Professor James Ladyman of Bristol University is leading a three-year AHRC sponsored project on The Foundations of Structuralism. The project aims to integrate work in philosophical logic, mathematics and physics concerning the nature of objects and individuality. The project is investigating various formulations of structuralism, paying special attention to its conceptual and logical foundations. A list of preliminary questions provides a starting point to the research.
Welcome to PhilSci-Archive, an electronic archive for preprints in the philosophy of science. It is offered as a free service to the philosophy of science community. The goal of the archive is to promote communication in the field by the rapid dissemination of new work. PhilSci-Archive invites submissions in all areas of philosophy of science, including general philosophy of science, philosophy of particular sciences (physics, biology, chemistry, psychology, etc.), feminist philosophy of science, socially relevant philosophy of science, history and philosophy of science and history of the philosophy of science.
The Neoplatonian philosopher Hypatia of Alexandria, Egypt, was the first well-documented woman in mathematics. Her actual date of birth is unknown, although considered somewhen between 350 and 370 AD. She was the head of the Platonist school at Alexandria and additionally taught philosophy and astronomy.
On August 11, 1464, German philosopher, theologian, jurist, and astronomer Nikolaus of Cusa (in latin: Nicolaus Cusanus) passed away. He is considered as one of the first German proponents of Renaissance humanism. His best known work is entiteled 'De Docta Ignorantia' (Of the Learned Ignorance), where also most of his mathematical ideas were developed, as e.g. the trial of squaring the circle or calculating the circumference of a circle from its radius.
On July 13, 1527, Welsh mathematician, astronomer, astrologer, occultist, navigator, imperialist and consultant to Queen Elizabeth I, John Dee was born. He is considered one of the most learned men of his age. Besides being an ardent promoter of mathematics and a respected astronomer, in his later years he immersed himself in the worlds of magic, astrology and Hermetic philosophy. One of his aims was attempting to commune with angels in order to learn the universal language of creation.
On July 11, 1382, significant philosopher of the later Middle Ages Nicole Oresme passed away. As for many historic people of the middle ages, his actual birthdate is unknown and can only be fixed to a period between 1325 and 1330. Nicole Oresme besides William of Ockham or Jean Buridan -- a French priest who sowed the seeds of the Copernican revolution in Europe -- is considered as one of the most influential thinkers of the 14th century and he wrote influential works on economics, mathematics, physics, astrology and astronomy, philosophy, and theology.
On April 8, 1859, German philosopher and mathematician Edmund Gustav Albrecht Husserl was born. He is best know as the founder of the 20th century philosophical school of phenomenology, where he broke with the positivist orientation of the science and philosophy of his day, yet he elaborated critiques of historicism and of psychologism in logic.
On March 31, 1596, French philosopher, mathematician, and writer René Descartes was born. The Cartesian coordinate system is named after him, allowing reference to a point in space as a set of numbers, and allowing algebraic equations to be expressed as geometric shapes in a two-dimensional coordinate system. He is credited as the father of analytical geometry, the bridge between algebra and geometry, crucial to the discovery of infinitesimal calculus and analysis. Descartes was also one of the key figures in the Scientific Revolution and has been described as an example of genius. He has been dubbed the 'Father of Modern Philosophy'. His Meditations on First Philosophy continues to be a standard text at most university philosophy departments.
On March 31, 1596, French philosopher, mathematician, and writer René Descartes was born. The Cartesian coordinate system is named after him, allowing reference to a point in space as a set of numbers, and allowing algebraic equations to be expressed as geometric shapes in a two-dimensional coordinate system. He is credited as the father of analytical geometry, the bridge between algebra and geometry, crucial to the discovery of infinitesimal calculus and analysis.
On February 17, 1600, Domonican friar, philosopher, mathematician and astronomer Giordano Bruno was burned on the stake after the Roman Inquisition found him guilty of heresy. His cosmological theories went beyond the Copernican model in proposing that the Sun was essentially a star, and moreover, that the universe contained an infinite number of inhabited worlds populated by other intelligent beings.
