I stopped reading as soon as the article said Einstein's strength was his physical intuition and not his mathematical prowess. Try explaining tensor calculus to someone that doesn't have proper understanding of Riemanian geometry. Einstein was indeed a very good mathematician and not just a physicist with "great physical intuition".
Einstein was hardly the best mathematician at the time, simply because there were people who spent as much time on theoretical math as he did on physics.
The parent poster claimed that one of Einstein's strengths was his mathematical prowess. Whether or not Einstein was among the best mathematicians of his day is irrelevant.
"I admire the elegance of your method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot" - Einstein to Levi-Civita
He could grok the math, which was difficult and very new at the time, because he had the best possible teachers.
Einstein was an exceptionally strong mathematician. He mastered integral and differential calculus by age 15, and gained ample knowledge in more advanced topics (topology, geometry, etc.) in the following years. Note that this was much more impressive back then than it is today. That being said, he did also command impressive intuition and observational skills.
By the way, the exact same can be said for Feynman, with the exception that Feynman was a stronger mathematician yet.
why do you say this was more impressive back then than it is today? I thought all the areas you mentioned were well-understood. if anything I'd think this kind of math is more daunting today because it just takes 1 click to get to a wall of text on wikipedia in impenetrable notation. I'd think back then any kid who picked up a text could just read it without potentially diving into 100 years of abstract math.
A normal school curriculum back then did not go so far as to teach calculus. When Einstein taught himself vector calculus, he used the textbook written by the guy who invented it because it was the only textbook. Then he personally wrote to the guy to clear up his misconceptions.
Today, you can learn this material in open university courses, and if you have a question, consult a plethora of freely available resources on the internet.
While Einstein was extremely good at math, he didn't invent the math that he used. For example tensor calculus was invented by Tullio Levi-Civita. Einstein learned it from Levi-Civita's textbook, and then Einstein and Levi-Civita had a lengthy correspondence in which a number of Einstein's misconceptions were cleared up.
But including Feynman in that list is an egregious joke. Richard Feynman was one of the top mathematical talents to grace physics. He was a lightning fast calculator. I know of two math competitions that he was in (one from NYU, the other the Putnam). He won both by wide margins. He was the only student ever to get a perfect score on both the math and physics sections of the Princeton graduate entrance exams. He invented a whole series of mathematical techniques, Feynman diagrams being only the most famous.
Einstein was merely extremely good at math. Feynman was phenomenal. Neither is a good example of physicists not needing to be good at math.
But that's not what the article says at all! Here's the quote:
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.
There is nothing there that attempts to imply in any way at all that Einstein was bad at math. Infact, it matches your characterization of him ("Einstein was indeed a very good mathematician" ~= "mathematics is definitely key to making advances in fields like particle physics") quite well.
I think what it is trying to say is that it is quite rare for a break through in Physics to require new maths. While we can all think of exceptions (and indeed the article mentions them) I think it's a reasonable point to make.
I actually think this is a pretty good article, and attempts to make subtle points. Dismissing it like this by mischaracterizing what it is saying does it a dis-service IMHO.
> 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.
Isn't that really what that paragraph is getting at? Modern physics is not intuitive so it doesn't matter how hard you try to imagine space-time or wave functions there is nothing to grab onto without the mathematics. In fact being imaginative in those fields means coming up with new mathematical theories and theorems. That is why I'm dismissive. The article is not making a subtle point and it is doing a disservice by presenting "imagining physical phenomena" as anything other than intuition guided by mathematics. Especially so when it comes to relativity and quantum mechanics.
So the answer is affirmatively yes. You do need very deep mathematical understanding to make strides like Einstein and Feynman and can not just "imagine" your way to the answer without having a proper grasp of the mathematical foundations.
I also don't know what new mathematics means. The interplay between modern physics and mathematics is very strong and Ed Witten and people like him constantly make new connection within mathematical fields that mathematicians did not think of making. Is that considered new mathematics? I think yes.
My interpretation might be wrong but then article isn't making much of a point.
I interpret that paragraph as being able to intuitively understand the relationship between the maths and the physics.
By 'new' math I mean development of new mathematical tool and techniques to deal with new physics theory. Generally physicists are applying existing techniques to new problems, which is why that understanding of the relationship between maths and your theory is important.
