Abstract
The rigor of mathematics lies in its systematic organization that supports conclusive proofs of assertions on the basis of assumed principles. Proofs are constructed through thinking, but they can also be taken as objects of mathematical thought. That was the insight prompting Hilbert’s call for a "theory of the specifically mathematical proof" in 1917. This pivotal idea was rooted in revolutionary developments in mathematics and logic during the second half of the 19-th century; it also shaped the new field of mathematical logic and grounded, in particular, Hilbert’s proof theory. The derivations in logical calculi were taken as "formal images" of proofs and thus, through the formalization of mathematics, as tools for developing a theory of mathematical proofs. These initial ideas for proof theory have been reawakened by a confluence of investigations in the tradition of Gentzen’s work on natural reasoning, interactive verifications of theorems, and implementations of mechanisms that search for proofs. At this intersection of proof theory, interactive theorem proving, and automated proof search one finds a promising avenue for exploring the structure of mathematical thought. I will detail steps down this avenue: the formal representation of proofs in appropriate mathematical frames is akin to the representation of physical phenomena in mathematical theories; an important dynamic aspect is captured through the articulation of bi-directional and strategically guided procedures for constructing proofs.
The objects of proof theory shall be the proofs carried out in mathematics proper.
The motto is a deeply programmatic remark in Gentzen (1936, 499), Gentzen’s classical paper in which he proved the consistency of elementary arithmetic by transfinite induction up to ϵ0. It fully coheres with remarks by Hilbert, as we will see.
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Notes
- 1.
That is a requirement of Kant’s. It is discussed in detail and with reference to the logicist tendencies of Dedekind and Hilbert in Sieg (2016).
- 2.
Dedekind (1932, 336) In German, “Diese Schrift kann jeder verstehen, welcher das besitzt, was man den gesunden Menschenverstand nennt; philosophische oder mathematische Schulkenntnisse sind dazu nicht im geringsten erforderlich.”.
- 3.
In English, “What is provable should not be believed in science without proof.” An underlying principled separation of analysis (leading to fundamental concepts) and synthesis (using those concepts as the sole starting-points for the development of a subject) is articulated for elementary number theory most clearly in Dedekind’s letter to Keferstein (Dedekind 1890). The significance of creating new concepts is dramatically pointed out in the Preface to (Dedekind 1888, 339). For the broader methodological context, Dedekind points there to his Habilitationsrede (Dedekind 1854).
- 4.
Hilbert viewed the decision problem, i.e., the problem of deciding a mathematical question in finitely many steps, as the “best-known and the most discussed” question. This issue, he asserts, “goes to the essence of mathematical thought”. (Hilbert 1918, 1113).
- 5.
Bernays started to work as Hilbert’s assistant in the fall of 1917. He was intimately involved in every aspect of Hilbert’s foundational work. In his 1922-paper, the integration of structural and formal axiomatics is expressed very clearly. He views, fully aligned with Hilbert’s perspective, the representation of mathematical proofs in formalisms as a tool for their investigation not as a way of characterizing mathematics as a formal game. About the logical calculus of “Peano, Frege, and Russell” he writes: these three logicians expanded the calculus in such a way “that the thought-inferences of mathematical proofs can be completely reproduced by symbolic operations.” (p. 98).
- 6.
- 7.
The call for such an investigation was not a whim for Hilbert. After all, we just saw that he had already formulated in 1900 the 24-th problem concerning “a theory of the method of proof in mathematics”.
- 8.
See my paper (Sieg 2014), in particular, the analysis of the Frege-Hilbert correspondence.
- 9.
- 10.
Grundgesstze der Arithmetik, p. 139 of Translations from the philosophical writings of Gottlob Frege, Peter Geach and Max Black (eds.), Oxford 1977.
- 11.
Poincaré’s 1902-review of Hilbert’s Grundlagen der Geometrie brings out this additional aspect in a quite vivid way, namely, through the idea of formalization as machine executability: “M. Hilbert has tried, so-to-speak, putting the axioms in such a form that they could be applied by someone who doesn’t understand their meaning, because he has not ever seen either a point, or a line, or a plane. It must be possible, according to him, to reduce reasoning to purely mechanical rules.” Indeed, Poincaré suggests giving the axioms to a reasoning machine, like Jevons’ logical piano, and observing whether all of geometry would be obtained. Such formalization might seem “artificial and childish”, were it not for the important question of completeness: “Is the list of axioms complete, or have some of them escaped us, namely those we use unconsciously? … One has to find out whether geometry is a logical consequence of the explicitly stated axioms, or in other words, whether the axioms, when given to the reasoning machine, will make it possible to obtain the sequence of all theorems as output [of the machine].”.
