# How important is computational skill to become a mathematician?

## Main Question or Discussion Point

I love doing proofs, but hate doing computational questions. If I become very good with proofs but remain just so-so with calculational problems, can I still become a good mathematician? My favourite subject is topology.

## Answers and Replies

mathman
I love doing proofs, but hate doing computational questions. If I become very good with proofs but remain just so-so with calculational problems, can I still become a good mathematician? My favourite subject is topology.
Yes you can. In my case, I am lousy in arithmetic, but I still managed to get a math Ph.D.

D H
Staff Emeritus
How many mathematicians does it take to balance a checkbook? I always screwed up on my timecard when it was on paper. The office manager thought it was funny how my whiteboard could be chock full of hairy math and my timecard could be chock full of simple addition errors.

Math Is Hard
Staff Emeritus
Gold Member
can I still become a good mathematician?
That depends on what you mean by "good."

If you mean "Can I one day get a PhD and publish some original research?", then the answer is yes.

If you mean "Can I attain sufficient mastery over mathematics such that the majority of other people with mathematics PhDs will consider me to be good at it?", then the answer to that is yes as well.

Having said that, remember that being poor at calculations is never something to be proud about. There is nothing especially impressive about getting a PhD in mathematics despite being poor at calculation. The truth is that many people are not very well suited to their chosen occupation.

JWHooper
I love doing proofs, but hate doing computational questions. If I become very good with proofs but remain just so-so with calculational problems, can I still become a good mathematician? My favourite subject is topology.
I think 80% of all gifted mathematicians has trouble with numbers, but they're good at variables, equations and geometric shapes. I don't know how this works, but for me, I love arithmetic.

My favorite subject in mathematics by far is calculus, and least favorite is statistics, of course

mgb_phys
Homework Helper
Isn't there a quote that junior school kids need to count upto 1000, high school needs to count to 100, a maths undergrad needs 1-10,e,pi, and a maths PhD needs to count to 0.

Isn't there a quote that junior school kids need to count upto 1000, high school needs to count to 100, a maths undergrad needs 1-10,e,pi, and a maths PhD needs to count to 0.
Would Newton, Euler, Gauss, or Hilbert respect such an attitude? Of course not.

My investigations trace this failure-embracing attitude to Hardy's essay, a mathematicians apology (See attached article, which has recently entered the public domain in Canada).

The impact of Hardy's essay on the modern image of a mathematician cannot be understated. The sentiments in that essay, along with the style of exposition crafted by the Bourbaki group, have had an inordinate role in defining the present day mathematical personality. Before the 1930s it was not considered impressive to be poor at calculations.

In fact, there was a time not too long ago when it was much more difficult to find permanent employment as a research mathematician. Back then it would not be favorable for one's livelihood to brag about being useless and selfish in various degrees, as Hardy does in his essay. And meanwhile the public thinks they have no choice but to accept this. Mathematics at present is like a ritualistic secret society whose membership is highly guarded and which is tolerated for it's rather difficult to pinpoint contribution to society as a whole but is also proud to claim its distance from any such contributions. I don't think it is possible or desirable for the mathematical community to let this stereotype continue to poison our up-and-coming talent.

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mgb_phys
Homework Helper
I believe that the point of the quote is that mathematics is not numbers and the further you go in maths the less important arithmetic becomes.

It is ironic though that Hardy's work is now very useful in cryptography and other military applications.

morphism
Homework Helper
mathboy mentioned computation in general, and not just arithmetic. And, in my opinion, computation is fairly important!

By computation, I did not mean just arithmetic. For example, being able to manipulate a complicated algebraic expression to get it in a desired form; computing that all overlaps in an atlas on a manifold are smooth; searching for a regular topological space that is not completely regular, and then computing (with the bizarre topology you've congured up) that it is indeed regular but not completely regular, etc...

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Gib Z
Homework Helper
I would say that a certain degree of computational skill is required even for proofs of certain things, it is often that desired form that is a key step in the proof. However, luckily, if it one of those things that you can practice at and get better, so don't just give it up, keep trying those types of questions.