Nah, advanced physics still requires a strong physical intuition. I'll wager that coming up with novel and interesting solutions to physics like amplituhedron [1] requires a large dose of intuition. It also requires formidable mathematical chops, but not nearly as much as similar methods in string theory and similar require. Basically there's a base level of "good mathematics" which helps with physics, after which it's handy but not necessary. Sort of like IQ vs income, where greater IQ tends to mean greater income up to IQ's of about 120-130. Afterwords it's a wash.
What the OP means by saying that Einstein (or Feynman) didn't excel at mathematics mean they didn't excel vis-à-vis noted mathematicians or other physicists at raw mathematical formalism. One example of this would be the different approaches to study quantum electrodynamics by Schwinger and Feynman [2]. There's a funny version Feynman recounts in one of his memoirs, but essentially he developed (Feynman) diagrams to calculate particle probabilities while Schwinger preferred (formal) algebras.
PS I'd recommend reading "Sure you're joking, Mr Feynman!" if you want anecdotal stories about physics and insight in Feynman's famed "intuition".
> Nah, advanced physics still requires a strong physical intuition.
True, but without mathematics, physical intuition can become a dead end, because it doesn't lead to theories expressed clearly enough to forge consensus among practitioners. Consider Nicola Tesla, who had plenty of physical intuition, but whose record in science is sparse because he didn't express his ideas mathematically to a degree sufficient to built lasting theories.
As to the present, string theory (granted its limited testability) is built entirely on novel mathematics, and physical intuition has a limited role.
Also, it can be argued that a physical intuition guided by mathematical concepts is more productive than the raw, unvarnished kind that Tesla showed.
Special relativity is pretty much math-free though (in Einstein's formulation, not in Minkowski's).
I think Einstein himself wrote that to move to general relativity he had to go to great lengths in studing fields of mathematics he didn't know - not least because the were just being discovered at the time.
This lends credence to the idea that is genius was foremost guided by physical (I would say also philosophical) intuition, though of course you can' t discount the role of math in GE.
> Special relativity is pretty much math-free though (in Einstein's formulation, not in Minkowski's).
True, but only because Einstein hadn't yet acquired the essential mathematical background. Instructive in this connection is a key equation from his first relativity paper (https://www.fourmilab.ch/etexts/einstein/specrel/www/):
β = 1/sqrt(1-v^2/c^2))
(sorry -- the original paper doesn't number its equations)
It should seem obvious that this equation describes a relation between orthogonal dimensions, something that Einstein didn't realize, but that Minkowski certainly did. Minkowski expanded and clarified Einstein's mathematics in order to express time as a separate dimension, then made his famous remark about spacetime: "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality".
Einstein then remarked, "Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore." But in subsequent years Einstein learned much more mathematics, and the general theory is much more dependent on mathematical ideas than the special theory was, indeed the general theory can't be meaningfully expressed without advanced mathematics.
I think you are a little hasty in assuming that the author had the popular misconception that Einstein was bad at math. Certainly, Einstein was much better at math than the average man, but among his fellow physicists, not to mention the handful of physicists as celebrated as he is, his mathematical skills pales in comparison.
In other words, his contributions to science arose from his exceptional physical insights, and not his mathematical brilliance.
> In other words, his contributions to science arose from his exceptional physical insights, and not his mathematical brilliance.
True about special relativity, certainly false about general relativity. Between 1905 and 1915, Einstein learned the mathematics required to formulate and write the general theory.
- You need to know some math to do great science. How much math depends very much on your field, and what you're looking into (even within my field, 'great science' can range from things that only need long division to things that need serious calculus or simulation chops).
- Knowing math does not mean you can do great science. The mathematically inclined occasionally fall susceptible to the stack fallacy, and assume that once the math is known, everything else falls into place. Some of the worst science I've seen in my field stems from that assumption.
There are two versions of "being good at mathematics".
1. You are good in solving mathematical problems like solving complex equations using various techniques. In this case you are working inside mathematics.
2. You are good at modeling systems using mathematics. Using mathematics as a language and create a mapping between the system you are researching and mathematical concepts.
I think the article point is that you can do good science if you have the second skill.
I would add some nuance, to make the list look more like:
1A) Ability to create new mathematical ways of talking about your systems. This means things like inventing new notations, conventions, or combinations of mathematical methods from disparate mathematical disciplines.
1B) Ability to recognize when a mathematical finding applies to your system. You may not be able to come up with it on your own, but you know it when you see it.
2) Ability to implement conventional methods quickly to your problem, even a complex one.
In general: there's an appreciation for basic logic that is necessary.
