# Wilfried SiegCarnegie Mellon University | CMU · Department of Philosophy

Wilfried Sieg

PhD

## About

109

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Citations since 2016

## Publications

Publications (109)

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...

Human-centered automated proof search aims to capture structures of ordinary mathematical proofs and discover human strategies that are used (implicitly) in their construction. We analyze the ways of two theorem provers for approaching that goal. One, the G&G-prover, is presented in Ganesalingam and Gowers (J Autom Reason 58(2):253–291, 2017); the...

There are many examples of failed strategies whose intention is to optimize a process but instead they produce worse results than no strategy at all. Many fall under the loose umbrella of the “no free lunch theorem”. In this paper we present an example in which a simple (but assumedly naive) strategy intended to shorten proof lengths in the proposi...

Human-centered automated proof search aims to capture structures of ordinary mathematical proofs and discover human strategies that are used (implicitly) in their construction. We analyze the ways of two theorem provers for approaching that goal. One, the G&G-prover, is presented in [Gane-salingam and Gowers (2017)]; the other, Sieg's AProS system,...

The incompleteness theorems constitute the mathematical core of Gödel's philosophical challenge. They are given in their "most satisfactory form", as Gödel saw it, when the formality of theories to which they apply is characterized via Turing machines. These machines codify human mechanical procedures that can be carried out without appealing to hi...

Hilbert’s programmatic papers from the 1920s still shape, almost exclusively, the standard contemporary perspective of his views concerning (the foundations of) mathematics; even his own, quite different work on the foundations of geometry and arithmetic from the late 1890s is often understood from that vantage point. My essay pursues one main goal...

Mathematical structuralism is deeply connected with Hilbert and Bernays’s proof theory and its programmatic aim to ensure the consistency of all of mathematics. That aim was to be reached on the basis of finitist mathematics. Gödel’s second incompleteness theorem forced a step from absolute finitist to relative constructivist proof-theoretic reduct...

DOI: http://doi.org/10.26333/sts.xxxiv1.03
The incompleteness theorems constitute the mathematical core of Gödel’s philosophical challenge. They are given in their “most satisfactory form”, as Gödel saw it, when the formality of theories to which they apply is characterized via Turing machines. These machines codify human mechanical procedures tha...

Natural Formalization proposes a concrete way of expanding proof theory from the meta-mathematical investigation of formal theories to an examination of “the concept of the specifically mathematical proof.” Formal proofs play a role for this examination in as much as they reflect the essential structure and systematic construction of mathematical p...

Dedekind's proof of the Cantor–Bernstein theorem is based on his chain theory, not on Cantor's well-ordering principle. A careful analysis of the proof extracts an argument structure that can be seen in the many other proofs that have been given since. I contend there is essentially one proof that comes in two variants due to Dedekind and Zermelo ,...

This essay examines the two central and equivalent ways of introducing rigorous notions of computation as starting-points. The first way explicates effective calculability of number theoretic functions as the (uniform) calculability of their values in formal calculi; Gödel, Church and Kleene initially pursued this way. The other way views mechanica...

Proof theory is not an esoteric technical subject that was invented to support a formalist doctrine in the philosophy of mathematics; rather, it has been developed as an attempt to analyze aspects of mathematical experience and to isolate, possibly overcome, methodological problems in the foundations of mathematics. The origins of those problems, f...

The Cantor-Bernstein Theorem (CBT) is a classical result of general set theory. We have formalized a number of its proofs in the system ZF of Zermelo-Fraenkel. The formalizations are carried out via the system AProS, expanded to a convenient logical and set theoretic inference mechanism. AProS serves as a proof assistant and allows the direct const...

IN MEMORIAM: SOLOMON FEFERMAN (1928–2016) - Volume 23 Issue 3 - Charles Parsons, Wilfried Sieg

Turing was awarded the Order of the British Empire in June of 1946. Most people thought that the award was a well deserved mark of recognition honoring the mathematician who had given a successful definition of mechanical procedure, had introduced the
universal machine
capable of simulating all mechanical procedures, and had settled in the negative...

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the fifteenth publication in the Lecture Notes in Logic series, collects papers presented a...

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the sixth publication in the Lecture Notes in Logic series, collects the proceedings of the...

