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In 1960, the physicist Eugene Wigner wrote the article "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" explaining how unexpected it is that mathematical formalism can make predictions about reality. Although his article was essentially restricted to what happens in modern Physics.

The picture described by Wigner is not one typically found in other natural and social sciences. In fact, there has been much joking in recent times about "unreasonable ineffectiveness of mathematics" in many sciences (from social sciences to natural sciences such as biology).

My questions are:

  1. what could be the reasons why mathematics seems to work very well in certain areas (physics or engineering), but is not as useful or accurate in in social sciences such as Economics?
  2. Is it reasonable to assume that economics or sociology will one day have as intensive and predictive a use of mathematics as is currently the case in physics, or can these sciences as we know them today simply not be formalized to that degree?
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    $\begingroup$ It would be nice if there would be any support for your premise. Please add it. $\endgroup$ Commented Jul 7, 2022 at 16:37
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    $\begingroup$ There is an article in wikipedia: en.wikipedia.org/wiki/… which gives a lot of published references pointing to this problem in economics and finance. Vela Velupillai, Sergio M. Focardi and Frank J. Fabozzi, and even the renowned Irving Fisher wrote extensively on the subject (the wikipedia article mentioned above has references to their original articles and books where this problem is addressed). $\endgroup$
    – Davius
    Commented Jul 7, 2022 at 16:43
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    $\begingroup$ You can describe lots of relationships well with math (gravity, energy, mass, density, intertia etc.) because they never change their "behaviour". People constantly change. They like cities, decide to move to suburbs, want to work part time or in home office, have several kids but "suddenly" very few kids.... You cannot describe something well that is unpredictable and changes constantly (human behaviour). People thought the earth is flat (and some still do!) $\endgroup$
    – Alex
    Commented Jul 7, 2022 at 19:20
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    $\begingroup$ @Alex "never change their behavior" like how two particles always attract each other irrespective of their charge? Or like the quantity of salt soluble in water irrespective of its temperature? Peoples choices are indeed inscrutable (if we do not add state variables). $\endgroup$
    – Giskard
    Commented Jul 7, 2022 at 23:58
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    $\begingroup$ @Giskard, even though the amount of solute that can dissolve in a solvent changes with temperature, it will still always be the same amount if you repeat this at whatever temperature. Likewise, if two particles attract or repel each other will always depend on the dame logic. On the other hand, lots of people who were "attracted" to each other actually ended up in divorce or war. $\endgroup$
    – Alex
    Commented Jul 8, 2022 at 0:41

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I think by today the arguments you mention are completely outdated. Nowadays, using combination of psychometry and econometrics companies can predict whether you are pregnant (from your shopping patterns) earlier than it is possible for humans to notice the pregnancy (see here). If companies being able to use data with combination of social sciences to predict pregnancy does not count as "unreasonably effective" I don't know what does.

there has been much joking in recent times about "unreasonable ineffectiveness of mathematics"

I do not know where you heard such jokes, but they must have come from some people who are not up to date on modern research. While economics or other social sciences are not yet as precise as some areas of physics they are not far behind. Such jokes would ring true before the 'credibility revolution' in economics (see Angrist and Pischke 2010). After the credibility revolution such jokes are certainly not accurate.

what could be the reasons why mathematics seems to work very well in certain areas (physics or engineering), but is not as useful or accurate in in social sciences such as Economics?

In past (pre 80s) social science models were very ineffective mainly for the following reasons:

  • lack of data to verify/quantify relationships.
  • there is more noise in social science data which requires large sample sizes for disentangling relationships from noise.
  • lack of computing power.
  • lack of statistical techniques crafted for social sciences and lack of experimentation.

lets take these points one by one:

  1. lack of data to verify/quantify relationships.

in the past it was very difficult to find high quality data. You will find a lot of social science papers from that era using sample sizes of 40-60, which by modern standards would be laughable. In fact even as recently as in 90s you could see published papers using sample sizes as small as 117 observations (e.g. see Dollar 1992).

For example, even in physics it would be difficult if not impossible to predict path of an asteroid if there are no good data on its velocity, position etc available.

