Does there exist applied mathematics research that involves primarily pencil-to-paper work, without the use of computers? Or, are computers quickly replacing everything that an applied mathematician can once do by hand / traditionally does by hand?

Anything in academia in applied math research that computers can't do?

For instance, do PDE theorists use computers or strictly pencil and paper / chalk and chalkboard -- and leave the numerics of PDEs to numerical analysts instead? Is there such a distinction, or is there instead always some sort of collaboration between the two types of applied mathematicians?

Another example that I can think of is the work of probability theorists.

Any insight to my very naive question would be greatly appreciated :)


  • 9
    Even in applied math you often need to prove things.
    – Bitwise
    Commented Jun 4, 2016 at 2:39
  • 4
    What do you mean by "use computers?" Essentially all papers are written using a computer these days.
    – Thomas
    Commented Jun 4, 2016 at 3:36
  • 10
    There's not really any clear definition of "applied mathematics". For instance, my own research essentially straddles the fields of PDE and probability theory (coincidentally the two that you named), yet I consider myself a pure mathematician and none of my papers have involved any digital computation. Commented Jun 4, 2016 at 4:28
  • 2
    Today's SMBC: smbc-comics.com/index.php?id=4130 Commented Jun 4, 2016 at 5:36
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    I'll add a data point which isn't enough for an answer: I have a couple of cosmologists in my group who do theory work, and they use a paper and pencil for everything. They have computers for answering emails, but everything they do is on-paper maths. Commented Jun 4, 2016 at 10:43

2 Answers 2


I am an applied mathematician and I can't work without paper and pencil and in most cases I can't get things done without a computer.

My computer can't tell me in advance if my algorithm will converge, at what rate and what the error will be. My computer can't figure out how to discretize some new problem (unless I tell him how). My computer (up to now) never had a great idea.

On the other hand, I can't invert matrices with pen and paper and I even can't multiply a large matrix like that. In fact I can't even store most data I use on paper... (and also access rates for data on paper are pretty slow).

tl;dr I basically always use paper, pen and computer.

  • You could do Gaussian elimination by hand on large matrices, but you're likely to make a mistake.
    – Bill Barth
    Commented Jun 4, 2016 at 14:16
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    If they fit on my paper and I have a cubic amount of time and patience...
    – Dirk
    Commented Jun 4, 2016 at 14:44
  • Newsprint comes on wide rolls of virtually infinite length.
    – Bill Barth
    Commented Jun 4, 2016 at 14:57

In addition to @Dirk's excellent answer, which I wholeheartedly concur with, I will also note that many types of applied mathematical research are done entirely without aid of a computer.

Consider, for example, this well-cited paper on Brewer's conjecture, which is all mathematical reasoning about the nature of distributed computations. At the more theoretical side of any applied field, you can find such papers, where the conclusions are in the symbolic mathematics that is still best carried out by hand, most typically on paper, whiteboard, or blackboard in its crucial early stages.

Of course, basically nobody submits physical paper to journals any more, so I guess a computer does get involved eventually...

  • Thanks so much for your answer and for the link to the paper, @jakebeal :)
    – User001
    Commented Jun 6, 2016 at 0:21

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