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After the publication of Begriffsschrift, a conflict erupted between Frege and Schröder regarding their respective logical systems which emerged around the Leibnizian notions of lingua characterica and calculus ratiocinator. Both of them claimed their own logic to be a better realisation of Leibniz’s ideal language and considered the rival system a mere calculus ratiocinator. Inspired by this polemic, van Heijenoort (1967b) distinguished two conceptions of logic—logic as language and logic as calculus—and presented them as opposing views, but did not explain Frege’s and Schröder’s conceptions of the fulfilment of Leibniz’s scientific ideal.
In this paper I explain the reasons for Frege’s and Schröder’s mutual accusations of having created a mere calculus ratiocinator. On the one hand, Schröder’s construction of the algebra of relatives fits with a project for the reduction of any mathematical concept to the notion of relative. From this stance I argue that he deemed the formal system of Begriffsschrift incapable of such a reduction. On the other hand, first I argue that Frege took Boolean logic to be an abstract logical theory inadequate for the rendering of specific content; then I claim that the language of Begriffsschrift did not constitute a complete lingua characterica by itself, more being seen by Frege as a tool that could be applied to scientific disciplines. Accordingly, I argue that Frege’s project of constructing a lingua characterica was not tied to his later logicist programme.
In practice, mathematical proofs are most often the result of careful planning by the agents who produced them. As a consequence, each mathematical proof inherits a plan in virtue of the way it is produced, a plan which underlies its “architecture” or “unity.” This paper provides an account of plans and planning in the context of mathematical proofs. The approach adopted here consists in looking for these notions not in mathematical proofs themselves, but in the agents who produced them. The starting point is to recognize that to each mathematical proof corresponds a proof activity which consists of a sequence of deductive inferences—i.e., a sequence of epistemic actions—and that any written mathematical proof is only a report of its corresponding proof activity. The main idea to be developed is that the plan of a mathematical proof is to be conceived and analyzed as the plan of the agent(s) who carried out the corresponding proof activity. The core of the paper is thus devoted to the development of an account of plans and planning in the context of proof activities. The account is based on the theory of planning agency developed by Michael Bratman in the philosophy of action. It is fleshed out by providing an analysis of the notions of intention—the elementary components of plans—and practical reasoning—the process by which plans are constructed—in the context of proof activities. These two notions are then used to offer a precise characterization of the desired notion of plan for proof activities. A fruitful connection can then be established between the resulting framework and the recent theme of modularity in mathematics introduced by Jeremy Avigad. This connection is exploited to yield the concept of modular presentations of mathematical proofs which has direct implications for how to write and present mathematical proofs so as to deliver various epistemic benefits. The account is finally compared to the technique of proof planning developed by Alan Bundy and colleagues in the field of automated theorem proving. The paper concludes with some remarks on how the framework can be used to provide an analysis of understanding and explanation in the context of mathematical proofs.
A globally expressivist analysis of the indicative conditional based on the Ramsey Test is presented. The analysis is a form of ‘global’ expressivism in that it supplies acceptance and rejection conditions for all the sentence forming connectives of propositional logic (negation, disjunction, etc.) and so allows the conditional to embed in arbitrarily complex sentences (thus avoiding the Frege–Geach problem). The expressivist framework is semantically characterized in a restrictor semantics due to Vann McGee, and is completely axiomatized in a logic dubbed ICL (‘Indicative Conditional Logic’). The expressivist framework extends the AGM (after Alchourron, Gärdenfors, Makinson) framework for belief revision and so provides a categorical (‘yes’–‘no’) epistemology for conditionals that complements McGee’s probabilistic framework while drawing on the same semantics. The result is an account of the semantics and acceptability conditions of the indicative conditional that fits well with the linguistic data (as pooled by linguists and from psychological experiments) while integrating both expressivist and semanticist perspectives.
Many students complete PhDs in functional programming each year. As a service to the community, twice per year the Journal of Functional Programming publishes the abstracts from PhD dissertations completed during the previous year.
This book will help readers understand fundamental and advanced statistical models and deep learning models for robust speaker recognition and domain adaptation. This useful toolkit enables readers to apply machine learning techniques to address practical issues, such as robustness under adverse acoustic environments and domain mismatch, when deploying speaker recognition systems. Presenting state-of-the-art machine learning techniques for speaker recognition and featuring a range of probabilistic models, learning algorithms, case studies, and new trends and directions for speaker recognition based on modern machine learning and deep learning, this is the perfect resource for graduates, researchers, practitioners and engineers in electrical engineering, computer science and applied mathematics.