The conceptual confluence of Post’s and Turing’s analysis of combinatory processes, respectively of mechanical procedures, is the central topic in Davis and Sieg’s [14]. Where Turing argued convincingly for the adequacy of his notion of machine computation in 1936, Post viewed his identical notion in the same year as being tied to a working hypothe...

We show that strategies implemented in automatic theorem proving involve an
interesting tradeoff between execution speed, proving speedup/computational
time and usefulness of information. We advance formal definitions for these
concepts by way of a notion of normality related to an expected (optimal)
theoretical speedup when adding useful informati...

While online instructional technologies are becoming more popular in higher education, educators’ opinions about online learning tend to be generally negative. Furthermore, many studies have failed to systematically examine the features that distinguish one instructional mode from another, which weakens possible explanations for why online instruct...

In 1936, Post and Turing independently proposed two models of computation that are virtually identical. Turing refers back to these models in his (The word problem in semi-groups with cancellation. Ann. Math. 52, 491–505) and calls them “the logical computing machines introduced by Post and the author”. The virtual identity is not to be viewed as a...

Dedekind's mathematical work is integral to the transformation of mathematics in the nineteenth century and crucial for the emergence of structuralist mathematics in the twentieth century. We investigate the essential components of what Emmy Noether called, his 'axiomatic standpoint': abstract concepts (for systems of mathematical objects), models...

SimpsonStephen G.. Friedman's research on subsystems of second order arithmetic. Harvey Friedman's research on the foundations of mathematics, edited by HarringtonL. A., MorleyM. D., S̆c̆edrovA., and SimpsonS. G., Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 137–159. - Volu...

It is a remarkable fact that Hilbert's programmatic papers from the 1920s still shape, almost exclusively, the standard contemporary perspective of his views concerning (the foundations of) mathematics; even his own, quite different work on the foundations of geometry and arithmetic1 from the late 1890s is often understood from that vantage point....

Alan Turing was an inspirational figure who is now recognised as a genius of modern mathematics. In addition to leading the Allied forces' code-breaking effort at Bletchley Park in World War II, he proposed the theoretical foundations of modern computing and anticipated developments in areas from information theory to computer chess. His ideas have...

TennantNeil. Natural logic. Edinburgh University Press, Edinburgh1978, ix + 196 pp. - Volume 48 Issue 1 - Wilfried Sieg

The following lectures, delivered during the Winter Semester 1917/18, are a pivotal event in the development of mathematical logic and mark the start of Hilbert1#x2019;s long and fruitful collaboration with Paul Bernays. Towards the end of his lectures on set theory in the preceding Summer Semester, Hilbert had stated (p. 146), 1#x2018;I hope to be...

During the two years following the 1917/18 lectures which form Chapter 1 of this Volume, Hilbert appears to have devoted little time to foundations, at least in public. A letter from Bernays to Russell on 8 April 1920 remarks, ‘As you may know, Professor Hilbert I am honoured to be his assistant has been working intensively for a number of years on...

The following set of lectures from the Winter Semester of 1924/25 (Hilbert 1924/25* ) has a different character from the other lecture notes published in this Volume. Hilbert’s logic lectures from the fall of 1917 to the spring of 1924, addressed to advanced students of mathematics, are a remarkable technical achievement.

The lectures in the Summer Semester of 1920 ended with consistency proofs for extremely weak fragments of arithmetic. The question, made explicit in the Introduction to Chapter 2 (see p. 296) was then this: Can these consistency proofs somehow be extended to establish the consistency of increasingly stronger and thus mathematically more interesting...

Church’s and Turing’s theses assert dogmatically that an informal notion of effective calculability is adequately captured by a particular mathematical concept of computability. I present analyses of calculability that are embedded in a rich historical and philosophical context, lead to precise concepts, and dispense with theses. To investigate eff...

Gödel’s incompleteness theorems had a dramatic impact on Hilbert’s foundational program. That is common lore. For some, e.g. von Neumann and Herbrand, they undermined the finitist consistency program; for others, e.g. Gödel and Bernays, they left room for a fruitful development of proof theory. This paper aims for a nuanced and deepened understandi...