  1. there is more noise in social science data which requires large sample sizes for disentangling relationships from noise.

In connection to the previous point there is usually much more noise in social science data, hence it goes without saying that social science requires much larger data sets to be precise. This is why the stories such as that about the prediction of pregnancy are popping up just now as they are result of large data being employed in social sciences.

lack of computing power.

Lack of computing power was large issue in the past, and it is still limiting nowadays. For example, analogue hydraulic computer (e.g. MONIAC) was used in economics even after regular computers were invented as early computers did not had computing power to model even simple macro models.

Modern computers are better but still computing power is big limiting factor. Recently I took graduate class from Ben Moll on distributional macro where we were building relatively simple macro models with multiple agents (e.g. see examples of the models here), yet even modern PCs have still some trouble running such models (they can take quite long to solve).

Nonetheless, outside area of simulations (that are usually very intensive computationally) the computing power now is sufficient to run wide array of statistical models which would be impossible to run in the past.

  1. lack of statistical techniques crafted for social sciences and lack of experimentation.

In the past big issue with social sciences was lack of appropriate statistical models. In physics most relationships are exogenous, you have typically very simple chains of causality. In social sciences most relationships are endogenous. Such relationships cannot be as easily analyzed with (comparatively) basic methods that are sufficient for many natural sciences.

As a result before the 'creditability revolution' most empirical research was in very bad state. However, the development of new statistical techniques such as diff-in-diff, synthetic control, TSLS, RD etc as well as greater focus on running randomized controlled trials brought credibility and much greater predictive power to social science research (see Angrist and Pischke 2010).

Even if social science research is still not as precise as some areas of physics, it would be absurd to say that mathematics is somehow 'unreasonable ineffective' at present day.

Is it reasonable to assume that economics or sociology will one day have as intensive and predictive a use of mathematics as is currently the case in physics, or can these sciences as we know them today simply not be formalized to that degree?

It is reasonable to assume that. In fact I would say that present day economics can be as accurate as many areas of physics were decades ago. With better data, more computing power, better statistical techniques social sciences can be frighteningly precise. For example, recently university of Chicago researchers developed algorithm that can predict crimes weeks in advance with about 90% accuracy (see Rotaru et al 2022 or here) which almost sounds like the "precogs from the minority report.

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    $\begingroup$ I like Angrist and Pischke as a source, but both of your success stories are non-scientific papers - Forbes for the pregnancy one and Sciencedaily (!) for the crime prediction one. IMO comparing this latter result about statistics to "precogs" overhypes the latter a way that is unfortunately not unusual in econ. $\endgroup$
    – Giskard
    Commented Jul 8, 2022 at 8:12
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    $\begingroup$ Might be better to change the reference from ScienceDaily to the original paper published in Nature or directly from the University $\endgroup$
    – James
    Commented Jul 8, 2022 at 14:06
  • $\begingroup$ @James thanks for the paper I added it to my answer $\endgroup$
    – 1muflon1
    Commented Jul 8, 2022 at 19:22
  • $\begingroup$ The MONIAC was created in 1949. Is there evidence that after the general availability of digital computers, analogue computers were still used seriously in economics? $\endgroup$ Commented Jul 9, 2022 at 18:40
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    $\begingroup$ I would distinguish success in causal explanation vs. prediction. As far as I know, the credibility revolution did not bring (much) predictive success. $\endgroup$ Commented Jul 10, 2022 at 9:20
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(Usually) In the places where mathematics does work really well, there are clear ways to quantify and the way we quantify can be used same for all situation, but, in the places were mathematics (usually) fails, there is an element of non-quantitative change.

What do I mean by that? Well the thing is that, what you use measure stuff by one day can't be used another day. Say for instance, a study like economics or political science, everyday through the 'force of the human mind' there are more and more variables and generative effects occuring.

Let me give you a story to explain this, once upon a time there was a copper mine somewhere and due to environment concerns of sulfur being produced when copper is extracted, the mine was pressured to reduce/ shut down operations by a local environmental authority. Some months later, the copper mine choose a fundamentally different strategy to the problem of excessive sulfur production, they decided to turn the Sulfur into Hydrosulfuric acid. This made a lot of profit and led to the business then being focused on selling Hydrosulfuric acid.

