### On Mathematics and Science

In the preceding post Eli pointed en passant to an article by EO Wilson where he let the cat out of the bag

For many young people who aspire to be scientists, the great bugbear is mathematics. Without advanced math, how can you do serious work in the sciences? Well, I have a professional secret to share: Many of the most successful scientists in the world today are mathematically no more than semi-literate.and described his own experience

I speak as an authority on this subject because I myself am an extreme case. Having spent my precollege years in relatively poor Southern schools, I didn't take algebra until my freshman year at the University of Alabama. I finally got around to calculus as a 32-year-old tenured professor at Harvard, where I sat uncomfortably in classes with undergraduate students only a bit more than half my age. A couple of them were students in a course on evolutionary biology I was teaching. I swallowed my pride and learned calculus.This, of course, brought forth a number of bleats. Eli would, if he were being kind, call these the cries of the inexperienced, if not, those of the clueless.

Fortunately, exceptional mathematical fluency is required in only a few disciplines, such as particle physics, astrophysics and information theory. Far more important throughout the rest of science is the ability to form concepts, during which the researcher conjures images and processes by intuition.

Wilson is making an important point that everyone is missing, that
it is more important to be able to formulate the problem and from the
formulation conceptualize the broad outline of the answer before doing the math.

At that point you can go out and learn the math, or learn who the mathematician is who can help you or buy Mathematica or hire a mathturbator. If you think about this in physics story terms, this is pretty much Einstein. His real strength was the ability (and the guts) to par the problem down to its basics not his skill as a mathemagician. Physicists worship elegance not algebra.

At that point you can go out and learn the math, or learn who the mathematician is who can help you or buy Mathematica or hire a mathturbator. If you think about this in physics story terms, this is pretty much Einstein. His real strength was the ability (and the guts) to par the problem down to its basics not his skill as a mathemagician. Physicists worship elegance not algebra.

## 17 comments:

I think EO is right here, but maybe that is because I am in that boat- not that I didn't have math through two years of calculus, it is that I don't really remember any of it, and in my work I think genetically in terms of pathways and function. Math is incredibly powerful, and can mean a lot to our biological understanding, but there have been massive contributions to biological science in the absence of math. Maybe this is suboptimal, but things get done and discoveries are made, and the right questions get asked.

Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.Albert Einstein

Eli's spelling is once again sub-par.

I think that EO Wilson's comments, and the counter piece at Slate are, paraphrasing the Three Bears story, "too simple" and "too harsh" repectively.

A good maths understanding is certainly a great asset in Physics and Engineering. I'm sure others will chip in on other diciplines.

However, and more fundamentally, what is needed is a good maths education, and a good "evangalizing" instinct in the maths community.

With the former, my maths ability was stunted through ineptitude and dogma at my primary school: jumped forward a year with no covering material provided - jumped into a class where New Math was seen as vital (in the mid 70's!). The latter, maths evangalization, gets hung up to dry by the fact that most proponents cannot or will not abandon terminology and methodology that render maths a frustrating attempt to get past the first paragraph for the unfamiliar. This is also the biggest problem of maths education: primary and secondary education should not rely on students reaching the "a-ha!" moment on their own as a basic precept - but (for me) it seems that maths does.

Do not worry about Eli's difficulties in mathematics. Greater ones were presented by the Monmouth Park parimutual machines that had Einstein and Godel stumped.

Anon

eMichael Faraday was another physics genius and brilliant experimentalist whose mathematics was weak.

Faraday thought up the idea of a magnetic field because he did not have the math to theorise about his experimental results. Others who came after supplied that.

I loved maths at school. I also revelled in statistics.

Unfortunately the jargon and notations have all changed since those days (pre 1970s). My work and higher ed studies only required basic arithmetic and perhaps a bit of algebra/geometry and rudimentary calculus - so maths skills languished and my enthusiasm for numbers took a back seat.

Later this year I'm doing a stats course to try to catch up on that side of things.