In Curious Wavefunction's field, chemistry (which is something I am, uh, somewhat familiar with), graph theory could probably be useful, but useful graph theoretic construction for retrosynthetic analysis seems to be persistently elusive since at least E. J. Corey's era.
But sometimes just being able to do math is important. Today I got into a minor dispute with someone who is running a sequencing/diagnostics startup. Their sequencing technique is, reported as of late 2015, getting an 8% error rate for an 8x coverage sequence. Asked him what the error would have been for a 1x coverage, and he couldn't answer. I would think that if you're going to be using sequencing to do diagnostics, being able to rapidly figure out things like this is important.
> "Most top chemists and biomedical researchers have little use for mathematics per se, except in terms of using statistical software or basic calculus. "
Now I never ended up working in the field, but I remember my organic chemistry Msc. being fairly math heavy. Solving the Schroedinger equation for a complex molecule or refining a stochastic simulation model involved a bit more than basic calculus. The whole course was more math than lab work.
Was this article written by someone who never took an advanced science class, or by a mathematician to whom all the math used in natural sciences is trite and simple?
I think Curious Wavefunction was a practicing chemist. Nobody really solves schroedinger equations (actually you can't really solve it for anything besides hydrogen) and first-pass "gut" understanding of kinetics is usually good enough to get most things done. As for kinetics: in practice, just do it five times at five temperatures to find the best yield where it goes fast and doesn't degrade.
This would be for organic synthesis.
Biochemists use kinetics more, one time my boss yelled at her grad students for using 1 uL pipettes because they have poor resolution for kinetic studies, but even so they're not necessary for many studies (for example, for my biochemistry pretty much everything was pushed to the saturating rate and we had linear kinetics).
You don't necessarily need great math to do productive science, but even if you are working in a "soft" field like psychology, strong math is a strong asset, such as in the field of causal learning, where it was found that mathematical models of associative learning doesn't sufficiently account for rats learning (can you do that without math?), or in behavioral variability.
Otherwise you're going to be stuck plugging in numbers into a model you don't understand.
Also, I think strong intuition is much harder to develop than a strong math foundation. I don't know if you can hard work your way into strong intuition.
Note the author taught biology. I think we would all agree that there are a variety of sciences, some of which rely heavily on advanced maths (particle physics) while in others (biology) it's really optional and there is plenty that can be done without a deep understanding of numbers theory.
My undergrad (UBC) used math (calc 101) as a weeding-out course. They needed 50% of students to fail. When computerized teaching was tested and pass rates went up, it was a problem. So they eliminated the computer-aided option (early 90s, probably different now). Math was a weapon for dissuading students. It worked. I probably would have given up had I not completed calc in a highschool with a teacher who actually cared.
I would say that the answer is trivially yes. You need to know math to do great science.
Why?
Science is all about reproducibility. Without a strong grasp of statistics, you do not know whether you've actually discovered an effect, or whether you just got (un)lucky with your data. And far from being a theoretical problem, numerous scientific disciplines, especially those in biology and medicine, are currently dealing with a "reproducibility crisis", as they find that prominent results don't recur when replication experiments are run.
Far from there being too much math in science, I think there's currently too little math. If researchers knew more about the limits of the statistical tests they were using, rather than focusing on the magic p=0.05 level, we'd have much higher quality science, with a much higher level of confidence that the results being reported were real results, rather than statistical abberations.
But (1) statistics is really not the kind of math that is being talked about in the article, (2) even if you study statistics in a graduate program, you'll be inundated with hypothesis tests and alpha=0.05 and any deeper insight just kind of has to develop on its own, (3) the only math you need to understand the shortcomings of statistical tests is P(A|B) != P(B|A) and (4) the reproducibility crisis is primarily due to publication bias and null hypothesis significance testing is at most a catalyst.
Can this really be said in all fairness? What of the flashes of brilliance, the striking insights into understanding seemingly inexplicable phenomena? The discovery of the structure of benzene, the creation of the periodic table comes to mind.
Not only that but the purpose of science is also to make quantifiable predictions.
If you treat math as a blackbox the best case scientific contribution you can make will be reduced to measuring numbers. New numbers might lead to some new major insight, however this new insight will most certainly not be delivered by you but by someone who uses your numbers to actually "do the math" hence develops a mathematical theory capable of delivering quantifiable predictions for a more general case including the new data.
In the sciences, mathematics serves as nothing less than a convenient notation for expressing (and manipulating) certain principles: it is an invaluable tool, both for analysis and exposition.
