Abstract
We give a concise presentation of the Univalent Foundations of mathematics outlining the main ideas, followed by a discussion of the UniMath library of formalized mathematics implementing the ideas of the Univalent Foundations (Sect. 8.1), and the challenges one faces in attempting to design a large-scale library of formalized mathematics (Sect. 8.2). This leads us to a general discussion about the links between architecture and mathematics where a meeting of minds is revealed between architects and mathematicians (Sect. 8.3). On the way our odyssey from the foundations to the “horizon” of mathematics will lead us to meet the mathematicians David Hilbert and Nicolas Bourbaki as well as the architect Christopher Alexander.
Work on this paper was supported by grant GA CR P201/12/G028, by the European Research Council Advanced Grant ALEXANDRIA (Project 742178) and by the Centre for Advanced Study (CAS) in Oslo, Norway.
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Notes
- 1.
The definition of the homotopy fibers of a map is given later in Sect. 8.1.2.1.
- 2.
The fundamental concept of contractibility is defined later in Sect. 8.1.1.2.
- 3.
This type is small with respect to a higher universe. This technical detail is unimportant for people unfamiliar with type theory.
- 4.
This analogy is explained in the quote from Voevodsky that ends the current Sect. 8.1.1.2.
- 5.
Given a small type A, see Sect. 8.1.1.2 for the definition of the type iscontr A.
- 6.
In the UniMath library any type whose name starts by the prefix “is” is a proposition.
- 7.
See the file DecidablePropositions of the package MoreFoundations.
- 8.
See the file AxiomOfChoice in the package MoreFoundations.
- 9.
A general theorem that “isomorphism is equality” for a large class of algebraic structures (assuming the Univalence Axiom) was proven by Thierry Coquand and Nils Anders Danielsson in 2013 (Coquand and Danielsson 2013). Closely related, is the formulation of the more abstract Structure Identity Principle due to Peter Aczel, see Chapter 9 of The Univalent Foundations Program (2013).
- 10.
See also http://www.cs.ru.nl/~freek/factor/
- 11.
- 12.
The key step towards the widespread use of formalized mathematics could be to start teaching mathematics with the help of proof assistants, not to try very hard to gain the support of the working mathematicians. Given that present-day students are the mathematicians of tomorrow, the latter could be a consequence of the former. Moreover, a mathematical education using, at least occasionally, proof assistants could be solidly grounded in rigorous proofs and the students could benefit from it. Unfortunately, most proof assistants might be still too difficult to use except for graduate students. Note that the Isabelle proof assistant, which offers more automation, was used recently by a group of undergraduate students in Germany, under the supervision of three advisers, to formalize partly the DPRM theorem motivated by Hilbert’s Tenth Problem. See their joint paper Aryal et al. presented during the FLOC 2018 conference in Oxford.
- 13.
I will give a few examples: “Je me sens faire partie, quant à moi, de la lignée des mathématiciens dont la vocation spontanée et la joie est de construire sans cesse des maisons nouvelles. Chemin faisant, ils ne peuvent s’empêcher d’inventer aussi et de façonner au fur et à mesure tous les outils, ustensiles, meubles et instruments requis, tant pour construire la maison depuis les fondations jusqu’au faîte, que pour pourvoir en abondance les futures cuisines et les futurs ateliers, et installer la maison pour y vivre et y être à l’aise. Pourtant, une fois tout posé jusqu’au dernier chêneau et au dernier tabouret, c’est rare que l’ouvrier s’attarde longuement dans ces lieux, où chaque pierre et chaque chevron porte la trace de la main qui l’a travaillé et posé. Sa place n’est pas dans la quiétude des univers tout faits, si accueillants et si harmonieux soient-ils - qu’ils aient été agencés par ses propres mains, ou par ceux de ses devanciers. D’autres tâches déjà l’appelant sur de nouveaux chantiers, sous la poussée impérieuse de besoins qu’il est peut-être le seul à sentir clairement, ou (plus souvent encore) en devançant des besoins qu’il est le seul a pressentir.” (Grothendieck, 2.5 Les héritiers et le bâtisseur); and “Comme le lecteur l’aura sans doute deviné, ces “théories”, “construites de toutes pièces”, ne sont autres aussi que ces “belles maisons” dont il a été question précédemment : celles dont nous héritons de nos devanciers et celles que nous sommes amenés à bâtir de nos propres mains, à l’appel et à l’écoute des choses. Et si j’ai parlé tantôt de l’ “inventivité” (ou de l’imagination) du bâtisseur ou du forgeron, il me faudrait ajouter que ce qui en fait l’âme et le nerf secret, ce n’est nullement la superbe de celui qui dit: “je veux ceci, et pas cela !” et qui se complaît à décider à sa guise; tel un piètre architecte qui aurait ses plans tout prêts en tête, avant d’avoir vu et senti un terrain, et d’en avoir sondé les possibilités et les exigences.” (Grothendieck, 2.9); and again “C’était peut-être là la principale raison pour laquelle les maisons que je prenais plaisir à construire sont restées inhabitées pendant le longues années, sauf par l’ouvrier maçon lui-même (qui était en même temps aussi l’architecte, le charpentier etc.).” (Grothendieck, 18.2.8.3 Note 135).
- 14.
(1) Levels of scale (2) Strong centers (3) Boundaries (4) Alternating repetition (5) Positive space (6) Good shape (7) Local symmetries (8) Deep interlock and ambiguity (9) Contrast (10) Gradients (11) Roughness (12) Echoes (13) The void (14) Simplicity and inner calm (15) Not-separateness.
- 15.
Some proof assistants like Isabelle have structured proofs (in the case of Isabelle thanks to an additional layer called the Isar language), but there is still room for improvement.
- 16.
The good practices of writing a short table of contents at the top of a file starting a new formalization and a bibliography at the end are surprisingly not even included in the style guide (https://github.com/UniMath/UniMath/blob/master/UniMath/README.md) of UniMath as of 6 September 2018.
- 17.
Again, the Isabelle theorem prover and its bundled editor jEdit, even if not perfect, have built a competitive advantage with search support and clickable keywords.
- 18.
See also the Footnote 7 on that point.
- 19.
I have discovered a truly remarkable answer to this question which this footer is too small to contain.
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The author would like to thank Benedikt Ahrens, Thierry Coquand, and an anonymous referee for their useful comments and suggestions.
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Bordg, A. (2019). Univalent Foundations and the UniMath Library. In: Centrone, S., Kant, D., Sarikaya, D. (eds) Reflections on the Foundations of Mathematics. Synthese Library, vol 407. Springer, Cham. https://doi.org/10.1007/978-3-030-15655-8_8
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