A full history of IL remains to be written. A commonly-stated position – so frequently presented that it has become a fundamental trope of the field – is that the concept was created in a 1974 report by Paul Zurkowski, presented to the US National Commission on Libraries and Information Science and entitled The Information Service Environment: Relationships and Priorities. In Kelly's eulogy to this work and its author, he writes (2014, 1): ‘The power of Information Literacy was unleashed in 1974 when Paul G. Zurkowski, Esq., founded the IL movement at the dawn of the Information Age.’ The statement is, at best, an exaggeration. Yet, many other writers (Rader, 2002; Eisenberg, Lowe and Spitzer, 2004; Markless and Streatfield, 2007) consider Zurkowski (1974) as such a seminal piece that their histories pay no attention to relevant work that comes before it.
Computerisation may have brought about significant changes to how we work with and learn about information landscapes, but the value of information per se has been recognised for millennia. Socrates recognised ‘that knowledge ought to be applied to conduct in the same way as it was already applied with such success to carpentry, shoemaking or medicine’ (Lindsay, 1906, xii). Plato would ‘persuade those who are to share in the highest affairs of the city to take to calculation, and embrace it in no amateur spirit… until they arrive by the help of sheer intelligence at a vision of the nature of numbers’ (Lindsay, 1906, 219). In other words, he affirms the importance of information, an ability to handle it effectively and an objective, dispassionate viewpoint, when it comes to taking effective decisions and planning (military) strategy.
Information, and by extension those trained in its effective use and application, has always been valuable to power holders. The history of espionage, propaganda and surveillance acknowledges this, whether in wartime and/or in asserting political strength at home. Control over information gathering and dispersal is a key element in how consent to dominant political orders is secured less through overt force and more through the manufacturing of consent, or what Gramsci (1971) called hegemony. Hegemonic processes include control over culture, education and the media. Dahl (1961) included ‘control over information’ in his analysis of key political resources.
Ground-breaking can be an overused and hackneyed phrase but it is entirely true in this case, and an apt pun, for a book that focuses on information landscapes in their infinite variety. But that's not to say that it is narrow in view, this is a much broader treatment on the discourse of information literacy – it is a sure-footed and forensic exploration of the topic. Drew's discussion on the history of information literacy is a very clear break with received wisdom of the past and especially questions the hegemony of thought embodied in powerful Western voices. Drew provides us with a critical overview, in a sense an information literacy of information literacy, which has been sadly lacking but now rectified in this volume. However, it is the notion of the information landscape that defines this work.
When Mark Hepworth and I wrote our book on information literacy, we were very conscious of the ‘information character’ of an information source (whatever that might be), but never fully developed what we meant by that. Drew gives us a clear notion of what information character is through this notion of an information landscape (not as metaphor but as embodied social practice); how we inhabit, map and navigate it to become, not just informed, but to make well-calibrated judgements of the information we encounter. It is notable that Drew draws our attention to the overlooked, and often misunderstood issue by those outside the information profession, that information literacy is not simply about technical competencies, no, it is much more than this. Indeed, it is about our practice as we move through, encounter and use information for whatever purpose.
Information literacy is a social and cognitive set of practices which help us locate ourselves in the world. These practices are contextual (work, education or play), negotiated and re-negotiated and, are not entirely rational. Therein lies a problem, which Drew highlights very eloquently, in effect the limited perspective that information literacy taught in higher education brings. Educators are excellent at enabling or even empowering students to become information literate within the educational context but this does not translate well to the workplace because, like the geographical landscape where mountains and valleys are very different places, the information landscapes of education and work are entirely different too.
I walk this regularly in both directions. I describe it here as done from my office to the station; from station to office I simply reverse the route.
Ellen Wilkinson Building — Oxford Road — University Place — National Graphene Institute — Booth Street East — Upper Brook Street — Princess Street — Canal Street — Abingdon Street [see note] — Portland Street — Nicholas Street — Booth Street — Pall Mall — King Street — Cheapside — Chapel Walks — Cross Street — Exchange Square — National Football Museum — Victoria Station.
Abingdon Street has been closed since summer 2018 due to building works and remains so at the time of writing. During this time I have omitted Canal and Abingdon Streets from my route and turned straight from Princess Street into Portland Street, but the day Abingdon Street is eventually reopened I will go straight back to using the route as defined.