That being said, in my opinion, great science is largely about insight, and this does not necessarily depend on maths. For example, consider the contributions Faraday made to our understanding of electromagnetism. Although he never mastered maths, he broke the ground for many that followed him, most notably Maxwell. With Maxwell, you have almost the opposite situation, as he was a highly skilled mathematician: his insights were largely borne of mathematical abstraction.
So I think the best answer to the question is sometimes. Faraday's insights were not (directly) dependent on advanced knowledge of mathematics, but Maxwell's were. We needed the genius of both these men, but mathematics was only required of the second.
What should we do, then? Definitely require hard science majors to learn mathematics, even more so then currently. Encourage mathematicians to think about science problems, and vice-versa. But let us not make the mistake of closing the scientific doors on the so-called 'mathematically illiterate.' For all we know, they carry the solutions to some of our most difficult problems.
> But let us not make the mistake of closing the scientific doors on the so-called 'mathematically illiterate.'
Fair enough, but it can lead to a very serious problem -- how do we compare the results of different studies? Day-to-day experimental results can be collected with only a little math, but any theories that might be shaped on the basis of those results, that might end up defining new fields, usually have a mathematical form that summarizes the experimental work and creates new principles and paradigms. Those theories that last the longest and have the largest effect on science, tend to be more mathematical than anecdotal.
> Faraday's insights were not (directly) dependent on advanced knowledge of mathematics, but Maxwell's were. We needed the genius of both these men, but mathematics was only required of the second.
This example supports the role of mathematics in science. Faraday described some results that arose in laboratory experiments, then Maxwell explained those results using mathematics. Faraday's results were fascinating, but Maxwell's results were portable and later served as a foundation for relativity -- which was also very mathematical.
>Faraday described some results that arose in laboratory experiments, then Maxwell explained those results using mathematics.
The issue is with expecting a single person to be able to do both; people with incredible insight who are also great at mathematics are inherently rare. If you require the latter you're going to miss out on much of the world's stock of the former. I think the future of science rests on the coordination of both.
I agree completely, but by 1915, Einstein had reshaped himself into exactly the multidisciplinary person required to write the general theory.
The real tragedy in science are those with only physical intuition and little mathematical foundation, like Nicola Tesla, who wasted much time on notions that flatly contradicted the mathematical underpinnings of physical theory.
Is great science done anymore? It seems like there were prolific scientists making incredible advances directly impacting the standard of living of humans for a period, and now we're coasting and making small incremental changes. I'm sure some folks here have examples of amazing things being done in science now, but I don't seem to be noticing the discoveries impacting the same way that seemed to be happening in 1900-2000 era. We're moving into another dark age. Kidding about that last part, and I actually do love science.
It just seems like before people were making amazing discoveries and now it's like "let's make facebook work better and get everyone addicted to smartphones and track 'em so we can sell stuff. Starbucks and iPhones. Donald Trump."
In terms of the article, I don't see how knowing math could be anything but helpful for doing great science. Maybe one doesn't have to be an expert, but to pretend it couldn't be helpful seems silly. Maybe I don't need to know how to spell words to do great science either, but conveying ideas in a clear manner is so important. Hence math.
"The more important fundamental laws and facts of physical science have all been discovered, and these are now so firmly established that the possibility of their ever being supplanted in consequence of new discoveries is exceedingly remote... Our future discoveries must be looked for in the sixth place of decimals." Albert A. Michelson, speech given in 1894 at the dedication of Ryerson Physics Lab, University of Chicago.
Death years of these guys: Maxwell (1879), Darwin (1882), Newton (1727), Mendeleev (1907). So it seems he was right. We have been just applying the science of the giants for the past 122 years.
Plenty from finding a Higgs-Boson compatible particle and PDS 456 to genome mapping to water on mars. Directly applicable would be the new antibiotic, name escapes me and well genome mapping has plenty of directly applicable implications etc.
Maybe it's the case of hindsight is 20-20 type of thing for me... the previous discoveries have a visible impact on my life. I'll have to give the more recent ones time to play out and see what happens.
However, these small, incremental changes are going to pile up and amount to what ends up being a qualitative change.
Also: I'd not fear a new dark age. Chinese or other East-Asians are bound to get to the business of modifying embryos for higher mathematical intelligence before 2060. That's going to induce quite a jolt for the sciences, even if 3/4 of those kids end up doing something else.
Yes. The problem is not a lack of excellent scientific results, the problem is the sheer number of mediocre people in the field and the amount of trash that's published as though it was either significant or science. These mediocre results tend to hide the high-quality work.
