All posts by Matt Garrod

I'm an MRes student at the Mathematics of Planet Earth Doctoral Training Center. I'm currently studying spatially embedded networks with applications in the study of wireless sensor networks and opinion dynamics.

More Than Just Symbols

 

When you think of a mathematics textbook you probably imagine a series of intimidating pages with a few words and a bunch of strange (often Greek) symbols. I don’t think I’m alone in thinking that the fact that a lot of modern mathematics is only presented in this form is a bit of a crime. For instance, Tristan Needham expresses a similar feeling in the pre-amble to his text “Visual Complex analysis.” Professional mathematicians can usually get some idea of what is going on in the pages of a paper or textbook. However, anyone who hasn’t had as much training in their past loses out. Particularly since, to the untrained eye, there is no way to associate these abstract symbols with anything visual or otherwise.

Reading a mathematics textbook is not like reading a novel–it can be a slow and arduous process. Despite this, someone with mathematical knowledge might eventually be able to understand what is going on. To draw an analogy with computers – it is as if you need the “correct software” installed in your mind to process the text. The same concept applies to reading novels written in other languages – if you don’t have the “correct software” installed in your mind then all that you will see is a series of random symbols. An education in mathematics allows you obtain this “software”, once you have this you can “speak” about things you have never spoken about before.

The important thing to draw from above is that the symbols are simply placeholders for various ideas and concepts just as they are in any other written language. In my opinion adding visuals or graphics to a piece of mathematics significantly helps us to tie down what the symbols are trying to suggest (even if it is just a crude analogy). On the other hand, it is probably true that even illustrations and graphs on their own are probably not enough. Without the proper context, graphs or illustrations may simply appear as static creations with no further meaning. I think additional understanding can be achieved by playing around with the image in your head (or by sketching variations with a pen and paper). This playful approach to imagining visualising mathematical concepts no doubt inspired artists such as M. C. Escher (M. C Escher was a Dutch graphic artist who is well known for his often mind boggling and mathematically inspired work).

Nowadays we can go significantly further than Escher with the power of computer graphics. As an example I have listed a few of my the coolest looking pages and blogs related to visualising mathematics and mathematical concepts below:

  • http://www.graphonaute.fr/ A selection of animations and images created by French engineering student Hugo Germain. Makes me think of what Escher’s work may have been like if he were born into the digital age!
  • http://blog.matthen.com/ Lots of nice visualisations and a few cool visual proofs of well known mathematical theorems as well. The code for the visualisations is also available allowing anyone to play around/ learn how to create their own!
  • http://imaginary.org/ Imaginary is an interactive platform which designed to showcase mathematical media content. The site contains plenty of pictures, videos and interactive demonstrations!
  • http://visualizingmath.tumblr.com/ A tumblr page full of math related visualisations.
  • http://geometric-aesthetic.tumblr.com/archive Tons of geometrical patterns and fractals.

In summary, the “beauty” of mathematics may be something “cold and austere” (as Bertrand Russell puts it), however, I believe everyone can gain if we do more to visualize the concepts involved. As well as helping our understanding, it allows us to think up strange new worlds (such as those depicted in other M.C Escher’s work – and potentially Einstein’s general relativity). Given the amount of maths out there I’m sure there is a lot of potential for mathematically inclined artists out there!

Matt
References and Further Reading
[1] This post is a continuation on themes I previously wrote about in about Maths and Visualization

Cover Image:

http://i.vimeocdn.com/video/458881089_1280x720.jpg

How Climate Model Uncertainty Should Influence Climate Policy

“All models are wrong; some models are useful” –  George Box

“I don’t believe anything, but I have many suspicions” – Robert Anton Wilson

Climate models are our primary method for predicting the future state of the Earth, and so are a crucial influence on climate policy.  Politicians often demand firm evidence that climate change is real. Scientific evidence of climate change has been around for decades, however, skeptics still manage to blow any uncertainty in scientists’ models out of proportion. In the following post I will discuss a short article by J. Norman et al. in which the authors argue for action to mitigate climate regardless of whether we have perfect models.

