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Goldbach’s Conjecture as a ‘Transcendental’ Theorem

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Abstract

Goldbach’s conjecture, if not read in number theory (mathematical level), but in a precise foundation theory of mathematics (meta-mathematical level), that refers to the metaphysical ‘theory of the participation’ of Thomas Aquinas (1225–1274), poses a surprising analogy between the category of the quantity, within which the same arithmetic conjecture is formulated, and the transcendental/formal dimension. It says: every even number is ‘like’ a two, that is: it has the form-of-two. And that means: it is the composition of two units; not two equal arithmetic units (two numbers ‘one’), but two different formal-transcendental units, which are, in arithmetic, two prime numbers.

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Notes

  1. For a review of the typical positions of the Philosophy of Mathematics see Shapiro (2007) and many other books and Handbooks on the subject. The position absent in these texts, because absent in the extraordinary era of research on the fundation of Mathematics between the end of the nineteenth century and today, is a ‘Thomasian’ position, that is, one that refers to the metaphysical conceptual structure of Thomas Aquinas (1225–1274), as we shall say.

  2. In the common language, it often happens to specify the reference of the proper name with a paraphrase similar to this: “Laurel who? Laurel of Laurel-and-Hardy?”

  3. The theoretical approach of Thomas Aquinas is absent in (almost) all the texts of contemporary Metaphysics, Logic, Philosophical Logic, even if an enormous amount of studies on his thought is being reborn, especially in reference to Philosophy of Law, Ethics and, obviously, Theology. In Italy, in the last century, there was a rediscovery of the so-called ‘essential Thomism’ by a great thinker named Cornelio Fabro (1911–1995), who had the merit and the effort to take up the texts of Aquino and free them from the historical interpretations of Thomism. The result is a metaphysical vision of the real, pregnant with considerable developments, yet to be explained. Beware, because ‘essential Thomism’ is not the ‘analytical Thomism’ of many contemporary english authors. We here try to decline those metaphysical principles on the fundamentals of mathematics.

  4. Here we will quote and translate the texts of Thomas Aquinas published in the critical edition available online at www.corpusthomisticum.org.

  5. Ens is the first and fountain concept of classical and medieval philosophical reflection. We modern people are used, even in the common non-philosophical speech, to using words such as ‘thing’, ‘object’… to refer as generically as possible, to any thing. The preferability of the medieval term over the more modern ‘thing’ will be partly understood precisely by the distinction of the transcendentals that we will see shortly. For a deep theoretical discussion see Basti (2011).

  6. The reference text is De Ver., q. 1, a. 1, co.

  7. “The being is not a genre” (Metaphysica, 998 b 22) and therefore it is ‘beyond’ all the formal specificities, in the sense that they do not divide and limit it how instead the differences are added to the genre (and how the accidents do to the subject); cf. In Met. lib. 5, lect. 9, n. 5–6.

  8. See De Ver., q. 1, a. 1, co.

  9. See S. Th. I, q. 30, a. 3, co; ivi, ad 1um; ivi, q. 39, a. 3, ad 3um.

  10. In I Sent., lec. 24, q. 1, a. 2, co.

  11. In I Sent., lec. 31, q. 3, a. 1, co.

  12. S. Th. I, q. 30, a. 3, co.

  13. See In I Sent., lec. 24, q. 1, a. 4, co.

  14. De Pot. q. 9. to. 7 co.

  15. S. Th. I, q. 11. a. 2 ad 2um; ivi, q. 30. a. 3 co.

  16. S. Th. I, q. 30, a. 3, ad 3um.

  17. Basti and Perrone (1996, p. 231).

  18. In Met., lib. 7, lec. 13, n. 24.

  19. In Met., lib. 7, lec. 13, n. 24.

  20. Quod. I, q. 4 a. 1, co.

  21. Of the being-machine, not of the denomination of it with “machine”.

  22. And the famous categories of substance (space, time, quantity, quality…) are both real and linguistic: “ways of being are proportional to ways of saying”, In Phys., lib. 3, lec. 5, n. 15.

  23. In Met., lib. 3, lec. 8, n. 15.

  24. In Met., lib. 7, lec. 13, n. 24.

  25. As introduction to ‘essential Thomism’, in English, see: Fabro (1966, 1967a, b). For a contemporary and fomal approach to Thomism, see Basti (2002, 2014) and Panizzoli (2014).

  26. In Met., lib. 1, lec. 10, n. 15.

  27. On this enormous issue too, we depart both from the usual Platonic position and from the more common one that assigns the problem of the nature of number to a vague “intuition’. Here we briefly explain the basic concept of the Aristotelian-Thomist imprint, referring to other writings and works.

  28. S. Th. I. q. 7. a. 4, co. See Metaphysics, 1057a, 3–4.

  29. Giuseppe Peano (1858–1932) in 1889, in Turin, published a small booklet in Latin entitled Arithmetices principia, nova methodo exposita, in which he exhibited the first symbolic axiomatization of the natural numbers.

  30. In Phys., lib. 7, lec. 8, n. 16. It is interesting to note that the opposite is not true, as Aristotle already said: “not everything that is one is also number: for example, an indivisible thing is not number”, Metaphysics, 1057a, 6–7. That’s why the number 1 is not considered as the other numbers.

  31. Even the summing operation ‘\(+\)’ comes out of an abstraction from the real, but here we do not care precisely. Let’s assume that it makes a certain composition (and decomposition) of the two.

  32. In Met., lib. 1, lec. 10, n. 15.

  33. See Basti (2004).

  34. The analogy with the material forms can help to understand these steps. Let’s just say: Socrates is a man like Plato is a man like Aristotle is a man…. They have humanity in common, that is, ‘the same’ humanity (as all peers are divisible by 2); yet they are all ‘different’ men because there is a proper way of being a man (analogy of proportionality: even numbers are different numbers). But all this is true because being-man ‘forms’ them all (just as all peers are divisible by two because the 2 it forms them all like a two).

  35. In Met., lib. 1, lec. 9, n. 16.

  36. In Met., lib. 1, lec. 10, n. 15.

  37. In I Sent., lec. 8, q. 4, a. 2, ad 3um.

  38. S. Th., I-II, q. 96, a. 1, ad 2um.

  39. This text is of a contemporary theologian of Aquinas: Thomas de Sutton, De pluralitate formarum, pars 1.

  40. One of the best attempts to do an impredicative math is Nelson (1986).

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    Francesco Panizzoli

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Panizzoli, F. Goldbach’s Conjecture as a ‘Transcendental’ Theorem. Axiomathes 29, 463–481 (2019). https://doi.org/10.1007/s10516-019-09429-y

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