The Unreasonable Ineffectiveness of Mathematics in most sciences

Question

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 human 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 other fields, such as social sciences or biology?
2. Is it reasonable to assume that biology, 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?

Answer 1

There's tons of mathematics in biology and economics. It's not as successful at predicting outcomes in those fields, because biological and economic systems are inherently more complex than the systems that physicists study; they involve more kinds of objects, and more complicated objects, interacting with each other. Also, they are harder to control and observe, so it's more difficult to set up experiments.

So that means that a simple model of a few equations is less likely to work. However, it also means any model is less likely to work. It's not that mathematics in particular fails to work in biology and economics, but that it's more difficult to produce accurate predictions at all, by any method, mathematical or otherwise, in those fields.

To the extent accurate predictions can be produced in those fields, statistics is the only way to verify that the predictions are in fact accurate.

Answer 2

Mathematics as a tool is progressively more effective as models become progressively simpler. The fewer the number of objects involved, the fewer the variables to be considered, the fewer the relationships being measured, the more precise mathematical analysis can be. In fact, there's a common phrase in statistics to the effect that 'power comes from valid assumptions'. Every assumption one can legitimately make about an object of study means one more thing that does not have to be controlled for, that does not create interferences or interaction effects, and that can in other ways be dismissed. That makes calculations much easier.

Take the classic physics example of ballistic trajectory. Ballistics calculations appear incredibly precise because in low-level textbooks they are presented as the interaction between three elements: projectile mass, impulse force, and launch angle. But consider all of the assumptions that are made in that model:

• Negligible air resistance
• Mass of uniform density and regular (spherical) shape
• Instantaneous and momentary force along the center of mass of the projectile
• Flat and uniform gravitational field
• No outside forces to contend with

If all of these assumptions held true, then the mathematics of ballistic motion would be absolute and perfect, such that one could always calculate precisely where a projectile would land. In the real world, of course, these assumptions are never perfectly true, but in most cases of interest — e.g., firing a cannon ball at an enemy — the large mass and strong initial impulse are sufficient to overcome most of the other variables. We can effectively ignore them and still get our cannonball close to where we aim, with some small error term involved.

But consider what happens if those assumptions do not hold. Say that instead of a nice round cannonball we fire something shaped like an anvil, something with an uneven density so that force is applied unevenly and spin is odd. Say the air is strange, with channels of gale force wind in some places, while other places are soupy and viscous. Even with the best knowledge of these conditions our targeting would be poor, because there are too many variables involved in the calculations. These simple, powerful equations are only simple and powerful because we can assume away all of the nasty, knotty, confounding complications.

Physics is easy(ish) because the material world generally allows us to make all sorts of assumptions and approximations, and most complexities can be overcome by sheer force: giving things more mass or impelling them harder smoothes out the error term. Most other sciences aren't as fortunate, and so mathematics becomes less precise within them. The social sciences in particular have the problem of choice: different people making different choices within a group produces strong interaction effects which are significantly unwieldy for mathematics to deal with. Trying to predict social tendencies is about as successful as trying to chart the trajectory of a cannonball fired into a tornado. One can do it, roughly, but there's no sense expecting high precision.

Answer 3

The answer is, symmetry. Especially, not just the more familiar ones like symmetry under rotation or reflection, but continous symmetries.

The whole-number line is an abstraction of translational symmetry, and objects considered indistinguishable. Physics generally focuses on indistinguishable or barely so objects, like bosons and fermions, moving in space. Noether's Theorem tells us conservation laws are directly equivalent to statements about contnous symmetries under transformations - these are then elaborated into mathematics in general, with mathematical operators drawn from analogy to types of transformation.

Taken from Cellular Landscape Cross-Section Through A Eukaryotic Cell

In biology the relationships are more complex, because entropy increase means far-from-equilibrium systems are always involved so they change dynamically, through eating and waste plus energy flowing in. Meaning the symmetry across time-translation is always going to be incomplete. And just as chemistry involves much more composite structures than particle physics, so biology does than chemistry.

There are still spatial & time relationships with conserved symmetry in biology. Fractal relationships are often involved between layers of organisation, which are symmetries of fractional dimensions. And intersubjectivity depends on a partial 'indistinguishability', and it underlies communication & so compounding of conceptual abstractions, as discussed here: According to the major theories of concepts, where do meanings come from? But they are generally less common. In order to make analysis tractable, we create 'overlays' or heuristic-explanatory-layers, that group together or 'chunk' a lot of information. Knowing a persons character is generally a lot better way to predict them, than knowing the position and momentum of their constituents - small errors in initial position down even to from the Uncertainty Principle, or in calculations from that initial state, will rapidly make it useless; whereas we expect a lot of time-translation symmetry of the composite 'character', though of course, some change. We picture causes in the language of one of the layers: 'His temper caused -' etc. But we expect those to be reducible in principle to atoms (ie, compatibilism).