In this case, the initial model of the company being about copper itself changed as time progressed. However, lets say we take a subject like physics, then the subject matter that it describes, it is so that whether we did it last year or yesterday, the experiments would show the same laws of physics are obeyed.

Essentially, the argument is that the high effectiveness of quantitative sciences, at least in it's current states, relies on high levels of regularities in it's universe of discourse to work.

This is not to say math doesn't totally work in the non quantitative sciences, we can use the mathematical ideas to extract out deeper meanings of observed phenomena rigorously. For example, have a look at this presentation where Eugeina Cheng discusses how Category Theory can help one understand about Racism.

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  • $\begingroup$ More than just businesses changing their methods, perhaps one issue unique to economics is that people use economics knowledge to make bets on the future and those bets influence the way economics happens - new knowledge ends up invalidating itself! $\endgroup$ Commented Jul 9, 2022 at 18:43
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What could be the reasons why mathematics seems to work very well in certain areas?

In physics or engineering we can observe which phenomena affects the studied problem, then make relations (equations) and then hide the lack of knowledge under constants. But there are phenomena like the tulip mania or bitcoin manipulations by tweets that cannot be described by equations, because we don't know what exactly caused them.

Statistics may predict such anomalies, but expert systems (AI) is better tool for this. The AI is mathematics in nature (it works with matrices and derivations, gradient descents), but it is empirical, unlike the "old school" mathematics which describes itself as purely rational.

There are areas where AI can "figure out" some rules that we don't know and make better decissions than old school math. But there are also "attacks" which may lead the AI to make surprisingly bad decission, see here.

Is it reasonable to assume that economics or sociology will one day have as intensive and predictive a use of mathematics as is currently the case in physics, or can these sciences as we know them today simply not be formalized to that degree?

Strict formalization may not be necessary when we can do heuristics with strong computers. For example, the iterations of setting weights in neural networks can be formalized, but the solution of the problem that the network solves (the weights of trained network) is pure heuristics, it is not formalized, not fully understood how and why the weight of particular perceptron was set for some particular value.

Today, mathematics merge into computing. Therefore its use will grow, but as shown in the link above, it still have limits where it is not the best tool, like aesthetics. One of these limits is the formalizations: what can not be formalized, can not be described by mathematics.

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  • $\begingroup$ Nice answer! It is an interesing perspective! $\endgroup$
    – Davius
    Commented Jul 7, 2022 at 22:42
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I will address only your second question, because I think there is something important here which has not been discussed by the other answers:

Is it reasonable to assume that economics or sociology will one day have as intensive and predictive a use of mathematics as is currently the case in physics, or can these sciences as we know them today simply not be formalized to that degree?

There is no reason I can imagine at all that these subjects should not be formalized to the degree where mathematical techniques are not extremely effective. However that is a different question as to whether mathematical techniques will be predictive in general.

To see the reason for this it's easiest to look at a system where there is no doubt that mathematical techniques are appropriate: the weather. We know a great deal about the underlying processes involved in weather: there are certainly things we probably don't yet fully understand such as some of the chemistry that goes on, but we know enough.

But, still, we can't predict the weather very well: we can predict it far better than we could (I think that forecasts have improved by about a day a decade for a while, so that a forecast four days out is now as good as one one day out thirty years ago), but there is no chance that we will ever be able to make any useful prediction of the weather three months out, for instance. The best we can do, and probably the best we will ever ba able to do is predict the climate three months out, which is much easier but not actually a weather forecast.

That's for two reasons.

  • Firstly the weather is just a vast system with a huge number of components, and predicting the behaviour of any such system requires enormous resources, even when you know the rules that govern it.
  • Secondly, and very famously, the weather is a nonlinear system and in particular it has sensitive dependence on initial conditions (and in fact it has chaos as well, although the two are not quite the same thing). That means, effectively, that it will never be possible to predict it very far into the future because the amount of resources you need to do so, as well as the quality of the measurements you would need to do so goes up so fast.