H. Freudenthal. Educational Studies in Mathematics, 1 (1/2):
3-8(May 1968)Systematization is a great virtue of mathematics, and if possible, the student has to learn this virtue, too. But then I mean the activity of systematizing, not its result. Its result is a system, a beautiful closed system, closed, with no entrance and no exit. In its highest perfection it can even be handled by a machine. But for what can be performed by machines, we need no humans. What humans have to learn is not mathematics as a closed system, but rather as an activity, the process of mathematizing reality and if possible even that of mathematizing mathematics..
P. Ernest. Why Learn Maths, London University Institute of Education, London, 1. To reproduce mathematical skill and knowledge based capability
The typical traditional reproductive mathematics curriculum has focused exclusively on this first aim, comprising a narrow reading of mathematical capability. At the highest level, not always realised, the learner learns to answer questions posed by the teacher or text. As is argued elsewhere (Ernest 1991) this serves not only to reproduce mathematical knowledge and skills in the learner, but to reproduce the social order and social injustice as well.
2. To develop creative capabilities in mathematics
The progressive mathematics teaching movement has added a second aim, to allow the learner to be creative and express herself in mathematics, via problem solving, investigational work, using a variety of representations, and so on. This allows the learner to pose mathematical questions, puzzles and problems, as well as to solve them. This notion adds the idea of creative personal development and the skills of mathematical questioning as a goal of schooling, but remains trapped in an individualistic ideology that fails to acknowledge the social and societal contexts of schooling, and thus tacitly endorses the social status quo.
3. To develop empowering mathematical capabilities and a critical appreciation of the social applications and uses of mathematics
Critical mathematics education adds in a third aim, the empowerment of the learner through the development of critical mathematical literacy capabilities and the critical appreciation of the mathematics embedded in social and political contexts. Thus the empowered learner will not only be able to pose and solve mathematical questions, but also be able to address important questions relating to the broad range of social uses (and abuses) of mathematics. This is a radical perspective and set of aims concerned with both the political and social empowerment of the learner and with the promotion of social justice, and which is realised in mainstream school education almost nowhere. However, the focus in the appreciation element developed in this perspective is on the external social contexts of mathematics. Admittedly these may include the history of mathematics and its past and present cultural contexts, but these do not represent any full treatment of mathematical appreciation.
4. To develop an inner appreciation of mathematics: its big ideas and nature
This fourth aim adds in further dimension of mathematical appreciation, namely the inner appreciation of mathematics, including the big ideas and nature of mathematics. The appreciation of mathematics as making a unique contribution to human culture with special concepts and a powerful aesthetic of its own, is an aim for school mathematics often neglected by mathematicians and users of mathematics alike. It is common for persons like these to emphasise capability at the expense of appreciation, and external applications at the expense of its inner nature and values. One mistake that may be made in this connection is the assumption that an inner appreciation of mathematics cannot be developed without capability. Thus, according to this assumption, the student cannot appreciate infinity, proof, catastrophe theory and chaos, for example, unless they have developed capability in these high level mathematical topics, which is out of the question at school. The fourth aim questions this assumption and suggests that an inner appreciation of mathematics is not only possible but desirable to some degree for all students at school..(2000)
J. Sowa. Proceedings of the 14th International Conference on Conceptual Structures (ICCS 2006), volume 4068 of Lecture Notes in Computer Science, page 54-69. Springer, (2006)
L. Wittgenstein. University Of Chicago Press, Chicago, (October 1989)characterizes mathematical propositions: - Do not have a temporal sense (pp. 34). - Are rules of expression. "the connection between a mathematical proposition and its application is roughly that between a rule of expression and the expression itself in use" (pp. 47). A rule of expression defines what is meaningful and what not, how a particular form should be used, etc. - Is invented to suit experience and then made independent of experience (pp. 43). "In mathematics we have propositions which contain the same symbols as, for example, "write down the integral of..", etc., with the difference that when we have a mathemaitical proposition time doesn't enter into it and in the other it does. Now this is not a metaphisical statement." (pp 34).