I want to add something here: great mathematicians compute too. They also know how to perform an algorithm. It is in performing, say, long division, that you start to notice things like when 10 is a primitive root modulo the divisor. Gauss spent his down time counting primes (in his head, he said). Riemann's notes were full of haphazard computations. Amidst his scratch work where the Riemann-Siegel formula appears, there's a computation of sqrt(2) to 38 decimal places for no discernible purpose. He probably was merely exercising, just like a musician practices scales. Erdős was able to memorise 10 phone numbers at a time from a glance at a phone book, amidst other calculating feats.
I hypothesise that offloading this brain work to a machine atrophies your cerebral muscles. Exercises left to the reader are really exercises in an almost kinesthetic sense.
The computations don't have to be purely numerical: even a diagram chase in abstract nonsense is a valuable exercise. Computation frequently leads to insight. As children are adding two-digit numbers, they start to notice shortcuts about how to perform the computations. These are their own little private theorems, so to speak. So when Devlin here is talking about "mathematical thinking" and the less importance that performing algorithms has today, I hope we don't forget that just because it's less important, it doesn't mean it's not important at all.
>Riemann's notes ..., there's a computation of sqrt(2) to 38 decimal places for no discernible purpose.
A while back I read about a mathematician taking an informal survey of other mathematicians. One of the questions was if they knew how to manually calculate square roots. (Apparently, the mathematician was part of an older generation that was taught this procedure in class.) Most mathematicians did not know how and a few responded with commentary such as "I don't need to know it in my work and if I had to, it's a procedure I could look up."
Unfortunately, I don't remember if I read about that in a book or a webpage (and google search doesn't find any hits) so I can't give a cite.
The mathematician doing the survey was including it as part of a larger essay. He was explaining a similar theme to the article's author: tedious manual computation exercises are overemphasized in the typical math education.
This is the method I also learned (at one time and then forgot). Apparently, this procedure was part of a typical math curriculum many decades ago. Like Latin and cursive handwriting, it's been deleted as a mandatory skill.
Not really. It's a generalization of the long-division algorithm. In that we remove lumps and then work on the remainder, ensuring that at each stage we've removed the largest lump possible given the number of digits we can have used so far. Then we recurse on the remainder.
In the square-root algorithm we remove the largest square possible at each stage, and work on the remainder. There's a hitch, because there's more detail to carry over at each stage. We do that with a linear factor of what we've already removed (that's the bit where we multiply the quotient so far by two) and then add the "epsilon" for the next lump to remove.
This can all be derived from first principles using the fact that (x+a)^2 = x^2 + 2ax + a^2. I did that when I was about 13.
There is a funny story about Gauss from the book Men of Mathematics, in which Gauss asked two of his students to compute a certain infinite series to 30 decimal places. After many days and nights, the students had a disagreement over what number was in the 30th decimal place, so they asked for help. After looking at the calculations for a few seconds, Gauss accurately pointed at the correct answer and went back to what he was doing.
That does not mean he did some tedious calculation fast. It is more likely that he understood some special property of those series to determine which number is incorrect.
I kinda agree with you, but there are counter-examples. For eg: Grothendieck.
> One striking characteristic of Grothendieck’s mode of thinking is that it seemed to rely so little on examples. This can be seen in the legend of the so-called “Grothendieck prime”. In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, “All right, take 57.”
Heh, I knew someone would come up with the Grothendieck prime example. That's why I said, "the computations don't have to be purely numerical: even a diagram chase in abstract nonsense is a valuable exercise." I think Grothendieck certainly was a master of these and performed them regularly.
> I hypothesise that offloading this brain work to a machine atrophies your cerebral muscles
Personally I think this is a myth and the brain is not like some muscle that needs to be kept in shape with a specific set of exercises. Certainly stimulating the brain has been shown to have positive effects in older people, but it hasn't been proven that there's a difference between learning a new language or playing bridge. And nothing proven was found to prevent Alzheimer.
Generally speaking, offloading computations to a machine allows one to concentrate his gray matter on other things and before we come to a conclusion that this leads to brain atrophy, evidence needs to be presented.
Trying to view the brain as "just a muscle" is probably simultaneously helpful and misleading in different ways.
But I believe the parent's point was more that what is taught is becoming fundamentally different. If you learn how to add using a calculator are you learning to add or learning how to use a calculator? It's a subtle distinction but in practice the ramifications become important.
