LLogical Methods in Computer ScienceVolume 15, Issue 3, 2019, pp. 7:1–7:6https://lmcs.episciences.org/ Submitted Oct. 01, 2018Published Jul. 29, 2019
A FORGOTTEN THEORY OF PROOFS?
E.ENGELERETH Zurich e-mail address : [email protected]
The Hilbert Program in G¨ottingen was winding down in the early 1930s. By then it wasmostly in the hands of Paul Bernays who was writing the first volume of
Grundlagen derMathematik . Hermann Weyl had succeeded David Hilbert. There were three outstandingdoctoral students in logic: Haskell B.Curry, Saunders MacLane and Gerhard Gentzen. Thesethree students are at the beginning of three threads in mathematical logic: CombinatoryLogic (Curry), Proof Theory (Gentzen) and Algebra of Proofs (MacLane), the last oneessentially forgotten, except perhaps for some technical results useful in computer algebra(cf. Newman’s Lemma).The present author, also a student of Bernays, when looking up this mathematicalancestry, was fascinated by the contrast between MacLane’s enthusiasm about the ideasin his thesis as expressed in his letters home , and the almost complete absence of anymathematical follow-up. Is it possible that mathematical development has passed somethingby, just because MacLane did not find resonance for this work and, back in the U.S., wassoon successful, as the strong mathematician he was, in other fields. His work on theconceptual structure of mathematics, category theory and its pervading influence throughoutmathematics, is well known [6].In this essay we attempt to revive the idea of an algebra of proofs, place MacLane’sthesis work and its vision in a new framework and try to address his original vision. Hebelieved that the mathematical enterprise as a whole should be cast into a grand algebraicframework. If completed, it would embody axiomatic, methodology of inventing and proving,and would underly mathematical exposition; a youthful dream.1. Prolegomena to an algebra of mathematical thoughts
Let us first talk about thinking. Thinking means to apply thoughts to thoughts, thoughtsbeing things like concepts, impressions, memories, activities, projects – anything that youcan think about, including mathematics. And, of course, the results of applying a thoughtto a thought. Thinking is free, all combinations of thoughts are admitted into the universeof thoughts. As a mathematician I perceive here the structure of an algebra: Thoughts are As Bernays was only Dozent (and was soon to be dismissed as foreign, Swiss and of jewish ancestry),Weyl was official thesis advisor, and of course took personal interest until he also left (for Princeton, Bernaysfor ETH.) Excerpts in [4]
LOGICAL METHODS l IN COMPUTER SCIENCE DOI:10.23638/LMCS-15(3:7)2019 c (cid:13)
E.Engeler CC (cid:13) Creative Commons :2 E.Engeler
Vol. 15:3 the elements of the algebra and applying a thought X to a thought Y is a binary operationwhich results in the element X · Y , again a thought.Mathematical thoughts are about sets of definitions, problems, theorems, proofs andproof-strategies . In the present context, to do mathematics means to make a selection ofsuch sets, states of knowledge and proof procedures as it were, and apply these sets to eachother. To mathematize this idea, we need to represent states of mathematical knowledgeand the pursuit of its development in a form that permits an application operation betweenthem. Let us first experiment with formalized mathematics and its states of knowledge.Mathematical logic aims to represent mathematics by a system based on a formallanguage. Formal mathematical thoughts thereby consist of sets of statements (axioms,theorems) and proof-trees. Take propositional logic. Let A be the set of propositionalformulas a, b, c, . . . composed from some atomic propositions by some connectives such as ∧ , ∨ , ⊃ , ¬ . A formal proof has the form of a tree such asa bc dg e fhkIn an obvious notation, this tree would be rendered as {{{ a, b } → c, d } → g, { e, f } → h } → k. Such a proof can be parsed differently in order to reflect the conceptual structure of theproof – which in fact originally may have progressed through the development or employmentof various auxiliary theorems and general lemmas. For example, g may be a lemma and theproof of k starts with this lemma and e and f : {{{ a, b } → c, d } → g } → ( {{ e, f } → h } → k ) . Another parsing would be: {{{ a, b } → c } → ( { d } → g ) } → ( {{ e, f } → h } → k ) . All of these denote the same tree but give different narratives/ interpretations of theformal proof. They are what will be called proof-expressions and denoted by