April 9, 2013 | 17

Ashutosh (Ash) Jogalekar is a chemist interested in the history and philosophy of science. He considers science to be a seamless and all-encompassing part of the human experience. Follow on Twitter Ashutosh (Ash) Jogalekar is a chemist interested in the history and philosophy of science. He considers science to be a seamless and all-encompassing part of the human experience. Follow on Twitter

Contact Ashutosh Jogalekar via email.

Follow Ashutosh Jogalekar on Twitter as @curiouswavefn. Or visit their website.

Follow Ashutosh Jogalekar on Twitter as @curiouswavefn. Or visit their website.

Writing in the Wall Street Journal, biologist E. O. Wilson asks if math is necessary for doing great science. At first glance the question seems rather pointless and the answer trivial; we can easily name dozens of Nobel Prize winners whose work was not mathematical at all. Most top chemists and biomedical researchers have little use for mathematics per se, except in terms of using statistical software or basic calculus. The history of science is filled with scientists like Darwin, Lavoisier and Linnaeus who were poor mathematicians but who revolutionized their fields.

But Wilson seems to be approaching this question from two different perspectives and by and large I agree with both of them. The first perspective is from the point of view of students and the second is from the point of view of research scientists. Wilson contends that many students who want to become scientists are put off when they are told that they need to know mathematics well to become great scientists.

“*During my decades of teaching biology at Harvard, I watched sadly as bright undergraduates turned away from the possibility of a scientific career, fearing that, without strong math skills, they would fail. This mistaken assumption has deprived science of an immeasurable amount of sorely needed talent. It has created a hemorrhage of brain power we need to stanch.”*

I do not know if this is indeed what students feel, but at least on one level it makes sense. While it’s true that chemists and biologists certainly don’t need to know advanced mathematical topics like topology or algebraic geometry to do good science, these days they do need to know how to handle large amounts of data, and that’s a trend that only going to grow by leaps and bounds. Now analyzing large amounts of data does not require advanced mathematics per se – it’s more statistics than mathematics – but one can see how mathematical *thinking* can help one to understand the kinds of tools (things like machine learning and principal component analysis) that are standard parts of modern data analysis. So while Wilson may be right that professors should not discourage students by requiring them to know mathematics, they also should stress the importance of abstract mathematical thinking that’s useful in analyzing data in fields ranging from evolutionary biology to social psychology. You don’t have to *be* a mathematician in order to *think* like a mathematician, and it never hurts these days for any kind of a scientist to take a class in machine learning or statistics.

At the same time Wilson is quite right that true success in science mostly does not come from mathematics. In many fields math is a powerful tool, but only a tool nonetheless; what matters is a physical feel for the systems to which it is applied. As Wilson puts it, *“Far more important throughout the rest of science is the ability to form concepts, during which the researcher conjures images and processes by intuition”*. In Wilson’s own field for instance, you can use all the math you like to calculate rising and ebbing populations of prey and predator, but true insight into the system can only come from broader thinking that utilizes the principles of evolution. In fact biology can claim many scientists like John Maynard Smith, J. B. S. Haldane and W. D. Hamilton who were excellent mathematicians, but the fact remains that these men’s great contributions came from their understanding of the biological systems under consideration rather than the mathematics itself.

In my own field of chemistry, math is employed as the basis of several physics-based algorithms that are used to calculate the structure and properties of molecules. But most chemists like me can largely get away by using these algorithms as black boxes; our insights into problems comes from analyzing the results of the calculations within the unique structure and philosophy of chemistry. Knowledge of mathematics may or may not help us in understanding molecular behavior, but knowledge of chemistry always helps. The use of mathematics in a field like quantum chemistry (which is perhaps closest to math among all chemical fields) also makes the distinction between “using” math and “knowing” it quite clear; I don’t really know the math behind many theoretical calculations on molecules, but I certainly use it on a regular basis in an implicit way.

