New: Mathematics And Philosophy – Gaps and Continuities

The world is a place of curious gaps and withdrawals, appearing fractured between here and there, then and now, idea and thing, thought and thinker. But, as thinkers ourselves, how can we tease out, both mathematically and philosophically, the covert patterns of connection and currents of continuity running through these and other such intimate separations?

Who is this course for

No specific mathematical knowledge is required - just curiosity! The philosophical level will be reasonably advanced, and should suit students who would like to be challenged by what a diverse mix of great thinkers - Spinoza, Hume and Peirce amongst others - themselves had to say.

What does this course cover

In his mathematically structured masterpiece, the Ethics, Spinoza proposes and, so he believes, proves that 'the order and connection of ideas is the same as the order and connection of things'. Should we read this to mean that there is a system of ideas and a separate system of things, and that somehow the one system mirrors the other? Or should we take Spinoza's words, as some have done, to imply a more intimate separation - a separation which subtly articulates a deeper current of metaphysical continuity?

'Nature makes no leaps,' declared Spinoza's near contemporary, the inexhaustible philosopher and mathematician Gottfried Wilhelm Leibniz - an intuition that became one of the principles guiding subsequent attempts to rationally grasp both the physical and the living worlds. But nature has a way of withdrawing from the gaze of reason - a reason that craves unbroken and uninterrupted attention - whilst the mathematics created to reflect nature's patterns withdraws in its turn to discover a nature all of its own, a pure and abstract realm untouched by practical limits and restraints. Or so it thinks.

Just what we ourselves think of some of the philosophical and mathematical efforts that have been made to uncover continuities in the fractured texture of things will be the main concern of this short course. Can, for example, the uneven disruptions of contingency be conceptually 'smoothed out' by an underlying Spinozist necessity? In what sense can or should change as such be accounted for by notions of causality - whether grounded in observational habit, as for Hume, or as a pre-condition of experience itself, as for Kant? Or is the search for connections better framed within a broader picture of signification, such as C.S. Peirce's, offering a flowing semiotic world of signs and meanings?

And, last but not least, we will blend into our philosophical reflections a flavour of some mathematical attempts to imaginatively and conceptually capture and abstractly extend our compelling intuition of a continuity, if not a unity, underlying diversity and multiplicity. We will sample, for example, the spatial generalisations of topology and follow the notion of number away from a merely one-dimensional progression into the two-dimensional complex plane - an abstract realm of quantification in which negation, a disruption if ever there was one, re-appears within an expanded context of continuous transformation.

Ultimately, since this is a philosophy course, your time will have been well spent if your appreciation of the questions raised has been enhanced, no matter which 'solutions' have been proposed

What will it be like

The course material will be introduced with the help of handouts provided at the start of each class. For a short course such as this, assessment of your progress can most appropriately be made on the basis of your participation in the discussions arising from the ideas and themes presented.

What else do you need to buy or do

The course is completely self-contained. Apart from pen and paper, no extra materials are required, and nor is any particular preparation. However, should you wish to whet your mathematical appetite, the following three books can be recommended, although none of them covers the course content as such. They can all be previewed on Google Books ( and are available to buy online for around £15.
How Mathematicians Think - William Byers - 2007 - Princeton University Press - ISBN 9780691145990
Meaning in Mathematics - ed. John Polkinghorne - 2011 - Oxford University Press - ISBN 9780199605057
Why Is There Philosophy of Mathematics At All? - Ian Hacking - 2014 - Cambridge University Press - ISBN 9781107658158
The open-access Stanford Encyclopedia of Philosophy ( contains a huge number of up-to-date articles covering topics at varying levels of generality and specialisation across the whole range of philosophical traditions past and present.

What this course could lead to

Further study in philosophy, including its overlaps and points of intersection with other related areas of interest and enquiry.

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