lower-caseletters such as x , y , z from the latter part of the alphabet. The set P of proof-expressions isbuilt up recursively from A : P = AP n +1 = P n ∪ { α → x : x ∈ P n , α ⊆ P n finite } P = (cid:91) n P n . Of course, these “proof-expressions” represent formal proofs only in the case that thearrows correspond to legal steps in a formal proof (here of propositional logic); of this later.The result of the proof denoted by a proof expression x is x itself if it is a propositionalformula, an element of A ; otherwise, if x is composite it is the root of the tree denoted by x .We let x (cid:96) denote this root, it is a propositional formula.Sets of proof-expressions are denoted by capital letters X , Y , . . . or by special symbolsintroduced as cases arise. Such sets represent “mathematical thoughts” in the sense of the ol. 15:3 A FORGOTTEN THEORY OF PROOFS? 7:3 introduction to this section, here restricted to the realm of formal propositions.– To completethe picture there, it remains to specify the operation of application, X · Y as follows: X · Y = { x : ∃ α ⊆ Y, α → x ∈ X } . This definition is best understood if X is considered as a sort of graph of a (partial andmany-valued) function, each of its elements α → x associating an argument(-set) α to avalue x . By this operation the set of subsets of P , i.e. the set of mathematical thoughts,becomes an algebraic structure, the algebra P of propositional thoughts. Modus Ponens is the thought which applied to the set of formulas { a ⊃ b, a } produces b . Correspondingly, [ modusponens ] as an element of the algebra P contains at least the oneelement { a ⊃ b, a } → b ; we posit that it consists of all elements of that form. Thus, if X isa set of propositional formulas, [ modusponens ] · X is the set of all propositional formulasprovable from the set of propositional statements in X in one step. Compare this with theusual notation a ⊃ b bb [ modusponens ] , specifying the proof-rule on the right.More to the point, Modus Ponens can also function as a proof-constructor . Thecorresponding element of P is[MP] = {{ x, y } → ( { x, y } → b ) : ∃ a ∃ b ∈ A such that x (cid:96) = a ⊃ b, y (cid:96) = b } . [MP] · X combines proofs of formulas a ⊃ b and a to a proof of b . An iteration startingwith X = X using [MP] produces the propositional theory T of X restricted to the oneproof-rule: X i +1 = X i ∪ [MP] X i ,And so on, to develop propositional logic as an algebraic proof-theory of P , a tentativepartial realization of MacLane’s Logikkalkul .Instead, we take another elementary example, finitely presented groups:Let A be the set of terms u, v, . . . built up from variables and a finite set of constants(“generators” ) denoting some elements of a group G by the operations of multiplication,inverse and the unit element. Finite sets of constant terms, called relations, constitute agroup-presentation. Based on A we construct a calculus of reductions R starting from a setof reduction-expressions x, y, . . . analogously to P above, (most of which of course wouldnot denote valid reductions). Valid reductions are based on group laws such as associativityand on the relations given by the presentation:The associative law, when applied to a reduction-expression x , replaces a sub-term ofthe final term x (cid:96) , assuming it has the form u ( vw ), by ( uv ) w , or ( uv ) w by u ( vw ). Let [ASS]denote this element of R , hence [ASS] is the set of all { x } → t , where t results from x (cid:96) bysubstituting some sub-term u ( vw ) or ( uv ) w of x (cid:96) by 1. Similarly for inverse law: [INV]replaces sub-terms uu − or u − u of x (cid:96) by 1. The identity law is realized as an operation[ID] on reductions, using replacements of u u by u .Relations r , . . . r n of the presentation give rise to reduction laws and therefore toreduction- constructors [ r i ]. For example, if r = g g − g with generators g , g , then [ r ] isthe set of all { x } → t , where t results from x (cid:96) by substituting some sub-term g g − g by 1 . :4 E.Engeler
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Example: To construct a reduction (by “normalization” ) of the term ( st − ) t we startwith the set X , consisting of this term, and use the three operators in succession, resultingin a linear reduction-tree x with x (cid:96) equal to 1:[ID] · ([INV] · ([ASS] · { ( st − ) t } )) (cid:96) = 1 . Taking the closure of [ASS] ∪ [INV] ∪ [ID] ∪ [ r ] · · · ∪ [ r n ] under iteration as above, weobtain an object [ALG] of R which, applied to X gives its normalization, [ALG] · X in thisfinitely generated group.2. MacLane’s Thesis and its Vision, Revisited
