Great Scientist ≠ Good at Math
wsj.com
wsj.com
> Fortunately, exceptional mathematical fluency is required in only a few disciplines, such as particle physics, astrophysics and information theory.
is embarrassingly incorrect. For one example of why this is wrong, and sticking to E O Wilson's field, consider statistics. No biologist can publish anything nowadays without including some statistical analysis in their paper. So there's a pretty simple choice: either publish statistical stuff you don't understand, or learn enough math to achieve a basic understanding. Integrating probability densities, maximum likelihood estimators, etc etc, you certainly need calculus, and linear algebra will be very helpful too. As a second example, what about theoretical biology? How is a student going to have a hope of understanding theoretical models in behavioral ecology or population genetics without some mathematical training?
What E O Wilson is saying is straight out of the bad old days which ended in the 1990s. Back then, at least in the UK, it was much more common to think that students studying biology didn't need math. Those days are over. Everyone recognizes that the statistical, computational, and theoretical parts of biology are essential for students to come to grips with. I'm a little sad to publicly criticize an old biologist who has made great contributions but no young scientist should read that article and come away with the impression that it's anything other than embarrassingly out of touch.
>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.
He doesn't have to dig on quantitative biology to recognize that there's lots of science that is largely founded on using reasoning that doesn't require lots of symbol manipulation.
The point is that to be a great scientist you don't need to be a math genius. Having the math ability of a von Neumann is not a prerequisite to being a great scientist.
And "Think twice, though, about specializing in fields that require ... quantitative analysis....as well as a few specialties in molecular biology." Come on! Even a simple DNA agarose gel requires your careful calculations about the gel percentage and some basic interpolation for DNA sizes.
Basically WSJ has been turned into the Rant Mountain and the rants like "Great Scientist ≠ Good at Math" or "To (All) the Colleges That Rejected Me” won't be the last two.
"Mathematical skill is not essential, and neither is a genius IQ, he says. Of more importance is creativity, deep thinking, confidence, commitment and allegiance to the small, informal experiments."
And what EO Wilson said himself: "I found out that advances in science rarely come upstream from the ability to stand at a blackboard and conjure images from unfolding mathematical proposition and equations. They are instead the product of downstream imagination leading to hard work, during which mathematical reasoning may or may not prove to be relevant."
There is nothing about Calculus, ODE/PDE, stochastic modeling, or whatever. He explicitly refers to mathematical reasoning and skills.
I can't wait to see this book promotion op appears on The Colbert Report.
[1] http://www.pbs.org/newshour/rundown/2013/03/post-27.html
I think Wilson is saying that whilst scientists need to be able to use mathematical tools effectively, they don't need to be able to come up with them.
(Though statistics is probably much easier to use in this way than, say, calculus. Mathematica is very good at differentiation, but I think in many situations where you need to know how to differentiate, rather than just the result?)
"OBJECTIVE To develop a mathematical model for the determination of total areas under curves from various metabolic studies.
RESEARCH DESIGN AND METHODS In Tai's Model, the total area under a curve is computed by dividing the area under the curve between two designated values on the X-axis (abscissas) into small segments (rectangles and triangles) whose areas can be accurately calculated from their respective geometrical formulas. The total sum of these individual areas thus represents the total area under the curve. Validity of the model is established by comparing total areas obtained from this model to these same areas obtained from graphic method Gess than ±0.4%). Other formulas widely applied by researchers under- or overestimated total area under a metabolic curve by a great margin.
RESULTS Tai's model proves to be able to 1) determine total area under a curve with precision; 2) calculate area with varied shapes that may or may not intercept on one or both X/Y axes; 3) estimate total area under a curve plotted against varied time intervals (abscissas), whereas other formulas only allow the same time interval; and 4) compare total areas of metabolic curves produced by different studies.
CONCLUSIONS The Tai model allows flexibility in experimental conditions, which means, in the case of the glucose-response curve, samples can be taken with differing time intervals and total area under the curve can still be determined with precision."
More information about it is available here. [2]
EDIT: I just looked at the full article and it is more bizarre than I had anticipated. For one, it says that "three formulas have been developed to calculate the total area under a curve", and then lists the works in which the formulas are given. One of the works listed is Irving Adler's A New Look at Geometry, which is a math book accessible to readers with a high school education (it's good book by the way — the MAA has given it a rating of BLL [3]). But Adler's name is misspelled "Alder" in the article and in the citation. The author also gives credit to a doctor at Yale for "his expert review". How embarrassing.
