# Pragmatism and mathematics

### The dichotomy

Mathematics may be considered in two parts: that which is a tool supporting other disciplines in their modelling, which is considered pragmatic; and that which seeks to test underlying assumptions in methods and models, which is not so well appreciated.

### Pragmatism and sustainability

Setting mathematics to one side for a moment, consider two courses of actions, S and P, with notional actual benefits as shown.

Sure and steady may beat Boom and bust

‘Boom and bust’ arises when (as is usually the case) the strategy is ‘closed loop’, with activity being adjusted according to outcomes (e.g. Ashby). Even a sustainable strategy would be subject to statistical effects and hence cyclic variations, but these will be small compared with the crashes that can arise when the strategy is based on some wrong assumption (Conant-Ashby). If something happens that violates that assumption then one can expect performance to crash until the defect is remedied, when performance can again increase. In this sense, the boom-bust strategy is pragmatic.

If one has early warnings of potential crashes then it can also be pragmatic to incorporate the indicators into the model, thus switching to a safer strategy when things get risky. But, to be pragmatic, the model has to be based on earlier experience, including earlier crashes. Thus, pragmatically, one can avoid crashes that have similar causes to the old ones, but not novel crashes. This is a problem when one is in a complex situation, in which novelty is being continually generated. Indeed, if you are continually updating your model and ‘the environment’ is affected by your actions and the environment can innovate, then one is engaged in cyclic co-innovation and hence co-evolution. This is contrary to an implicit assumption of pragmatism, which seems (to me) to be that one has a fixed ‘external world’ that one is discovering, and hence one expects the process of learning to converge onto ‘the truth’, so that surprises become ever less frequent. (From a Cybernetic perspective in a reflexive situation ‘improvements’ to our strategy are likely to be met by improvements in the environmental response, so that in effect we are fighting our own shadow and driven to ever faster performance until we hit some fundamental limit of the environment to respond.)

### Rationalising Pragmatism

The graph shows actual benefits. It is commonplace to discount future benefits. Even if you knew exactly what the outcomes would be, a heavy enough discounting would make the discounted return from the boom-bust strategy preferable to the sustainable one, so that initially one would follow boom-bust. As the possible crash looms the sustainable strategy might look better. However, the Cybernetic view (Conant-Ashby) is that a sustainable strategy would depend on an ‘eyes open’ view of the situation, its possibilities and the validity of our assumptions, and almost certainly on a ‘multi-model’ approach. This is bound to be more expensive than the pragmatic approach (hence the lower yield) and in practice requires considerable invest in areas that have no pragmatic value and considerable lead times. Thus it may be too late to switch before the crash.

In complex situations we cannot say when the crash is due, but only that a ‘bubble’ is building up. Typically, a bubble could pop at any time, the consequences getting worse as time goes on. Thus the risk increases. Being unable to predict the timing of a crash makes it less likely that a switch can be made ‘pragmatically’ even as the risk is getting enormous.

There is often also an argument that ‘the future is uncertain’ and hence one should focus on the short-run. The counter to this is that while the specifics of the future may be unknowable, we can be sure that our current model is not perfect and hence that a crash will come. Hence, we can be sure that we will need all those tools which are essential to cope with uncertainty, which according to pragmatism we do not need.

Thus one can see that many of our accounting habits imply that we would not choose a sustainable strategy even if we had identified one.

### The impact of mathematics

Many well-respected professionals in quite a few different complex domains have commented to me that if they are in trouble the addition of a mathematician often makes things worse. The financial crash brought similar views to the fore. How can we make sense of this? Is mathematics really dangerous?

In relatively straightforward engineering, there is sometimes a need for support from mathematicians who can take their models and apply them to complicated situations. In Keynes’ sense, there is rarely any significant reflexivity. Thus we do believe that there are some fundamental laws of aerodynamics which we get ever closer to as we push the bounds of aeronautics. Much of the ‘physical world’ seems completely unresponsive to how we think of it. Thus the scientists and engineers have tended to ‘own’ the interesting problems, leaving the mathematicians to work out the details.

For complex situations there appear to be implicit assumptions embedded in science,  engineering and management (e.g. pragmatism) that are contrary to the mathematics. There would thus seem to be a natural (but suppressed) role for mathematics in trying to identify and question those assumptions. Part of that questioning would be to help identify the implications of the current model in contrast to other credible models and theories. Some of this activity would be identical to what mathematicians do in ‘working out the details’, but the context would be quite different. For example, a mathematician who ‘worked out the details’ and made the current model ‘blow up’ would be welcomed and rewarded as contributing to that ever developing understanding of the actual situation ‘as a whole’ that is necessary to sustainability.

### Multi-models

It is conventional, as in pragmatism, to seek single models that give at least probabilistic predictions. Keynes showed that this was not credible for economics, and it is not a safe assumption to make for any complex system. This is an area where practice seems to be ahead of current mainstream theory. A variant on pragmatism would be to have a fixed set of models that one only changes when necessary, but the objections here still stand. One should always be seeking to test one’s models, and look for more.

It follows from Conant-Ashby that a sustainable strategy is a modelling strategy and that there will still be bubbles, but they will be dealt with as soon as possible. It may be possible to engineer a ‘soft landing’, but if not then a prediction of Conant-Ashby is that the better the model the better the performance. Thus one may have saw-tooth like boom and busts, but finer and with a more consistent upward trend. In practice, we may not be able to choose between two or more predictive models, and if the available data does not support such a choice, we need to ‘hedge’. We can either think of this as hedge across different conventional models or as a single unconventional model (such as advocated by Keynes). Either way, we might reasonably call it a ‘multi-model’. The best strategy that we have identified, then, is to maintain as good as possible a multi-model, and ‘hedge’.