My theory is that there are just as many original thinkers as ever lived, but the noise level is much higher now, and scientific publications tend to publish page fillers with little or no significance or that are simply wrong:
In my opinion, the problem is not that science is over, the problem is that it's become to a large extent a business instead of an exploration of nature for the sake of exploration.
Your theory makes sense to me -- it seems the number of PhDs produced yearly increased greatly since around 1960 [1][2]. Since maintaining a career in science requires journal publication, I'm sure this correlates with incredible increases in published results over the past (say pre-1960). Perhaps the mediocre results are published by those wanting to 'look like a productive scientist' without adding much to the knowledge-base. The result being what you're mentioning, and maybe what I'm noticing too.
Quote: "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."
This is very misleading -- it tries to hold mathematics responsible for what in fact was a mixture of superficial mathematical reasoning and an inability to grasp that the mathematical models had little to do with reality. It was an example of using mathematics to conceal rather than reveal.
If the mathematical model doesn't accurately model reality, don't blame the math, blame the mathematician.
"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"
Nonsense. Feynman, for example, was famous for his ability to formulate and to solve mathematical problems in hours that his colleagues had spent months working on.
Mathematics is a fundamental tool to convey your imaginations to others in a way they can test it for approval or disapproval. IMO if you take away mathematics from Science, the only thing left are novel ideas (which you can also find in sci-fi novels).
Of course a person can have deep imaginations and solutions to the great science problems, but if she can't express the ideas then how to prove them right or wrong??
> IMO if you take away mathematics from Science, the only thing left are novel ideas (which you can also find in sci-fi novels).
Math is an important part of nearly all scientific fields, but to say that there is nothing left but ideas when the math is taken away is too broad. There are many fields of science that traditionally have been developed without a lot of need for math, like for example biology and earth sciences. These days mathematics is a large part of these fields too, but in some subfields, like in ecology, many researchers use little math even today.
As I see mathematics as somewhat like the act of traversing, exploring, and constructing a hypergraph built fundamentally of pointers where the cliques and common paths are named simply for economical reasons for quick addressing, it clearly can increase the probability of doing great science no matter what field you are.
I'd still say that you can still do great science although you do not know the formal language of mathematics, but the probability of getting to the position of doing the science today is harder because we have more aggressive naked alpha monkeys protecting their turfs from those they see less worthy of joining the adventure.
I find it troubling that students are put off by the notion that knowing mathematics is required to do X.
Mathematics needs a PR campaign (and maybe better teachers?).
I've noticed that a large number of our students are scarred of taking any courses that could possibly be related to math.
Both. I'm worried that the way education is being conducted is increasingly turning people off of math, science, and literature. These are all wonderful and highly useful subjects of study and even recreation, but the school systems have turned then into drudgery, and have been using them for aversion therapy for years. It's a wonder society functions at all.
>I'm worried that the way education is being conducted is increasingly turning people off of math, science, and literature.
I disagree. Looking globally (and to some extent historically) I don't see evidence that more successful education systems were taught in more engaging ways by insightful teachers. I do see cultural differences, I see modern children believing they are there to be entertained and not to simply learn and do as they are told.
Blaming the teachers and the schools is easy. Blaming the parents and the society that we have created is more honest though imo.
65 comments
[ 3.0 ms ] story [ 132 ms ] threadHe could grok the math, which was difficult and very new at the time, because he had the best possible teachers.
By the way, the exact same can be said for Feynman, with the exception that Feynman was a stronger mathematician yet.
Today, you can learn this material in open university courses, and if you have a question, consult a plethora of freely available resources on the internet.
Accessibility of information matters.
But including Feynman in that list is an egregious joke. Richard Feynman was one of the top mathematical talents to grace physics. He was a lightning fast calculator. I know of two math competitions that he was in (one from NYU, the other the Putnam). He won both by wide margins. He was the only student ever to get a perfect score on both the math and physics sections of the Princeton graduate entrance exams. He invented a whole series of mathematical techniques, Feynman diagrams being only the most famous.
Einstein was merely extremely good at math. Feynman was phenomenal. Neither is a good example of physicists not needing to be good at math.
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.
There is nothing there that attempts to imply in any way at all that Einstein was bad at math. Infact, it matches your characterization of him ("Einstein was indeed a very good mathematician" ~= "mathematics is definitely key to making advances in fields like particle physics") quite well.
