Student mathematical proofs by Edward Stephen (?) Wigglesworth, 1788

Sheet with two handwritten mathematical proofs signed "Wigglesworth, 1788," likely referring Harvard student Edward Stephen Wigglesworth. The first proof, titled "Problem 1st," examines a prompt beginning, "Given the distance between the Centers of the Sun and Planet, and their quantities of matter; to find a place where a body will be attracted to neither of them." The second proof, titled "Problem 2d," begins "A & B having returned from a journey, had riden [sic] so far that if the square of the number of miles..." and asks "how many miles did each of them travel?"

An Innovative Approach to Teaching Structural Induction for Computer Science

Proofs by induction are central to many computer science areas such as data structures, theory of computation, programming languages, program efficiency-time complexity, and program correctness. Proofs by induction can also improve students’ understanding of and performance with computer science concepts such as programming languages, algorithm design, and recursion, as well as serve as a medium for teaching them. Even though students are exposed to proofs by induction in many courses of their curricula, they still have difficulties understanding and performing them. This impacts the whole course of their studies, since proofs by induction are omnipresent in computer science. Specifically, students do not gain conceptual understanding of induction early in the curriculum and as a result, they have difficulties applying it to more advanced areas later on in their studies. The goal of my dissertation is twofold: 1. identifying sources of computer science students’ difficulties with proofs by induction, and 2. developing a new approach to teaching proofs by induction by way of an interactive and multimodal electronic book (e-book). For the first goal, I undertook a study to identify possible sources of computer science students’ difficulties with proofs by induction. Its results suggest that there is a close correlation between students’ understanding of inductive definitions and their understanding and performance of proofs by induction. For designing and developing my e-book, I took into consideration the results of my study, as well as the drawbacks of the current methodologies of teaching proofs by induction for computer science. I designed my e-book to be used as a standalone and complete educational environment. I also conducted a study on the effectiveness of my e-book in the classroom. The results of my study suggest that, unlike the current methodologies of teaching proofs by induction for computer science, my e-book helped students overcome many of their difficulties and gain conceptual understanding of proofs induction.

An Infinite Pebble Game and Applications

We generalize the well-known pebble game to infinite dag's, and we use this generalization to give new and shorter proofs of results in different areas of computer science (as diverse as "logic of programs" and "formal language theory"). Our applications here include a proof of a theorem due to Salomaa, asserting the existence of a context-free language with infinite index, and a proof of a theorem due to Tiuryn and Erimbetov, asserting that unbounded memory increases the power of logics of programs. The original proofs by Salomaa, Tiuryn, and Erimbetov, are fairly technical. The proofs by Tiuryn and Erimbetov also involve advanced techniques of model theory, namely, back-and-forth constructions based on a variant of Ehrenfeucht-Fraisse games. By contrast, our proofs are not only shorter, but also elementary. All we need is essentially finite induction and, in the case of the Tiuryn-Erimbetov result, the compactness and completeness of first-order logic.

Canonical proof nets for classical logic

Proof nets provide abstract counterparts to sequent proofs modulo rule permutations; the idea being that if two proofs have the same underlying proof-net, they are in essence the same proof. Providing a convincing proof-net counterpart to proofs in the classical sequent calculus is thus an important step in understanding classical sequent calculus proofs. By convincing, we mean that (a) there should be a canonical function from sequent proofs to proof nets, (b) it should be possible to check the correctness of a net in polynomial time, (c) every correct net should be obtainable from a sequent calculus proof, and (d) there should be a cut-elimination procedure which preserves correctness. Previous attempts to give proof-net-like objects for propositional classical logic have failed at least one of the above conditions. In Richard McKinley (2010) [22], the author presented a calculus of proof nets (expansion nets) satisfying (a) and (b); the paper defined a sequent calculus corresponding to expansion nets but gave no explicit demonstration of (c). That sequent calculus, called LK∗ in this paper, is a novel one-sided sequent calculus with both additively and multiplicatively formulated disjunction rules. In this paper (a self-contained extended version of Richard McKinley (2010) [22]), we give a full proof of (c) for expansion nets with respect to LK∗, and in addition give a cut-elimination procedure internal to expansion nets – this makes expansion nets the first notion of proof-net for classical logic satisfying all four criteria.

Prophetical Extracts.