[1] http://care.diabetesjournals.org/content/17/2/152.full.pdf+h...
[2] https://fliptomato.wordpress.com/2007/03/19/medical-research...
[3] http://mathdl.maa.org/mathDL/19/?pa=reviews&sa=viewBook&...
[‽]http://scholar.google.com/scholar?q=A+Mathematical+Model+for...
If people are blacked from seeing it, here's a commentary on it: http://www.nashturley.org/2012/09/27/pinguin-poop-paper/
As a grad student, I did work in experimental quantum physics. Emphasis on experimental. There were only a handful of places on Earth where theorists could go to test the sort of stuff we could do in the lab. In a very specific corner of scientific knowledge, we were the choke point. We collaborated with theorists from all over the world. Many of them could do things on paper that I could barely understand. I had to understand them to test them though, and they helped me greatly. I'd be rather disappointed in Wilson if he was happy to test theories he couldn't understand!
I think the message shouldn't be that "there are people to do math/stats for you if you want to do science". Perhaps you could get by that way, and perhaps Wilson has been able to con people into thinking he's good when he's testing things he doesn't understand. That takes a completely different skillset than math/stats I suppose, but I don't find it admirable.
Perhaps a better message is that you don't have to be a hardcore theorist to contribute to scientific research. You need to be able to understand what you're doing, but you don't need to be a genius who covers every surface around him in incomprehensible symbols while gibbering to himself gleefully. (Yes, these people do exist, and they are sometimes awesome to have around.) Theorists are just one kind of scientist. Experimentalists are every bit as important, but their skill set is different. Theorists can wave their hands and make assumptions when solving problems. Experimentalists have to think things through very pedantically because assumptions usually bite them in the ass. Theorists can tackle a different problem every day and move between wildly disparate fields on a whim. Experimenalists have to stick with one problem for months or years at a time. Experimental work involves a lot of craft and patience.
So no, you don't need to be a mathematical genius to be a scientist, but I wouldn't want to be an experimentalist who can't understand what he's doing!
I was writing out how I don't agree with the author, but then I took in the bigger picture and realized that indeed I do think that a lot of the experimentalists in the field have either forgotten or just aren't that great at their math skills. I think this shows most in classes, where it's easy for them to get tripped up over details.
That being said, most of the time it's just simple things. I think there is still clearly a higher bar (in general) in physics than in other sciences (i.e., biology) as far as math goes, and it's far too much of a stretch to say you don't need to know math well to do well. We aren't talking about algebraic topology here, I think Wilson is commenting even on just multivariable calc and linear algebra. Quantum Mechanics without linear algebra? At the very least, it would be very hard to learn (if at all).
If anyone from non-physics fields has some comments, I would be interested to hear.
http://www.ted.com/talks/e_o_wilson_advice_to_young_scientis...
I did the really elementary to high school materials in the past two years on and off. Math concepts didn't really get harder to learn. They're easy or hard to learn at time, but there's no difficulty ramp to speak of. Then what happens afterward is that you start forgetting about how to do things like logarithmic operation, how to simplify trig identities, and so on after a week or two, or sometime days. So you need to continuously review them everyday or do enough real world problems that you know how to do them for life.
The skills and knowledge that we master requires daily maintenance, or otherwise you lose your ability to do them. It's just like fitness. A person who stop long distance running for a while will no longer be able to run as long.
Most people's concept of what "math" is, isn't advanced math. Advanced math is incredibly complex. (I honestly kind of chuckle when people tell me they are "really good at math".) Read Wiles' proof of Fermat's Last Theorem (http://www.cs.berkeley.edu/~anindya/fermat.pdf), or Godel's incompleteness theorem and tell me if you understand any of that.
Math much easier than his proof, yet vastly more difficult than calculus, is used in many areas of physics and chemistry. And I think it is this kind of math to which the original article was referring.
People like to quip that Einstein wasn't good at math, because he had mathematicians helping him with his work. What they don't understand is that Einstein's skills in math blows everyone else's out of the water; it's just that they weren't up to the level of an actual mathematician.
I guess what I am doing and learning is really elementary. But I like to think my point about having foundational knowledge and the willingness to practice everyday still stand.