Walk 2, Crowborough
Day one: Railway station — Crowborough Hill — path behind the social club — Forest Rise — Palesgate Lane [Limekiln Football Ground] — through the woods, down to the Ghyll — Forest Rise — Burdett Road — Crowborough Hill — Church Road — Beacon Road — Melfort Road — Warren Road — Fielden Road [B&B].
Day two: Fielden Road — Beacon Road — Sheep Plain — footpath — Manor Way — Southridge Road — Whitehill Road — Gladstone Road — Queens Road — Blackness Road — Luxford Road — Old Lane [my house]— Poundfield Road — footpath [Beacon school: Fermor school] — Crowborough Hill — New Road — Beacon Road — Mill Lane — Croft Road — High Street [lunch at Crowborough Cross pub] — St Johns Road — Goldsmiths Avenue — Glenmore Road East — Rannoch Road — Warren Road — Fielden Road.
Walk Three, Seaford
Seaford station — Station Approach — Dane Road — Esplanade — past the Martello Tower and beach huts — up onto the cliffs — along cliff edge — past end of golf course [Viewpoint] — retrace steps a couple of hundred yards — path to South Hill Barn — (unnamed?) road down to houses — Chyngton Way — Arundel Road — Sutton Avenue — Bramber Road — Bramber Lane — East Street — High Street [The Old Boot] — Church Street — Dane Road and back to station.
Appendix 2: Concept map of this book's key ideas
I sketched this concept map as a way of breaking out of a bout of ‘writer's block’ which held up progress on the book during May 2019.
The aim of the next three chapters is to report on studies of mapping and navigation in terms of how they play out in real landscapes and what can be learned by applying certain mapping techniques, both in a research sense and educationally. Chapters 5 and 6 report on research projects that were conducted in a more traditional way than the investigation presented in this chapter. Here, I engage in a self-reflective and autobiographical exercise in order to generate a rich description of my own navigational practice.
This approach is similar to autoethnography. As a scholar, I am not here trying to view the landscape from a detached point, but (re-)inserting my mind and body into it, ‘as the site from which the story is generated’ (Spry, 2001, 708). In autoethnography, the researcher ‘is the epistemological and ontological nexus upon which the research process turns’ (ibid., 711). Spry notes that for many this is ‘scholarly treason and heresy’ (ibid., 709). But it is also a way of assessing affect in a given setting and accumulating a rich, thick description of practices within a landscape and the chronotopes that comprise it, in a form that would be difficult to replicate experimentally (although not practically impossible, as theirwork suggested).
Autoethnography appears periodically in studies of IL. For example, Purdue (2003, 654–5), considers the old ACRL IL standards and self-reflects on his performance as a researcher against each, finding himself wanting when it comes to most of them. A number of published works on IL pedagogy are effectively autoethnographies, the authors self-reflexively exploring their experiences in the IL classroom in depth, to produce rich, thick descriptions of practice. Books by Downey (2016) and Badke (2012) are examples. However, the concept also has a (self-)critical angle that is not present in this chapter. Here, I do not set out to make observations about the sociopolitical contexts in which I exist and am engaging in informational practice. I am not critiquing myself, nor my positioning with respect to these contexts (cf. Spry, 2001, 710).
If this chapter is not a true autoethnography, then what is it?
Mapping as a practice is wholly intertwined with conceptions of power, the ability to impose one's practices and perspectives on the world and on others. An argument in favour of this stance is constructed, in detail, by David Harvey in his book Justice, Nature and the Geography of Difference (1996), and this chapter's argument is significantly based on that work. In addition, Harvey offers the important notion of discursive mapping, an essential foundation for the chapters to follow. The discussion in Chapter 2 hinted at the idea that mapping processes can be occurring without an image, or cartograph, actually being produced (as with, for example, the rutter). Understanding this distinction is essential for appreciating how mapping – not just maps as a product, but the processes of dialogue, abstraction, organisation, representation and communication that create them – fundamentally shapes the information practices within social settings and is thus a locus of power.
What is being created through such processes are not just maps in the graphic sense – the Hereford Mappa Mundi, MOLA, etc. – but cognitive schema. Schema are ways in which we organise the knowledge in our minds and are thus part of the background against which we make judgements about information selection and the media of dissemination; they are learned, stored ways of thinking and:
… function to pick out relevant, ‘schema-consistent’ data from the rush of information we regularly confront. As such, they are pre-existing selection criteria that manage cognitive overload and enhance the capacity to solve problems.