Kurt Gödel (1906–1978) did groundbreaking work that transformed logic and other important aspects of our understanding of mathematics, especially his proof of the incompleteness of formalized arithmetic. This book on different aspects of his work and on subjects in which his ideas have contemporary resonance includes papers from a May 2006 symposiu...

Work in the foundations of mathematics should provide systematic frameworks for important parts of the practice of mathematics,
and the frameworks should be grounded in conceptual analyses that reflect central aspects of mathematical experience. The
Hilbert School of the 1920s used suitable frameworks to formalize (parts of) mathematics and provide...

We present strategies and heuristics underlying a search procedure that finds proofs for Gödel’s incompleteness theorems at an abstract axiomatic level. As axioms we take for granted the representability and derivability conditions for the central syntactic notions as well as the diagonal
lemma for constructing self-referential sentences. The strat...

Church’s and Turing’s theses assert dogmatically that an informal notion of effective calculability is captured adequately by a particular mathematical concept of computabilty. I present analyses of calculability that are embedded in a rich historical and philosophical context, lead to precise concepts, and dispense with theses.
To investigate effe...

The paper discusses tools for teaching logic used in Logic & Proofs , a web-based introduction to modern logic that has been taken by more than 1,300 students since the fall of 2003. The tools include a wide array of interactive learning environments or cognitive mini-tutors; most important among them is the Carnegie Proof Lab . The Proof Lab is a...

Turing’s notion of human computability is exactly right not only for obtaining a negative solution of Hilbert’s Entscheidungsproblem that is conclusive, but also for achieving a precise characterization of formal systems that is needed for the general formulation of the incompleteness theorems. The broad intellectual context reaches back to Leibniz...

Identifying the informal concept of effective calculability with a rigorous mathematical notion like general recursiveness
or Turing computability is still viewed as problematic, and rightly so. In a 1934 conversation with Church, Gödel suggested
finding axioms for the notion of effective calculability and “doing something on that basis” instead of...

The identification of an informal concept of ‘effective calculability’ with a rigorous mathematical notion like ‘recursiveness’
or ‘Turing computability’ is still viewed as problematic, and I think rightly so. I analyze three different and conflicting
perspectives Gödel articulated in the three decades from 1934 to 1964. The significant shifts in G...

In the book Grundlagen Der Mathematik, Hilbert and Bernays systematically present their proof-theoretic investigations and a wide range of current results, such as Herbrand's theorems and Gödel's incompleteness theorems. Some specialized topics are also discussed, such as the development of mathematical analysis and the unsolvability of the decisio...

Church’s and Turing’s theses dogmatically assert that an informal notion of computability is captured by a particular mathematical
concept. I present an analysis of computability that leads to precise concepts, but dispenses with theses.

Two young logicians, whose work had a dramatic impact on the direction of logic, exchanged two letters in early 1931. Jacques Herbrand initiated the correspondence on 7 April and Kurt Gödel responded on 25 July, just two days before Herbrand died in a mountaineering accident at La Bérarde (Isère). Herbrand's letter played a significant role in the...

We present strategies and heuristics underlying a search procedure that finds proofs for Gödel's incompleteness theorems at an abstract axiomatic level. As axioms we take for granted the representability and derivability conditions for the central syntactic notions as well as the diagonal lemma for constructing self-referential sentences. The strat...

We provide a theoretical framework that allows the direct search for natural deduction proofs in some non-classical logics,
namely, intuitionistic sentential and predicate logic, but also in the modal logic S4. The framework uses so-called intercalation calculi to build up broad search spaces from which normal proofs can be extracted,
if a proof ex...

The notion of effective calculability is central for logic and the philosophy of mathematics, not to speak of computer science, artificial intelligence, and cognitive science. Turing gave in 1936 what is viewed as the most convincing analysis of this informal notion:1 he was led to the mathematical concept of computability by idealized machines and...

A little more than three quarters of a century ago, in the Spring of 1921, Hilbert gave the first presentation of his new investigations concerning the foundations of arithmetic in Copenhagen and some months later in Hamburg. Hilbert’s 1922 paper Neubegründung der Mathematik is based on these talks.2 The paper is important for a variety of systemat...

Hilbert's finitist program was not created at the beginning of the twenties solely to counteract Brouwer's intuitionism, but rather emerged out of broad philosophical reflections on the foundations of mathematics and out of detailed logical work; that is evident from notes of lecture courses that were given by Hilbert and prepared in collaboration...

