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The Purpose of Computing Is Not Numbers But Insight
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This famous quote by Richard Hamming (of the Hamming code and distance fame) was in the preface of a numerical analysis text he authored. I briefly glanced through the text several decades ago but have largely forgotten it except for this sentence above. Today I happened to re-read one of my earlier blog articles on “advices to student pursuing ph.d studies http://www.sciencenet.cn/blog/user_content.aspx?id=11667” and noticed one of the reader comments “ . . . 我们单位有这样的博士和教授，硕士研究生时还学了一点实验技能，包括硬件和软件的，读了博士之后，反而什么技能都没有了，只剩下matlab仿真的技能，为了写论文而仿真，为了SCI和EI检索而写论文，写的论文都是仿真的结果。仿真是科学研究的一个方法，但是永远仿真如何得了。而我们的很多教授就是靠做仿真写检索论文而生存的，因为靠这个也可以申请到国家自然科学基金项目，也能结题 . . .”I did not choose to respond at the time. However, the comment clicked with the above quote.
Yes, this is a problem – what the reader described is a bit like a physicist or chemist doing endless experiments and accumulating enormous amount of data without ever deriving a theory or conjecture concerning the underlying structure that generated these data. Another way to dramatize this view is to say that if you had a good understanding of the problem most of the times, you should already know the results you will get from your experiment. The data merely confirm what you have guessed or theorized. For the few rare instances that the experimental results SURPRISE you, you should also feel happy for this also signals the possibility of new discoveries. In my opinion, it is this sense – a computing experiment either confirming what you theorized or leading to new discoveries – that Hamming meant to imply by his quote.
Science typically proceeds iteratively along the following paradigm.
· Observations
· Conjecture
· Design of experiments
· Modify conjecture if needed as a result of the experiment
· More experiments and iterate until
· Confirmation of theory
Until the advent of computer, development of mathematics are largely missing the “experimental” part of the above cycle. This is particularly evident for Applied Mathematics which was the reason that it used to be synonymous with theoretical physics in the old days. More recently as I advocated in my blog article “应用数学与理论工程学(原文及译文）Applied Mathematics and Theoretical Engineering, http://www.sciencenet.cn/blog/user_content.aspx?id=38683 “mathematics began to be applied to man-made phenomena. In this realm, observation and direct experimentation in the scientific sense above may not often be possible or feasible. Computer simulations become the logical substitute.
It is in this sense, I sympathize with the above complaint by the student and urge the title quote on practicing simulation scientists.
I close by giving a personal experience illustrating this interplay of experimentation and theorizing. In the 1980s, I initiated the study of perturbation analysis of discrete event system which purports to answer efficiently the following question – WHAT will be the effect on system performance IF we change one of the system parameter by a small amount? (see http://www.sciencenet.cn/blog/user_content.aspx?id=38683Why Is Science Conservative? - 科学为何是保守的（一）). This was at that time an entirely new method of computing which caused a fair amount of controversy initially. In the course of developing this method and confirming the theory, we constructed a computer simulation model to test it experimentally . The result confirmed our analysis with a small error of 3% numerically. Ordinarily, the story could have ended here accepting the 3% as acceptable experimental error. But I had a nagging unease as to why I cannot reduce this error to an even smaller number despite various careful numerical and statistical remedies. After much thinking and detective work with the simulation model, it suddenly dawned on us that we made an unimportant but serious conceptual error in the programming of the model. As discussed above, we wanted to see the effect of varying a system parameter by a small amount ,say 5%. The parameter in the experiment happens to be the mean of a random variable use to represent the service time distribution of a machine. The distribution we happened to use is exponential (i.e., the sample values of the distribution can be from zero to infinity with probabilities following an exponential curve). To change the mean of this distribution in the computer program we simply added 5% of the mean value to every sample value generated by the original distribution for convenience – a seemingly innocent short cut which will give a new mean 1.05 times larger than the original by construction. What we did not realize initially is that we have fundamentally changed the distribution from an exponential type to a non-exponential type (note in the new distribution, there can be no possibility to get any sample with value smaller than 5% of the original mean value since this value is added to every sample originally generated). With this hindsight, the 3% error was eliminated by simple quick change in the computer program. The lesson of this episode is how an experimental result forced us to look closer at our analysis and enabled us to have a deeper understanding of the theory we are creating.
https://blog.sciencenet.cn/blog-1565-49124.html
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