The above example is from MacLane’s thesis “Abgek¨urzte Beweise im Logikkalkul” . It is“abgek¨urzt” , shortened – but more importantly it is a proof-template, a formal object ina proof-manipulating system for elementary group theory, a “Reduktionsbeweis” . In theoriginal, it reads: Anfang Th, Sub (4), Sub (2), Ende (3).The Logikkalkul of MacLane takes its examples from the formal logic of
PrincipiaMathematica [9].The main technical development in the thesis shows how MacLane convinces himself thathis approach to proof theory suffices to treat all of mathematical logic, whose main corpusat that time was
Principia . The basic insight is that proofs are built up from individual proof operations by composition.Correspondingly, short descriptions of logic-proofs use operators corresponding to the familiarintroduction/elimination of logical connectives.By introducing names for proofs of auxiliary theorems MacLane enriches the totality ofproof-operators by names for proof-plans. It is clear that he develops the rudiments of acalculus of such expressions for proof-operators.Admittedly, MacLane’s algebraisation of deduction processes at first sight does notlook very impressive. Today; but to actually complete the project, there were tedious andoccasionally delicate technical details of substitution, replacement etc. to be handled. Infact, what MacLane did was at the start of a mathematics of symbol manipulations systemswhich later became computer algebra and computational logic, (cf. normal forms, confluence,etc.). Later in life, MacLane was aware of this [5].But in 1932 MacLane’s aim was broader, it was to study all formal and informal proofactivities as a mathematical subject. “... great ideas and magnificent generalizations,...suggest all sorts of vague possibilities of applications to other fields...” . He submittedthese ideas in a first draft of his thesis to Weyl who apparently found it too ambitious aproject. Upon the wise counsel of Bernays he then confined himself to “the kind of ideas [of]the professors here ” , namely the formal manipulation and specifically the simplification ofproofs. This is in fact the recently unearthed 24th problem that Hilbert had prepared forthe famous 1900 list [8]! Some of the original targets remain as remarks in the thesis, in G¨ottingen, Huber & Co.1934. Reprinted in Kaplansky [3], p.1 - 62. In the order of applications, reversed from the operational notation above; “Th” denotes the equation tobe derived, “Sub (4)” denotes the application of associativity [ASS], etc. As it was for G¨odel three years before, (he strangely is not mentioned in the thesis). Quote in [4], p.59 Ibid.p.60 ol. 15:3 A FORGOTTEN THEORY OF PROOFS? 7:5 particular a formal analysis of the beautiful lectures of Weyl with its transparent non-formalproofs, a book by E.H.Moore, and intuitionism.But now I’m puzzled.2.1.
Puzzle: Relation to Curry.
As shown above, proofs, including proof plans, can beviewed as algebraic objects with an operation of composition. These form an algebraicstructure which is in fact a combinatory algebra , a model of Curry’s combinatory logic.Moreover, the formation of arbitrary proofs by combination of proofs conform to the basicaxiom scheme of combinatory logic. Curry was a “Kommilitone” ( roughly: a fellow-student)of MacLane, whom he remembered in his autobiography as “a good friend of mine fromG¨ottingen”. He is not mentioned in the thesis. Had MacLane presented his “Logikkalkul”in the form proposed above, he would have found a model, and therefore a consistencyproof, of combinatory logic just by looking at the structure P , the elementary algebraicproof system of section 1. Whether he would have constructed the combinators S and K as explicit objects in P is questionable. This had to wait almost fifty years to thisauthors construction [2], which bases the Plotkin-Scott model of the Lambda Calculus onarbitrary set (useful for applications to non-numeric modeling interactive systems, logicprogramming, and more recently to neuroscience.) K and S are examples of a generalalgorithm that compiles expressions φ ( X , . . . X n ) to yield an explicit ”combinator” [phi]with (( . . . ([phi] · X ) · X ) · · · · X n ) = φ ( X , . . . X n ).Applied to φ ( X, Y ) = X , respectively φ ( X, Y, Z ) = ( XY )( XZ ) this compilation yields K = {{ y } → ( ∅ → y ) : y ∈ P } , and S = {{ τ → ( { r , . . . , r n } → s ) } → ( { σ → r , . . . , σ n → r n } → ( σ → s )) : n ≥ , r , . . . r n ∈ P, τ ∪ i σ i = σ ⊆ P, σ finite } . Puzzle: Relation to Gentzen.