I think what it is trying to say is that it is quite rare for a break through in Physics to require new maths. While we can all think of exceptions (and indeed the article mentions them) I think it's a reasonable point to make.
I actually think this is a pretty good article, and attempts to make subtle points. Dismissing it like this by mischaracterizing what it is saying does it a dis-service IMHO.
Isn't that really what that paragraph is getting at? Modern physics is not intuitive so it doesn't matter how hard you try to imagine space-time or wave functions there is nothing to grab onto without the mathematics. In fact being imaginative in those fields means coming up with new mathematical theories and theorems. That is why I'm dismissive. The article is not making a subtle point and it is doing a disservice by presenting "imagining physical phenomena" as anything other than intuition guided by mathematics. Especially so when it comes to relativity and quantum mechanics.
So the answer is affirmatively yes. You do need very deep mathematical understanding to make strides like Einstein and Feynman and can not just "imagine" your way to the answer without having a proper grasp of the mathematical foundations.
I also don't know what new mathematics means. The interplay between modern physics and mathematics is very strong and Ed Witten and people like him constantly make new connection within mathematical fields that mathematicians did not think of making. Is that considered new mathematics? I think yes.
My interpretation might be wrong but then article isn't making much of a point.
By 'new' math I mean development of new mathematical tool and techniques to deal with new physics theory. Generally physicists are applying existing techniques to new problems, which is why that understanding of the relationship between maths and your theory is important.
What the OP means by saying that Einstein (or Feynman) didn't excel at mathematics mean they didn't excel vis-à-vis noted mathematicians or other physicists at raw mathematical formalism. One example of this would be the different approaches to study quantum electrodynamics by Schwinger and Feynman [2]. There's a funny version Feynman recounts in one of his memoirs, but essentially he developed (Feynman) diagrams to calculate particle probabilities while Schwinger preferred (formal) algebras.
PS I'd recommend reading "Sure you're joking, Mr Feynman!" if you want anecdotal stories about physics and insight in Feynman's famed "intuition".
[1]: https://www.quantamagazine.org/20130917-a-jewel-at-the-heart... [2]: https://en.wikipedia.org/wiki/Julian_Schwinger#Schwinger_and...
True, but without mathematics, physical intuition can become a dead end, because it doesn't lead to theories expressed clearly enough to forge consensus among practitioners. Consider Nicola Tesla, who had plenty of physical intuition, but whose record in science is sparse because he didn't express his ideas mathematically to a degree sufficient to built lasting theories.
As to the present, string theory (granted its limited testability) is built entirely on novel mathematics, and physical intuition has a limited role.
Also, it can be argued that a physical intuition guided by mathematical concepts is more productive than the raw, unvarnished kind that Tesla showed.
I think Einstein himself wrote that to move to general relativity he had to go to great lengths in studing fields of mathematics he didn't know - not least because the were just being discovered at the time.
This lends credence to the idea that is genius was foremost guided by physical (I would say also philosophical) intuition, though of course you can' t discount the role of math in GE.
True, but only because Einstein hadn't yet acquired the essential mathematical background. Instructive in this connection is a key equation from his first relativity paper (https://www.fourmilab.ch/etexts/einstein/specrel/www/):
β = 1/sqrt(1-v^2/c^2))
(sorry -- the original paper doesn't number its equations)
It should seem obvious that this equation describes a relation between orthogonal dimensions, something that Einstein didn't realize, but that Minkowski certainly did. Minkowski expanded and clarified Einstein's mathematics in order to express time as a separate dimension, then made his famous remark about spacetime: "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality".
Einstein then remarked, "Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore." But in subsequent years Einstein learned much more mathematics, and the general theory is much more dependent on mathematical ideas than the special theory was, indeed the general theory can't be meaningfully expressed without advanced mathematics.
https://en.wikipedia.org/wiki/Relativity_priority_dispute
In other words, his contributions to science arose from his exceptional physical insights, and not his mathematical brilliance.
True about special relativity, certainly false about general relativity. Between 1905 and 1915, Einstein learned the mathematics required to formulate and write the general theory.
- You need to know some math to do great science. How much math depends very much on your field, and what you're looking into (even within my field, 'great science' can range from things that only need long division to things that need serious calculus or simulation chops).
- Knowing math does not mean you can do great science. The mathematically inclined occasionally fall susceptible to the stack fallacy, and assume that once the math is known, everything else falls into place. Some of the worst science I've seen in my field stems from that assumption.
