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population graphBlack Swans and Civilisation Collapse

A really interesting branch of mathematics is involved in making future predictions about how civilisation will evolve in the future – and indeed looking at how robust our civilisation is to external shocks.  This is one area in which mathematical models do not have a good record as it is incredibly difficult to accurately assign probabilities and form policy recommendations for events in the future.

Malthusian Catastrophe

One of the most famous uses of mathematical models in this context was by Thomas Malthus in 1798.  He noted that the means of food production were a fundamental limiting factor on population growth – and that if population growth continued beyond the means of food production that there would be (what is now termed) a “Malthusian catastrophe” of a rapid population crash.

As it turns out, agrarian productivity has been able to keep pace with the rapid population growth of the past 200 years.


Looking at the graph we can see that whilst it took approximately 120 years for the population to double from 1 billion to 2 billion, it only took 55 years to double again.  It would be a nice exercise to try and see what equation fits this graph – and also look at the rate of change of population (is it now slowing down?)  The three lines at the end of the graph are the three different UN predictions – high end, medium and low end estimate.  There’s a pretty stark difference between high end and low end estimates by 2100 – between 16 billion and 6 billion!  So what does that tell us about the accuracy of such predictions?

Complex Civilisations

More recently academics like Joseph Tainter and Jared Diamond have popularised the notion of civilisations as vulnerable to collapse due to ever increasing complexity.  In terms of robustness of civilisation one can look at an agrarian subsistence example.  Agrarian subsistence is pretty robust against civilisation collapse –  small self sufficient units may themselves be rather vulnerable to famines and droughts on an individual level, but as a society they are able to ride out most catastrophes intact.

The next level up from agrarian subsistence is a more organised collective – around a central authority which is able to (say) provide irrigation technology through a system of waterways.  Immediately the complexity of society has increased, but the benefits of irrigation allow much more crops to be grown and thus the society can support a larger population.  However, this complexity comes at a cost – society now is reliant on those irrigation channels – and any damage to them could be catastrophic to society as a whole.

To fast forward to today, we have now an incredibly complex society, far far removed from our agrarian past – and whilst that means we have an unimaginably better quality of life, it also means society is more vulnerable to collapse than ever before.  To take the example of a Coronal Mass Ejection – in which massive solar discharges hit the Earth.  The last large one to hit the Earth was in 1859 but did negligible damage as this was prior to the electrical age.  Were the same event to happen today,  it would cause huge damage – as we are reliant on electricity for everything from lighting to communication to refrigeration to water supplies.  A week without electricity for an urban centre would mean no food, no water, no lighting, no communication and pretty much the entire breakdown of society.

That’s not to say that such an event will happen in our lifetimes – but it does raise an interesting question about intelligent life – if advanced civilisations continue to evolve and in the process grow more and more complex then is this a universal limiting factor on progress?   Does ever increasing complexity leave civilisations so vulnerable to catastrophic events that their probabilities of surviving through them grow ever smaller?

Black Swan Events

black swan

One of the great challenges for mathematical modelling is therefore trying to assign probabilities for these “Black Swan” events.  The term was coined by economist Nassim Taleb –  and used to describe rare, low probability events which have very large consequences.  If the probability of a very large scale asteroid impact is (say) estimated as 1-100,000 years – but were it to hit it is estimated to cause $35 trillion of damage (half the global GDP) then what is the rational response to such a threat?  Dividing the numbers suggests that we should in such a scenario  be spending $3.5billion every year on trying to address such an event – and yet which politician would justify such spending on an event that might not happen for another 100,000 years?

I suppose you would have to conclude therefore that our mathematical models are pretty poor at predicting future events, modelling population growth or dictating future and current policy.  Which stands in stark contrast to their abilities in modelling the real world (minus the humans).  Will this improve in the future, or are we destined to never really be able to predict the complex outcomes of a complex world?

If you enjoyed this post you might also like:

Asteroid Impact Simulation – which allows you to model the consequences of asteroid impacts on Earth.

Chaos Theory – an Unpredictable Universe?  – which discusses the difficulties in mathematical modelling when small changes in initial states can have very large consequences.

IB Revision

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If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

Screen Shot 2018-03-19 at 4.42.05 PM.pngThe Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

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The Practice Exams section takes you to ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

Website Stats


IB HL Paper 3 Practice Questions (120 page pdf)

IB HL Paper 3 Practice Questions 

Seventeen full investigation questions – each one designed to last around 1 hour, and totaling around 40 pages and 600 marks worth of content.  There is also a fully typed up mark scheme.  Together this is around 120 pages of content.

Available to download here.

IB Maths Exploration Guide

IB Maths Exploration Guide

A comprehensive 63 page pdf guide to help you get excellent marks on your maths investigation. Includes:

  1. Investigation essentials,
  2. Marking criteria guidance,
  3. 70 hand picked interesting topics
  4. Useful websites for use in the exploration,
  5. A student checklist for top marks
  6. Avoiding common student mistakes
  7. A selection of detailed exploration ideas
  8. Advice on using Geogebra, Desmos and Tracker.

Available to download here.

Modelling Guide

IB Exploration Modelling Guide 

A 50 page pdf guide full of advice to help with modelling explorations – focusing in on non-calculator methods in order to show good understanding.

Modelling Guide includes:

Linear regression and log linearization, quadratic regression and cubic regression, exponential and trigonometric regression, comprehensive technology guide for using Desmos and Tracker.

Available to download here.

Statistics Guide

IB Exploration Statistics Guide

A 55 page pdf guide full of advice to help with modelling explorations – focusing in on non-calculator methods in order to show good understanding.

Statistics Guide includes: Pearson’s Product investigation, Chi Squared investigation, Binomial distribution investigation, t-test investigation, sampling techniques, normal distribution investigation and how to effectively use Desmos to represent data.

Available to download here.

IB Revision Notes

IB Revision Notes

Full revision notes for SL Analysis (60 pages), HL Analysis (112 pages) and SL Applications (53 pages).  Beautifully written by an experienced IB Mathematics teacher, and of an exceptionally high quality.  Fully updated for the new syllabus.  A must for all Analysis and Applications students!

Available to download here.

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