Does the desire for elegant equations bias scientific research?

[CTRandall]

Here's a question for all of you scientists and mathematicians out there. Given that there is physical evidence that scientists and mathematicians derive aesthetic pleasure from some equations (Euler's identity e^iπ+1=0 often making the top of the list), is it possible that maths-heavy research is affected by a sense of aesthetics? In other words, might researchers be motivated to pursue a path they perceive as 'elegant' over a more messy option? And might this direct research away from potentially important, yet inelegant, findings?

I suspect the gut instinct of most scientists and mathematicians would be, 'Don't be silly. We go where the evidence/maths takes us, elegant or otherwise.' I suspect also that, much of the time, this is true. But do you think that a particularly alluring equation might lure you in, even for just a short time? That you might be tempted to turn first to the problem that looks pretty? That you might spend more time wooing it, hoping in vain to find the key to its chest of treasure, and only turn to its ugly stepsister/brother after all of your flirting and cajoling came to naught?

In other words, at the very least could the desire to find an 'elegant' solution delay scientific progress because, in the absence of other clear evidence, people make choices about research based on aesthetics?

Finally, is anyone aware of any research on how the aesthetics of equations might affect research? I couldn't find anything on JSTOR (though I only did a fairly cursory search).

 

[hitmouse]

Elegance in this context really applies to the process of solving a problem: clever, novel, efficient steps in reaching a robust conclusion in mathematics, or similar when proving a hypothesis through practical science.

the solution as in the example you give, can be beautiful, interesting, and pleasing, but I would argue that it is not in itself elegant.

so: does this lead to bias? The Euler equation, e=mc^2, and suchlike express basic truths, so are not biased.

Scientific method can be misguided , inefficient, or wrong, and rarely is it as easy to judge as pure maths. Elegance is a term more often applied retrospectively when looking at how a conclusion was reached.

 

[Brian G Turner]

Not so much equations, but it really disturbs me how many current theories in science are based on statistics and models, both of which are subject to bias and IMO are little better than "we guessed stuff on a computer".

 

[hitmouse]

In defence of stats and models:

If used badly, statistics are rubbish, pretty much like anything in life. Statistical tools are however fundamental in science, particularly in medicine and biological sciences where they are needed to understand problems and uncertainties around sampling. We would not have any decent medical research ( e.g covid vaccines) or epidemiology (e.g. modelling the pandemic) without stats.

Models are an essential part of hypothesis generation, and as such are central to scientific method, which largely exists to try to disprove them.

 

[Dave]

I think that the natural world is naturally described by mathematics - the reflective and rotational symmetry, Fractals, the Fibonacci series spirals, concentric circles and ellipses, interlocked hexagons - so I don't think it is a surprise really that it's often beautiful equations that describe natural systems more adequately than words can.

I agree with @hitmouse about statistics and models. Statistics are heavily misused, deliberately cherry-picked to fit desired arguments, which is NOT the fault of the statistics. The point I'd make about predictive models is that they are wrong from the moment they are made, because people react to them and change their behaviour accordingly, if they want to moderate the outcome. They are always would could/would/might happen rather than what will/must happen.

 

[Biskit]

In other words, might researchers be motivated to pursue a path they perceive as 'elegant' over a more messy option?

From my own experience, 'elegant' is the last step, after you've figured out the answer, putting a polish on the solution that hides the mathematical equivalent of the pile of wood shavings, metal turnings and five failed prototypes in the corner of the workshop. Thirty-plus years ago, I had something that boiled down to an equation that looked something like y = I(a)/I(b), really nice and simple, although probably not really out into elegant. When I wrote it up, I needed a couple of pages to get down to that (along with an assumption that I would probably question these days), and what it hides is the complicated system of integrals represented by I(). The reason for boiling it down to something so simple was to make the point of the relationship between y, a and b. To actually make use of it required all the dirty details of I() and some numerical modelling.

the solution as in the example you give, can be beautiful, interesting, and pleasing, but I would argue that it is not in itself elegant.

I think it depends on what you mean by elegance. Visually, Euler's identity doesn't have much going for it so far as I am concerned, but the fact that it sits there, eight characters in the representation given (but only seven if you don't need the '^' to indicate the exponent) and links four fundamental things = e, i, pi and 1. That connection on it's own is a bit of a wow, but to be able to express that in such an apparently simple equation probably deserves to be called elegant.