I think the article point is that you can do good science if you have the second skill.
1A) Ability to create new mathematical ways of talking about your systems. This means things like inventing new notations, conventions, or combinations of mathematical methods from disparate mathematical disciplines.
1B) Ability to recognize when a mathematical finding applies to your system. You may not be able to come up with it on your own, but you know it when you see it.
2) Ability to implement conventional methods quickly to your problem, even a complex one.
In Curious Wavefunction's field, chemistry (which is something I am, uh, somewhat familiar with), graph theory could probably be useful, but useful graph theoretic construction for retrosynthetic analysis seems to be persistently elusive since at least E. J. Corey's era.
But sometimes just being able to do math is important. Today I got into a minor dispute with someone who is running a sequencing/diagnostics startup. Their sequencing technique is, reported as of late 2015, getting an 8% error rate for an 8x coverage sequence. Asked him what the error would have been for a 1x coverage, and he couldn't answer. I would think that if you're going to be using sequencing to do diagnostics, being able to rapidly figure out things like this is important.
> "Most top chemists and biomedical researchers have little use for mathematics per se, except in terms of using statistical software or basic calculus. "
Now I never ended up working in the field, but I remember my organic chemistry Msc. being fairly math heavy. Solving the Schroedinger equation for a complex molecule or refining a stochastic simulation model involved a bit more than basic calculus. The whole course was more math than lab work.
Was this article written by someone who never took an advanced science class, or by a mathematician to whom all the math used in natural sciences is trite and simple?
This would be for organic synthesis.
Biochemists use kinetics more, one time my boss yelled at her grad students for using 1 uL pipettes because they have poor resolution for kinetic studies, but even so they're not necessary for many studies (for example, for my biochemistry pretty much everything was pushed to the saturating rate and we had linear kinetics).
Otherwise you're going to be stuck plugging in numbers into a model you don't understand.
Also, I think strong intuition is much harder to develop than a strong math foundation. I don't know if you can hard work your way into strong intuition.
My undergrad (UBC) used math (calc 101) as a weeding-out course. They needed 50% of students to fail. When computerized teaching was tested and pass rates went up, it was a problem. So they eliminated the computer-aided option (early 90s, probably different now). Math was a weapon for dissuading students. It worked. I probably would have given up had I not completed calc in a highschool with a teacher who actually cared.
Why?
Science is all about reproducibility. Without a strong grasp of statistics, you do not know whether you've actually discovered an effect, or whether you just got (un)lucky with your data. And far from being a theoretical problem, numerous scientific disciplines, especially those in biology and medicine, are currently dealing with a "reproducibility crisis", as they find that prominent results don't recur when replication experiments are run.
Far from there being too much math in science, I think there's currently too little math. If researchers knew more about the limits of the statistical tests they were using, rather than focusing on the magic p=0.05 level, we'd have much higher quality science, with a much higher level of confidence that the results being reported were real results, rather than statistical abberations.
Can this really be said in all fairness? What of the flashes of brilliance, the striking insights into understanding seemingly inexplicable phenomena? The discovery of the structure of benzene, the creation of the periodic table comes to mind.
If you treat math as a blackbox the best case scientific contribution you can make will be reduced to measuring numbers. New numbers might lead to some new major insight, however this new insight will most certainly not be delivered by you but by someone who uses your numbers to actually "do the math" hence develops a mathematical theory capable of delivering quantifiable predictions for a more general case including the new data.
That being said, in my opinion, great science is largely about insight, and this does not necessarily depend on maths. For example, consider the contributions Faraday made to our understanding of electromagnetism. Although he never mastered maths, he broke the ground for many that followed him, most notably Maxwell. With Maxwell, you have almost the opposite situation, as he was a highly skilled mathematician: his insights were largely borne of mathematical abstraction.
So I think the best answer to the question is sometimes. Faraday's insights were not (directly) dependent on advanced knowledge of mathematics, but Maxwell's were. We needed the genius of both these men, but mathematics was only required of the second.
What should we do, then? Definitely require hard science majors to learn mathematics, even more so then currently. Encourage mathematicians to think about science problems, and vice-versa. But let us not make the mistake of closing the scientific doors on the so-called 'mathematically illiterate.' For all we know, they carry the solutions to some of our most difficult problems.
Edit: changed 'where' to 'were'
Fair enough, but it can lead to a very serious problem -- how do we compare the results of different studies? Day-to-day experimental results can be collected with only a little math, but any theories that might be shaped on the basis of those results, that might end up defining new fields, usually have a mathematical form that summarizes the experimental work and creates new principles and paradigms. Those theories that last the longest and have the largest effect on science, tend to be more mathematical than anecdotal.
