Useless Arithmetic?

Can we predict the future? Orrin Pilkey and Linda Pilkey-Jarvis say we can’t. They blame the complexity of the real world alongside a political preference to rely on the predictive results of models. I’m largely in agreement with them on many of their points but their popular science book doesn’t do an adequate job of explaining why.

The book is introduced with an example of the failure of mathematical models to predict the collapse of the Grand Banks cod fisheries. The second chapter tries to lay the basis of their argument, providing an outline of underlying philosophy and approaches of environmental modelling. This is then followed by six case studies of the difficulties of using models and modelling in the real world: the Yucca Mountain nuclear waste depository, climate change and sea-level rise, beach erosion, open-cast pit mining, and invasive plant species. Their conclusion is entitled ‘A Promise Unfulfilled’ – those promises having been made by engineers attempting to apply methods developed in simple, closed systems to those of complex, open systems.

Unfortunately the authors don’t describe this conclusion in such terms. The main problems here are the authors’ rather vague distinction between quantitative and qualitative models and their inadequate examination of ‘complexity’. In the authors’ own words;

“The distinction between quantitative and qualitative models is a critical one. The principle message in this volume is that quantitative models predicting the outcome of natural processes on the surface of the earth don’t work. On the other hand, qualitative models, when applied correctly, can be valuable tools for understanding these processes.” p.24

This sounds fine, but it’s hard to discern, from their descriptions, exactly what the difference between quantitative and qualitative models is. In their words again,

Quantitative Models:

  • “are predictive models that answer the questions ‘where’, ‘when’, ‘how much'” p.24
  • “if the answer [a model provides] is a single number the model is quantitative” p.25

Qualitative Models:

  • “predict directions and magnitudes” p.24
  • do not provide a single number but consider relative measures, e.g “the temperature will continue to increase over the next century” p.24

So they both predict, just one produces absolute values and the other relative values. Essentially what the authors are saying is that both types of models predict and both produce some form of quantitative output – just one tries to be more accurate than another. That’s a pretty subtle difference.

Further on they try to clarify the definition of a qualitative model by appealing to concepts;

“a conceptual model is a qualitative one in which the description or prediction can be expressed as written or spoken word or by technical drawings or even cartoons. The model provides an explanation for how something works – the rules behind some process” p.27.

But all environmental models considering process (i.e. that are not empirical/statistical) are conceptual, regardless of whether they produce absolute or relative answers! Whether the model is Arrhenius’ back of the envelope model of how the greenhouse effect works, or a Global Circulation Model (GCM) running on a Cray Supercomputer and considering multiple variables, they are both built on conceptual foundations. We could write down the structure of the GCM, it would just take a long time. So again, their distinction between quantitative and qualitative models doesn’t really make things much clearer.

With this sandy foundation the authors examine suggest that the problem is that the real world is just too complex for the quantitative models to be able to predict anything. So what is this ‘complexity’? According to Pilkey and Pilkey-Jarvis;

“Interactions among the numerous components of a complex system occur in unpredictable and unexpected sequences.” p.32

So, models can’t predict complex systems because they’re unpredictable. hmm… A tautology no? The next sentence;

“In a complex natural process, the various parameters that run it may kick in at various times, intensities, and directions, or they may operate for various time spans”.

Okay, now were getting somewhere – a complex system is one that has many components in which the system processes might change in time. But that’s it, that’s our lot. That’s what complexity is. That’s why environmental scientists can’t predict the future using quantitative models – because there are too many components or parameters that may change at any time to keep track of such that we couls calculate an absolute numerical result. A relative result maybe, but not an absolute value. I don’t think this analysis quite lives up to it’s billing as a sub-title. Sure, the case-studies are good, informative and interesting but I think this conceptual foundation is pretty loose.

I think the authors’ would have been better off making more use of Naomi Oreskes’ work (which they themselves cite) by talking about the difference between logical and temporal prediction, and the associated difference between ‘open’ and ‘closed’ systems. Briefly, closed systems are those in which the intrinsic and extrinsic conditions remain constant – the structure of the system, the processes operating it, and the context within which the system sits do no change. Thus the system – and predictions about it – are outside history and geography. Think gas particles bouncing around in a sealed box. If we know the volume of the box and the pressure of the gas, assuming nothing else changes we can predict the temperature.

Contrast this with an ‘open’ system in which the intrinsic and extrinsic conditions are open to change. Here, the structure of the system and the processes operating the system might change as a result of the influence of processes or events outside the system of study. In turn, where the system is situated in time and space becomes important (i.e. these are geohistorical systems), and prediction becomes temporal in nature. All environmental systems are open. Think the global atmosphere. What do we need to know in order to predict the temperature in the future in this particular atmosphere? Many processes and events influencing this particular system (the atmosphere) are clearly not constant and are open to change.

As such, I am in general agreement with Pilkey and Pilkey-Jarvis’ message, but I don’t think they do the sub-title of their book justice. They show plenty of cases in where quantitative predictive models of environmental and earth systems haven’t worked, and highlight many of the political reasons why this approach has been taken, but they don’t quite get to the guts of why environmental models will never be able to accurately make predictions about specific places at specific times in the future. The book Prediction: Science, Decisions Making, and the Future of Nature provides a much more comprehensive consideration of these issues and, if you can get your hands on it, is much better.

I guess that’s the point though isn’t it – this is a popular science book that is widely available. So I shouldn’t moan too much about this book as I think it’s important that non-modellers be aware of the deficiencies of environmental models and modelling and how they are used to make decisions about, and manage, environmental systems. These include:

  • the inherent unpredictability of ‘open’ systems (regardless of their complexity)
  • the over-emphasis of environmental models’ predictive capabilities and expectations (as a result of positivist philosophies of science that have been successful in ‘closed’ and controlled conditions)
  • the politics of modelling and management
  • the need to publish (or at least make available) model source code and conceptual structure
  • an emphasis on models to understand rather than predict environmental systems
  • any conclusions based on experimentation with the model are conclusions about the structure of the model not the structure of nature

I’ve come to these conclusions over the last couple of years during the development of a socio-ecological model, in which I’ve been confronted by differing modelling philosophies. As such, I think the adoption of something more akin to ‘Post-Normal’ Science, and greater involvement of the local publics in the environments under study is required for better management. The understanding of the interactions of social, economic and ecological systems poses challenges, but is one that I am sure environmental modelling can contribute. However, given the open nature of these systems this modelling will be more useful in the ‘qualitative’ sense as Pilkey and Pilkey-Jarvis suggest.

Orrin H. Pilkey and Linda Pilkey-Jarvis (2007)
Useless Arithmetic: Why Environmental Scientists Can’t Predict the Future
Columbia University Press
ISBN: 978-0-231-13212-1

Buy at

[June 3rd 2007: I just noticed Roger Pielke reviewed Useless Arithmetic for Nature the same day as this original post. Read the review here.]

2 thoughts on “Useless Arithmetic?

  1. Thank you for this review! I actually was considering purchasing this from, but I’ve heard mixed reviews so I figured I would hold off on it. I’ll probably still purchase it at some point, but I’m glad that you’ve helped elucidate what the authors did not.


  2. Yes, thank you. As a non-specialist I was worried that this idea of ‘qualitative’ modelling might be some recognised and radically different system that I’d missed…. but as you say the main difference between the two seems to be .. well, quantititative. One has more accurate numbers than the other. To demonstrate that some people misuse ‘quantitative’ modelling doesn’t prove that it never works.


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