So your second question falls into two parts, I think.

  • Can these subjects be formalized? I am sure they can, at least in principle.
  • Would such formalization lead to being able to make useful predictions, and in particular useful predictions over longer timescales? Very possibly it would not, since these are enormously complex systems and very likely have at least SDIC and quite possibly chaos.

Personally I would strongly assume that formalization would not lead to useful long-term predictions.

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  • $\begingroup$ Upvoted. One question. Your basic argument is that the economy might be a chaotic system like the weather. Has Chaos theory been seriously applied to economic questions? How successful have they been? $\endgroup$ Commented Jul 10, 2022 at 12:15
  • $\begingroup$ @IshanKashyapHazarika: yes, it has, see for instance this. No idea how successful, sorry. $\endgroup$ Commented Jul 11, 2022 at 8:32
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I think many complex phenomena are initially understood by observation and systemic organization of observations. For example superconductivity was initially observed and progressively described. But a microscopic explanation was not done until the 1950s. Even after then, one often can get useful insights/intuitions from some of the earlier (flawed) models or trends (e.g. isotope effect) that do not occur (intuitively) if you just think about BCS.

Another example is in chemistry. Knowing the characteristics of an -OH functional group (polar, tending to water solubility, etc.) is very useful in doing chemical synthesis design or even commercial chemical process design. These are features that would not be intuitive if you were a physicist who just thinks/says it's quantum mechanics. For one thing, the equations are almost always not analytically soluble. And often even computational approaches are limited. Then, even if they exist, you don't get the immediate intuitive benefit of understanding tendencies and trends. It will slow you down a LOT if you don't have intuitive concepts of oil and water solubility and instead have to reach for a computational approach or a reference book. There's a reason why physicists don't make advances in natural products synthesis (IN GENERAL, pedants).

Within economics, you have (1) complex, multivariable, stochastic systems, and (2) limited ability for experimentation--much less than in bench chemistry, and (3) lack of time scale (often) for checking hypotheses by out of sample testing (and then the normal human disinterest in being reminded of out of sample failures...for example peak oil pushers of c. 2005.)

I think another issue is that very abstract mathematicians (and theoretical physicists) tend to cleave to axiomatic and "always true/false" logic. But in the real world, even small correlations are interesting and useful. We use them in our daily lives. And they are useful in practical natural and social sciences as well.

Furthermore, I would emphasize that statistics is a mathematical topic. So knowledge of regressions and the like is important and a tool of math, used in econ (really in some ways coming from ag studies!)

Finally, I suggest taking a look at Stanley Lieberson "Making It Count: The Improvement of Social Research and Theory" (1987). (Thoughtful monograph on some of the issues in social sciences as opposed to natural sciences.) Very highly cited. Gives you some useful humility and perspective on the multiple regression approach. And I particularly enjoyed the point about irreversibility of some effects (taking a bullet out of the heart doesn't bring a dead man back to life). Maybe a similar concept is magnetic hysteresis. I wonder if such is observed in some social science situations.

P.S. Ethaka...S to HCl? (Also, of course, disposal of waste products, including by sale/transformation is a very common concept in the chemical/refining industry. Many, many examples. And the optimal choice may differ based on transport costs. I'm thinking of a certain classic case example that Ron Braeutigam used to give, where the key insight is to realize you can think of yourself as being in the chloride business and selling NaOH is just partially offsetting a disposal cost.

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Many practical applications of science do not require distinguishing causation from correlation unless something can happen that would cause the factors in question to become "detached". In many scientific fields, correlations which are observed will persist unless or until one performs actions which break them.

Fields like ecomomics become difficult because many of the things that practitioners would like to know can't be measured directly. If some desirable thing X can't be measured directly, but people who do Y also do X, then one may be able to estimate the extent to which different groups of people do X by measuring the extent to which they do Y, but only until such estimations are used in ways that affect the people involved. If people who do Y are rewarded because of the presumption that they will also do X, this will likely result in an increase in the number of people who do Y but not X, thus making Y cease to be a good proxy measurement.

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