Adding to @hitmouse's point, statistics and models are often the only way to deal with hugely complex problems. The first one that springs to mind is the nineteenth century work of Ludwig Boltzmann developing statistical mechanics to describe macro thermodynamics using a statistical description of microscopic processes. My second experience comes from just before I switched from physics to financial software, when I worked in a group where Monte Carlo simulations were an essential tool for modelling processes involving large numbers of moving particles.

Statistics and modelling of random processes can be an invaluable tool, with the caveats that you have to use the right tool and in the right way.

The danger of statistics is putting them in the hands of politicians, PR consultants and, worst of all, the general public, who (on average ) have no idea what they mean or how to interpret them. During one of my first courses at uni, the lecturer illustrated basic statistical hypothesis testing using polling data for the local elections at the time. Based on the information available, and with careful choice of hypotheses, he was able to show that the polls clearly indicated that every party was going to win.

Statistics are a tool a bit like a hammer. Once upon a time, I was quite skilled with a hammer. I could put 3-4 inch nails in with two taps - one to get the tip in with the shaft straight, and one rather harder to drive the nail in. However, if you don't care too much about a good outcome, a hammer can put screws in, fit the square peg in the round hole and trim your finger nails.

 

[CTRandall]

The Euler equation, e=mc^2, and suchlike express basic truths, so are not biased.

I am not trying to suggest that there is anything wrong with the equations themselves and I take your point about the potential for poor practice on the part of researchers as being more significant than any possible influence from aesthetics. @Biskit mentioned making an assumption that he would question today (thanks for being open about that!). For most people, that is a remarkably easy thing to miss and certainly has a larger impact on the quality of research than my little question here.

The reason for boiling it down to something so simple was to make the point of the relationship between y, a and b. To actually make use of it required all the dirty details of I() and some numerical modelling.

In this case, at least, it sounds like you used the final equation as a kind of rhetorical device, a means of making a specific point. That strikes me as point where aesthetic judgment might come into play. That isn't a criticism, per se, as an ability to frame an equation effectively--even beautifully--could help make its purpose or usefulness more clear. In that sense, an aesthetic sensitivity might help communication between researchers.

 

[Venusian Broon]

Here's a question for all of you scientists and mathematicians out there. Given that there is physical evidence that scientists and mathematicians derive aesthetic pleasure from some equations (Euler's identity e^iπ+1=0 often making the top of the list), is it possible that maths-heavy research is affected by a sense of aesthetics? In other words, might researchers be motivated to pursue a path they perceive as 'elegant' over a more messy option? And might this direct research away from potentially important, yet inelegant, findings?

To my mind, the first level of 'beauty' in physical theories is quite simple. The more of the physical world you can explain with fewer axioms, the more elegant the theory is. This view is, I believe, uncontroversial among scientists, but says little about the actual mathematics. I can explain the quite simple elements that make up the axioms of Relativity, but trying to explain to a lay person how the end product, a differential equation using tensors actually works in practice...

Of course a proper physicist is constrained by the evidence of experiments and observations. A theory may have one axiom and be exquisitely simple and elegant...yet explain bugger all.

As for 'messy equations'.... well, for example, we cannot solve exactly the dynamical equations (whether Newtonian, with General Relativity or QM) for systems with three or more bodies. Using our equations they extremely quickly become intractable and a mess when applied to the real world.

I suspect the gut instinct of most scientists and mathematicians would be, 'Don't be silly. We go where the evidence/maths takes us, elegant or otherwise.'

Mixing these two is a bit wrong in my book - physicists need proof of their theories through real world data through experiments, mathematicians are not constrained by this.

I suspect also that, much of the time, this is true. But do you think that a particularly alluring equation might lure you in, even for just a short time? That you might be tempted to turn first to the problem that looks pretty? That you might spend more time wooing it, hoping in vain to find the key to its chest of treasure, and only turn to its ugly stepsister/brother after all of your flirting and cajoling came to naught?

I don't see it this way. A physicist builds a theory (perhaps based on an observation), makes a prediction, finds or observes more data...and if it doesn't fit, then clearly something is wrong, in terms of their theory. A mathematican is looking to prove a theorem (usually) - and it either does or doesn't!

In other words, at the very least could the desire to find an 'elegant' solution delay scientific progress because, in the absence of other clear evidence, people make choices about research based on aesthetics?

I kinda reject the premise of what you've written here, namely the 'absense of other clear evidence'. In that case surely any theorising, no matter how elegant or how messy, gets you nowhere.

The problem is our ability to get the 'right' evidence. New evidence.