JWHooper
By computation, I did not mean just arithmetic. For example, being able to manipulate a complicated algebraic expression to get it in a desired form; computing that all overlaps in an atlas on a manifold are smooth; searching for a regular topological space that is not completely regular, and then computing (with the bizarre topology you've congured up) that it is indeed regular but not completely regular, etc...
So, you're saying that for example, simplifying 3x - 2x + (2 + 2x + 3x^6)^2 kind of stuff is calculation stuff, right?

So, you're saying that for example, simplifying 3x - 2x + (2 + 2x + 3x^6)^2 kind of stuff is calculation stuff, right?
That's still arithmetic.

By computation, I did not mean just arithmetic. For example, being able to manipulate a complicated algebraic expression to get it in a desired form; computing that all overlaps in an atlas on a manifold are smooth; searching for a regular topological space that is not completely regular, and then computing (with the bizarre topology you've congured up) that it is indeed regular but not completely regular, etc...
Which subject do you prefer between algebra and analysis?

I think that algebra and category theory is where most of the modern theory-building research is going on in topology. My suggestion to you would be to go back into the 1960s journals and follow the trail blazed by Grothendeick. This would be a good example of a theoretical progression in contemporary mathematics; see if the calculations are too much for you.

Daniel Y.
That's still arithmetic.
Hmm? I always thought arithmetic was considered the basic operations of addition, inverse addition, multiplication, and inverse multiplication for numbers only, not variables. Where does the arithmetic line stop and the more advanced stuff begin?

That's still arithmetic.
Hmm? I always thought arithmetic was considered the basic operations of addition, inverse addition, multiplication, and inverse multiplication for numbers only, not variables. Where does the arithmetic line stop and the more advanced stuff begin?
Mathboy is using the word "arithmetic" in a broader sense then the American usage. This includes what we normally think of as arithmetic (basic operations on literal numbers) as well as the algebra we do in high school (basic operations on letters that are placeholders for numbers). The foreign usage of the word "algebra" is reserved for what we call "abstract algebra" or "modern algebra" i.e. the study that is usually introduced to undergraduates through groups, rings, fields, etc. I have also heard foreigners use the term "school algebra" interchangeably with "arithmetic", but this seems to be less common.

There was a time when number theory was called "higher arithmetic" and the basic calculations we think of today as arithmetic were called "logistic" (comes from the Greek logos $\lambda \acute{o} \gamma o\sigma$ meaning "order").

Note that whenever I have encountered the foreign use of the terms "arithmetic" and "school algebra" it always seems to be intended to belittle the subject. It seems that the things we humans are quickest to distance ourselves from, and that we are the most ashamed of, are the points of development that we ourselves have recently outgrown.

Math and arithmetic is kind off two different skills. If you want to get good at arithmetic. Begin practicing with an abacus and you will soon become faster at it in the head than on the calculator.

AlephZero
Homework Helper
You don't need to be good at doing "standard" arithmetic, algebra or calculus by hand. That's what Mathematica is for.

Remember the quote from Hammimg, "the purpose of computation is not numbers, but insight". Nobody has yet made a machine to automatically generate the insight, so far as I know. Insight generation is what you need to be good at, and the same applies to science and engineering just as much as to mathematics.

I'm going to agree with the majority here and say that it is not detrimental to be a less-than-stellar calculator. I'm not good in this way at all. In fact, I hated maths in High School for this reason, and only was successful in the field once I saw a proof.

However, I will also say that I wish sometimes that I was better at "computational" mathematics, because during undergraduate exams it can be a source of great stress. Often I'd breeze through the proof portions of exams and struggle on the computational ones. This always stung more for me, as usually I'd leave the exam and know that I made stupid mistakes in my arithmetic or something. On the other hand, I've been told that the way that I think, while not very well suited for computation, is ideal for a mathematician. This has often set me apart from the majority in the undergraduate courses, and I'm thankful for that. Sometimes those students that were stellar in HS and are great at things like algebra and arithmetic get into University level maths and are intimidated by the sheer abstractness of it all, a lot end up changing majors. I never had this problem, so I guess in a way there is a plus side to the issue. I would say, though to try and get better at computation; certainly don't ignore it or write it off, but be proud that you have been successful with proofs and other aspects of real mathematics.

gb7nash
Homework Helper
If you want to be a pure/research mathematician, no. You really don't need computer programming in a lot of fields. If you want to be an applied mathematician on the other hand, you'll be at a disadvantage if you don't know anything about programming.

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mathwonk