The two cultures of mathematics and biology
liorpachter.wordpress.com
liorpachter.wordpress.com
When I was a biology grad student, I was the only person in the department that tried to do active collaboration with the math department. I took more math classes during my graduate program than biology classes.
On the biology side, I got ridiculed for all the 'hand waving' that seems to happen with the math. Biologists want to see concrete experiments and results.
On the math side I found people to be much more open and fascinated by the biology, but they had a tough time explaining what they were doing to a lay audience.
No easy answers, but I think more programs should graduate people with dual skills in both subjects(and of course have job opportunities for those grads, instead of having them jump into industry like I did).
How interesting. Usually, it's the math folk who ridicule everyone else for being sloppy with their reasoning. For example, physics = hand-wavery :D
However, they can also be pretty free with their assumptions (which can be wild, though if reasoned from carefully usually turn out to be useful even if nobody anticipated that they would).
People working with mathematicians probably spend a lot of time wondering why they're so persnickety about some things most people would consider obvious ("but how do you really know 2+2=4?") but sometimes not so much about whether the constructions they're working with actually reflect a particular concrete reality.
Biologists (or in my case epidemiologists) may, on the other hand, be very concerned about why the value you chose for a particular parameter is 6 instead of 7, and whether or not it's really exponentially distributed.
Another way of putting it is "A mathematician might be interested in the general properties of dynamical systems, and a biologists entire research agenda at the moment may very well be this set of equations with a very specific parameter set."
I surely won't disagree, but I want to add the point that when the model turns out to be wrong (and most will), also the biologist will have an interest to know, whether just some parameters (say values of constants, wrong probablility assumption) are wrong or the fundamental model has to be reconsidered.
Physics today does have some remarkably talented mathematical thinkers, like Edward Witten, but most of them are involved in very fanciful theories like string theory.
And the Dirac delta is the limit of a function with integral 1, not a function.
Maybe you need to hang out with a better class of physicists? There are certainly plenty of mathematicians who believe odds things, but I'd be leery of tarring the whole lot with the same brush.
On the other hand, it really is a useful result. The negative part is, for example, why the Casimir force is attractive.
Here's an article about the whole thing, with links to the various Numberphile videos and Phil Plait articles that have been done about it: http://physicsbuzz.physicscentral.com/2014/01/does-1234-112....
> In the Wikipedia article, they have an equation that looks like this \zeta(-3)=1/120, but the stuff on the left hand side is just another way of writing 1^3+2^3+3^3 +4^3+...
If you can't explain why the above claim is screwy, then every time you claim, "the sum of all positive integers is -1/12," you need to qualify it with "I have no idea why, but I know there's some complicated mathematics that removes the word 'sum' and 'positive integers' and replaces them with more complicated concepts I'm don't have the time to learn about, and we physicists abuse it to say what I just said."
> If QED is correct ... then I would argue that the things that go into QED calculations must be just as correct.
Also not how logic works.
Sure sure, but do know that a lot of all that hacky regularization stuff can be made mathematically sound after all --see for instance [1,2].
[1] http://www.maths.qmul.ac.uk/~tp/talks/casimir.pdf#page55
[2] http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin...
For example, many labs have been studying an important gene and how other genes are functionally related to it for more than 10 years. The research involved are simply tedious, but indispensable, biological experiments. It is a waste of time to study math for this work, because it doesn't apply, except some basic statistics on data analysis.
Disclaimer: I have advanced degrees in both biology and computer science, and has multiple years of biomedical research experience. I have met exceptionally smart people working in both biology, CS, math, and physics. Yes, these smart biologists don't understand advanced topics in math and physics. I believe, however, if they had studied math or physics, they would have been excellent mathematicians or physicists.
For example, for much of the work I do, you could get away with never using anything more sophisticated than ANOVA.
There are some other points to consider. First, the dataset sizes for most biomedical research are very small. Most advanced statistical methods don't apply. Due to curiosity, I took some advanced stat courses and tried to apply the methods to our lab's data. It didn't provide any significant improvement compared to basic ones, like linear regression, logistic regression, etc.
Second, biomedical research is highly collaborative these days. For some research that generate a large amount of data, either the researchers themselves understand statistics very well, or they collaborate with statisticians very closely. There is a field called biostatistics. Most biostatistics professors are either math or stat major, and many of them are adjoint professors in biomedical departments.
Biomedical research is really tedious and time-consuming. The professors I knew when I was doing biomedical research worked more than 60 hours a day, and they wish they had more time. One young woman professor came to the lab at 8am, left at 6pm, spent some time with her 4 children, and came back to lab at 9pm again, and worked until midnight, on every weekday. She brought her children to the lab on Saturday, and worked the whole day. IMHO, it is better for her to focus all her energy on the biomedical part, which she is best at, and collaborate with statisticians.