Currently General Relativity and Quantum mechanics both explain a great deal of our universe. But we know both are incomplete and both have holes. So we know that we should be trying to puzzle out some sort of Quantum Gravity theory that somehow encompasses both (at least that's our assumption). The issue and problem we have is that to build a theory like this we need to experiment with energies and masses that are way beyond our current means. In essence we really need to be experimenting next to real black holes . (Or coming up with very, very clever low-energy experiments or hopefully stumble across some observational anomaly that is big enough to show that the existing theories just can't 'cover' the anomaly!)

Unfortunately both established theories have been incredibly successful, it makes it much, much harder to find gaps and anomalies.

I could talk about some examples - String theory or Dark Matter/MOND and all the various issues they represent and have - but I'll leave it there for the moment!

Elegance in this context really applies to the process of solving a problem: clever, novel, efficient steps in reaching a robust conclusion in mathematics, or similar when proving a hypothesis through practical science.

No, this is a very narrow definition of elegance in mathematics or physics. A great many theorems or theories can have quite horrendous levels of calculation after starting with some simple axioms, but will end up with extremely elegant solutions.

the solution as in the example you give, can be beautiful, interesting, and pleasing, but I would argue that it is not in itself elegant.

Balderdash. It's not a solution, it's an Identity. And if you don't think the connection of five of mathematics most fundamental constants is not elegant, then you must live in another universe to me.

Not so much equations, but it really disturbs me how many current theories in science are based on statistics and models, both of which are subject to bias and IMO are little better than "we guessed stuff on a computer".

All theories in science are based on models. And many use statistics which are required to order the data and evidence to see if the model and it's assumptions are justified, surely??? Could you do it another way? We're also squirty, smelly humans - bias is something we are likely to have with us for ever

 

[Wayne Mack]

I argue that elegance is the root of science. Science is about taking the complex and finding the commonalities. There is a beauty in a pattern that provides a sense of order in one's mind. The challenge in science is not in finding elegant explanations, but in the hesitancy to abandon one elegant explanation for another that is perhaps better.

 

[Wayne Mack]

Not so much equations, but it really disturbs me how many current theories in science are based on statistics and models, both of which are subject to bias and IMO are little better than "we guessed stuff on a computer".

Though most of the calculations used in statistics can be easily replicated by anyone with a spreadsheet on a computer, the sampling theory behind the equations is rarely understood. This is the failing behind many, if not most, opinion polls that are published in the news. The people sampled are simply not representative of the population as a whole and thus the results are skewed.

The one area that gives me pause concerning statistics is handling of highly infrequent events. This may be my lack of knowledge, but I have an underlying question as to whether infrequent events can achieve a level of statistical confidence. This may be offset by experiment set up to isolate the infrequent events in order to increase their likelihood.

 

[hitmouse]

Though most of the calculations used in statistics can be easily replicated by anyone with a spreadsheet on a computer, the sampling theory behind the equations is rarely understood. This is the failing behind many, if not most, opinion polls that are published in the news. The people sampled are simply not representative of the population as a whole and thus the results are skewed.

The one area that gives me pause concerning statistics is handling of highly infrequent events. This may be my lack of knowledge, but I have an underlying question as to whether infrequent events can achieve a level of statistical confidence. This may be offset by experiment set up to isolate the infrequent events in order to increase their likelihood.

It depends what you mean by an infrequent event. If it is something very distinctive and unique compared to a background population, then statistical confidence may not be a problem. It is more of an issue in practice when trying to understand with confidence whether there is a difference in events between an experimental and a control group. You are correct that trials/ experiments are generally set up to optimise this so as to reduce the sample size required as far as possible.

 

[hitmouse]

Balderdash. It's not a solution, it's an Identity. And if you don't think the connection of five of mathematics most fundamental constants is not elegant, then you must live in another universe to me.

I suspect we are splitting hairs here. Euler was the first time I ever saw anything in maths ( age about 14) that produced a real sense of awe and wonder.

 

[CTRandall]

I kinda reject the premise of what you've written here, namely the 'absense of other clear evidence'. In that case surely any theorising, no matter how elegant or how messy, gets you nowhere.

The problem is our ability to get the 'right' evidence. New evidence.

Fair point!

 

[Wayne Mack]

It depends what you mean by an infrequent event. If it is something very distinctive and unique compared to a background population, then statistical confidence may not be a problem.

The challenge is that if an event is not easily repeatable, it may be categorized as a random occurrence or due to error.

Unless something can be shown to be statistically meaningful, any thing 'distinct and unique' will be viewed as an anomaly.