How much math you need for programming (2014)

If you're programming a basic business app then you won't need too much math. If your app is simulating the big bang origin of the universe you would probably need a bit more.

How much? Depends. In some contexts essentially nothing. In other contexts up to and including some of the most advanced known so far with advances wanted.

First, Military

During the Cold War, around DC a large fraction of the interest in programming was for some of the math for the work of the Cold War.

Then I was programming at the David Taylor Model Basin in numerical fluid flow, that is, solution of the Navier-Stokes equations. Some of the work involved an early computer algebra package Formac; I used it for local series solutions to the equations.

Later there was work with the fast Fourier transform (FFT): A guy had wired up a hardware box for the algorithm. I showed him a version of the FFT that used the same signal flow graph at each algorithm stage. His jaw hit the floor: That version of the FFT could have saved him a big fraction of all the circuitry in his box.

Later I was in a software house bidding on a software project from a Navy lab. In part the lab wanted to be able to analyze and simulate ocean waves. They wanted to regard the waves as a second order (covariance) stationary stochastic process. For the analysis, calculate the power spectrum of data collected at sea on ocean waves. Then for the simulation, generate sample paths of the stochastic process with that power spectrum. So, I dug into

Blackman and Tukey, The Measurement of Power Spectra,

saw the statistical basics, and quickly wrote some illustrative software that did the work. One of the lessons was, to get an accurate power spectrum takes, roughly, some number of cycles of data at the frequencies of interest so that at low frequencies, i.e., for ocean waves, need possibly a lot of data, hours, where in radio engineering can need just seconds. I showed the results of my software, and our company got sole source on the software development project!

Second, Academic

At one point, I was consulting in programming, data analysis, and algorithms at Georgetown U. and teaching courses in computer science. One prof had programmed some polynomial curve fitting and was getting bad numerical accuracy. There are connections with the notorious Hilbert matrix. I programmed for him a version based on constructing polynomials orthogonal over the given input data; that approach had much better numerical accuracy.

Third, Commercial

E.g., GE was running a computer time sharing service, and there I was the main GE HQ guy for math, numerical analysis and statistics. So, it was statistical hypothesis testing, classic regression analysis, analysis of variance, factor analysis, polynomial curve fitting, some non-linear curve fitting, random number generation, sorting algorithms, all of standard numerical linear algebra, interval arithmetic (for guaranteed numerical accuracy), digital filtering, the fast Fourier transform, solution of ordinary differential equations, and more.

Later I was programming at FedEx. A first little bit of math, hard for the other FedEx people, was how to calculate great circle distances. A simple solution is to use the law of cosines for spherical triangles. The software I wrote to schedule the fleet used that solution.

Later FedEx wanted some revenue projections. We knew (a) our current revenue and (b) the revenue at the planned full service. So, the projections were a version of an interpolation between those two. So, how was the interpolation to go? I argued that the growth would be from word of mouth advertising, that is, viral growth and that growth would be directly proportional to (i) the number of current customers talking about the service and (ii) the number of target customers to be hearing about the service. This led to the first order, linear, ordinary differential equation initial value problem

d/dt y(t) =

y'(t) = k y(t) ( b - y(t) )

where t is time (in days), y(t) is the revenue (dollars) a time t, b is the planned capacity of the service, k is the constant of proportionality, and y(0) the current revenue. There is a simple closed form solution. We picked a value for k that gave a reasonable graph, and that was our projections.

Uh, with more details, each of those two FedEx projects saved the company once.

It may be that currently in the commercial world there is interest in principal components analysis and various cases of curve fitting, linear and non-linear.

Fourth, Startups

Currently I have two startup projects going: In some contrast with

Sam Altman, How to Start a Startup

at

https://startupclass.samaltman.com/

key to both is some relatively advanced, original math I derived. The math is the crucial key to being able to deliver the service and is the secret sauce, "unfair advantage", technological barrier to entry, difficult to equal.

Broadly, the power of current computing is a wide open invitation to use the computing to do the data manipulations for some math that can be powerful for some real problems. Then that computing needs programming so that it can help if the programmer knows the math.

Net, in some contexts it can help if a programmer knows some math.

What math you need has changed over time. Today, probability and statistics are used more, while discrete math and number theory are less essential. CS used to have a lot of automata theory, and that's not important much now unless you have to write regular expression parsers.

If you do anything with graphics, you need a subset of matrix math, and there are graphics books which cover what you need. If you get into robotics, you need differential equations.

I didn't get a CS degree precisely because I bombed out of math so bad in high school that I couldn't get admission to any program. Here in Canada a good grade in academic-stream high school math was simply mandatory for admission to any BSc program. Took a philosophy BA instead, though didn't finish it... dropped out to go get a job programming during the .com boom, and now 25 years later I work at Google.

Was a long road, and would have been a lot less torturous if I had been better in math. Not because I've needed math much in my programming career, but because my bad math became a reason that gatekeepers closed doors on me over and over again. Perhaps the strongest cause of imposter syndrome I have at work is less related to not having a CS degree -- because I feel like I know about 99% of the content of any CS program -- but more to do with the fact that I work around a lot of maths geniuses.

In reality the "math" I failed in high school was more arithmetic than mathematics per se. I feel like I might have a disability around numbers but I've rarely had problems with the higher level symbolic manipulation involved in algebra or programming or symbolic logic, etc. though my probably-ADHD means I have to really step back and work through things at my own pace in my own environment, and so whiteboard coding type interviews really piss me off.

You know honestly, we as an entire whole of the world should just go back to hands on training. Forget studying the same crap for 15-20 years of our lives to have a damn calculator to solve basic adding and subtracting that's done on a cash register in our daily lives. How many of you solve out of this world equations in your daily life? 95% of the world doesn't. We all use basic add and subtract. Hardly anyone knows how to start a fire, or kill game to survive. If the internet goes out and you can't check that daily habit of yours, your life is turned upside down. Its a waste to study all these years to find out you don't need to figure out a right side of triangle to drop a tree. If you can count to 10 on your fingers, what's the point of an in depth educational system.