There's a difference between the training you need to be a world-class mathematician and what the rest of the world thinks of as "mathematics."
One of the key differences (being a mathematician myself) is that the big leaps of progress often come at the high level, often talking with others, and ignoring computations. Then when you have three hours to sit down and calculate, you go back and make sure your high-level ideas pan out. And you worry that you're making mistakes the whole time :)
The point is that you don't just sit and compute for its own sake, nor is there a time crunch, nor is there even a single "right" way to get your computations to go through. Often you can choose one of many routes to bound some quantity, or many different proof techniques that involve very different computations.
I think if you're already doing computations all the time as a matter of course, there's no need to specifically set time aside to do them. But would you agree that it's necessary to know how to do them? Perhaps when working in an unfamiliar area, you take some time to learn how to compute some things in this field?
The computations just get more technical. So in research it's rare to "know" a computation in the sense that I think you mean because it's different almost every time you apply it. A common example is rounding linear programs to integer solutions for the purpose of approximation (or exact solutions, as the case may be). There is almost no way to tell in advance if it will work, and most problems require significantly different versions of this technique. There are many textbooks showcasing the many variations. So you might need days or weeks before you figure out if it will work on your problem, and then it's a "standard calculation" only in retrospect.
People really don't compute things "on the fly" in mathematics. When you hear stories about Gauss these are folk legends about people with extraordinary practiced abilities, and the majority of the mathematical world doesn't work like that. People do the grunt calculations offline so they can spend their time in talks/discussions doing actual work.
Historically that's true. But, I suspect there will eventually be a great mathematician who has also suffers from something like severe dyslexia which makes accurate computation vary difficult yet still makes significant contributions.
Mathematicians I knew tended to make minor arithmetical errors as much as anyone else, some of them a lot. Ability to do accurate computation is not much relevant in theoretical math. What is relevant is your ability to reason and abstract thinking.
Of course, that does not mean kids should not be taught arithmetic. It just mean that we should distinguish between mistakes caused by not understanding the topic and those that are just result of doing a lot of operations with similar numbers. the former should be dealt with as bigger deal then latter.
I had a college professor like that. He was nearly unable to perform arithmetic, and would ask for the answer from the class. I wouldn't say he was dyslexic (I have no idea), but he certainly couldn't perform mental arithmetic. But he was fabulous at math. We'd often ask him questions far outside the realm of the textbook and class just for the privilege of seeing an extremely good mind tackle a problem.
So I disagree with the OP. The fact that some mathematicians brains work that way is not evidence that they all do, nor is it evidence that it is helpful or preferable.
sqrt(2) sounds like a doodle to me, something you do to distract yourself while considering another problem.
For some reason I developed the habit of sketching an Escher Cube* (with a cross in the middle) when I was in highschool, I still do it when my mind is blank.
*Perhaps it isn't Escher after all, I cannot find an image online. It is an impossible 3d cube where each corner crosses through the centre.
> I want to add something here: great mathematicians compute too.
I totally agree.
To me, there are 2 kinds of algorithms that one should get used to. 1. is computing fast enough (that doesn't mean super fast, but sometimes it means hours instead of months), and 2. is spotting out errors, both evident and subtle.
This said, I'm certainly not in the set of great mathematicians.
I have a three year old son, and it is absolutely fascinating to watch his mathematical understanding develop. I've been a math teacher my entire adult life so I've had plenty of experience watching older students develop their understanding. It's entirely different watching your kid develop their understanding from scratch.
I recently looked for some kids' books that would focus on the more interesting problems in math, rather than just counting. I was happy to find a few books that have helped him see math as more than just counting. My favorite so far is The Boy Who Loved Math: The Improbable Life of Paul Erdos. [0] I knew of Erdos, but I didn't know much about him. I learned from reading this book, and my kid loves it as well. He is fascinated with aging, and he now sees it as normal that someone would spend their whole life focusing on numbers.
We are also starting to enjoy Bedtime Math: A Fun Excuse to Stay Up Late. [1] The idea is to give your kid some interesting math problems to think about at bedtime. We've found that it's a good way to help him think about things other than the dark, and strange noises while he's falling asleep.
It's fascinating to watch this development. A few nights ago: "Did you know that one of the oldest questions people have asked is, How many stars are there in the sky?"
