Some descriptions of mathematics

I. Mathematics is an exact language for description, calculation, deduction, modeling, and prediction — more a systematic way of thinking than a set of rules.

Mathematics is the language in which it is impossible to make a nebulous or imprecise statement.

Using a legal analogy, mathematics is a language for writing contracts with Nature that Nature accepts as legally binding.

II. The practical importance of mathematics lies in its ability to describe the real world.

The real world consists of what matters. The word “matter” as a noun is used for what the physical world is made of. But if we ask, “What’s the matter with Anne?” we may be asking about a physical ailment, or we may be asking about an idea that is causing Anne to behave strangely. Ideas matter.

The whole point of mathematical education is to make ideas real for students, ideas that were not real for

them before. Ideas like fractions, for example. The fact that 2/3 is smaller than 3/4 matters in the real world.

III. Mathematically educated people are stem cells of a technologically advanced society. Because of the universality of mathematics, mathematicians and well educated users of mathematics are flexible in applying and inventing tools for work in technological environments which never existed before.

IV. Learning mathematics involves the profound assimilation of intellectual and aesthetic criteria as well as practically orientated ones. The very difficulty in learning mathematics makes it a personality-enhancing experience.

[With contributions and borrowings from David Corfield, Tony Gardiner, Michael Gromov, Niall MacKay, Henri Poincare, Frank Quinn, David Pierce.]


Mathematical Cultures

The UK Arts and Humanities Research Council has agreed to fund a research network on mathematical cultures.  Here, I describe this project and what we hope to learn from it.

Why study mathematical cultures?  Why now?

Mathematics has universal standards of validity.  Nevertheless, there are local styles in mathematics.  These may be the legacy of a dominant individual (e.g. the Newtonianism of 18th century British mathematics).  Or, there may be social or economic reasons (such as the practical bent of early modern Dutch mathematics).

These local mathematical cultures are scientifically important because they can affect the direction of mathematical research.  They also matter because of the cultural importance of mathematics.  Mathematics enjoys enormous intellectual prestige, and has seen a growth of popular publishing, films about mathematicians, at least one novel and plays.  However, this same intellectual prestige encourages a disengagement from mathematics.  Ignorance of even rudimentary mathematics remains socially acceptable.  Policy initiatives to encourage the study of mathematics usually emphasise the economic utility of mathematics (for example the 2006 STEM Programme Report).  Appeals of this sort rarely succeed with students unless there is a specific promise of employment or higher remuneration.
What these political anxieties call for is a re-presentation of mathematics as a human activity, which means, among other things, that it is part of culture.  The tools and knowledge necessary for this have been developing in recent years.  Historians of mathematics have begun to consider mathematics in its social, political and cultural contexts.  There is now an established sociology of science and technology, published in journals such as Science as Culture and the Journal of Humanistic Mathematics.  Mathematics educationalists have begun to draw on some of these developments (particularly historical research).

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Aaron Sloman: Is education research a form of alchemy?

This piece by Aaron Sloman in ALT News Online may be of interest:

First three paragraphs:

Alchemists did masses of data collection, seeking correlations. In the process they learnt a great many useful facts – but lacked deep explanations. Searching for correlations can produce results of limited significance when studying processes with an underlying basis of mechanisms with astronomical generative power. But this correlation-seeking approach characterises much educational research.

Accelerated progress in chemistry came from developing a deep explanatory theory about the hidden structure of matter and the processes such structure could support (atoms, subatomic particles, valence, constraints on chemical reactions, etc.). Thus deep research requires (among other things) the ability to invent powerful explanatory mechanisms, often referring to unobservables.

My experience of researchers in education, psychology, social science and similar fields is that the vast majority of the ones I have encountered have had no experience of building, testing, and debugging, deep explanatory models of any working system. So their education does not equip them for a scientific study of education, a process that depends crucially on the operations of the most sophisticated information processing engines on the planet, many important features of which are still unknown. [Read more…]

Mathematical Cultures

Mathematics has universal standards of validity.  Nevertheless, there are local styles that result from national policies, charismatic individuals, historical circumstances, intellectual contexts and practical needs.  These differences matter, not least because they can affect the pace and direction of mathematical research.

The Arts and Humanities Research Council has made available funds to explore these mathematical cultures under its Science in Culture highlight notice.  This project will span three symposia, to take place at De Morgan House.

Project Website