Natural deduction (for short: nd-) calculi have not been used systematically as a basis for automated theorem proving in classical logic. To remove objective obstacles to their use we describe (1) a method that allows to give semantic proofs of normal form theorems for nd-calculi and (2) a framework that allows to search directly for normal nd-proo...

Alonzo Church's mathematical work on computability and undecidability is well-known indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Church's Thesis" put forward publicly only in April 19...

In the current discussion on philosophy of mathematics some do as if systematic foundational work supported an exclusive alternative between Platonism and Constructivism; others do as if such mathematical and logical research were deeply misguided and had no bearing on our understanding of mathematics. Both attitudes prevent us from grasping insigh...

Does the presentation and use of the search space matter for complex problem solving tasks? We address these questions for the construction of proofs in sentential logic. Using a fully computerized logic course, we isolated crucial features of computer environments and assessed their relative pedagogical effectiveness. After being given a pretest f...

A ”linear — style” sequent calculus makes it possible to explore the close structural relationships between primitive recursive
programs and their inductive termination proofs, and between program transformations and their corresponding proof transformations.
In this context the recursive — to — tail — recursive transformation corresponds proof the...

The report details a facet of current logical work, namely characterizing the provably total functions of weak theories for
analysis. That is described paradigmatically for a theory introduced by H. Friedman. However, the connection to broader foundational
concerns is emphasized throughout.
Tra le sfaccettature della ricerca logica contemporanea vi...

Herbrand's Theorem, in the form of
$\underset{\raise0.3em\hbox{$\underset{\raise0.3em\hbox{
-inversion lemmata for finitary and infinitary sequent calculi, is the crucial tool for the determination of the provably total function(al)s of a variety of theories. The theories are (second order extensions of) fragments of classical arithmetic; the class...

SmithRick L.. The consistency strengths of some finite forms of the Higman and Kruskal theorems. Harvey Friedman's research on the foundations of mathematics, edited by HarringtonL. A., MorleyM. D., S̆c̆edrovA., and SimpsonS. G., Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp....

Hilbert’s Program deals with the foundations of mathematics from a very special perspective; a perspective that stems from Hubert’s answer to the question “What Is Mathematics?”. The popular version of his “formalist” answer, radical at Hubert’s time and shocking to thoughtful mathematicians even today, is roughly this: the whole “thought-content”...

A Symposium on Hilbert's Program - Volume 53 Issue 2 - Wilfrid Hodges, Wilfried Sieg

On June 4, 1925, Hilbert delivered an address to the Westphalian Mathematical Society in Miinster; that was, as a quick calculation will convince you, almost exactly sixty years ago. The address was published in 1926 under the title Über das Unendliche and is perhaps Hilbert's most comprehensive presentation of his ideas concerning the finitist jus...

We establish by elementary proof-theoretic means the conservativeness of two subsystems of analysis over primitive recursive arithmetic. The one subsystem was introduced by Friedman [6], the other is a strengthened version of a theory of Minc [14]; each has been shown to be of considerable interest for both mathematical practice and metamathematica...

The formal theories considered here are all subsystems of second order arithmetic with the full comprehension principle, briefly (CA). They are theories for classical analysis: Hilbert used a theory equivalent to (CA) as a formal framework for mathematical analysis in lectures during the early twenties; an extensive portion of analysis had already...

The title of my paper indicates that I plan to write about foundations for analysis and about proof theory; however, I do not intend to write about the foundations for analysis and thus not about analysis viewed from the vantage point of any “school” in the philosophy of mathematics. Rather, I shall report on some mathematical and proof-theoretic i...

Strategic thinking is at the intellectual core of the calculus for the 21 st century, which is not the mathematical calculus that emerged in the 17 th century, but rather the logical calculus that was conceived in the same period by Leibniz, one of the two inventors of the mathematical calculus. Leibniz put great emphasis on a universal language to...

What is it that shapes mathematical arguments into proofs that are intelligible to us, and what is it that allows us to find proofs efficiently? – This is the informal question I intend to address by investigating, on the one hand, the abstract ways of the axiomatic method in modern mathematics and, on the other hand, the concrete ways of proof con...