What’s interesting is that mathematics is not even a game changer in the world of physics, the one field where its application is considered to be essential. The physicist Eugene Wigner did write an essay named “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, but even the greatest theoretical physicists of the twentieth century including Einstein, Fermi, Feynman and Bohr were really known for their physical intuition than for formidable mathematical prowess. Einstein’s strength was to imagine thought experiments, Fermi’s was to do rough back-of-the-envelope calculations. So while mathematics is definitely key to making advances in fields like particle physics, even in those fields what really matters is the ability to imagine physical phenomena and make sense of them. The history of physics presents very few examples – Paul Dirac’s work in quantum mechanics and Hermann Weyl’s work in group theory come to mind – where mathematical beauty and ability alone served to bring about important scientific progress.

This use of mathematics as little more than an elegant tool relates to Wilson’s second point concerning math, this time in the context of collaboration. To me Wilson confirms a quote attributed to Thomas Edison who is purported to have said, “I can hire a mathematician but a mathematician cannot hire me”. Most non-mathematicians can collaborate with a mathematician to firm up their analyses, but without a collaborator in the physical or social sciences mathematicians will have no idea what to do with their equations, no matter how rigorous or elegant they are.

The other thing to keep in mind is that an over-reliance on math can also seriously hinder progress in certain fields and even lead to great financial and personal losses. Finance is a great example; the highly sophisticated models developed by physicists on Wall Street caused more harm than good. In the words of the physicist-turned-financial modeler Emanuel Derman, the modelers suffered from “physics envy”, expecting markets to be as precise as electrons and neutrinos. In one sense I see Wilson’s criticism of mathematics as a criticism of the overly reductionist ethos that some scientists bring to their work. I agree with him that this ethos can often lead one to miss the forest for the trees.

The fact is that fear of math often dissuades students and professionals from embarking on research in fields where data analysis and mathematics-type thinking are useful. Wilson’s essay should assure these scientists that they need not fear math and do not even need to know it too well to become great scientists. All they need to do is to use it when it matters. Or find someone who can. The adage about mathematics being the “handmaiden of the sciences” sounds condescending, but it’s not, and it’s fairly accurate.

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Science is done with collaboration of scientists from different fields. So, you don’t have to have a deep math knowledge to be a great scientist. But, you definitely need to know when and where you need someone else in another field to help you. I think Biophysics is a great example of this. There are physicists involved to fine tune the lasers etc., although they (some of them) know very little or nothing about biology and chemistry. They also deal with the math of the research and come up with equations to help to explain the data obtained.

Platon was not a mathematician but he said “Let none ignorant of geometry enter here.” Erwin Schrodinger did NOT solve his own equation, but his mathematician friend help him to solve it. But, he absolutely knew what that equation meant. He was able to interpret it physically. So, a scientist should either know enough math himself/herself or consult a mathematician if any help needed.

Link to thisI can certainly agree with the non-necessity of advanced math for Wilson’s generation and even decades younger, but heading into the future I suspect math will become increasingly important for success in high-level science work… similar to computer programming/coding — a non-necessity for past generations but increasingly vital for the work of future generations.

Link to thisFaraday did some rather great physics, but could not do math. He referred to himself as “amathematical” and quipped that “the only mathematical operation I ever performed was turning the crank of the calculator.

He learned about electromagnetism via direct experiments on the physical world.

Our heroic theoretical physicists who have given us the barren creations of string theory, supersymmetry, “WIMPs”, anthropic pretzel logic, and multiverse fantasies might learn something from Faraday.

Study nature, not untestable theoretical thought-bubbles.

Robert L. Oldershaw

Link to thishttp://www3.amherst.edu/~rloldershaw

Discrete Scale Relativity/Fractal Cosmology

Let me add the point that some of Einstein’s greatest contributions were not mathematical discoveries. Rather they were conceptual discoveries about how nature actually worked.

Once he had the correct conceptual natural philosophy, then he would subsequently develop the mathematical formalism that gave the concepts the rigor and predictive power by which the conceptual ideas could be further developed and scientifically tested.

That is how real science is done.

As Einstein put it: “Books on physics are full of complicated mathematical formulas, but thought and ideas are the beginning of every physical theory.”