"No, I didn't know that!"
"How many stars do you think there are in the sky?"
I think that if you want to understand how ordinary people use math, a good place to start would be by looking at their Excel spreadsheets. Excel is arguably the most widely used tool for computation and even programming. I've seen some impressive spreadsheets for solving complex problems, written by folks with no more than a high school education.
With that said, I'd endorse a move away from solely algorithm based (standardized test based) learning.
And yet the way math is traditionally taught in schools and colleges, there is almost always is a unique right answer.
There are alternative approaches to teaching math such as problem-based learning, active learning, and inquiry-based learning, that help students not only understand the math better, but learn why the math is useful and how to apply it in the real-world (transfer of learning). Students also are much more likely to continue on and succeed in future courses and graduate.
It is refreshing to find an article with a false title that contains it's own refutation within the first five paragraphs:
"The only career in which a high school graduate can expect to continue to work on [problems with a unique correct answer] is academic research in pure mathematics"
Unfortunately for this article, the premise that mathematics is the same thing as engineering is false.
If an engineering problem does not have a unique solution, its because of complications introduced by the real world. Any engineering problem which can be well-posed as a pure math problem, does of course have a unique solution; as the author concedes.
It depends on what you mean by a math problem. It seems a common idea is: A math problem is a question asking for the set of solutions to some mathematical equation, and only the actual solution is useful.
But I would include all of the following as math problems:
-Prove that some equation actually has solutions
-Find some bounds on solutions to that equation
-How can I compute an approximate solution to that equation?
-How good is that approximation going to be?
-What's a way to try to separate large data sets into clusters?
With regards to the first four: many equations people are interested in simply have no hope of getting a solution you can write down. Simple example: sqrt(2). A more complicated example: solutions to the Navier-Stokes equations.
I agree with the author entirely. But I still feel it's important to be able to do computation in your head if just to develop that "number sense". As a physicist, I use computers to do all my real calculations. But when I get the answer that seems wrong, I can mentally do some basic calculations that convince myself the computer-answer I got is probably off by a factor of two, and look for the bug.
How is adding armor to the engine area not a unique right answer? I'll leave you to figure out why that is the best solution. If there is only a single best solution, isn't that then the unique right answer?
It depends on the assumptions you make. For example, are the planes you're measuring a uniformly random sample of all planes you're interested in? The answer is different depending on how you answer this question.
That sounds like different problems have different solutions. "Most math problems are (initially) poorly defined" would have been a better title, I guess.
I agree with your title suggestion. The author phrases the problem in an intentionally misleading way: "They examined the fuselages of all the damaged planes that had flown back after being hit to see where the most damage was. It turned out that the engines had an average of 1.11 bullet holes per square foot, the fuel system had 1.55, the fuselages 1.73, and the rest of the plane 1.8."
He says that they checked to see where the most damage was, but really they were checking to see where the least damage was. If the planes were able to fly back with terrible damage to everything except for one part, then adding armor to that part is a pretty obvious solution.
On the other hand, figuring out exactly how much armor to add is much more complicated, because you have the constraints of weight, production time, etc, plus you have to have some kind of estimate for the effectiveness of the armor in actual combat. I would probably go with "Most Real-World Math Problems Do Not Have a Unique Right Answer".
And, perhaps surprisingly, most mathematical work in the real world is spent figuring out which assumptions are useful to model the problem and which are not. You sound like you are claiming that modeling the problem is not part of the problem, but somehow "before" the problem is posed. It sounds like a fantasy to me.
I'm a senior in college. I've learned about a host of mathematical concepts that are misunderstood by popular media (N dimensions, linear algebra, linear functions). Wanting to correct these topics, I wrote a blog post[1] about them.
But this blog post blossomed into much more than I was expecting. I was expecting to cover only those specific details, but then I got into nonlinear problems that have no closed form solution and got into what mathematicians do. While I'm still learning and they're an experienced mathematician, I like to think I can understand what everyone else thinks and what mathematicians think.
If we randomly choose a matrix A of NxN coefficients, we are quite likely to end up with a linear independence, meaning that Ax' = 0 where x' is the vector [x^(n-1) x^(n-1) ... x 1] is a system of equations with a unique solution.
This is an example in which most problems derived from some real data in fact have a unique solution.
