Infinitesimal Knowledges

The notion of indivisibles and atoms arose in ancient Greece. The continuum—that is, the collection of points in a straight line segment, appeared to have paradoxical properties, arising from the ‘indivisibles’ that remain after a process of division has been carried out throughout the continuum. In the seventeenth century, Italian mathematicians were using new methods involving the notion of indivisibles, and the paradoxes of the continuum appeared in a new context. This cast doubt on the validity of the methods and the reliability of mathematical knowledge which had been regarded as established by the axiomatic method in geometry expounded by Aristotle’s younger contemporary Euclid. The teaching of indivisibles was banned within the Society of Jesus, the Jesuits. In England, indivisibles were used by the mathematician John Wallis, and there was an acrimonious and extended feud between Wallis and the philosopher Thomas Hobbes over legitimate methods of argument in mathematics. Notions of the infinitesimal were used by Isaac Newton and Gottfried Leibniz, and were attacked by Bishop Berkeley for the vagueness of the concept and the illegitimate reasoning applied to it. This article discusses aspects of these events with reference to the book Infinitesimal by Amir Alexander and to other sources. Also discussed are wider issues arising from Alexander’s book including: the changes in cultural sensibility associated with the growth of new mathematical and scientific knowledge in the seventeenth century, the changes in language concomitant with these changes, what constitutes valid methods of enquiry in various contexts, and the question of authoritarianism in knowledge. More general aims of this article are to widen the immediate mathematical and historical contexts in Alexander’s book, to bridge a gap in conversations between mathematics and the humanities, and to relate mathematical ideas to wider human and contemporary issues

A cause of chaos

An abstract dynamical system consists of a collection S of points together with a transformation, or function, f which maps points of S into points of S. The points in S stand for all possible states of the system. The transformation is a “process of change” over one time unit, that changes each state x in S into another state f(x). Then we can interpret the equation f(x)=y as meaning that if the system is in state x, over the next time unit it will change into the state y. Alternatively, x is a “cause” of y, or x is an “antecedent” of y. There is no reason to consider the elapse of only one time unit – the transformation can be applied over and over again. Thus, if we start off in an initial state x, the next state is f(x), then the next state is f(f(x)), and so on. The sequence of states x, f(x), f(f(x)), f(f(f(x))), …. . is called the orbit of x, and it describes the evolution of the system from an initial state x. Now the actual state x of the system may not be known, but may only be approximated, by the state y. Then, if the orbits of x,y are very different, this would mean that the behaviour of the system cannot be predicted. This inability to predict is an intrinsic feature of chaotic systems. In this paper, chaotic behaviour is linked to a property that a dynamical system may have: given a state y, it may have more than one antecedent or, alternatively, any state may have more than one cause. A proliferation of possible causes may lead to chaos

Properties of Orbits and Normal Numbers in the Binary Dynamical System

In 1930 G. H. Hardy and J. E. Littlewood derived a result concerning the rate of divergence of certain series of cosecants. In more recent terminology, their result can be interpreted as a result about the behaviour of orbits in dynamical systems arising from rotations on the unit circle. In general terms, this behaviour is related to the question of `how often' a point under successive rotations gets `close' to $1$. Now, the expansion of numbers in $[0,1)$ to the base $2$ can be associated with a different system -- the binary dynamical system. This article considers orbit behaviour in the binary system that corresponds to the behaviour that was, in effect, observed by Hardy and Littlewood in systems involving rotations. Now, except for a countable set, the sequence of binary digits of a number in $[0,1)$ may be arranged as an infinite sequence of consecutive, finite blocks, each block consisting of all zeros or all ones. The relationships between the lengths of these blocks determine Hardy-Littlewood types of behaviour in the binary system. This behaviour is considered and results relating to normal and simply normal numbers are obtained. There also are suggestions for further investigation.Comment: 18 page

Can the love of learning be taught?

