Nominal techniques

This is the author accepted manuscript. The final version is available from the Association for Computing Machinery via http://dx.doi.org/10.1145/2893582.2893594 Programming languages abound with features making use of names in various ways. There is a mathematical foundation for the semantics of such features which uses groups of permutations of names and the notion of the support of an object with respect to the action of such a group. The relevance of this kind of mathematics for the semantics of names is perhaps not immediately obvious. That it is relevant and useful has emerged over the last 15 years or so in a body of work that has acquired its own name: nominal techniques. At the same time, the application of these techniques has broadened from semantics to computation theory in general. This article introduces the subject and is based upon a tutorial at LICS-ICALP 2015 [Pitts 2015a]. </jats:p

Igbo Numerical System and Mathematics: Towards Harnessing Potential for Business, Governance, Science and Technology

This study explored the nature of Igbo numerical system and mathematics towards harnessing its potential for business, governance and science and technology. Like other civilizations, Igbo mathematics evolved from its numerical system which is aboriginally vegesimal number system (of base 20). The Igbo number system was advanced because it has number names for all the numbers within its vegesimal system although the numerals were not symbolized. Outside its general and conventional use, the Igbo numbers do have social and religious implied meaning which influenced the use of the numbers and the people. The cultural influence of this number system to a large extent determined the utility and application of same in mathematical principles and later in its outcome as human development. Some humanistic aspects were implicated such as business enterprise, governance and science and technology as areas where the application potentials of indigenous numerical system and mathematical expressions in daily usage and science can be harnessed for greater benefits

Lassie: HOL4 Tactics by Example

Proof engineering efforts using interactive theorem proving have yielded several impressive projects in software systems and mathematics. A key obstacle to such efforts is the requirement that the domain expert is also an expert in the low-level details in constructing the proof in a theorem prover. In particular, the user needs to select a sequence of tactics that lead to a successful proof, a task that in general requires knowledge of the exact names and use of a large set of tactics. We present Lassie, a tactic framework for the HOL4 theorem prover that allows individual users to define their own tactic language by example and give frequently used tactics or tactic combinations easier-to-remember names. The core of Lassie is an extensible semantic parser, which allows the user to interactively extend the tactic language through a process of definitional generalization. Defining tactics in Lassie thus does not require any knowledge in implementing custom tactics, while proofs written in Lassie retain the correctness guarantees provided by the HOL4 system. We show through case studies how Lassie can be used in small and larger proofs by novice and more experienced interactive theorem prover users, and how we envision it to ease the learning curve in a HOL4 tutorial

The Place of Philosophy in European Culture

In this paper the author investigates the place of philosophy in European culture. Philosophy has taken a considerable time to be recognised, or to recognise itself, as distinct from other disciplines. Although philosophy gave birth to physics and, more recently, to other sciences, it is not seen as a “technical” subject, like mathematics and natural sciences, or even social sciences such as economics. Philosophy is available to the general (educated) public while the technical subjects are not. All educated people know the names of the great Western philosophers. Less people know the names of the great mathematicians (other than those such as Descartes and Leibniz which were, at the same time, philosophers). Therefore, philosophy has not lost its place as part of high culture, as have the natural sciences and mathematics. Philosophy continues to exert a pervasive effect upon European culture in general. However, according to the author, two tasks lie before philosophers; two gulfs are for us to bridge. The fi rst one is the gulf between philosophers of all schools and scientists (particularly physicists); the other one is that between divergent philosophical schools – between analytical philosophy and so called “continental” philosophy. If it solves these problems, philosophy will remain what it has been in the past – a shining component of European culture

Exploring the landscapes of "computing": digital, neuromorphic, unconventional -- and beyond

The acceleration race of digital computing technologies seems to be steering toward impasses -- technological, economical and environmental -- a condition that has spurred research efforts in alternative, "neuromorphic" (brain-like) computing technologies. Furthermore, since decades the idea of exploiting nonlinear physical phenomena "directly" for non-digital computing has been explored under names like "unconventional computing", "natural computing", "physical computing", or "in-materio computing". This has been taking place in niches which are small compared to other sectors of computer science. In this paper I stake out the grounds of how a general concept of "computing" can be developed which comprises digital, neuromorphic, unconventional and possible future "computing" paradigms. The main contribution of this paper is a wide-scope survey of existing formal conceptualizations of "computing". The survey inspects approaches rooted in three different kinds of background mathematics: discrete-symbolic formalisms, probabilistic modeling, and dynamical-systems oriented views. It turns out that different choices of background mathematics lead to decisively different understandings of what "computing" is. Across all of this diversity, a unifying coordinate system for theorizing about "computing" can be distilled. Within these coordinates I locate anchor points for a foundational formal theory of a future computing-engineering discipline that includes, but will reach beyond, digital and neuromorphic computing.Comment: An extended and carefully revised version of this manuscript has now (March 2021) been published as "Toward a generalized theory comprising digital, neuromorphic, and unconventional computing" in the new open-access journal Neuromorphic Computing and Engineerin

The Effect of Gender in the Publication Patterns in Mathematics

Despite the increasing number of women graduating in mathematics, a systemic gender imbalance persists and is signified by a pronounced gender gap in the distribution of active researchers and professors. Especially at the level of university faculty, women mathematicians continue being drastically underrepresented, decades after the first affirmative action measures have been put into place. A solid publication record is of paramount importance for securing permanent positions. Thus, the question arises whether the publication patterns of men and women mathematicians differ in a significant way. Making use of the zbMATH database, one of the most comprehensive metadata sources on mathematical publications, we analyze the scholarly output of ~150,000 mathematicians from the past four decades whose gender we algorithmically inferred. We focus on development over time, collaboration through coautorships, presumed journal quality and distribution of research topics -- factors known to have a strong impact on job perspectives. We report significant differences between genders which may put women at a disadvantage when pursuing an academic career in mathematics.Comment: 24 pages, 12 figure

Willard Van Orman Quine's Philosophical Development in the 1930s and 1940s

As analytic philosophy is becoming increasingly aware of and interested in its own history, the study of that field is broadening to include, not just its earliest beginnings, but also the mid-twentieth century. One of the towering figures of this epoch is W.V. Quine (1908-2000), champion of naturalism in philosophy of science, pioneer of mathematical logic, trying to unite an austerely physicalist theory of the world with the truths of mathematics, psychology, and linguistics. Quine's posthumous papers, notes, and drafts revealing the development of his views in the forties have recently begun to be published, as well as careful philosophical studies of, for instance, the evolution of his key doctrine that mathematical and logical truth are continuous with, not divorced from, the truths of natural science. But one central text has remained unexplored: Quine's Portuguese-language book on logic, his 'farewell for now' to the discipline as he embarked on an assignment in the Navy in WWII. Anglophone philosophers have neglected this book because they could not read it. Jointly with colleagues, I have completed the first full English translation of this book. In this accompanying paper I draw out the main philosophical contributions Quine made in the book, placing them in their historical context and relating them to Quine's overall philosophical development during the period. Besides significant developments in the evolution of Quine's views on meaning and analyticity, I argue, this book is also driven by Quine's indebtedness to Russell and Whitehead, Tarski, and Frege, and contains crucial developments in his thinking on philosophy of logic and ontology. This includes early versions of some arguments from 'On What There Is', four-dimensionalism, and virtual set theory