In case the above isn't clear what I simply mean is that, for instance, if we randomly choose the coefficients for a system of three equations in three unknowns, we are in fact statistically unlikely to end up with an under-determined system (multiple solutions). Two or more of the vectors in the matrix would have to point in the same direction, which is unlikely for randomly chosen 3-vectors.
Since uncountably many problems have unique right answers, I'd guess the cardinality of the set of problems with unique right answers is the same as the cardinality of the set of problems without. :)
Careful there. If you require one to be able to express a problem via some finite string (and via some universally fixed method for encoding problems as strings), then there are only countably many problems.
This, too, requires care. I think that we can all agree that, for each real number x, "is x normal?" is a problem (or at least a question that a mathematician might ask). That's uncountably many problems right there! (The fact that only countably many specific instances of it can be written down is, I think, a different matter.)
The question is, "What do you define as a 'problem'?" You need to answer that before you can make claims like "we can all agree..." and give an example of something that I don't agree is a problem (if you're saying what I think you're saying).
My weak, but (I think) practical, implicit definition of 'problem' was:
> a question that a mathematician might ask
I agree that this is not a very useful definition, by virtue of its extreme and probably excessive inclusiveness, but I think that it's hard to do any better without using words like 'interesting' that themselves need careful definition. (It's fair to argue that my definition in turn requires clarification of the term 'mathematician', but I can weasel my way around that by replacing it with 'person', or else just declaring that anyone interested in trying to ascertain the normality of a number is mathematically minded enough to be called a mathematician.)
By this definition, I think that it is hard to argue with my claim to have produced an uncountable family of problems—simply because, at least classically, to do so you'd have to produce a specific number x about whose normality no mathematician could ever ask. I could then demolish that counterexample by asking you if that particular number x was normal. :-)
You're falling into the interesting number paradox here, regardless of not using the word interesting. You can't make a statement about the cardinality of a set if your definition of that set isn't expressed using mathematics. In other words, you're saying "I'm not going to give a mathematical definition of this thing, but I can nevertheless give a mathematical proof of a mathematical statement about that thing."
For example, why is the set of all questions utterable by humans not a countable set? By that reasoning there are more numbers than questions!
Though, if we apply your requirement, the set of problems without a unique answer must be countable as well. So, the cardinality of problems with a unique answer is still equal to the cardinality of problems without one.
You're just using the terminology imprecisely is all. (Indeed, two things being countable does not mean they have the same cardinality; finite sets are also countable)
I'm not making claims about the truth or falsity of your statement, just critiquing the logic and terminology you're using (the lack of a definition for a "problem," the minor misuse of the term "countable"). Regardless, I really hope you saw the article as something besides a mathematical claim, because it clearly did not pretend to be one.
He linked to DragonBox Elements which I had not heard of. It looks really amazing. This one is for geometry, similar to how the previous DragonBox was for algebra. Awesome find!
As for the armor plating question, did the plate the engine because of its size? The engines combined make up a smaller surface area than the other places but it was still shot up quite a bit. It would be less weight and not to mention it would be more 'mission critical'.
Bullets can be considered almost random in where they will hit, so, if it gets back with lots of holes in the fuselage, it tells you that you can have lots of holes in the fuselage and it still gets back. If almost none return with hits to the engine, then almost every engine hit is bringing down a plane.
I've read this story before, but asserting it was the RAF rather than the USAAF, which triggered my urban-legend detector.
In fact, the mathematical work was done by Abraham Wald¹, a Jewish refugee from Romania, at Columbia University in 1943. A reprint of his memos is available², as is a more accessible article about his work³; the specific one is Part V Subdivision of the Plane⁴ Into Several Equi-Vulnerability Areas:
“Thus, for the observed data of this hypothetical example [italics added], the engine area is the most vulnerable in the sense that a hit there is most likely to down the plane. The fuselage has a relatively low vulnerability.”
I haven't seen evidence that his work was applied to actual aircraft design using real data.
No I have read the story elsewhere, the reasoning is that if they were shot any more they wouldn't have returned at all (hence skewing the statistics). So they can take lots of hits on other spots and come back safely, but not so much on the engine so we protect the engine
Most Math Problems Do Not Have a Unique Right Answer
The title is both right and wrong. You are actually comparing school or college math with math applied in real world. School or college math works with few variables, for instance, and we consider most others remaining constant. Rarely have I seen school or college level students working with, say, derivatives of more than three variables. School or college math is an exercise to establish the rule, the rigour and in most cases to create an appreciation of what math can achieve.
