This dissertation addresses a variety of foundational issues pertaining to the notion of algorithm employed in mathematics and computer science. In these settings, an algorithm is taken to be an effective mathematical procedure for solving a previously stated mathematical problem. Procedures of this sort comprise the notional subject matter of the subfield of computer science known as algorithmic analysis. In this context, algorithms are referred to via proper names of which computational properties are directly predicated )). Moreover, many formal results (...) are traditionally stated by explicitly quantifying over algorithms. These observations motivate the view that algorithms are themselves abstract mathematical objects -- a view I refer to as algorithmic realism. The status of this view is clearly related to that of Church's Thesis -- i.e. the claim that a function is computable by an algorithm just in case it is partial recursive. But whereas Church's Thesis is widely accepted on the basis of several well-known mathematical analyses of algorithmic computability, there is presently no consensus on how we can uniformly identify individual algorithms with mathematical objects. In the first chapter of this work, I attempt to illustrate the theoretical significance of algorithmic realism. I suggest that this view is not only of foundational significance to computer science, but also worth highlighting due to the role algorithms play in the fixation of mathematical knowledge and their relationship to intensional entities such as propositions and properties. Chapter Two presents a formal framework for reducing computational discourse to mathematical discourse modeled on contemporary discussion of mathematical nominalism. Chapter Three is centered on a case study intended to illustrate the technical exigencies involved with defending algorithmic realism. Chapter Four explores various ways in which algorithms might be identified with models of computation. And finally, Chapter Five argues that no such identification can uniformly satisfy all statements of algorithmic identity and non-identity affirmed by computational practice. I suggest that the technical crux of algorithmic realism lies in the necessity of reducing recursive specifications of algorithms to iterative models of computation in a manner which preserves algorithmic identity. (shrink)

A prized property of theories of all kinds is that of generality, of applicability or least relevance to a wide range of circumstances and situations. The purpose of this article is to present a pair of distinctions that suggest that three kinds of generality are to be found in mathematics and logics, not only at some particular period but especially in developments that take place over time: ‘omnipresent’ and ‘multipresent’ theories, and ‘ubiquitous’ notions that form dependent parts, or moments, of (...) theories. The category of ‘facets’ is also introduced, primarily to assess the roles of diagrams and notations in these two disciplines. Various consequences are explored, starting with means of developing applied mathematics, and then reconsidering several established ways of elaborating or appraising theories, such as analogising, revolutions, abstraction, unification, reduction and axiomatisation. The influence of theories already in place upon theory-building is emphasised. The roles in both mathematics and logics of set theory, abstract algebras, metamathematics, and model theory are assessed, along with the different relationships between the two disciplines adopted in algebraic logic and in mathematical logic. Finally, the issue of monism versus pluralism in these two disciplines is rehearsed, and some suggestions are made about the special character of mathematical and logical knowledge, and also the differences between them. Since the article is basically an exercise in historiography, historical examples and case studies are described or noted throughout. (shrink)

In his essay Thinking Colours and/or Machines Kittler hints at a key point in the emergence of modern European culture: the point at which ‘letters and numbers no longer coincide’. In this essay - first published in 2003 as Zahl und Ziffer - Kittler traces the split between numerals and numbers in sweeping historical detail. This is part of a much larger project, the aim of which is to think about technology, history and culture anew by considering the ways in (...) which ‘letters, numbers, images and tones’ have been differentiated and re-integrated by developing notation systems and media technologies. In this present essay, Kittler is concerned specifically with the question of number. His argument is that numbers and numerals have not always stood apart. In Old Hebrew and even nursery rhymes, for example, numbers are in fact words. This might seem like a banal observation, but for Kittler it is crucial as historically, mathematics proper only developed ‘in cultures in which numbers are present as numerals’, a development which entailed the transformation of numbers from signifiers (‘a matter of hearing’) into signifieds (‘a matter of reading and writing’) and which rested on the emergence of storage and transmission media that Kittler calls ‘the media of mathematics’. This connection between media and mathematics is explored through a wide range of philosophical sources: Plato, Philolaus, Aristotle and Aquinas, to name but a few. Kittler is fascinated by the inscription technologies that make mathematics possible, and which at the same time structure cultural forms as well as our bodily experience of them. As he puts it in a programmatic aside, ‘media studies only make sense’ if they focus on how ‘media make senses.’ Hence his focus on the Greek phonetic alphabet: for Kittler, its superiority has less to do with its ability to reproduce the spoken words of any language, than with the fact that at one point it was used to handle language, music, and mathematics - that is, one and the same set of signs was used to encode letters, tones and numbers. This, however, was not an abstract undertaking but developed in constant feedback with specific instruments or media, especially the lyre and the bow. It was here that fundamental concepts such as logoi were first developed that were subsequently distorted, misunderstood and deprived of their musico-technical origins by philosophers such as Aristotle. Kittler’s essay is thus also part of a larger cultural project, indebted in particular to Martin Heidegger, whose aim it is to ferret out the different, as yet unrevealed beginning of occidental culture. Moreover, while it was necessary for the evolution of modern mathematics that numbers receive a notation system of their own that will allow for ratios and decimals, among others, it is obvious that Kittler sees the computer (as first envisaged in Alan Turing’s mathematical modelling) as a return of universal alphabet that operates in constant feedback with a medium that shapes our senses: ‘In the Greek alphabet our senses were present - and thanks to Turing they are so once again.’. (shrink)

In this article, we come back to the seminal role of epistemology in didactics of sciences and particularly in mathematics. We defend that the epistemological research on the interactions between mathematics and informatics is necessary to feed didactical research on today’s mathematics learning and teaching situations, impacted by the development of informatics. We develop some examples to support this idea and propose some perspectives to attack this issue.