In the agriculture dept where I worked for a bit there was a resident biometrician who had to "pass" all the field trials etc before they were allowed to go ahead. Not sure what happened after he retired. These days they employ some people with very fancy skills in informatics for complex systems analysis.

In my mind, math is an incredibly precise language equivalent. Its value is to find when an idea does not match the information about it. Just as one does not create a short story by flinging around sentences first, one must first know what the question is.

Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.Albert Einstein

Einstein was a bit modest (to say the least), given that he actually beat one of history's great mathematicians (David Hilbert) to the punch on the field equations of general relativity.

Einstein had been corresponding with Hilbert about GR and both were madly working on the equations.

Though Hilbert also came up with the equations shortly after Einstein, he gave Einstein full credit on the theory, since all the physical ideas were his alone.

Most pertinent to the topic of this post, Einstein actually pursued two approaches to the field equations of GR -- one primarily physical and the other purely mathematical. We know this because he actually had two separate notebooks.

In the end, the mathematical approach was triumphant.

So, though it was physical ideas that were at the root of his theory, it was ultimately mathematics that expressed it.

The lesson in that case seems to be that one actually needs BOTH the concepts and the math and that neither is alone sufficient.

It's true enough that math is more important to some sciences (like physics) than others, but it does seem a bit odd that someone like Wilson would be saying anything other than "learn as much math as you can because you never know when you might need it."

~@:>

The history of planetary orbits and the discovery of Neptune is useful example as well, there are several good books written about it. They got lost in a 'blizzard of algebra' but finally almost serendipitously pulled a large planet out of the mess. Without the solid anchor of physics and reality mathematics is very messy and almost pointless.

Mathematical training is important not simply to learn how to "do math", but in my view more critically to learn to think logically and precisely, and especially to learn how to tackle complex problems through a wide variety of techniques.

Fundamental notions about equivalence, proof, representation, models, etc. can be learned well even with just a good course in geometry; they are hard to learn well without doing any math at all though. I think those skills are fundamentally important to scientists in general. But knowing how to integrate fractional-order Bessel functions, probably not so critical...

My Physics career died after first year undergraduate. It is true I struggled with the math, but I probably could have got past that. What really killed, was my utter inability to get my head around the non absolute nature of time.

Rabid Doomsaying little mouse.

Rabid,

You might want to watch this

It might actually help.

You can find more of Julian Barbour's ideas about time (or the lack thereof) at Platonia.com

I don't know enough about this stuff to have a feel for whether he is right, but his ideas about time sure are interesting.

And his ideas don't seem any more outlandish than some others that the physics community takes seriously -- eg, string theory, many worlds quantum theory, neither of which is supported by ANY experimental evidence (not a single shred)

I understand that people have trouble with math. I sort of have the opposite problem--I have a very hard time understanding things unless I can reduce them to a mathematical model.

Many is the time, I have been discussing a matter with colleagues. I will come up with a model and say, "There," only to be met with blank stares. Try as I might, I cannot understand how you can understand a phenomenon unless you can see how the variables play with each other.

Rabid,

Here's a talk that Barbour gave at the Perimeter Institute that puts his ideas into historical context.

Barbour has been working on his ideas for a long time (ha!) but I suspect that many physicists are not even aware of them.

~@:>

Math is a language for describing what is, and is not, reality. Correct formulation (modeling) of things is the purpose of good math. Good modeling describes describes something with useful accuracy, precision, and correctness. Poor math skills tend to produce models with less accuracy, precision, and correctness. Good math leads to a model that can be fully communicated and which provides useful insight to the problem.

Learning math after formulating the answer is like learning French after you come home from Paris. Paris is easier to navigate if you speak French. Reality is easier to formulate if you do math.

Too few biologists receive adequate training in statistics and other quantitative aspects of their subject

From a recent Nature editorial

http://www.nature.com/news/announcement-reducing-our-irreproducibility-1.12852

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