I know what I am doing and learning isn't what mathematican really do day to day. They make up new stuff, prove things, and solve problems in creative ways.
I just commit to memory or practice steps on how to solve exercise sets, although the way I do is way smarter and time efficient than what people do when I was in high school. I don't know if all that practice will eventually pay off in some way, but I know that I just really hate forgetting stuff that I learned in high school math class. ( Yes, I really do "incredibly boring" math homework everyday despite finishing high school three years ago)
Proving Poincare conjecture or Riemann hypothesis is insanely hard and completely beyond the grasp of us mere mortals.
Early programming languages created by mathematicians? They were terrible (for the most part). Then CS people started formalizing stuff, grammars were invented, and finally, the situation began to improve in the late 60s.
But not math. Nope, it's the same archaic, imprecise, terrible notation that's been around for centuries. AFAIK there's nothing in math comparable to the development of programming languages in CS. It simply doesn't improve. Imagine if the first language ever created in CS was also the last. That's math.
In CS, we've kept inventing new, better programming languages and conceptual foundations. Not math. There, they have a bad foundation, but they just keep building up and up and up, each paper even more hand-wavy and imprecise than the next.
It's to the point where things that are actually trivial mathematical concepts to understand are damn near impossible to learn by actually reading a math paper about the topic -- all because of the notation, implied rules, etc.
IN my experience, mathematical objects and concepts are simple, at the same level of difficulty as, say, quicksort or an AVL tree. All of math is that way. What's hard about math is the notation, not the concepts or mathematical objects that are presented.
And because the notation is so bad, it's very hard for people to make use of math in an À la carte manner. You either spend years of drudgery with little payoff, or you don't use math at all.
Contrast that with open source software (the CS equivalent, I would argue, to reading math papers), which is easy to use, because, hey, consistent notation, no hidden rules, no assumed set of knowledge or information. Everything has a definition. There is an answer for everything in the source code itself.
Math papers are the opposite, including the math articles on Wikipedia. Math is damn near unusable for that reason by anyone but mathematicians who can intuit the implicit rules, notational deviations, etc. after a lifetime of study.
It doesn't have to be this way.
[EDIT: unlike the other replies I don't actually disagree with the parent, I just want some examples!]
In CS, it's a notation we inherited from math and, literally, only use because of familiarity (the aforementioned introduction to children at an early age).
Infix sucks. It's ambiguous to parse, requiring the introduction of parenthesis. No one likes using the parenthesis, because it's ugly and takes up space, so on top of this (arguably bad foundation) we add in operator precedence. Genius -- except no one can remember all the rules. And when you leave out the parenthesis, and hand it to someone else, and it's ambiguous (i.e. has a complicated application of the operator precedence rules), how can you be sure the person meant what they wrote down?
The simple fact is that this:
2 2 +
2 + 2
Now, this is super simple example, and math can't even get this much right. With more complex mathematics, we see additional symptoms of the above problems:
1. complex, implicit rules associated with the notation
2. a tendency to "leave important stuff off" because the notation is too verbose
The latter happens all the time. I work in 3D rendering, and basically no one writes out full equations for things, ever, because it's so verbose. So to save space, we just leave it out.
Now, I know it's missing, because one time, I just happened to see the full, correct equation. But what about that A la carte math user? Can they just read the paper and then go ahead and write some code implementing it? Not even close. Laughably not even close.
Of course, I cheated here, because even this equation is presented in dozens of different ways (I'm thinking here of the Rendering equation[1]). Depending on who's writing the paper, you'll have have to decipher it out -- including dealing with all of the "missing" stuff they've left out to save space.
All of math is like this, though. There's is no such thing as a self-contained math paper. Things are rarely "written out" at the level of detail we expect in CS.
[0] http://en.wikipedia.org/wiki/Infix_notation [1] http://en.wikipedia.org/wiki/Rendering_equation
Regarding infix, in most mathematics that I've seen there tend to be only two levels of operator precedence: some form of multiplication taking precedence over some form of addition. It's in computer programming where you get multiple levels and it becomes hard to remember.
1. complex, implicit rules associated with the notation
Again I would appreciate examples.
the Rendering equation
That's not really mathematics. It's a mathematical result of engineering presented in mathematical notation. If you want to argue that engineers horribly abuse mathematical notation then I'm right with you on that point!