A way that we can think about how the sciences have been unifying, is that the symmetry underlying energy conservation, and the assymetry involved in entropy change and the Arrow of time, provide a 'language' to interface together and check the consistency of different areas of science, because of associated conservation laws. As discussed here: Is the idea that "Everything is energy" even coherent? It's important to note the slipperiness of 'reducible in principle': With perfect information we might be able to fully predict a mind, but that might require such a detailed model it would require making another copy of a persons brain, a functional one. That would mean irreducibility, in practice. And be a fundamental limit as such to abstracting complex phenomena into simpler forms (even if consciousness is digitised, and in a way 'mathematised').

When we gain insights using mathematics, it is often by identifying a symmetry property. Consider the ideal gas laws, which identify the relevant properties of interaction, and allow a lot of other data to be dropped. We have Effective Field Theories, manifestations of general theories that appear at particular length scales and with specific degrees of freedom. Comparably, we might find there is a set of predictable behaviours in some circumstances, like say laminar flow or a liquid, but unpredictable chaotic turbulant behaviour after some transition of interaction. You might call particle physics, chemistry and biology, different effective field theories seperated by transitions like that between laminar and turbulent flow, with difference that could described in general terms by how reducible the system ordering is.

Previous related discussions

About Wigner's view on the relation between mathematics and physics?

Does reality have axioms?

Answer 4

The answer is a combination four factors.

1. Simplicity vs. complexity

As @causative pointed out, the systems that can be predicted are relatively simple. By contrast, the subjects of the social sciences are much more complex. Whether you are using mathematics or not, it's much easier to precisely predict simple systems than complex systems. You don't need mathematics to predict the trajectory of a ball when you throw it. With a bit of practice you can learn to control the trajectory very accurately just by judgment. By contrast, even the most astute national leaders and CEOs, using their judgment, often make extremely bad predictions about the economy.

2. Idealization

As @TedWrigley pointed out, the systems that are easy to predict are not only highly simplified but highly idealized. They are so simple that we can idealize them to even simpler systems consisting of just a few parameters. Non-physical sciences cannot generally be simplified to this extent. In fact, even a lot of physical systems are too complex to handle this way. The weather, for example, is a physical system that cannot be predicted well.

There are simplifications of certain non-physical systems that do produce mathematically precise answers. Game theory, for example, can be seen as economics, extremely simplified and idealized. In game theory, we can accurately predict the outcome of various behaviors, just like in physics we can accurately predict the properties of ideal gases. But when you take game theory to economics or ideal gases to weather, the idealization no longer works; there are too many factors to account for.

3. The illusion of success

The success of mathematics in physical systems is to a large extent an illusion caused by a tradition of successful engineering. As Ted Wrigley mentioned, the idealized equations of ballistics don't actually work very well in real situations on Earth. When you are actually targeting artillery you have to include additional factors such as air pressure and wind, and the accuracy varies a great deal depending on conditions.

When an engineer builds a new kind of device, he uses mathematics to get close, but then he has to do lots of experimenting to get the behavior he really wants. The mathematics doesn't give a complete prediction until after you have done enough experiments to give specialized equations for whatever kind of device you are building. For example, the performance of internal combustion engines can be very precisely predicted these days, but they don't use the basic equations of thermodynamics; they use detailed charts and equations derived from building lot of other engines.

4. Well-behaved systesms

In the real world, there are no exact predictions or exact results, only approximations. For example, I want to predict the pressure change in a gas that will derive from a specific temperature change. I can measure the temperature change and predict the pressure change using an equation. The prediction will be an exact mathematical solution, but neither the temperature nor the pressure will be exact measurements. What we are doing is taking an inexact measurement of temperature to product an exact prediction of a pressure that we can only confirm inexactly.

This only works in well-behaved systems, systems where small differences in the predicting parameters correspond with small differences in the predicted parameters, so measurement errors in the input lead to differences in the output that are within the margin of error. Not all systems have this character, and in particular, not many in the social or biological sciences.

Answer 5

I'd contest the view that you can do science without any kind of math. Math includes areas such as logic and statistics. As such, a paper devoid of math would ignore logic and be devoid of any scientifically useful evidence. It would be a mere rambling of words without any possible claim to truth, scientific rigor, or even factuality.

The truth is, that you can only ever put faith in results that rely on mathematics. Without mathematics, you may make guesses, only. And your guesses may be informed by your intuition and your experience, and thus they may have a higher than average chance of being right. But they remain guesses, and they don't become science until you apply some scientific rigor to them. And that means that you have to use math.

Examples of "science" without math are legion, and their failures are well-known. Like the view of earth being flat. Like the sun orbiting earth. Like the element of fire. Etc, pp. Even today, esotheric theories like these are legion, and people not trained in using math to their advantage fall for them. Only the scientific method, which requires applied math, can weed out the beautiful, misleading words and collect the hidden gems of knowledge. Whoever ignores it, ignores it at their own cost.