If we think of modelling in terms of assumptions then, like Keynes, we end up with a graph-like structure of models, not just the side-by-side alternative of some multi-modelling approaches. We have a trade-off between models that are more precise (more assumptions) or those that are more robust(less assumptions) as well as ones that are simply different (different assumptions). If a model is violated we may be able to revert to a more general model that is still credible. Such general models in effect hedge over the range of credible assumptions. The trick is to invest in developing techniques for the more general case even when ‘everybody knows’ that the more specific case is true, and – if one is using the specific model – invest in indicators that will show when its assumptions are particularly vulnerable, as when a bubble is over-extended.

### Duality

A traditional approach is to have two separate strands of activity. One – ‘engineering’ – applies a given model (or multi-model), the other – ‘science’ – seeks to maintain the model. This seems to work in complicated settings. However, in complex reflexive settings:

• The activity of the engineers needs to be understood by the scientists, so they need to work closely together.
• The scientists need to experiment, and hence interfere with the work of the engineers, with possible misunderstandings and dire consequences.
• Im so far as the two groups are distinct, there is a need to encourage meaningful collaborations and manage the equities between their needs. (Neither ‘on top’ nor ‘on tap’.)

One can see that collaboration is inhibited if one group is pragmatic, the other not, and that pragmatism may win the day, leading to ‘pragmatic scientists’ and hence a corruption of what ought to be happening. (This is in addition to a consideration of the reward system.)

It may not be too fanciful to see signs of this in many areas.

### The possible benefits of crashes

Churchill noted that economic crashes (of the cyclic kind) tended to clear out dead-wood and make people more realistic in their judgements, compared with the ‘good times’ when there would be a great deal of investment in things that turned out to be useless, or worse, when the crash came. From Keynes’ point of view much of the new investment in ‘good times’ are band-wagon investments, which cause the bubble which ought to be pricked.

We can take account of such views in two ways. Firstly if the apparent boom is false and the apparent crash is beneficial then we can take this into account in our measure of benefit, so that ‘boom’ becomes a period of flat or declining growth, the crash becomes a sudden awakening, which is beneficial, and the post-crash period becomes one of real growth. The problem then becomes how to avoid ‘bad’ band-wagons.

Either way, we want to identify and avoid fast growth that is ill-founded, i.e., based on an unsustainable assumption.

### Conclusion

It is well recognized that mathematics is extremely powerful, and the price for that is that it is very sensitive: give it the slightest mis-direction and the result can be far from what was intended. Mathematics has made tremendous contributions to the complicated disciplines, and seems quite tame. In contrast, my experience is that for complex subjects the combination of mathematicians and numerate professionals is risky, and requires an enormous up-front investment in exchanging views, which sometimes can be nugatory. Perhaps there is some mis-direction? If so, where?

From my point of view, the problem often seems to be one of ‘scientism’. That is, certain types of method are characteristic of the complicated professions, and so people expect problems that are almost the some but complex to be addressed in the same sort of ways. Anything else would not be ‘proper science’. The mathematician, on the other hand, has the habit of rooting out assumptions, especially un-acknowledged ones, and seeking to validate them. If they can’t be then they would rather not make them. (A cynic might say that the other professionals want to work with the tools they are familiar with while the mathematician wants to develop an all-purpose tool  so that he can move on to something more interesting.)

Numerous headline failures tend to reinforce the mathematician in his view that other professionals profess certain beliefs, while he is uniquely equipped to be productively cynical. But here I focus on one belief: in pragmatism. Often, when pressed, people do not actually believe that their assumptions are literally true, only that they are ‘pragmatic’. But, as previously noted, the mixture of literal mathematics and conventional pragmatism is unsafe. But in my view mixtures of pragmatisms from different domains (without any ‘proper’ mathematics) seems to lie behind many headline problems. I have shown why pragmatism is inappropriate for solving complex problems and briefly sketched some reforms needed to make it ‘mathematics friendly’.

Dave Marsay

Mathematician with an interest in avoiding 'bad' reasoning.

### 10 Responses to Pragmatism and mathematics

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3. sfnhltb says:

As well as converting the benefits of the boom/bust cycle into some sort of synthetic metric, aren’t there other adjustments that need to be done to know which is better – for example WWII had a massive and fairly sustained growth in economic output compared to the 30s, but it is highly dubious that it was better even compared to what is normally considered a fairly bad economic situation. Another example would be most forms of advertising – which overall give no benefit to anyone, apart from erecting barriers to entry and forming oligopolies that can compete with the advertising dollars (or sterling or yen) of the other large players. The canonical example being the US ban on tobacco advertising which significantly boosted tobacco companies profitability. It makes me wonder if a true measure of the strength of the economy can be derived, even before the step of trying to tease out what strategies will then optimize its growth.

• djmarsay says:

Good points. Economically WWII might well have been ‘a good thing’. My current thinking is that we should put more emphasis on things like life expectancy, taking account of ‘quality of life’. Taking GDP/GNP as surrogates seems to assume that we are addicted to growth, in which case at some point we may need to dry out.

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