How about this? This is copied and pasted from the first sentence of the Wikipedia page[0] on the Guass-Newton algorithm:
Given m functions r = (r1, …, rm) of n variables β = (β1, …, βn), with m ≥ n,
This sentence, to me, is an integral part of "math notation". Personally, the CS way of saying what functions and variables are, along with their signatures, along with a precondition on a couple of length properties, is far superior to the above math notation, even though it's, perhaps, more verbose depending on the programming language.
The problem with the math notation is it's going to exclude anyone from understanding it who is not an abstract thinker even though nothing about it is actually abstract. It's become abstract only because we've been given the concrete ingredients, along with a written algorithm to construct the (now abstract) result. And we really need to understand that result to even finish the sentence!
That's math notation in a nutshell.
In CS, it'd be right there, concrete, unambiguous, sitting there on the screen (or page). In fact, you'd barely even need to say anything about it at all because there's nothing else, really, to say. CS notation is concrete and unambiguous.
These kinds of abstract, inline definitions are integral to math notation, and constitute the bulk of the problem people have with math: you force simple, concrete things to be abstract just by the notation itself.
[0] https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm
Given m real valued functions r = (r1, …, rm) of n real variables β = (β1, …, βn), with m ≥ n,
what is the signature of those functions?
Well, they're each of type R^n -> R.
Do they have return values?
Yes, of course. They're functions. That's what functions have.
What type?
R
Are there multiple return values?
No, functions don't have "multiple return values".
Are the "n variables" the same for every function? Different?
Do you mean is the number of variables n the same for every function? Yes. Otherwise I don't understand your question.
Is it a matrix, indexed by the function?
Huh?
Why is m greater than or equal to n?
Good question. I don't know.
In CS, it'd be right there, concrete, unambiguous, sitting there on the screen (or page). In fact, you'd barely even need to say anything about it at all because there's nothing else, really, to say. CS notation is concrete and unambiguous.
Perhaps you can give me a further example, because I don't know what you mean by "CS notation".
Maths is more like a human language - the writing necessarily does not contain a complete picture, just like a book can never unambiguously describe a scene. This is what sets it apart from the computer programs you have compared it with since a computer program is a complete, unambiguous definition of the thing it is describing.
All that math notation does is give us a language that sits somewhere between description and precision to let us talk about concepts that are very hard to formulate in natural language.
If it was made completely precise too much would be lost. Context is everything, and sometimes that requires you to understand the previous work in the field. I don't see a way around that.
Okay. So can you give some examples of better notation for an operad or a quasitriangular quasi-hopf algebra? Or even tracking the indices on tensors as in Riemannian geometry?
I'm not saying your wrong, the book Structure and Intepretation of Classical Mechanics makes some minor notation changes for greater clarity, or instance. But I don't think it terribly helpful to state that "math notation is horrible; no one can understand it but mathematicians".
Not really, no (although I don't think this proves anything one way or the other).
Obviously there's a difference between mathematics and computer science, as well as a lot of overlap.
My intuitive sense of things is that math "appears" to be more complex/novel/difficult than it actually is precisely because of the notation. More specifically, because math adopts notations that are useless without the accompanying text (see [0] for an example).
In CS, this would be like code that only "worked" if you also parsed and executed the comments!
My point is more or less that in math, this CS idea: List<List>, would be given a special name, it's own notation, along with a bunch of properties that, when presented as List<List>, clearly apply.
In the List<List> example, for instance, it's obvious I can iterate on each inner list -- I don't need to define a special "property" on this "object" telling me that.
In math, you would, along with special notation you invented just for your paper (yay!). When you define obvious properties of things, your mind starts to break down: the reason he's saying this is, presumably, because somewhere else some other obvious property doesn't hold. I need to pay attention.
Except...you don't. It's just an artifact of how math is done.
Of course, it's worse than that, because we wouldn't be content to refer to it as a List<List>. No, we'd need to give it a name, probably a random person's name, which you'd then have to learn.
So much of mathematics, to me, is like this. Pointless complexity, invented notation, and random names for mathematical "objects" that are no more interesting than List<List>. Wow, a group of groups. Here's your PhD.
Despite the snark, I do like math results, but I just really feel like they are needlessly unusable due to the notation. When you make something different, it feels different to people. They lose that concrete connection in their mind.
