Abstract
This chapter discusses the topic of mediation, number and mathematics. It aims to find a way to think of mathematics as the soul of reality by means of Wittgenstein and Cantor. This chapter reflects on the modern attempts to rethink mathematics in terms of logics. In Milbank’s view, a revision of this could support the ‘third way’ of a theistic metaphysics, which must, after Erich Przywara and William Desmond, be a metaphysics of the analogy of being or metaxology.
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Notes
- 1.
This is why Fichte’s doctrine , with his notion of the positing mind’s confrontation with a limiting anstoss could be readily inverted into realism by Novalis and Friedrich Schlegel. By comparison, the incorporation of the objective as a moment within dialectical, or else ‘artistic’ reasoning, by Hegel and Schelling is, paradoxically, more purely idealist.
- 2.
In disagreement here with Heller-Roazen (2011).
- 3.
Fowler (1987, pp. 25–27, 31–66, 191–192 and 364–371). Anthyphairesis, a procedure also found in Euclid, literally means ‘reciprocal subtraction’ because the resultant remainder was then further reduced ad infinitum—again by multiplying fragmentation of a unity—by measure of the difference between the first remainder and the initially given unit. Repeated application of this process is also carried out by modern mathematics but in terms of pure division and divisors. But already Aristotle, in denying Plato’s ‘great’ as well as ‘small’ infinite—arguing that in progress upwards we always transit by definite steps and do not encounter an infinitesimal interval that never arrives at nothing—removed the homology between infinite subtraction or division on the one hand and infinite addition on the other, and so opened the way for denying that the former can be ‘reciprocally’ conceived in a full sense as also a positive asymptotic progress. See Aristotle, Physics, III,6, 206b 3–206b 35. The riposte to Aristotle here would be to decide to treat the upwards infinite limit as an actuality of which zero (whose unattainable full stop Aristotle cannot deny) was but the echo. In the end, Aristotle himself has only decided against this outlook, not decisively argued against it.
- 4.
The primacy of the actual and the positive in premodern western mathematics shows us how naturally Christian theological theses such as the privative theory of evil could be grafted onto what the student would have learnt in the quadrivium.
- 5.
Plotinus, Enneads V.5.10: “[the Good’s] being is not limited; what is there to set bounds to it? … All its infinitude resides in its power: it does not change and will not fail; and in it all that is unfailing finds its duration”; V.5.11: “It is infinite also by right of being a pure unity with nothing towards which to direct any partial content”; VI, 9, 6: “We must … take the Unity as infinite not in measureless extension or numerable quantity but in fathomless depths of power”. [Stephen Mackenna’s translation.]
- 6.
See Jorge Luis Borges, ‘The Fearful Sphere of Pascal’ in Labyrinths (London: Penguin, 1971), 189–192. But for a corrective of Borges, see Harries (1975, pp. 5–15). Harries rightly says that Cusa preceded Bruno—that early modern cosmology altered in the wake of this transference of metaphorical application rather than the reverse, and that the shift itself is not a secularizing cosmic appropriation of a divine attribute but rather a following through of the full cosmological implications of this attribute of an infinite creative God. As with Grosseteste, if God is infinite, then his productions, though finite, cannot themselves be finitely bounded. The closed universe was pagan, not uninflectedly religious.
- 7.
Nicholas of Cusa, De Docta Ignorantia, I, 12.33; 23; De Visione Dei, 13–15.
- 8.
For the ontological role of problema in Proclus, see Part One, §§ 201, 243–244. His insistence on the essential initial role of problematic in producing the geometric field can be seen as consonant with his overall ‘theurgic’ perspective which, in contrast to Plotinus, stressed the full descent of the human soul into the human body and consequently the need for sensory and material mediation and the merciful descent of the gods to our realm, drawn down through and as myriad modes of ritual attraction.
- 9.