Robert L. Oldershaw

Link to thishttp://www3.amherst.edu/~rloldershaw

Discrete Scale Relativity/Fractal Cosmology

Mathematics is the language of nature. Or, as many suspect, nature is mathematics. I have always thought that the primitive semi-science state of biology, geology, etc. is related to ignorance of math.

Link to thisrloldershaw Faraday did rather primitive science. Not Einstein, but his ideas were not science until he found the mathematical form. And you may be happy to know that WIMPs just seem to have been found.

Link to thisYes – You Ash and Wilson are right to say maths is often just an adjunct.

In fact I’ve seen some (but not all) good mathematicians being a nuisance by trying to insist that nascent theories should be couched in terms of mathematical concepts they are familiar with. This often causes vital subtleties to be squashed into pre-existing forms, when a new form is needed.

An understanding of science is vital – which is why the lack of understanding amongst scientists and journalists is such a serious issue. And it’s why I prepared sciencepolice-14.

However some sciences, for example dinobird palaeontology, are ruined by widespread ignorance of statistics too, as well as ignorance of the nature of science.

Link to thisGreetings Julia,

I think you should inform yourself better regarding the actual results from the AMS-02 experiment you refer to.

A good source of unbiased scientific analysis can be found at

http://resonaances.blogspot.com/

also some discussion at Peter Woit’s http://www.math.columbia.edu/~woit/wordpress/

The truth may surprise you.

RLO

Link to thisFractal Cosmology

Math is a functional tool to perform calculations

Math lends a philosophical approach

Math is a roadmap

Math is a lens from which to observe and

Math is a language that can be used to share ideas

Math isn’t science

Math isn’t necessary to understand science

Math isn’t necessary to draw scientific conclusions

Science requires the application of math, but it does not require that every scientist apply math or that it’s applied to every situation. Perhaps science even ‘enjoys’ a diversity of approaches and ‘benefits’ from math and non-math perspectives.

That’s what I think…..

Link to this“The fact is that fear of math often dissuades students and professionals from embarking on research in fields where data analysis and mathematics-type thinking are useful. Wilson’s essay should assure these scientists that they need not fear math and do not even need to know it too well to become great scientists.”

I don’t think Wilson or Ash has done a sufficient job of defining exactly what they mean by students don’t need to know math too well. In the Wilson quote he is talking about students and their fear of, and complete confusion about, mathematics can occur on the first day of trig class. I have seen students drop out of the first semester of chemistry because of the mathematics requirements. The physics requirements for an undergraduate science degree are a huge stumbling block to many. I think you should be very clear when you say that you don’t need to know math too well when trying to encourage students to embark on a science career. What level of mathematics is required? What level of mathematics can you pass on?

If Wilson is considering students in an introductory biology class then those students can have a peak at the course catalog and see for themselves the mathematics reality. I think it is the reality of three semesters of calculus and a statistics class that puts off many. It is our carppy lack of emphasis on fundamental mathematics in secondary school that puts college freshman at a disadvantage. When we have high school teachers saying that algebra need not be studied by all then we find ourselves with 18 and 19 year olds who consider trig and calculus as advanced mathematics.

Link to thisThere’s Einstein’s “now famous” initial publication of faulty field equations in Oct. 1915, one month before his scheduled presentation of GR to the Prussian Academy of Science in Nov. 1915.

http://en.wikipedia.org/wiki/General_relativity#History

Einstein suddenly realized the equations “were inconsistent with the local conservation of energy-momentum unless the universe had a constant density of mass-energy-momentum. In other words, air, rock and even a vacuum should all have the same density. This inconsistency with observation sent Einstein back to the drawing board. However, the solution was all but obvious, and in November 1915 Einstein published the actual Einstein field equations…”

http://en.wikipedia.org/wiki/History_of_general_relativity#The_development_of_the_Einstein_field_equations

So a less lucky Einstein might have had his theory of general relativity roundly discredited before it was ever seriously considered – we might otherwise now have Hilbert’s theory of general relativity…

This anecdote also demonstrates that Einstein’s theory of gravity was extensively developed using conceptual analysis methods (‘thought experiment’) rather than being derived through careful mathematical analysis… <%)

Link to thisAlso, please see http://blogs.scientificamerican.com/talking-back/2013/04/10/new-study-neuroscience-research-gets-an-f-for-reliability/ regrading applied sciences…

Link to thisWell, Einstein had a little trouble getting started with the mathematical description of general relativity. He had to go to his friend and mathematician Marcel Grossmann for direction and tutoring in some advanced non-Euclidian mathematical ideas. Einstein did not want the mathematician to do the work but Einstein did need a mathematician to provide insight and direction.