The general rule in biology is if you need to use statistics you did the wrong experiment. The reason for this rule is it is all too easy to use clever statistical methods to solve a flawed experimental design.
The study of biology is further complicated by the large number of confounding factors that muddle experimental results. Because of this, it is hard to know exactly when it is appropriate to bring in mathematics. Without a proper understanding of all the variables, math can only get you so far.
Another way to consider this is that biologists have not pushed back hard enough to mathematicians in the sense of asking for some tools which would allow for just slurping up a vast amount of unstructured, unprocessed data and getting something out of it.
It is certainly true that mathematical modeling as it is done now currently will indeed only get you so far.
But spirit of math in conjunction with physics has been to create tools that allow leaps and bounds. If we want to follow that spirit, it seems appropriate to ask for tools to help with messy things that now can't easily be dealt with. It may not be possible but it seems worthwhile to go all the way to the brick wall and pound on it.
[1] https://www2.hu-berlin.de/biologie/theorybp/docs/dipl_scharp...
And genomics in general has forced the combination of biology, stats, and math. Classical genetics and evolution departments are often seeded with lots of appreciation for mathematics, as well.
However, pure molecular biology doesn't often have that much to say to non-applied math, and vice versa.
The odd thing is that this difference in attitudes between physicists and biologists seems to me to be at least as much historically contingent as really dependent on the subject matter. Physics is hugely sprawling too, and a great deal of it is only passingly-mathematical.
If Darwin had been more mathematically minded he could have easily come at the problem of the origin of species from a viewpoint as strongly idealized and mathematically compact as Newton did with regard to physics. Newton was admittedly blessed with a very-nearly ideal system to study (the solar system) but was able to apply his ideas to other systems using a wide variety of increasingly sketchy approximations.
But I'm still pretty sure a mathematical mind on the order of Newton's could find a way to state evolution by variation and natural selection as a theorem rather than a theory, that follows necessarily from the laws of probability and the facts of chemistry (many of which Darwin didn't know, but still). It is at the very least a fun idea to speculate about: http://www.amazon.com/Darwins-Theorem-TJ-Radcliffe-ebook/dp/...
Apart from the chemistry part (where I don't understand what you mean), biology has had this for 85 years, with numerous extensions and generalizations [1].
- Analysis of variance (ANOVA)
- Maximum likelihood estimation (MLE)
- Fisher's exact test
- The Fisher-Yeats shuffle algorithm
There is age-old question of, "What should I major in if I want to go to medical school?" Turns out that mathematics majors have the following statistics:
1. Highest average MCAT Physical Sciences Scores 2. Highest average MCAT Biological Sciences scores (higher than biosci majors) 3. Second-highest MCAT Verbal Reasoning scores (second only to Humanities majors) 4. Highest overall average MCAT scores. 5. Second-Highest average Science GPAs (biosci majors are 0.02 higher) 6. Highest average Overall GPAs
Source: https://www.aamc.org/download/321496/data/factstable18.pdf
'Course, math majors make up < 1% of medical school applicants (0.81% to be exact), so this very well may be selection bias. Still, it seems as though what medical schools are looking for are individuals with analytical (mathematical) reasoning skills.
EDIT: It's also worth noting that in many countries an undergraduate education is not a prerequisite for medical school, so there's likely to be even less math-major physicians outside of the US.
I'm not going to let this slide. This would be scandalous if it were true; however:
http://www.mi.uni-koeln.de/Bringmann/ https://www.math.psu.edu/wli/ https://web.math.princeton.edu/~smorel/ http://www.math.tamu.edu/~ptretkoff/ http://en.wikipedia.org/wiki/Maryam_Mirzakhani http://math.uchicago.edu/~wilkinso/ http://www.math.wisc.edu/~lsmith/ http://www.maths.ox.ac.uk/people/frances.kirwan http://math.stanford.edu/~ionel/ http://gauss.math.yale.edu/~ho2/
I could easily keep going.
Edit: As ninguem2 points out, apparently the author was referring to the percentage of female math professors. Nevertheless, if you do not count only doctoral-level departments, even that appears to not be true:
http://www.ams.org/profession/data/cbms-survey/chapter4.pdf
(This study is counting all tenured professors, rather than full professors only, but the proprortion is well enough north of 10% that I feel it's safe to extrapolate.)
Of course, the proportion of female math professors is terribly and inexcusably low, but the situation is at least slightly less bleak than is painted here.