[No. III] Prophetical extracts. Containing a very scarce prophetic piece, intitled [sic], A cry from the desart [sic], or, testimonials of the miraculous things lately come to pass in the Cevennes, or Southern parts of France. Verified upon oath, and by other proofs. [by Maximilian Misson] (1794).--No. IV. Prophetical extracts. Relative to the revolution in France, and the decline of the papal power in the world: selected from Fleming, Usher, Jerieu, Goodwin, Gill, Love, Daut, Brown, Knox, Willison, More, Newton, Lacey, Owen, Marion, Cavalier, and many more (1794).--No. V. Prophetical extracts. Relative to the revolution in France...The lives, prophecies, visions, and revelations, of Christopher Kotterus, and Christina Poniatonia, two eminent prophets, in Germany, containing predictions concerning the Pope, the King of France, and the Roman Empire ; with the sudden destruction of the papal power, the miraculous conversion of the Turks, the calling in of the Jews, and the uniting all religions into one universal visible church. Many of which prophecies being desired by the then King of Bohemia, were by the learned Comenius presented to him [by Johann Amos Comenius] (1794).

An innovative approach to teaching structural induction for computer science

Proofs by induction are central to many computer science areas such as data structures, theory of computation, programming languages, program efficiency-time complexity, and program correctness. Proofs by induction can also improve students’ understanding and performance of computer science concepts such as programming languages, algorithm design, and recursion, as well as serve as a medium for teaching them. Even though students are exposed to proofs by induction in many courses of their curricula, they still have difficulties understanding and performing them. This impacts the whole course of their studies, since proofs by induction are omnipresent in computer science. Specifically, students do not gain conceptual understanding of induction early in the curriculum and as a result, they have difficulties applying it to more advanced areas later on in their studies. The goal of my dissertation is twofold: (1) identifying sources of computer science students’ difficulties with proofs by induction, and (2) developing a new approach to teaching proofs by induction by way of an interactive and multimodal electronic book (e-book). For the first goal, I undertook a study to identify possible sources of computer science students’ difficulties with proofs by induction. Its results suggest that there is a close correlation between students’ understanding of inductive definitions and their understanding and performance of proofs by induction. For designing and developing my e-book, I took into consideration the results of my study, as well as the drawbacks of the current methodologies of teaching proofs by induction for computer science. I designed my e-book to be used as a standalone and complete educational environment. I also conducted a study on the effectiveness of my e-book in the classroom. The results of my study suggest that, unlike the current methodologies of teaching proofs by induction for computer science, my e-book helped students overcome many of their difficulties and gain conceptual understanding of proofs induction.

Inductive Theorem Proving and Computer Algebra in the Mathweb Software Bus

Reasoning systems have reached a high degree of maturity in the last decade. However, even the most successful systems are usually not general purpose problem solvers but are typically specialised on problems in a certain domain. The MathWeb SOftware Bus (Mathweb-SB) is a system for combining reasoning specialists via a common osftware bus. We described the integration of the lambda-clam systems, a reasoning specialist for proofs by induction, into the MathWeb-SB. Due to this integration, lambda-clam now offers its theorem proving expertise to other systems in the MathWeb-SB. On the other hand, lambda-clam can use the services of any reasoning specialist already integrated. We focus on the latter and describe first experimnents on proving theorems by induction using the computational power of the MAPLE system within lambda-clam.

Attempted Proofs of the Differentiation of Magmas by Syntexis

This investigation was an effort to obtain proof of the much disputed principle that most of the observed mag­netic differentiations have been due to the assimilation, or syntexis, of the intruded country rocks.

Forty-seven identifications of the British nation with the lost Ten Tribes of Israel : founded upon five hundred scripture proofs. Dedicated to the (so-called) British people by the kinsman /

Mode of access: Internet.

Printers' proofs; the methods by which they are made, marked, and corrected, with observations on proofreading, illustrated /

"Supplementary reading": p. 42.

An abridgment of the Christian doctrine : with proofs of scripture on points controverted, by way of question and answer /

Mode of access: Internet.

A new translation of Aristotle's Rhetoric; with an introduction and appendix, explaining its relation to his exact philosophy, and vindicating that philosophy, by proofs that all departures from it have been deviations into error.

Mode of access: Internet.

Elements of medical logic, illustrated by practical proofs and examples.

"Goodsell & Wells, - Printers, Hartford."

The history of Oregon and California : and the other territories of the North-West coast of North America, accompanied by a geographical view and map of those countries, and a number of documents as proofs and illustrations of the history /

Includes index.

The remnant found; or, The place of Israels̓ hiding discovered. Being a summary of proofs, showing that the Jews of Daghistan of the Caspian Sea are the remnant of the Ten Tribes.: The result of personal investigation during a missionary tour of eight months in Georgia, by permission of the Russian government in the years 1837 and 1838.

With this is bound his An appealon behalf of Jews scattered in India, Persia and Arabia. London, 1840.