Answer 6

It's obvious but I think someone should point out that natural sciences largely deals with dead matter while social sciences largely deal with living subjects.

That makes it easier to analyze and experiment with. So if things are static, regularly dynamic, can be broken down into pieces and analyzed and where you can control the interaction between them and other things. It's generally easier to obtain data and observe patterns. And cycle through primitive theories until you get better ones.

But it's not necessarily just the complexity, I mean once you understood simple concepts your models and the things that you study can get arbitrarily complex and it's not that anybody would argue that physics is simple.

It's also about that dead or alive feature of your test "objects". Like you don't need to ask your "samples" for consent, you don't need to consider harm being done to your samples (beyond wasting resources), you don't have to consider that your samples have a will of their own and play tricks on you. If your samples resist your experiment then it's assumed that they had no chance but to do that and that it's a general effect, not the consequence of them being bored and having better things to do.

So you can't do the same intensity of experiments and you quite frankly don't want the same intensity of experiments being done on you or for that matter any living thing. Primitive models can very well end up being brutal torture and wrong assumptions can kill people. It's often not "just an experiment" with relatively low stakes (just money and resources).

Also it's not as easy to reduce the sample to it's components. Like if you took a corpse and apply some charge you might have the same components as a living thing, but it's not the same thing as a living person. It's somewhat working like biology and medicine have identified organs and subsystems that they can explain and if necessary repair or swap out, but it's a much slower process and much more often you're taking the role of the observer rather than the experimenter. And the stakes are much higher, you're not just "destroying a sample" if you fail, you're killing someone. And not just that, humans have complex social relations with each other, so you're also likely disrupting the social environment as well. So it has a much more immediately felt impact on these living things.

Which includes the application of scientific ideas to people. The impact largely doesn't stem from the experiments themselves but from the idea to test them on the larger public. Which also usually has much stricter regulations than experimenting with samples.

The point is that the whole ethical angle of the thing comes into that and I don't mean purely as a road block, but as something that is important to consider but which is largely ignored in the natural sciences.

Answer 7

Forget Mathematics, it is not the problem. You can invent a language/tool (equivalent to Mathematics, likewise, based on Reason and causality) to predict facts: For example, say:

Stupidmathics

            reality: I-see-all
       define-cause: +
 define-consequence: -
      trigger-cause: <<
predict-consequence: >>
---start-reality---
                   + I-close-eyes
                   - I-can't-see
                  << I-close-eyes
>>                I-can't-see
                  << I-close-eyes [NOT]
>>                I-can't-see [NOT]

---stop-reality---

Following the logic of the question, even if this far from real mathematics (notice the tollendo-tollens error in the negatives, which can horribly fail in darkness, but will never fail in the predefined reality) we have proven the unreasonable effectiveness of Stupidmathics predicting facts of reality.

So, what is the problem?

The assumption that Mathematics has an unreasonable effectiveness predicting facts is subjective and arbitrary. Math works in some sciences, doesn't work in others.

1. Mathematics predicts ONLY WHAT OCCURS WITHIN the Mathematical domain.
2. Finding correspondences between mathematical objects and reality is the real problem.

The problem is how to describe reality in the language of Mathematics, and the hermeneutics about what happens in Mathematics regarding reality.

For example, you can find simple correspondences in some disciplines (e.g. free fall), and you will say that Mathematics has an unreasonable effectiveness predicting facts. In other disciplines (e.g. aerodynamics or meteorology), you cannot predict facts with precision, and you will say that Mathematics has an unreasonable ineffectiveness predicting facts.

But it is the same Mathematics!

So, it is not Mathematics that is/is not effective predicting the future: it is how Reason works, which sometimes can be expressed with Mathematics.

Answer 8

Here is a quote by David Enoch, “it really is impossible to deliberate sincerely without believing in irreducibly normative truths”.

I see deliberation at least on par with science as far as indispensability to society goes, and part of science itself.

I also take it that to formalize a discipline, there must be irreducible normative truths. And certain disciplines by their very nature are easier to find foundational truths. The laws of physics hold in every inertial frame, and perhaps no such universality holds for any biology. The truths of biology or morals may be much more multiplicitous and non-general.

Mathematical truths are an ideal form of irreducible normative truths (even if they are abstract they still serve this purpose). But they are perhaps not ideal for every discipline and nor are they they only form of irreducible normative truths.

So we may get away with just saying certain disciplines are harder for us to find irreducible truths, and if needed, math isn’t the only or best irreducible normative truth for some studies. For purposes of sincere deliberation and comparing scientific theories, normative truths are indispensable. But we still have to find them. Unfortunately (or not) this may be a very unsatisfying answer.

