Scientists struggle with mathematical details

Jun 25, 2012

(Phys.org) -- Scientists would like to believe that the popularity of new theories depends entirely on their scientific value, in terms of novelty, importance and technical correctness. But the Bristol study, published in the Proceedings of the National Academy of Sciences USA, shows that scientists pay less attention to theories that are crammed with mathematical details.

Dr Tim Fawcett and Dr Andrew Higginson, researchers in Bristol's School of Biological Sciences, found that scientific articles presenting many equations on each page are seldom referred to by other scientists. The most maths-heavy articles are referenced 50 per cent less often than those with little or no maths.

Many scientists, including the celebrated Stephen Hawking, have worried about how mathematics will affect the impact of their work, but the Bristol study is the first to measure the extent of the problem.

Dr Fawcett said: "This is an important issue because nearly all areas of science rely on close links between and experimental work. If new theories are presented in a way that is off-putting to other scientists, then no one will perform the crucial experiments needed to test those theories. This presents a barrier to scientific progress."

So is there any way to overcome the communication barrier between theory and experiments? One long-term remedy would be to improve the mathematical training of science graduates.

But there could be more immediate solutions, as Dr Higginson explained: "Scientists need to think more carefully about how they present the mathematical details of their work. The ideal solution is not to hide the maths away, but to add more explanatory text to take the reader carefully through the assumptions and implications of the theory."

But the authors of the study fear that this approach will be resisted by some scientific journals, for which page space is at a premium.

"The top journals want articles to be extremely concise, with many of the technical details going in an online appendix," said Dr Fawcett.

"Fortunately, our study suggests that equations in an appendix have no effect on citation rates. So moving some of the equations to an appendix may be the most pragmatic solution."

Explore further: Hyperbolic homogeneous polynomials, oh my!

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User comments : 21

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A2G
1.5 / 5 (15) Jun 25, 2012
Those who are hung up on math details are the ones who are truly the problem and not the other way around. I do not mean to imply that math is not important, of course it is. But it is more important that something works than that the math for it can be explained.

For instance, magnets work. They both repel and attract beyond a doubt. Now explain that mathematically for certain. The important thing is that you can show easily that the system actually works in the real world and not just in math.

The math guys sometimes act like the math is more important than things actually being proven to work.

You can fly on the first flight of a mathematically new concept of a flying machine. I'll stick to the one that has been proven to actually fly safely time after time.
foofighter
4.5 / 5 (15) Jun 25, 2012
Those who are hung up on math details are the ones who are truly the problem and not the other way around. I do not mean to imply that math is not important, of course it is. But it is more important that something works than that the math for it can be explained.

For instance, magnets work. They both repel and attract beyond a doubt. Now explain that mathematically for certain. The important thing is that you can show easily that the system actually works in the real world and not just in math.

The math guys sometimes act like the math is more important than things actually being proven to work.

You can fly on the first flight of a mathematically new concept of a flying machine. I'll stick to the one that has been proven to actually fly safely time after time.


Spoken like a true life scientist.
Vendicar_Decarian
4 / 5 (9) Jun 25, 2012
A2G confuses many things. In this case he confuses playing around with Science.
Parsec
5 / 5 (7) Jun 25, 2012
The question presented here in the comments is an important one. Is a demo of work-ability more important that a mathematical explanation? It depends. If you are trying to both explain why something works with the potential towards predicting the behavior of systems that may seem unrelated but are not, the math is vastly more important. In fact it is the only thing that's important.

If your trying to make a machine that crushes apples, a cup of cold cider suits.
dunning_kruger
3.3 / 5 (8) Jun 25, 2012
Those who are hung up on math details are the ones who are truly the problem and not the other way around. I do not mean to imply that math is not important, of course it is. But it is more important that something works than that the math for it can be explained.

[deleted for brevity]

You can fly on the first flight of a mathematically new concept of a flying machine. I'll stick to the one that has been proven to actually fly safely time after time.


Spoken like an engineer whose comments have little to do with science.
julianpenrod
2.9 / 5 (7) Jun 25, 2012
The essence of hard science theory is mathematical legitimacy. And it is considered, or was considered, essential for someone to be a legitimate theoretical, and even applied, physicist to be able to make sense of the mathematics behind a statement. It's eminently sinmple to write doggerel claiming even an invalid theory works; supposedly the mathematics worked or didn't, depending on the validity of the theory, and it was that which established its truthfulness! For example, simple mechanical interactions that may follow a non-linear formula take on a linear form over a small enough range! If, "it works" is the only criterion you're looking at, you can say they follow a linear realtionship, focusing on just a small range of parameters and be accepted! Epicycles "works", too. Just another demonstration of the fact that what calls itself science today, is not.
antialias_physorg
4 / 5 (6) Jun 25, 2012
A rigorous mathematical framework is the backbone of any useful theory. That said: mathematics is a huge field. And unless you are working on just the same niche it is unlikely that you have all the tools 'at hand'(so to speak) to fully understand highly specialized papers. With the high numbers of papers being published in any field, time per paper is limited (and don't forget: you also have your own work to contend with).

