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Arts and Sciences (BASc)

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# BASC0003 Quantitative Methods and Mathematical Thinking

Complex systems dominate our interconnected world. Whether it is the internet, the human brain, individuals travelling to and from a shopping centre or molecules in the latest nanotechnology device, all such examples involve individual ‘agents’ interacting as part of a complex network.

We seek to understand such complexity using quantitative methods. In seeking this understanding we will learn a variety of tools that have applicability across many areas of human knowledge in Arts and Sciences. These tools can be used to deepen our existing knowledge as well as to make predictions about future developments. A few examples of where the techniques are applied give some idea of their range and power:

• Medics can study the spread of disease to enable better understanding of prevention or the success rates of surgical procedures to decide on health policy
• Cultural historians can take a quantitative approach to complement connoisseurship in comparing the texts of writers to decide on questions of authorship
• Judges can assess evidence in quantitative ways to give different weightings to different aspects of a case

#### Course details

The overall approach in this module is to proceed from an analysis of simple situations to complex ones.

For example, we look first at how an individual (a non-interacting ‘agent’) contracts an illness, or we look at the structure and behaviour of an individual atom. We then analyse what happens as things interact – for example, we study the spread of diseases that can transmit from human to human, or what happens to atoms when we put them together in a nanotechnological device.

The study of ‘interaction’ leads us to complexity, as we consider more and more agents interacting in different qualitative networks. Such notions of complexity, interaction, networks and interconnectedness are major themes in modern thought.

Underpinning the module is the unifying idea that some forms of quantitative reasoning are powerful enough to bring these many areas of thought under one conceptual scheme.

This unity will be emphasised by keeping as an objective the understanding of one particular mathematical equation. The importance and range of applicability of this equation will be emphasised and repeated as we work through the techniques required to understand and apply it. Having grasped the importance of this simple equation, we can then take the next steps to confronting complexity more fully.

The limits of our current quantitative understanding and the range and limits of computers in this analysis will also be emphasised.

These include, but are not limited to:

• An understanding and appreciation of the wide-ranging use of quantitative techniques in both work and academic spheres (examples from archaeology, anthropology, business, medicine, finance, literature, geography and engineering will be given)
• An ability to set up and analyse simple systems in a quantitative manner
• An understanding that mathematics is the language used in quantitative analysis
• An appreciation of the limits of applicability of quantitative methods

This course is taught in Term 2 of Year 1.

• Weeks 1-2 Qualitative examples of complex networks: cities, the internet, the human brain, business organisations. Moving from analysing the simple to the complex.
• Weeks 3-4 Introduction of equation needed to describe non-interacting systems. Description of tools necessary for this analysis: differential equations, exponentiation, matrices.
• Weeks 5-7 Complex systems. Worked example: cities – both contemporary examples in urban planning, shop design etc and examples from Archaeology and History. The role of computers. The limits of quantitative analysis.
• Weeks 8-10 Quantitative reasoning: a contemporary approach. What is probability? How are statistics used in e.g. health policy and financial industry? Tools needed for this sort of analysis: distributions, Bayes’ theorem, the concept of entropy.

#### Teaching

The course runs for three hours per week in Term 2 of Year 1 as follows:

 Lecturer: Dr Thomas Oléron Evans Lectures: 1-2pm on Tuesdays and 1-2pm on Fridays Seminars*: 4-5pm on Tuesdays or 5-6pm on Tuesdays or 10-11am on Wednesdays or 11am-12pm on Wednesdays or 12 - 1pm on Wednesdays or 9-10am on Thursdays or 10am-11am on Thursdays or 3-4pm on Thursdays or 4-5pm on Thursdays or 10-11am on Fridays or 11am-12pm on Fridays Module level: Level 4 Credit value: 15 credits

* Students are automatically allocated to a seminar class, so it is not possible to select one. Students will only be permitted to change their seminar class if they have a clash with another module.

#### Assessment

• 2,000 word project - 50%
• 3 x 200 word Homeworks - 40%
• 5-10 minute presentation - 10%