen (1871) and Was sind und was sollen die Zahlen (1888), Russell wrote his PhD on the Foundations of Geometry (1897) and Whitehead published a comprehensive volume entitled Universal Algebra (1898). All of this before the writing and appearance of Russell‘s and Whitehead‘s Principia Mathematica in 1910/1913.

(22)  Cayley [(1996) [1883]], as reprinted in collected mathematical papers volume 11, p. 784.

(23)  Cayley [(1996) [1883]], as reprinted in collected mathematical papers volume 11, p. 784.

(24)  Cf. how Deleuze [1968] speaks about the possibility of intellectual intuition, by reference to an „ideal cause of continuity“: „the limit must be conceived not as the limit of a function but as a genuine cut [coupure], a border between the changeable and the unchangeable within the function itself. […] the limit no longer presupposes the ideas of a continuous variable and infinite approximation. On the contrary, the notion of limit grounds a new, static and purely ideal definition of continuity, while its own definition implies no more than number, or rather, the universal in number. Modern mathematics then specifies the nature of this universal of number as consisting in the ‘cut’ (in the sense of Dedekind): in this sense, it is the cut which constitutes the next genus of number, the ideal cause of continuity or the pure element of quantitability.“ p. 171.

(25)  This is indeed a very old meaning of symbols – symbols evoke an immaterial presence in our thinking of something which lacks manifest presence. Symbols are place-holders, indexes, and they enforce a certain immediacy upon us. Hence our associations with symbols tend to center around mystical or sacral contexts. Or around contexts of undisputable control, when we think of our passports as symbols of our identity, for example.

(26)  Husserl referred to them as „Anschauungsthatsachen“, as intuitive facts (although he means it in a different sense than the intuitionist schools of Herman Weyl or Luitzen Egburtus Jan Brouwer), and Russell‘s main preoccupation remained the quest for how logical quantification is possible.

(27)  It is in this sense that Deleuze speaks of „the universal in number“ as „the next genus in number, the ideal cause of continuity or the pure element of quantitability“ ([1968] p. 171); and in which Jules Vuillemin proposes an „Ontologie Formelle“ to complement a „Critique Générale de la Raison“ in the end of his book La Philosophie de l‘Algèbre, Epiméthée, Paris 1962, pp. 465ff. The crucial point is that the notion of the universal is related to the learning made possible by mathematics, and not to a realm of logical representation of the achievements of such learning. Hence, instead of universal logical and ontological quantification, these philosophers suggest to consider the quantification of the general in relation to a formal ontology of the general, by viewing in it a kind of philosophical auxiliary structures for learning within an experimental empiricism in the realm of abstraction. Cf. footnote [28].

(28)  In physics and Engineering we more or less casually calculate spaces with algebraic numbers which relativize the assumed unproblematical rootedness of all numerical values in positive natural integers or a homogenous spatiality; and with regard to the formalization of language in logics, we very well know meanwhile about all the problems related both, to universal and to existential quantification. For a discussion of this issue in analytical philosophy after Kant and Frege, cf. Michael Potter, Reason’s Nearest Kin. Philosophies of Arithmetic from Kant to Carnap. Oxford University Press, Oxford, 2000. Potters book is a great overview and introduction over the issues involved and approaches presented, but its limits are within the author‘s decision to exclude any discussion concerning the challenges introduced by algebraic quantities. The framing question in his book is: „Can we give an account of arithmetic that does not make it depend for its truth on the way the world is? And if so, what constrains the world to conform to arithmetic?“ (p.1). Potter takes it as a given that such an account is possible, and he assumes that all of the figures he discusses do as well. Certainly for Wittgenstein and for Dedekind, this seems to me like a crucial misreading. Cf. for a critique in a similar direction also the review by Richard Zach, „Critical study of Michael Potter’s Reason’s Nearest Kin“, Notre Dame Journal of Formal Logic 46 (2005) 503-513 (online: http:// people.ucalgary.ca/~rzach/static/ndjfl-potter.pdf).

(29)  Algebraic terms are polynomials, they embody unequal potentials. The liberty of engendering solutions as cases comes in because every algebraic solution requires a depotentialization of its terms, such that an order can be established which is shared by all the components. If number spaces that may extend into the imaginary and algebraic ideality are allowed for solving polynomial equations, there are many different ways of achieving this depotentialization; hence the vastness of the possible solution spaces.

(30)  Algebraic reasoning means specifying what outcome you will want to have, and producing it by taking an infinitary approach to deal with this probabilisitc element. Infinitary means, by eliminating from the vast and non-controllable possibility space of your solutions anything you do not want to feature in the result. This elimination procedure works by injecting into equations, as a kind of doping, whatever is necessary and sufficient for common denomination and factorization of the terms involved, such that the two sides of an equation may be transformed and balanced in ways that are neither fully necessary nor arbitrary.

(31)  Alfred North Whitehead, Treatise on Universal Algebra with applications, Cambridge University Press, Cambridge 1910, p. vii.