(Blaug, 2007, 30)
In other words, schema are substructures of the collective matrix of interpretation (ibid., passim). Schema – and the mapping processes from which they emerge and which help define their form – act as media through which power can be used to impose information practices on subordinate groups and the information landscapes they steward. But the same processes can also be used by communities to better understand and organise their own landscape, share information about how that landscape can be nurtured for the collective benefit of the community which inhabits it, and ultimately, distribute authority over it, and the associated information practices, among members of the group.
This book is a study of how educators and learners can use the practice and process of mapping in developing their information and digital literacy, and of the centrality of place and time to these notions. Mikhail Bakhtin used the term chronotope (1981), derived from the Greek words for time and place, to describe the unique moments that we all continuously experience, the differing qualities of which irrevocably shape our encounters with information. The information landscapes (Lloyd, 2010a) which we inhabit can be and, due to our individuality, must be viewed from a multitude of different chronotopes, and thus from a multitude of differing perspectives. And in any decision-making process, whether individual, organisational or societal, judgements are more informed when decision-makers can effectively navigate the information they must draw upon, discern the contours and nuances of these landscapes, map them and learn how to acknowledge difference in how these processes are perceived, and how they subsequently manifest in practice.
The effective use of information to learn and to take informed decisions, and the effective navigation of information landscapes, have been termed information literacy, though as Chapter 1 will discuss, this term has by no means achieved universal acceptance and can mean quite different things depending on the perspective from which one attempts to describe it. Nevertheless, informed judgements are always made against the background of information landscapes that intersect at the here and now – that is, at given chronotopes – and which are organised across both the physical, geographical realm and the virtual, informational one. One of this book's principal arguments is that these two realms are indivisible and the techniques we use to navigate each are essentially similar.
Based on this fundamental indivisibility of the geographical and information landscape, I present mapping as an educational approach that can help us, as individuals and as members of groups, better learn how to navigate the information landscapes that are relevant at our (constantly shifting) chronotope and, as a result, develop and sustain information literacy (IL). To explore this proposition requires defining IL as more than just a set of technical competencies, but as a practice that is embodied in a multitude of social sites that shape how we relate to, and use, information.
So far I have discussed mapping as both a graphic and discursive practice, but the case studies used have been mostly concerned with the mapping and navigation of geographical space. While I hope I have already shown that all mapping is a mapping of information as well as geographical landscapes, what I focus on in this chapter are mappings that, on the surface, have no relation to physical geography: the concept or mind map.
This chapter brings in the notion that the mapping techniques explored so far can be applied to informational space, at least at some level and without losing basic principles: such as maps’ boundedness and materiality, the social nature of the mapping process and maps as a reification of agreements made at particular places and times. Through discussing several prior research studies, I investigate how concept mapping has been used in two principal ways vis-à-vis information literacy: first, the generation of data regarding research subjects’ information horizons and landscapes; second, as teaching tools to raise learners’ awareness of resources and pathways within a landscape.
The educational interventions discussed here are not necessarily exemplars. I agree with Jane Secker (in her foreword to Forster, 2017): given that IL interventions must always be customised to (and within) the context, I should be wary of generalising from these cases and concluding that they will be applicable in different, or all, contexts. Rather, the aim is to use these case studies to ask what we can learn from them vis-à-vis mapping specifically and the development of IL practice more broadly. There are lessons to be learned about not just how these prior studies engaged participants with creating a map (graphical or discursive), but what participants learned about mapping as a practice; how their maps were talked into being, via a discursive and/or dialogic process; and social aspects of mapping, affect and power.
Non-geographical maps
First, though, it is necessary to establish that maps do not have to be geographical. Consider the organisation chart in Figure 5.1: this is an imaginary organisation but it is a prevalent and familiar type of image.
Growing up in Crowborough (see Chapter 4), the English Lake District felt far away. And in English terms, it is, lying in the opposite corner of the country from Sussex. Yet despite this geographical separation, and before I had ever visited there, the Lake District felt like a familiar place, it having been brought into my life by my parents’ copy of the seven-volume Pictorial Guide to the Lakeland Fells. Even when young I appreciated that these books, written between 1952 and 1965 by Alfred Wainwright, are different from the norm. Rather than being conventionally typeset, the pages are photo-reproductions of a manuscript that, in its entirety – the text, illustrations, maps and the whole layout – Wainwright had hand drawn. This is impressive enough, even before one takes account of the scale and detail of this undertaking. For example, across more than 1,000 pages, all the text is fully justified to both margins without any use of hyphenation. To make the diagrams of mountain ascents most useful, Wainwright altered the perspective as the reader's eye moved up the representation, a technique that is innovative even today (Hutchby, 2012, 26–9). And in addition to his skills as a draughtsman, Wainwright's Guide is exhaustive in terms of its coverage of this area of natural beauty, erudite and possessing a dry wit. Little surprise that the books have become classics of their genre. Those who seek to climb all the fells, or mountains, described in the books are now said to be ‘doing the Wainwrights’.