> Faraday's insights were not (directly) dependent on advanced knowledge of mathematics, but Maxwell's were. We needed the genius of both these men, but mathematics was only required of the second.
This example supports the role of mathematics in science. Faraday described some results that arose in laboratory experiments, then Maxwell explained those results using mathematics. Faraday's results were fascinating, but Maxwell's results were portable and later served as a foundation for relativity -- which was also very mathematical.
The issue is with expecting a single person to be able to do both; people with incredible insight who are also great at mathematics are inherently rare. If you require the latter you're going to miss out on much of the world's stock of the former. I think the future of science rests on the coordination of both.
The real tragedy in science are those with only physical intuition and little mathematical foundation, like Nicola Tesla, who wasted much time on notions that flatly contradicted the mathematical underpinnings of physical theory.
It just seems like before people were making amazing discoveries and now it's like "let's make facebook work better and get everyone addicted to smartphones and track 'em so we can sell stuff. Starbucks and iPhones. Donald Trump."
In terms of the article, I don't see how knowing math could be anything but helpful for doing great science. Maybe one doesn't have to be an expert, but to pretend it couldn't be helpful seems silly. Maybe I don't need to know how to spell words to do great science either, but conveying ideas in a clear manner is so important. Hence math.
"The more important fundamental laws and facts of physical science have all been discovered, and these are now so firmly established that the possibility of their ever being supplanted in consequence of new discoveries is exceedingly remote... Our future discoveries must be looked for in the sixth place of decimals." Albert A. Michelson, speech given in 1894 at the dedication of Ryerson Physics Lab, University of Chicago.
However, these small, incremental changes are going to pile up and amount to what ends up being a qualitative change.
Also: I'd not fear a new dark age. Chinese or other East-Asians are bound to get to the business of modifying embryos for higher mathematical intelligence before 2060. That's going to induce quite a jolt for the sciences, even if 3/4 of those kids end up doing something else.
Yes. The problem is not a lack of excellent scientific results, the problem is the sheer number of mediocre people in the field and the amount of trash that's published as though it was either significant or science. These mediocre results tend to hide the high-quality work.
My theory is that there are just as many original thinkers as ever lived, but the noise level is much higher now, and scientific publications tend to publish page fillers with little or no significance or that are simply wrong:
http://www.nature.com/news/over-half-of-psychology-studies-f...
In my opinion, the problem is not that science is over, the problem is that it's become to a large extent a business instead of an exploration of nature for the sake of exploration.
[1] http://www.nsf.gov/statistics/nsf06319/figures/fig02-04.htm
[2] http://www.nsf.gov/statistics/infbrief/nsf12303/
This is very misleading -- it tries to hold mathematics responsible for what in fact was a mixture of superficial mathematical reasoning and an inability to grasp that the mathematical models had little to do with reality. It was an example of using mathematics to conceal rather than reveal.
If the mathematical model doesn't accurately model reality, don't blame the math, blame the mathematician.
Nonsense. Feynman, for example, was famous for his ability to formulate and to solve mathematical problems in hours that his colleagues had spent months working on.
How is this in Scientific American?
Not worth reading.
Of course a person can have deep imaginations and solutions to the great science problems, but if she can't express the ideas then how to prove them right or wrong??
Math is an important part of nearly all scientific fields, but to say that there is nothing left but ideas when the math is taken away is too broad. There are many fields of science that traditionally have been developed without a lot of need for math, like for example biology and earth sciences. These days mathematics is a large part of these fields too, but in some subfields, like in ecology, many researchers use little math even today.
I'd still say that you can still do great science although you do not know the formal language of mathematics, but the probability of getting to the position of doing the science today is harder because we have more aggressive naked alpha monkeys protecting their turfs from those they see less worthy of joining the adventure.
I've noticed that a large number of our students are scarred of taking any courses that could possibly be related to math.
I disagree. Looking globally (and to some extent historically) I don't see evidence that more successful education systems were taught in more engaging ways by insightful teachers. I do see cultural differences, I see modern children believing they are there to be entertained and not to simply learn and do as they are told.
Blaming the teachers and the schools is easy. Blaming the parents and the society that we have created is more honest though imo.
But if you want to publish in a respectful magazine and you don't know how to formalize (in maths) your accomplishment, you will not get accept.