Then, when general relativity was in its final stages Einstein was having trouble with the field equations. The great mathematician David Hilbert then began to work on the problem and the controversy continues as to how much help Hilbert provided Einstein.

It should be noted that Einstein had already shown that his basic theory could explain the orbit of Mercury and that, if correct, his theory could be verified by measuring the deflection of light passing close to the Sun. In fact Einstein enlisted the help of astronomer Erwin Freundlich in 1911 and he went off to Crimea to observe the 1914 eclipse. WWI interrupted the measurement.

The theory was all Einstein but he needed some help from mathematicians.

Link to thisAs a mathematician I agree with the main thrust of this argument. Certainly mathematics is a valid field of study unto itself, but the sort of mathematical sophistication that one might apply to topology is not necessary for biology and shouldn’t be imposed as an impediment to research. The math should come before as part of inspiration, as it can open up fruitful directions for those already interested, or later, once some hypothesis needs to be confirmed or refuted.

The problem with our thinking nowadays starts earlier than college. I recall being thoroughly bored by high school math until my first calculus course. We imagine math classes teach problem solving, but they don’t, they teach only algebraic manipulation. The constant question of all but the most dutiful of high school students, ‘what am i ever going to use this for?’ is still not being answered in high school courses, and as long as it’s not, the vast majority of college students who graduate from U.S. high schools will continue believing that math is just a game of manipulating symbols.

We should be teaching problem solving and critical thinking in all fields of endeavor rather than the mere use of tools. If we present people with problems to be solved in educational contexts, the use of math will be natural when it is necessary or desirable.

The way we teach math now is a bit like teaching someone who has never sailed to trim a sail or never built to hammer a nail, all theory without any purpose. such things can’t be assimilated.

Link to thisAbraham Pais goes through the evolution of General Relativity in detail in his book ‘Subtle is the lord’, from conceptual beginnings to the full mathematical physics, and he identifies the many people who helped with the mathematical intricacies of the theory.

To me, the key point is that Einstein knew the correct conceptual path well before he went to his mathematician friend Marcel Grossmann and asked him to find a mathematics (Riemannian geometry) that would be well suited to expressing his conceptual vision for how nature worked.

Link to this“an over-reliance on math can also seriously hinder progress in certain fields and even lead to great financial and personal losses. ” I fully aggree from where I am sitting, which is as a mathematician working with biology researchers. There is a lot of what I like to call sub-prime science done these days, over-reliant on sophisticated mathematical models that are poorly understood and highly error prone, leading to lots of bad science. Smaller emphasis on the maths and more emphasis on a falsifiable prediction would be great.

Link to thisWhen I was a child I used to be good at math, but then calculus become kind of difficult for me; however, I like math.

And I’m think I have a good example of have you can success making a good combination of minds.

The last year I finished high school, and my last semester was interesting for me, I had a project certificated for a science fair in Brazil, and with some friends we could pass the filters for an important event called IGARSS.

So at high school I had a good background of how to do a research ans passions for science, and my friend is very good at math, so we achieve have a publications at the IEEE, while being high school students.

Maybe most of you are successful professional, and this for you is kind simple, but for us it’s really cool.

Now I’m studying civil engineering for many reason, and one of them is because I’m not pretty good at math, and I want to get over it.

The last semester my math grade was awful, but I’m keep working on that.

I have 19 years old and not so much experience as all of you, but what I can tell you based on what I lived, is that you don’t need so much math for do science; however, when you dominate it you have another amazing vision of your world.

**And sorry If some of my ideas are not correctly written, English is not my native language.

Link to this