In real world, variables are plenty. If you take a handful to solve a problem considering other important ones to be constants, you will end up with one set of answers as against others if you had taken a different set of variables. Real world applied math is contextual. You remove context from the problem and real world math looks like school or college math. As demonstrated by the engine-armour-plate example, without the context of the airplanes returning after taking hits, the mathematicians would probably have gone with a statistical answer and would have been proven wrong!
However, I do agree that most math problems may not have unique right answer. Of course, we are not talking of,say, square-root-of-two having two different answers. However, take an instance where the problem is:"Find a number that is a sum of two infinitesomely large numbers one ocurring at an infinitely large interval of time from the other. Does it essentially fall on the numberline?" Well the first reaction to this question is: well, yes. Because if we are sure to find those two numbers then we are more likely to find their sum which has to fall on the numberline. Now, a more discerning reader might pause and ask: can you define infinitely large number and infinitely large interval. Hence, a question like this may not have a unique right answer. If you allow philosophers in, you will definitely not have a unique answer :)
Coming to a more basic argument: With math we are striving to arrive at a single agreeable solution. Whether it is statistics or calculus, we are interested in modelling the world to arrive at a set of recognizable pattern or a set of patterns. We apply the templates we learnt in school and college. For instance in arithmetic, numerals -- which are nothing but symbols -- help us reduce our problems into an expression which we can solve. The operations allow us to take these symbols through a set of processes that helps us model the problem.
But thanks to the author, what is clear is that applied math is contextual and answer may vary with the change in context. While school math is merely an exercise in familairising ourselves with a template.
> Hence, a question like this may not have a unique right answer. If you allow philosophers in, you will definitely not have a unique answer :)
I'm not sure if this is what you're describing, but many nonlinear[1] math problems have no closed form solution[2]. That means you can't use any regular function, all the operators and the infinitely real numbers to describe every solution: you can only use the infinitely real numbers to describe one solution.
I've written a blog post on this topic[3]; that blog post works through all the underlying stuff before getting to these closed form solutions.
Right - mathematical insight is what tells you which factors to measure in order to make a prediction, and which factors to ignore. A great example I've seen recently was in the context of the solar road concept, where there was a discussion around how much pressure is exerted by a vehicle on the road surface, in the context of determining if solar panels could realistically be manufactured to stand that pressure. Someone argued that the pressure would be roughly equal to the tire pressure of the vehicles passing over; eminently sensible people then tried to make mathematical arguments that that was nonsense, because the weight of a vehicle had to have some bearing on it - it stands to reason trucks must exert more pressure than cars or bikes, right? So the assumption is that the formula for road pressure must depend in some way on the vehicle weight.
Well, yes it does - but so does the tire pressure. If you add weight to a car, the tire pressure increases. So does the pressure it exerts on the road. Noticing that vehicle weight is a common scaling factor in two places tells you there's probably a simple relationship between the tire pressure and the road pressure. And it leads you to the counterintuitive conclusion that yes, if you increase the pressure in a tire, and keep vehicle weight constant, it increases the pressure the tire exerts on the road. Pressure's tricky and counterintuitive like that.
For sure, the road pressure and tire pressure aren't necessarily equal - not all of the weight of a vehicle is borne by the column of air between the contact patch and the wheel hub, some is transferred through the sidewalls, some through the tire rim to the air above the hub, and so on, and if you are a tire manufacturer or a formula one race engineer you will want to take those things into account. But for arguing about what the pressure on the surface of the solar panels in a road surface would be, tire pressures are -a- right answer.
I hypothesise that offloading this brain work to a machine atrophies your cerebral muscles. Exercises left to the reader are really exercises in an almost kinesthetic sense.
The computations don't have to be purely numerical: even a diagram chase in abstract nonsense is a valuable exercise. Computation frequently leads to insight. As children are adding two-digit numbers, they start to notice shortcuts about how to perform the computations. These are their own little private theorems, so to speak. So when Devlin here is talking about "mathematical thinking" and the less importance that performing algorithms has today, I hope we don't forget that just because it's less important, it doesn't mean it's not important at all.