In both cartographic and discursive terms, Wainwright's Guide is a map of great depth and, thanks to revisions undertaken in the 2000s (by Chris Jesty) for a second edition, still with much relevance to the contemporary walker as a guide for practice. Its classification schema influences this practice, with the chapter titles defining some fells as ‘Wainwrights’ and others as mere supporting players, perhaps less worthy of time and effort as a result. A wide range of emotional reactions are evident in the book (Wainwright does not even try to be an objective chronicler of his pathways and perspectives) and can be provoked simply by reading its pages, let alone following its guidance and immersing oneself in the landscapes it articulates.
The previous chapter focused on learners creating graphical maps of information landscapes, whether individually or collectively. But the value of this to the learners was seen as much in the mapping as the maps; the process, as much as the product. The discussions so far have suggested that mapping has value in learning to use, nurture and steward information landscapes because it is a means by which representations of relationships between relevant landscape elements can be developed, communicated and scrutinised within communities. It has already been suggested that these processes do not have to result in the production of some kind of cartograph or other graphical map (like a concept map) for mapping to nevertheless be occurring – and, therefore, for these kinds of representational and communicative processes to still be taking place.
The aim of this chapter is therefore to look more specifically at the practices involved in discursive mapping and how these are manifested in dialogue between learners; an outcome of which is a more effectively, or at least differently, organised information landscape. Two key themes underpin this discussion. The first is how a learning environment, designed around the principles of informed learning (Bruce, 2008) and incorporating techniques and supporting technologies that promote the discursive mapping of an information landscape and the modelling of good practices, can build and validate the information literacy of small groups of learners in an HE setting. According to Bruce et al. (2017), effective design for informed learning has three key principles:
It takes into account learners’ existing experiences of informed learning, using reflection to enhance awareness;
It promotes simultaneous learning about disciplinary content and the information using process;
It brings about changes in learners’ experience of information use and of the subject being learned.
Chapter 4 was based around a real but wholly informal ‘educational’ situation. In Chapter 5 the focus was on either workplace learning and/or the use of mapping to generate data for academic researchers. In this chapter, the IL intervention being discussed is a formal one, part of an organised programme of study. This formalisation adds elements of power and authority that must also be discussed: this being the second key theme of the chapter.
This chapter explores how mapping has been used over the last two millennia as a technique for documenting, representing and communicating understandings of the world around us, and how it has consequently impacted on our perception of place and time and our ability to navigate the landscapes (mapped and unmapped) that we encounter. It examines mapping as a genre of communication, a methodology, and a social, material knowledge-forming practice. Starting here, but continuing more explicitly in Chapter 3, it also investigates mapping as a source of power: maps can empower their users, but also disempower them (and others), particularly if one is left ‘off the map’ for any reason (Bonnett, 2015).
This is not a story written by a cartographer or geographer. Scholars in these fields have documented the rich history of mapping both by examining the documents that it has produced (that is, the maps themselves) and by analysing mapping as a practice that helps us both produce the world and consume it (Kitchin et al., 2009, 4; Cosgrove, 1999). I am not making any substantial additions to these histories. But as my goal is to establish the educational value of mapping and how it can help learners make informed judgements about information, I must explore mapping as an epistemological and methodological practice; a way of developing, and validating, knowledge about information landscapes and the social sites with which they are integrated. Put simply, what can we learn about our information landscapes through mapping them? How can mapping help learners engage with the different modalities of information (epistemic, social and corporeal), and the different stages of information processing (selection, organisation and communication)?
Lloyd mentions this possibility on several occasions. Early on in Information Literacy Landscapes she says:
Information landscapes are the communicative spaces that are created by people who co-participate in a field of practice. As people journey into and through these landscapes they engage with site-specific information. This engagement allows them to map the landscape, constructing an understanding of how it is shaped. It is through this engagement that people situate themselves within the landscape.