For this reason, he thought that binding gravity might be an intrinsic feature of three dimensionality, rendering space itself a vital force field. This seems to recover the Stoic conception of geometry as ‘phoronomic’ or more primarily concerned with moving than with static figures. Of course, it was his conception of the manifold that allowed him to think non-Euclidean geometries and to conjecture that our three dimensionality could itself be but the ‘surface’ of a concealed, more multidimensional ‘sphericity’.
- 10.
See Fowler (1987, p. 14): “A much more faithful impression of the very concrete sense of the Greek arithmoi is given by the sequence: duet, trio, quartet, quintet”.
- 11.
For example, Wittgenstein (1981, II, §§ 18–38).
- 12.
For a clear and simple summary of Cantor’s diagonalisation proof and its immediate intellectual aftermath, see Seife (2000, pp. 147–153).
- 13.
Marion wrongly equates Wittgenstein’s reduction of the potential infinite to the logical or grammatical rule to ‘carry on’, making it “the property of a law, not of its extension” (King and Lee 1980, p. 13) with Aristotle’s denial that potential infinity would ever be actually realized, in the way that a potential statue can eventually come about [Physics, III, 6, 206a 18–20]. But for Aristotle, potential infinity still clearly denotes an indefinite power that is extensionally ‘out there’ in the world, something that can be ever further actualized, though never completed in its full actuality, which for him is impossible. Thus, time and human generation are both actually without limit, although this lack of limit “is not (like the statue potentialities of the bronze) all actualised at once but is in course of transit as long as it lasts” [Physics III, 6, 206a 22–24]. The same applies to division of a magnitude, except that in this case, the discarded parts remain to rebuke in their persisting actuality the infinitely destructive ambition of the divider and do not vanish down the abyss of the more successful destroyer, time [III, 6, 206a 30–206b 2]. Marion equally fails to see that for Aristotle apeiron is ontological chaos and not just heuristic instruction.
- 14.
Waissmann (1979, p. 63): “In mathematics it is just as impossible to discover anything as it is in grammar” and p. 34: “What we find in books on mathematics is not a description of something, but the thing itself. We make mathematics”. However, the real Platonic tradition concerning mathematics, which culminated in Cusa and Vico’s Christian Trinitarian radicalisation, was able to regard making as also a seeing, also a describing. Wittgenstein is too conservative to be able to question this alternative. See also Wittgenstein (1981, § 5): “The Proposition: ‘It is true that this follows from that’ means simply ‘this follows from that’”. Here, the redundancy of the word ‘true’ redounds to the benefit of a convention that must be forever reiterated if it is to remain in force. In addition, see Wittgenstein (1978, § 185), where Wittgenstein argues that the instruction to ‘add 2’ to 1000 will not necessarily produce 1002, either according to rule-following or continuing to follow the rule in the same way. In either case, there remains a margin of interpretation which only brute imposition of the standard mode of reiteration can prevent . Finally, see Marion (1998, pp. 1–20) and on p. 22: “[Wittgenstein] insists that we never discover facts about structures that we have already set up: any new theorem is in fact a new extension of mathematics”; see also Klenk (1976, pp. 8–18).
- 15.
Badiou (2011, p. 75) and passim. The sophistic label is not meant to be entirely negative.
- 16.
As Jacques Lacan showed, even the sign-operation, in order to avoid anarchy, has to occur within certain loosely ‘setted’ parameters. In this way, number interferes with the field of sign, ensuring that it concerns always ‘numbers of things’ just as, in the case of number, sign and reality coincide, though in ontologically thin air. See Milbank (2009, pp. 118–120).
- 17.
I am grateful to discussions with my son, Sebastian Milbank, on Priest and the ontological reality of paradox.
- 18.
The fact that Gödel did not see his demonstration of the undecidability of the continuum hypothesis as problematic for his Platonism might suggest its genuine character.
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Milbank, J. (2018). Number and the Between. In: Vanden Auweele, D. (eds) William Desmond’s Philosophy between Metaphysics, Religion, Ethics, and Aesthetics. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-319-98992-1_2
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