Gerhard Gentzen was more than just MacLane’scontemporary at G¨ottingen; indeed he translated MacLane’s thesis from his English intothe required German, at least in part. And he worked on his own famous thesis at justthis time, in which he does also treat of normalisation and operations on proofs. What didMacLane know or foresee and communicate on all this with Gentzen, (who is not mentionedin the thesis)?In fact Gentzen was the one person who could have furthered MacLane’s originalprogram; if he only would have made the connection to Church’s Lamba Calculus, publishedat the period, and cast his work in that framework. This was finally accomplished byBarendregt and Ghilezan [1] receiving the predicate “Theoretical Pearl” .There may be a personal resolution of this puzzle: MacLane, in the last months ofhis stay in Germany was freshly married and looked toward home eagerly, and with somerelief from his exposure to the frightening rise of Nazism and its disruptive incursion intoacademia.The present reconsideration of his youthful dream may perhaps revive some of its partsin view of the generality of our notion of “mathematical thoughts” and of their compositionalstructure and its algebra. Letter of MacLane to Menzler [7]. :6 E.Engeler
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References [1] Henk Barendregt and Silvia Ghilezan. Lambda terms for natural deduction, sequentcalculus and cut elimination in classical logic.
Journal of Functional Programming , 012000.[2] Erwin Engeler. Algebras and combinators.
Algebra Universalis , 13(1):389–392, Dec 1981.[3] Irving Kaplansky, editor.
Selected Papers – Saunders MacLane , 1979. Springer-VerlagNew York.[4] S. MacLane.
Saunders MacLane: A Mathematical Autobiography . A.K. Peters. 2005.[5] Saunders MacLane. A late return to a thesis in logic. In Kaplansky [3], pages 63 – 66.[6] Colin McLarty. The Last Mathematician From Hilbert’s G¨ottingen: Saunders MacLaneas Philosopher of Mathematics.
British Journal for the Philosophy of Science , 58(1):77–112, 2007.[7] Eckart Menzler-Trott.
Logic’s Lost Genius: The Life of Gerhard Gentzen . History ofMathematics. American Mathematical Society, 2007.[8] R¨udiger Thiele. Hilbert’s twenty-fourth problem. In
The American Mathematical Monthly ,number 110, pages 1 – 24, 2003.[9] Alfred North Whitehead and Bertrand Russell.
Principia Mathematica . CambridgeUniversity Press, 1913. Reprinted in 1962.
Dedication
This is to remember my good old friend Corrado Boehm.Our mathematical beginnings overlapped at the ETH in Zurich: Corrado was an assistantof Prof.E.Stiefel in applied mathematics in the late 1940s. As such he was sent to Bavariato inspect the hidden Zuse Z4 computer in view of its operability. With him was AmbrosSpeiser, who went on to found the Zurich IBM Research Laboratory (which in his timeproduced four Nobel prizes; he was one of the first presidents of IFIP). The machine wasthen smuggled into Switzerland and became the first programmable scientific computer onthe continent.We must have met around Z4 and in Bernays’ logic seminar in the early 1950’s. WithRichard Buechi, we were another triplet of his doctoral students. We diverged somewhat, Iwent into model theory and universal algebra and the USA, while he stayed with theoreticalcomputer science and returned to Italy. Corrado and I connected again on the semantics ofprogramming languages and, of course, on Combinatory Logic.
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