So I agree with the gist of the article: Papers should be allowed to be a bit more elaborate in explaining their methodology. With publishing being done more and more online (where paper space isn't much of an issue) that could even become a reality without much fuss.

Terriva
1.6 / 5 (8) Jun 25, 2012
The contemporary science is simply over-flooded with formal abstract studies without connection to real world. As everything in math, it may be or may not be true - so they're essentially a random noise. They're serving merely as a jobs and grant generators.

From this point of view it's not accidental, the common illustrations of modern physical theories (like string theory) are mostly quite schematic and pathetic. Such drawings illustrate nothing, but the fact, their authors have no true physical insight into real situation - so they cannot imagine/picture even their own models. This is pity in my opinion, because many hyperdimensional models are actually quite interesting both geometrically, both esthetically. But you can never see it in scientific publications, only the equations.
antialias_physorg
4.3 / 5 (12) Jun 25, 2012
They're serving merely as a jobs and grant generators.

You keep rehashing this nonsense, but it just isn't true.
Papers don't generate jobs (or grants). They are something you do while working (i.e. unpaid overtime in most cases).

It is expected of a researchers to let others know what results (s)he generates. The times when you went into a secluded room and came out with a great result 5 years later are gone since Newton's day. Publishing (and reading stuff( is the lifeblood of science. Only by being exposed to what others think do you sometimes come up with the best ideas. Science today is interdisciplinary and without publishing that wouldn't be possible at all.
Terriva
1 / 5 (2) Jun 25, 2012
Papers don't generate jobs (or grants). They are something you do while working (i.e. unpaid overtime in most cases).
Without publications you cannot get any grant money and the publications are actually the main output of scientific research. The research without publication of results is not serious work but a toying and waste of resources invested into research.
ERW
not rated yet Jun 25, 2012
At this point, the technical journals have no excuse for forcing conciseness in lieu of clear explanations. On-line papers remove the constraints that may have existed in paper based publications. Clear explanations benefit everyone, and will deter snow jobs where unexplained mathematical details are not easily checked.
gralp
3.7 / 5 (3) Jun 25, 2012
Actually, it less the authors, but more the readers to be blamed for the poor citation rate of mathematically oriented articles. Readers are impatient to just skim the general idea, and move on to a next paper, then a next, and to yet another. It is enough to read a forum discussion following a publication of a "breakthrough" article, to quickly realize that most of the participants stopped reading at the headlines, some went through the abstract, handful reached the conclusions, and out of those only couple understood the equations. Shouldn't that be called a blatant laziness?

Publications avoiding math, avoid rigor, and as such often qualify as mere hand-waving (despite hyperventilating media). Excising the equations to an appendix, as many "respectable journals" recommend, is not making the exposition more understandable, but instead denies importance of mathematical reason. Just how many readers trouble to check the supplementary information published online?
Silverhill
5 / 5 (4) Jun 25, 2012
@Terriva
Without publications you cannot get any grant money and the publications are actually the main output of scientific research.
There is a lot of information out there that needs reporting, whether it be positive or negative. (Negative results help others steer clear of fruitless areas of inquiry, as you should know. But which areas are fruitless cannot be discovered without research -- and publication.)

The research without publication of results is not serious work but a toying and waste of resources invested into research.
Nonsense. Researchers will not publish unless there is enough information to be worth reporting. Doing otherwise -- continually presenting micro-amounts of material -- would just be a waste of resources invested into research.
Meyer
3 / 5 (2) Jun 26, 2012
My perspective from programming:
I sometimes need to translate someone's mathematical description of an algorithm into source code, and I often have to work with code written by mathematicians. I can always spot a mathematician's code when I see variables named x, yy, yy2, phi, gamma_k, etc. Seemingly obtuse code can almost always be translated into something quite intuitive with good variable names (taken straight from the paper, but not used in its own equations!) and a bit of restructuring.

