Earlier in my day, I ran across a book review for Computer Science Reconsidered: The Invocation Model of Process Expression. The premise of the book, at least garnered second hand from the review:
Mathematicians and computer scientists are pursuing fundamentally different aims, and the mathematician's tools are not as appropriate as was once supposed to the questions of the computer scientistI had a number of reactions to that immediately. Most of them were, frankly, emotional. Certainly most of the code that gets churned out these days has very little conscious basis in Mathematics. But I would argue it doesn't have much of a conscious basis in Computer Science, Algorithms, or Finite Automata either which are all definitely critical some of the time. That is largely because so much of it is so well understood, and so abstracted away from first principles, that the underlying rigor isn't required to get the day to day bits churned out. The more important the code is, the more it moves down that spectrum of rigor.
But when we're really reaching for something new, something interesting, and something that isn't just an incremental change from the conventional wisdom, then my instincts say that Math provides a very valuable framework for describing something out of nothingness.
My gut was validated just hours later when reading an IEEE journal article on the so-called FastTCP active queue management TCP congestion control algorithm. The approach looks at congestion control as "a distributed algorithm over the Internet to solve a global optimization problem [.. to ..] determine the equilibrium and performance of the network"
Moreover, the underlying optimization problem has a simple structure that allows us to efficiently compute these equilibrium properties numerically, even for a large network that is hard to simulate.It seems clear here that math provides very strong underpinning for what the article needs to describe and achieve. To be fair to the author of the original book, he was trying to promote another basis for expressing key Computer Science thoughts: "the invocation model of process expression". Which from casual glance looks interesting, I just don't get why you have to tear down something old (e.g. "The Problem: Why the underlying theory of contemporary computer science is not helpful") in order to build up something new.
Specifically, we can regard each source as having a utility function that measures its “happiness” as a function of its data rate. Consider the problem of maximizing the sum of all source utility functions over their rates, subject to link capacity constraints. This is a standard constrained optimization problem for which many iterative solutions exist. The challenge in our context is to solve for the optimal source rates in a distributed manner using only local information. A key feature we exploit is the duality theory. It says that associated with our (primal) utility maximization problem is a dual minimization problem. Whereas the primal variables over which utility is to be maximized are source rates, the dual variables for the dual problem are congestion measures at the links. Moreover, solving the dual problem is equivalent to solving the primal problem. There is a class of optimization algorithms that iteratively solve for both the primal and dual problems at once.
TCP/AQM can be interpreted as such a primal-dual algorithm that is distributed and decentralized, and solves both the primal and dual problems. TCP iterates on the source rates (a source increases or decreases its window in response to congestion in its path), and AQM iterates on the congestion measures (e.g., loss probability at a link increases or decreases as sources traversing that link increase or decrease their rates). They cooperate to determine iteratively the network operating point that maximizes aggregate utility. When this iterative process converges, the equilibrium source rates are optimal solutions of the primal problem and the equilibrium congestion measures are optimal solutions of the dual problem. The throughput and fairness of the network are thus determined by the TCP algorithm and the associated utility function, whereas utilization, loss, and delay are determined by the AQM algorithm.
Maybe being shocking is good for selling books. Though, I'm not sure the labeling of math as not helpful is all that shocking to the general book buying population.
Check out www.fastsoft.com where some of the authors of that paper have created a clever hardware bridge to seamlessly migrate a legacy TCP data center into one that sends with FastTCP congestion control algorithm.
Karl Fant's book ... rumors of the death of Math exagerated? -
08-04-2007, 04:43 AM
Why is it that professed revolutionaries are so often so old-fashioned in their approaches to publishing? 
Re: Primary school maths is sufficient for most day-to-day activities for most people -
08-04-2007, 04:54 PM
The pub traditions of English and Math teachers -
08-08-2007, 05:42 AM
I know, and am proof of the fact, that math is not critical to have excellent computer skills. Logic is important, but logic is not the same as mathematics. I excelled in geometry (the only math class that I can say that), but that was because it was so much to do with logic problems. Math, on the other hand, is just out there... somewhere... and has little to do with everyday computing tasks.
Would I expect software developers for NASA and ESA missions into outer space to fully understand advanced calculus? Of course! But network integration specialists, senior network engineers, system administrators, and developers of general purpose software rarely need to know more than ratios and college-level algebra.
Instead of forcing computing graduates to learn math, the colleges and universities should require training in business and basic accounting, interpersonal relations skills, and electives in marketing, graphic design, business law, and foreign languages. Those are the skills that most computer professionals need to do their job well in our modern society.
As an IT manager, I make hiring decisions based on real-world skills, applicable experience, and a solid foundation of business and people-oriented coursework. The only thing I'm looking for when it comes to math is that an employee can fill out work orders, invoices, timesheets, and expense vouchers properly. Most of the IT managers I know, except those working for State and Federal government and educational institutions, seem to look for the same skills.
Is math necessary for good computer skills? Nope, except in a few special situations and when applying for jobs with employers who believe the hype.
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