Answer 9

The success in mathematics in helping to model the physical is not unreasonable but reasonable. It's what we ought to expect.

This is because both subjects amplify what is necessary. One in numbers and the other in the physical world. The other subject where the necessary is important is logic and its no surprise to find that there are close relationships between all three subjects.

On the other hand, the world of life, and in particular our human world is characterised by contingency, or to use a more simple and direct term - freedom and liberty. Liberty of thought, liberty of action. Hannah Arendt termed this as natality: the birth of new ideas and new action.

Its because of the lack of necessity in the human world and its overflowing with its opposite, contingency in its aspect of freedom, that the quantitative sciences have found to replicate their success in the natural sciences here.

At it's most basic level, Wigner's question can be construed as why the universe should be intelligible. But surely the answer to this question is obvious enough. We are not alien or foreign to this universe, we were born in it, brought up in it, evolved in it. What would it mean for an conscious being to not find this world intelligible? It would mean he or she could not operate in it, that is live in it. This makes no sense. Far from Heideggers supposition that we are 'thrown into the world', we are instead, at home in it. It is our world, our universe.

In many ways, this is merely an outcome of Plato's Naturalism thesis - or was is Sokrates?

Answer 10

Mathematics describes the world in terms of distinct, finite units. However, the world is fluid and frankly never able to be defined perfectly by distinct, finite units. Mathematics is the best way the human mind can comprehend many things. But the concept of 1, 2, and 3 objects means nothing to reality, only to us.

That being said, yes, of course, mathematics can one day be applied to things like biology and economics as effectively as i.e. physics - humans just have more work to do breaking down those more complex systems to fit into pure mathematical definitions. Other answers have already outlined this effectively.

Answer 11

Short Answer

There are some informed answers here. I'm going to try to rephrase drawing on some philosophical terminology, and to put my own spin on it. Off the top of my head, I'm not sure that you can say that mathematics is more "successful" in one scientific field more than another. I think you have to take it on a model-by-model basis given your standards of evaluation of success. Effective is a normative claim.

Long Answer

Philosophically speaking, mathematics is a method of quantifying predictability and describing occurrences and dispositions. The revolution that Galileo helped usher in was connecting mathematics to physics where reasoning about physics mathematically became essential to adding rigor to the natural philosophy of the day. Modern science presumes mathematical methods heavy in deduction to draw conclusions about knowledge, because mathematics is mechanism whereby we can objectify and justify belief as is generally a means of determining justified, true belief, which should be understood as a mental model of the exterior world.

I would argue that mathematics might be more effective in some disciplines in others not because of the discipline, but because of the preference of the phenomena being studied. This is where complexity and emergence comes in. A physicist can study a falling apple, and use mathematical physics to great effect, and claim, see, physics is the quintessence of science! And yet, if you hold that claimant to a simple three-body problem, the physicist with the same tools and methods will fall flat on their face. Can't the same situation be found in economics?

Imagine if a behavioral economist wanted to study supply and demand. Couldn't she delineate simple experiments where people and their purchasing choices would almost always be predictable? I would argue that just as a physicist selects a simple situation that is well modeled by math, an economist could choose to do the same. It's the fact that the physicist chooses a falling apple over a three-body problem that makes their disciplines appear to be more predictable. So, why would economics appear to be less successful mathematically? Because there are different standards of evaluation, that is a difference in psychological normativity when comparing.

A physicist or chemist may spend their whole lives focusing on isolating a small set of variables in a very repeatable context to get a desired outcome, but economist is tasked with making sense of the entire system. In a way, it's like comparing apples to oranges. If economists created theses on the spending habit of the same person over their life, and physicists focused on the probabilistic concerns of masses of quantum particles or ten-body problems, I suspect who "succeeds" mathematically would drastically reverse itself in the eyes of the evaluator.

Answer 12

Very interesting subject and highly philosophical, however mathematics in my opinion is just a description and solution tool, actauallya language, but with descriptive limitations - from the theorem of Hopital to other - these days almost obsolete. If properly fitted to physics, chemistry etc it accelerates experimentation and engineering and the definition of practical frameworks. It is also a very powerful theoretical research tool.

Answer 13

The reason that mathematics works unreasonable well, is that it represents perfect order which is part of the Universe (as defined by religion).

Where it fails is where other dimensions of logic, you could say, intersect and confound the ability for order and reason to prevail.

Such is the case with life and emotions -- there are simply other dimensions impending onto the classical 3d +1 spacetime.

If you find this to "otherwordly", consider that science never had an explanation to the order of space in a disordered universe. In fact, it is an anomaly -- an artifact of evolution (of GOD, you could say).

Also, you won't understand high-dimensional science without an understanding of the nature of GOD. It just doesn't work. This bias of science make it fail the immense possibilities of understanding that would otherwise be available to it.