Great article! I think a big problem in math is a certain insularity that suspiciously pushes away people who do interdisciplinary or education work unless they're also publishing "real math", narrowly defined.
http://www.nytimes.com/2014/11/25/science/the-lives-of-alexa...
Compare it to:
''The proper foundations of the enlarged view of algebraic geometry were, however, unclear and this is how Grothendieck made his first, hugely significant, innovation: he invented a class of geometric structures generalizing varieties that he called schemes. In simplest terms, he proposed attaching to any commutative ring (any set of things for which addition, subtraction and a commutative multiplication are defined, like the set of integers, or the set of polynomials in variables x,y,z with complex number coefficients) a geometric object, called the Spec of the ring (short for spectrum) or an affine scheme, and patching or gluing together these objects to form the scheme. The ring is to be thought of as the set of functions on its affine scheme.''
Sorry, this writing is just awkward. I can see why the editors of Nature rejected it.
I think the audience of Hacker News is closer to mathematics than the audience of nature (programming being more closely related to math than biology). I doubt that anyone in this thread unfamiliar with schemes was able to get much useful from Mumford's obituary. I don't even see many comments on the exposition.
This is similar to the insight in the tech world that customers don't buy technology because it is built well, they buy it because it solves a problem for them.
An example:
"This is the field where one STUDIES the locus of
solutions of
sets of
polynomial equations
by combining the algebraic properties of
the rings of
polynomials with
the geometric properties of
this locus,
known as a variety.
Traditionally, this HAD MEANT complex solutions of
polynomials with
complex coefficients but
just prior to Grothendieck's work,
Andre Weil and Oscar Zariski HAD REALIZED that
much more scope and insight WAS GAINED by
considering solutions and
polynomials over
arbitrary fields,
e.g. finite fields or
algebraic number fields."
........ This is "almost lisp code" in its structure.
The inner flow here is really dislocated here with all those interruptions and changes of direction and meaning. There is also a problem with the timeline. They talk in the same paragraph about at least four, maybe five different moments in time, shown in this order: 5,1,4,3 and maybe 2 (being 1 the oldest and 5 the current time). This is driving to distraction to the reader probably (I find it pretty annoying at least)
I quit soon after finding out that my (mathematics) supervisor had been actively blocking me from communicating with biologists, including blocking meetings with my (biology) co-supervisor.
I gather they thought real, practical concerns would be a distraction from the purity of theoretical problems.
That said, I was a mathy undergrad and now am in a neuro PhD program. The gulf is large indeed. I think the largest difference for me is the relation to science in general. As a mathy person, we are all about the predictive powers of science. I do A, then B happens at time T. In bio, it is not that at all. Bio is an observational science. I see A, then I see B at time T. Sure, you can make predictions, but what these events all have to do with each other is almost impossible to predict in a living organism/environment. As such, when bio people hear Partial Differential Equation, they go running for the hills.
Case in point, PDEs are no big deal for me, I took an entire class on them. But in one class we had to read a paper on using PDEs to model genetic interactions with a sugar input and then write up 1 single page on it (with some guidelines). Oh man, the riot! 59 of the other people in the class were up in arms about this. They tried to get the points on the paper halved, then eliminated, then the teacher to rescind the assignment, which they were all successful in doing. Then the non-stop complaining ensued for weeks in the halls. All because we had to read a paper with PDEs written out in it. My lord.
On the plus side, it leaves a huge hole that none of the bio people want to crawl down. This is a positive for mathy people, as the bio is ore memorization than anything. The bio field is rocking and rolling already from the intrusion that mathy people are mediating. Now, if your lab does not have a CS major in it, you are going to fall behind. The idea that quants and big data people are necessary is just starting to grab hold of the bio world. Now is a good time to get into grad school in the bio field if you have a math background as they are just now starting to realize they need you. It's just hard to get through classes though.
I have seen some very interesting math papers with appalling biology in them.
I think the issue I have with this post, and the article in general, is the very one sided "You need us, take more math" tone. With funding being harder and harder to get in pure math, math also needs applications - biology among them as it potentially unlocks NIH funding. It's not a one-sided failure.
Being a mathematician, I don't know any other way than to say: "Read this and this text and understand it. Then you'll grasp what I'm talking about.". I can tell you that even for many math students, say, PDEs are intimedating and not exactly terribly approachable at the beginning. But the difference is that by reading texts, hearing lectures etc. they simply get over it and get quite used to them. So, I believe, the mathematicians you talk about are quite honest in the sense that they are really doing their best, but simply don't know a better way to talk about their topic.
In my experience non-mathematicians don't like listening to loooong mathematical talks (although those would be necessary for introducing the topic in a more smooth way).