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 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.