If, instead of List<List>, I called that a Domanixly torus (note: not the geometric torus[1]), and then I went ahead and defined a bunch of "properties" of this exotic object I've created, people would (rightly) assume this was some new and exotic object.
Except it's not, but the math-notation approach, and really, the whole way the field is conducted, makes it impossible to see that. Complexity is introduced where none was needed.
Anyway, that's my intuitive sense of things: that math-notation obscures what is actually there, and in the process, makes its own results very difficult to apply and use, while also scaring away people who don't want to clutter their mind with "Domanixly torus" when List<List> would have worked just as well, and required far less mental gymnastics, but sadly, won't get you a PhD.
[0] http://www.mtm.ufsc.br/~cmdoria/Notices/what-is-column/16-op... [1] http://en.wikipedia.org/wiki/Elliptic_curve , last paragraph of intro for an example
If I was to pick a programming language most like mathematical notation I would go for lisp. You define the language to suit the problem.
A nice example is the Wikipedia page on least squares: https://en.wikipedia.org/wiki/Least_squares#Problem_statemen...
Everything is concisely described provided that you know the notation for calculus and summation. One definition builds on the previous.
<TLDR> The only difference between math and English, or any other language, is a degree of precision. In math, the symbols and connotations are more refined, but the results are still interpreted. There is no absolute truth. Mathematical truth is still defined by agreement among leading scholars that something is correct. Physical truth in the sciences also depends upon agreement. In the ephemeral world of communication, words and their meanings are relevant only to our own perception. We can try to better agree upon the starting ground, the definitions, but definitions still depend upon the flimsy symbols and what they signify in each person’s mind. There is no perfect line.
I enjoy math as a dilettante communicator. And while math can greatly enhance the precision of our communications, it can also be used to deceive. My non-schizoid self agrees that the probability of pulling the “Ace of Spades” from a deck of cards is 1 in 52, but this depends upon our understanding of the definitions, our belief that the logic of probability is reasonable, and our intuition that the results agree with the world we perceive. My skeptic self, however, disbelieves that the esoteric formulas of macroeconomics accurately predict the world around us. How much "advanced math" is being conducted at banks, universities, and government institutions in pursuit of the illusory goal of modeling the economy? This math serves the purpose of providing the illusion of understanding, appropriate for politicians justifying desired actions, bankers selling products, and economics professors advancing careers. In my perception of reality, this math too often blows up when events it describes as one in a trillion come to fruition on a much more frequent basis.
While I believe understanding math can prove beneficial to any endeavor that is trying to more precisely communicate a perception of the world, that mathematical understanding does not trump the necessity of generating agreement upon the definitions and the results. With little math but plenty of logic, Darwin was able to generate an insight into the world with which many people agree (including myself). Yet with plenty of math and very little logic, many areas of "science" travel down rabbit hole after rabbit hole where the community coalesces in complicated group think, with insights that fail to agree with most peoples' perception of reality, but results that are nonetheless justified as correct based upon the "math." In the end, math is a method of communication, often more precise than other languages, but still transitory in its ability to convey meaning and generate agreement. It is a helpful tool but not an indispensable instrument to science. But I guess this all depends upon the definition of science. </QED>
The mathematics is correct. 10 divided by 5 really is 2. The problem looks something more like 10 kg divided by 5 lb isn't 2 kg. It's 2 kg/lb. It's not the mathematics' fault that you got your translation into reality wrong.
Instead of denying the obvious, it would be better to think of ways to not scare people away from mathematics - he makes it seem like some people simply do not have the mathematical ability and there is nothing to do about it except finding a field that doesn't need it.
Of course I went back and finished the analysis after lunch.
As for the original article, the implication that somebody having trouble with basic algebra can be "great scientist" sounds dubious to say the least. Like I said, there is math, and then there is math... there is stuff that anybody claiming any meaningful level of intelligence should be able to handle.
A rabbit will be woefully unprepared to learn integrals if it forgot derivatives before it starts learning integral.
Something about that is quintessentially Russian.
However, given a ‘random’ expression, integration will be much more difficult than differentiation.
Well, you've just called many dyslexics stupid. Does your assertion extend to basic arithmetic? Because if so, you've just called everyone with dyscalculia and/or acalculia stupid, and that group includes a lot of autism-spectrum engineers.