The Provisional Nature of Mathematical Models

Since my Blog on the unusual weather in Northern Europe at the end of last year and its possible implications for the climate change debate (see ‘Freak Weather? Or Something Someone’s not Telling Us?’), a number of readers have accused me of accusing climatologists of dishonesty. If I gave this impression, I apologise. It was not my intention. If I thought I was accusing them of anything, it was simply of being human – about which I shall write more in my next article on this subject. My more central point, however, concerned the very nature of climate science, itself, and the rather obvious truth – to which I wished to draw people’s attention – that where a science doesn’t lend itself to testing its hypotheses by experimentation, the output of mathematical or computer models, used to forecast future events, should not in any sense be seen as ‘proving’ the hypotheses upon which these models are built, but should rather be regarded as strictly provisional, pending the outcome of the events themselves. 

To illustrate this more clearly, consider, for example, another well known if somewhat simpler mathematical model: the Drake Equation. Devised in 1960 by Frank Drake to estimate the number of technologically advanced civilizations likely to be found in our galaxy, it has seven variables and states that:

N = R* · f_p · n_e · f_l · f_i · f_c · L

where:

R*        is the number of stars formed in the galaxy each year.

f_p             is the percentage of those stars with planets.

n_e            is the number of planets in each of those planetary systems capable of supporting life.

f_l             is the percentage of those planets on which life actually develops.

f_i             is the percentage of those planets on which intelligent life then evolves.

f_c             is the percentage of those planets on which advanced communications technology then develops.

L          is the length of time such technological civilizations then endure.

There is another version of the equation in which R* is replaced by N*, the number of stars in the galaxy, and L is replace by L/T_g, where T_g is the age of the galaxy. The results, however, are exactly the same, and I have here therefore retained the original formulation. Of far greater consequence are the values one chooses to assign to each of the variables; and while the model, itself, has become more or less universally accepted, it is in this regard – as one would expect – that the debate has ebbed and flowed over the last fifty years, with the resulting calculated values of N ranging from a possibly apocryphal 50,000 to less than 1.

I say ‘possibly apocryphal 50,000’, because, for a long time it was generally believed that this was the figure that Frank Drake and his team originally arrived at when they first devised the model. Indeed, Frank Drake, himself, quoted this figure in a television programme made for the BBC last year. It is also thought to be what led NASA to fund the SETI project – the Search for Extra-Terrestrial Intelligence – which Frank Drake founded. Officially, however, it is now said that the original value assignments were as shown in Table 1, producing a value for N of just 10, which while certainly a lot closer to what one might now consider realistic, given the fifty years that SETI has so far spent searching the heavens without result, would probably not have won it the funding to undertake this search in the first place.

Table 1: Drake Equation with Original Values Assigned

If one looks at some of these value assignments more closely, however, and especially at the rough and ready way many of them were arrived at, even this much-reduced figure lacks a certain credibility. The original assessment of 2, for the number of planets thought to be capable of supporting life in the average planetary system, for instance, was based solely on our own solar system, the only planetary system that was known at that time. Understandable, though this may therefore have been, it was still, however, a statistically rather unreliable sample of just one. Moreover, the inclusion of Mars as well as the Earth in this category, seems to fly in the face of reason. For although it was thought at the time that, in principle, Mars could have supported life; the plain fact is that it doesn’t, which rather seems to contradict the team’s next assumption that all planets capable of supporting life would inevitably do so.

In fact, we now know that Mars actually falls a long way short of fulfilling either of the two primary conditions for a planet to be life-supporting. The first of these is that it should maintain a temperature range in which water is liquid for at least part of the time. It cannot therefore be too close to the sun or too far away from it. It also has to have an atmosphere and therefore a molten iron core. For without the latter it wouldn’t have a magnetic field; and being so close to the sun, without a magnetic field, its atmosphere would simply be blown away by the solar wind. In order to keep its core molten, however, a planet also has to be of a certain size. Mars, for instance, has an iron core, but being smaller than the Earth, it doesn’t generate enough gravitational pressure to keep the core molten. If it ever met the requirements for supporting life, therefore, it did so only until its core solidified, which was probably only a relatively short time, a fact which the model, quite significantly, is incapable of actually representing. 

Time, however, is not the only issue to be factored into the equation as to whether a planet, theoretically capable of supporting life, will eventually do so. One also has to take into account the conditions required to transform inorganic compounds into a single celled organism. In May 2010, researchers at the J. Craig Venter Institute in the United States inserted artificial genetic material – chemically printed, synthesized and assembled – into cells that were then able to grow naturally. This they then described as creating life from scratch. But even they would admit, I think, that designing DNA on a computer, and then manufacturing it, using carefully selected chemicals, is a long way from having it spontaneously emerge from a random chemical soup. The fact that this extraordinarily unlikely event did in fact happen here on earth means, of course, that, given the right conditions and enough time, it could in principle happen on any planet in the life supporting category. On the other hand, it is far from certain that all such planets will have either the right conditions or enough time. More realistic estimates, therefore, place the probability of it happening at around 13% rather than a 100%. If we also reduce the estimates for some of the other variables, including the number of planets capable of supporting life in each planetary system, one of the more plausible calculations of N now comes out at just 0.000065, as shown in Table 2. This would not only mean that we are, without doubt, alone in our galaxy, it might even mean that we are alone in the universe, an idea which almost doesn’t bear thinking about.

Table 2: Drake Equation with Pessimistic Values Assigned

Regardless of the implications of all this for man’s place in the cosmos, however, the more immediately relevant point is that since 1960, when those who devised the Drake equation were young, idealistic and excited by the prospect of life on other planets, the forecasts it has generated have gone first 50,000 to just 10, and now from 10 to a number so infinitesimally small that it casts the existence of life on earth in wholly new light. Of course, part of this transformation in our expectations is almost certainly due to the fact that, for all our efforts over the last fifty years, no sign of intelligent extra-terrestrial life has yet been found. Youthful enthusiasm has thus been tempered by disappointment and, and this respect, one might say that the money spent on SETI has not been wasted. It also, to some extent, justifies the exercise of both formulating the model in the first place and assigning some provisional values to it. For it gave us something against which we could measure reality. Not only does the failure of SETI’s mission emphasise the provisional nature of all such exercises, however, reminding us how often reality fails to conform even to those expectations in which we have the highest confidence, it also demonstrates the extent to which emotional states, like optimism and enthusiasm or, indeed, dread foreboding, can colour and inform scientific judgement, particularly when combined with the need to secure funding. Scientists may not like it, but science is a human activity and is subject to all the usual human frailties. Ironically, it is also when it seeks to deny this that it is at its most human and most dangerous… particularly when the rest of us start to believe it, as I shall explore in my next Blog.