In the article they pose a powerful question which helps us think critically about what factors come into our decisions about climate policy. This question can be phrased as: “what would the correct policy be if we had no reliable models?” Thinking carefully about this question allows them to dismantle the assumption that it is only worth acting to combat climate change if you believe in climate scientists predictions. This is important to consider since many climate skeptics arguments rely on pointing out the flaws in scientists models. Furthermore, this question represents an interesting limit case to think about since it encourages us to think about how we should behave given that there are uncertainties in our models.

Their argument is based on the so called precautionary principle which, from a risk management perspective, posits that if something is potentially harmful to the public, the burden of proof on the people who want to carry it out rests on establishing that it isn’t, not the other way round. In short: we shouldn’t dabble with things when we don’t know what the unintended consequences might be. This is especially true when there is even a small risk of a complete catastrophe. Ignore this principle at your own risk in everyday life, but when we alter things that might affect whole ecosystems or planets it is definitely worth being extremely cautious about the consequences of our actions, which may not be reversible.

Any predictive model will have uncertainty in its outcome. In addition to this uncertainty, we must also strive to remember that the model isn’t reality, no matter how hard we try, meaning that there will always be events which are out of the range of our predictive powers. The importance of events which are out of our predictive range was popularized by Nassim Nicholas Taleb in the 2007 book The Black Swan. “Black Swans” are extreme events that lie out of our predictive range (no matter how good our supercomputer is). In the management of risk, the impact of these events may also be referred to as Knightian Uncertainity, this is a risk that is essentially uncomputable.

Just as an unpredicted market crash will render forecasts produced by economists for the future obsolete, there may also be unexpected events which will have a dramatic effect on current climate predictions. For example, a huge volcanic eruption might occur or some new carbon capturing technology might conceivably allow us to remove a large amount of CO2 from the atmosphere in a matter of months. In fact, we even have examples of relatively sudden changes in atmospheric composition happening in the past, as is the case with the “Paleocene-Eocene Thermal Maximum”.  Taleb’s message in the Black Swan is not that we should try and predict these events, but that we should instead be aware that, good or bad, unknown unknowns are out there in the future, whether we like it or not.

Figure 1. [3] Forecasting Skill plotted against year for different forecasts. This demonstrates the improvement of our ability to predict weather patterns over relatively short time scales. We shouldn’t naturally assume that this transfers over to the quality of our climate predictions. However, these types of observations provide evidence of the plausibility of physical models in the prediction of future states of the climate system.
That being said, we shouldn’t be totally pessimistic about our ability to predict. Weather and climate models can and have been statistically tested to perform fairly well [2]. In addition, the accuracy of weather forecasts and our ability to “see” longer into the future has indeed improved over the years (see figure 1). Given this, we should be careful not to misinterpret Norman et al.’s article as saying “Policy makers should never use climate models because we simply can’t predict things.” The main point is summarized in the final two sentences of the article:

“The popular belief that uncertainty undermines the case for taking seriously the ’climate crisis’ that scientists tell us we face is the opposite of the truth. Properly understood, as driving the case for precaution, uncertainty radically underscores that case, and may even constitute it.”

There will be plenty more to come on the blog about this difficult situation! For now, the main thing to take away is that we need to keep an open mind about the different possible outcomes which might arise in the future, predicted or not.

In summary:

  • We shouldn’t blame scientists for failing to include the presence of certain rare events in their models. After all, with limited computing power there is only a certain amount one can include in a model of such a massive system.
  • However, we can blame scientists or policy makers if they attempt to implement policies which ignore the possibility of anything that the model doesn’t explicitly predict.
  • Uncertainty is reason to act, not a reason to not act.

“The ancients knew very well that the only way to understand events was to cause them.”  – Nassim Nicholas Taleb

Matt

References and Further Reading:

[1] Taleb, Nassim Nicholas. The black swan: The impact of the highly improbable. Random House, 2007. Highly recommended for anyone who wants to read more about the impact of rare events on the course of history. Taleb goes into detail about the reasons why we should be very careful when transferring predictive models inspired by those in the physical sciences into domains such as the social sciences and economics.

[2] For more on how good climate models and their predictions have been for us so far: https://www.skepticalscience.com/climate-models.htm

[3] Image Reference: http://www.nature.com/nature/journal/v525/n7567/images/nature14956-f1.jpg

[4] Cover Image: http://www.billfrymire.com/gallery/weblarge/global-network-earth-space-night-sky.jpg