If this study reveals a real problem, I would look at the notation itself, not the supporting text. In programming circles, rigor is still a priority, but much emphasis is also placed on readability and maintainability, while math notation seems most concerned with cramming as much as possible into a small space - a valuable goal in 1670 when paper was expensive, but maybe it's time for a modernized "language of math".
axemaster
not rated yet Jun 26, 2012
The reason for this is pretty simple. When I am looking for information in papers, I don't want to slog through pages of math only to find out it wasn't relevant to my needs. So I tend to look for papers that explain in clear terms what they're doing. That probably results in me using less math heavy papers as references.

Also, in my experience, math heavy papers very often are math heavy because the writer doesn't know how to properly explain himself. A good writer can make things very clear in words, and just use a few equations/derivations to prove his point.
TkClick
1 / 5 (2) Jun 26, 2012
Dense aether model makes rather clear the situation of contemporary physics, the evolution of which resembles the spreading of ripples at the water surface during splash. At first these waves are chaotic, but they soon organize itself into form of less or more regular circles. This illustrates the evolution of physics of the last century, which can be characterized with relative success of low-dimensional deterministic models. But as the spreading of ripples continues, it becomes hyper-dimensional and chaotic again due the scattering of ripples in the (hidden dimension of) underwater and the low-dimensional strictly deterministic models aren't sufficient for their description. They're poorly conditioned, prone to errors and misunderstanding and they tend to fragment the human understanding. I'm not sure, whether the mainstream physicists realize it, because they're forming an overgrown community, for which it's advantageous to maintain as many theories and models as possible.
antialias_physorg
5 / 5 (3) Jun 26, 2012
Without publications you cannot get any grant money

No. You apply for a grant by writing an application. In this application it is advantageous (though not required) to show a section which is called 'previous work'.
Having publications that show that you have already woked on the subject is, of course, a good way to show that you are qualified to work on the subject the grant aimed at. But you publish not for future grants but because of present results.

Actually, it less the authors, but more the readers to be blamed for the poor citation rate of mathematically oriented articles.

I think it has more to do with WHY you cite a paper. You cite a paper if the (test)results of the paper corroborate your statement. Highly mathematical papers tend to be about theoretical frameworks. As such they are more often the basis for entirely new sets of tests rather than corroboration for test/method/algorithm developments under certain conditions (which are the majority of papers).
antialias_physorg
4 / 5 (2) Jun 26, 2012
some went through the abstract, handful reached the conclusions, and out of those only couple understood the equations

Actually scientists read papers by other scientists a different way.

First you read the headline and the abstract. Then you skip to the back for the conclusions, results and the discussion/limitations section

If the above was of interest and relevant to your field (and the limitations didn't disqualify it for use in your own work) THEN you read the methods section. If it's so relevant that you want to build on it you're motivated enough to slog through the math.

Seemingly obtuse code can almost always be translated into something quite intuitive

Each discipline has their own lingo. For a mathematician those terms would be clear (and putting them into 'intuitive' language would obfuscate the code). Papers in a field are written for others working in that field - not as PR pieces for laymen. We forget that all to often.
Jotaf
not rated yet Jun 26, 2012
Actually, when "skimming" a paper (to see if it's worth reading or not), besides reading the abstract, one usually stops to look at figures and tables. Intuitive figures with concise captions are a very important factor in deciding relevance.

There's a reason for the current state of affairs in presentation of mathematical proofs. You can't describe everything in words because every paper would then be several dozen pages long. It would be a great read for a novice, but terrible for an expert, for whom it would be a long sequence of self-evident statements. Short and terse papers are the other extreme. Researchers always try to strike a balance between the too, often pointing to good introductory material for newcomers.

As a rule of thumb, interesting math goes in the paper, and I leave the more boring proofs that have lengthy algebraic manipulation to the appendix.

I prefer the appendix to be at the end of the paper, and not downloaded separately, so it's self-contained.
ziphead
1 / 5 (1) Jun 26, 2012
So what's the consensus here?
Do we need smarter scientists or dumbed-down science?
Mike_Massen
1 / 5 (3) Jun 30, 2012
We need maturity and in so many overlapping technical and social arenas.

We also need to cast off the dim-witted deterministic ideals and instead embrace a fundamentally dynamic probabilistic paradigm and appreciate that above all, details matter and dogma and craving for (static) certainties is as prevalent in any evidentiary context as it is in any religion and just as damaging long term to understanding.

Honest self-observation and managing emotional attachments should also be a key aspect of training (early) in all branches of Scientific endeavour, from where I sit, there are significant changes coming in diverse chaotic circumstances with few ready to position themselves considerately.

Observed adherence to any sort of dogma (whether it has appearance of political influence or not) should be seen with concern, psychological understanding and if there is no progress, outright suspicion...

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