This paper is an expanded version of a talk given at a Generic Skills Workshop at the University of Wollongong, and was intended for academic staff from any discipline and general staff with an interest in teaching. The issues considered in the paper include the capacity of all to learn, the distinction between learning as understanding and learning as information, the interaction between the communication and content of ideas, the tension between perception and content in communication between persons, and the human functions of a love of learning. In teaching, the creation of a fear-free environment is emphasised, as is the use of analogy as a means of breaking out of one discipline and making connections with another, with mathematics and history being used as a possible example. Some of the issues raised are explored in more depth in the notes at the end of the paper, to which there are references in the main text. About the author. Rodney Nillsen studied literature, mathematics and science at the University of Tasmania. He proceeded to postgraduate study at The Flinders University of South Australia, studying mathematics under Igor Kluvánek and, through him, coming into contact with the European intellectual tradition. He held academic positions at the Royal University of Malta and the University College of Swansea, Wales. Upon returning to Australia, he took up a lecturing position at the University of Wollongong, where he continues to teach and conduct research in pure mathematics. At the University he is a member of Academic Senate and is the Chair of the Human Research Ethics Committee. He received a Doctor of Science degree from the University of Tasmania in 2000. His interests include literature, classical music and the enjoyment of nature.YesThe Journal of University Learning & Teaching Practice is an international higher education journal hosted at University of Wollongong, NSW, Australia. It has over 90 reviewers from around the world providing a broad –based disciplinary approach. It is listed with EBSCO database. Submissions undergo a double blind peer review and resubmitted papers undergo a further check with one of the original reviewers

Integrity in and beyond contemporary higher education: What does it mean to university students?

Research has focused on academic integrity in terms of students' conduct in relation to university rules and procedures, whereas fewer studies examine student integrity more broadly. Of particular interest is whether students in higher education today conceptualize integrity as comprising such broader attributes as personal and social responsibility. We collected and analyzed qualitative responses from 127 students at the National University of Singapore to understand how they define integrity in their lives as students, and how they envisage integrity would be demonstrated in their lives after university. Consistent with the current literature, our data showed that integrity was predominantly taken as "not plagiarizing (in school)/giving appropriate credit when credit is due (in the workplace)", "not cheating", and "completing tasks independently". The survey, though, also revealed further perceptions such as, in a university context, "not manipulating data (e.g., scientific integrity)", "being honest with others", "group work commitments", "conscience/moral ethics/holding true to one's beliefs", "being honest with oneself", "upholding a strong work ethic", "going against conventions", and "reporting others", as well as, in a workplace context, "power and responsibility and its implications", "professionalism", and "representing or being loyal to an organization". The findings suggest that some students see the notion of integrity extending beyond good academic conduct. It is worthwhile to (re)think more broadly what (else) integrity means, discover the gaps in our students' understanding of integrity, and consider how best we can teach integrity to prepare students for future challenges to integrity and ethical dilemmas

An application of quadratic functions to Australian Government policy on funding schools

In the Sydney Morning Herald newspaper of 23rd March 2005 the economics writer Ross Gittins argued that the funding arrangements for private schools in Australia positively encourage parents to move their children from the state system to the private system. The Federal minister, Dr Brendan Nelson, responded by saying that the policy of subsidising pupils who go to a private school results in taxpayer savings of $4 billion. However, the minister\u27s response did not address the extent to which more funds could possibly be saved by having a different subsidy from the one currently offered by the government. Now, there are two conflicting factors in offering subsidies to private school pupils. On the one hand, the greater the subsidy per pupil, the more pupils will enroll in private schools. On the other hand, the greater the subsidy per pupil the less money will be saved each time a pupil enrolls in a private school. How do these factors balance out, and where would an optimal subsidy occur? The problem is closely related to other problems of optimisation that arise in business, industry and public policy. Mathematically, the problem can be modelled, at the level of school mathematics, by means of a quadratic function that describes how the savings to the taxpayer change as the subsidy changes. Further details are on the author’s website at http://www.uow.edu.au/~nillse