- 12.
It is a normative requirement (to insure intersubjectivity on a minimal cognitive basis), but it is also a practical one: if an inferential step required a proof that its premises imply its conclusion, we would circle into an infinite regress. Frege pursued also the particular philosophical goal of gaining “a basis for deciding the epistemological nature of the law that is proved.” (Grundgesetze, 118).
- 13.
The adequacy of these notions for capturing the informal concepts is still discussed today under the special headings, Church’s Thesis, or Turing’s Thesis or the Church-Turing Thesis. In my Sieg (2018), I gave a structural-axiomatic characterization of the concept of computation. That turns the “adequacy problem” into a standard problem any mathematical concept has to face when confronted with the phenomena it is purportedly capturing.
- 14.
Proofs officially are sequences of formulas, but for the proof theoretic investigations, Hilbert and Bernays turn them into proof trees by a process they call “Auflösung in Beweisfäden”.
- 15.
It is unfortunate that “Kalkül des natürlichen Schließens” has been translated as “natural deduction calculus”. “Calculus of natural reasoning” would express better that it is a calculus reflecting in a formal way patterns of natural, informal argumentation.
- 16.
Gentzen’s Urdissertation (Gentzen 1932–33) contains a formulation of a natural deduction calculus for intuitionist logic. Gentzen proved for that logic a normalization theorem; see (von Plato 2008) and (Sieg 2009). The systematic investigations of both classical and intuitionist natural deduction calculi was taken up in (Prawitz 1965); Prawizt established normalization theorems and discovered important structural features of normal proofs.
- 17.
“Intercalate” does not only mean “interpolate an intercalary period in a calendar” but also “insert something between layers in a crystal lattice or other structure”. So, I apply it to the insertion of formulas between layers in logical proof structures. (“Interpolate” could not be used in this logical context for obvious reasons.).
- 18.
For the NIC calculus one can prove a strengthened completeness theorem for both classical and intuitionist logic: for any ϕ and ℾ, there is either a normal proof of ϕ from ℾ or a counterexample to ℾ, ϕ. The central considerations in the completeness proof have been organized into efficient logical strategies for automated search and have been implemented in the system AProS. The completeness proofs are found in Sieg and Byrnes (1998) and Sieg and Cittadini (2005).
- 19.
Lemmas have been used as rules in classical texts but, of course, also for the standard structuring of mathematical expositions. Classical examples are found in Book I of Euclid’s Elements (for example, in the proof of the Pythagorean Theorem the earlier, very important theorem I.4 is being used in just that way), but also in the development of the theory of systems and mappings in Dedekind (1888).
- 20.
It makes explicit the informal notion of objects obtained from elements in a by finitely iterating f; it is defined as the intersection of all sets that contain a as a subset and are closed under f.
- 21.
That point is at the center of my paper (Sieg 2019). Aries Hinkis’ book Proofs of the Cantor-Bernstein Theorem: A mathematical excursion is a comprehensive discussion of proofs and their history. I have examined most of the proofs in that mathematical excursion, with special care given to the proofs of König and Banach. They all fall into one of these categories.
- 22.
Gowers’ and my work actually have a common motivating ambition: a pedagogical one. Gowers’ is articulated in Diaz-Lopez (2016); mine has propelled the development of a fully web-based course Logic & Proofs; see Sieg (2007). The AProS website has this URL http://www.phil.cmu.edu/projects/apros/. AProS forms the basis of a dynamic, interactive proof tutor in this introduction to logic. The course has as its primary goal teaching students strategic proof construction. Logic & Proofs has been completed by more than 12,000 students for credit at their home institution.
- 23.
- 24.
Implicitly, I argued against an artificial opposition of informal and formal proofs. By incorporating the strategic, dynamic aspect of interactive proof construction into a fully automated search procedure, one obtains the means for exploring structural features of proofs and their construction.
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Sieg, W. (2022). Proofs as Objects. In: Ferreira, F., Kahle, R., Sommaruga, G. (eds) Axiomatic Thinking I. Springer, Cham. https://doi.org/10.1007/978-3-030-77657-2_9
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