On 11/12/2013 12:06 PM, Herman Rubin wrote: > On 2013-11-12, Hetware <email@example.com> wrote: >> I'm working through a 1953 edition of Thomas's _Calculus And Analytic >> Geometry_. When I work problems, I use Mathematica to type my >> transformations, and to check my results. I use it for far more, as >> well; graphing, numerical solutions, etc. > >> Many years ago I found computers to be a nuisance when it came to math, >> and more importantly physics. I was contented to have a piece of chalk >> or a pencil and an eraser, than to have all the computing power in (the) >> Universe. Time was the only resource I found in short supply. > >> Now that I have used them for years, I realize that computers can do a >> whole lot. They can find integrals for equations which I cannot >> integrate by hand. They can produce graphics which a human could never >> produce, etc. > >> I've used a pocket calculator since the 1970's. But, I feel as if I >> should have learned to work the same problems on my own. I feel >> somewhat crippled by using it as a crutch. > >> I'm in a conundrum twixt the use of computers to do my thinking for me, >> and learning to think for myself. Should a child learn his times >> tables, or learn to use a computer to do it for him? > > > Computers can do things which they have beeen programmed to do > and nothing more. I have found that it can be somewhat unwise > to trust computer programs too much. I have found errors in them, > most recently yesterday.
Along these same lines...
There have been two articles within, say, the last three years talking about mathematics in relation to the sciences upon which we depend. Both appeared in prominent science magazines directed to the general public.
One of them, appearing in Science News, had been directed at the unsound use of statistics in general scientific study. The other, appearing in Scientific American, considered the consequences of using highly accurate computerized calculations. The premise of its underlying question had been how the elimination of "fudge factors" previously introduced at intermediate points in calculations might impact the reliability of the physical systems built by engineers.
Since it has been some time, I have certainly oversimplified the articles with these statements. But in both articles, the primary issue questioned the use of computationally simplified numerical methods without judgement based on the underlying mathematical principles. One can introduce covert error where the computer programs themselves have no errors.
In fact, this has been the defense of numerous financial analysts who developed the programs leading to the debacle of securitized mortgages. Complex computer programs rely on numerous rigid assumptions. On their account, the changing laws surrounding housing markets invalidated assumptions upon which those mathematical models had been constructed. Of course, this is a "he said"-"she said" thing. But, if true, those people who relied on those programs and who would not purposely choose to engage in fraud did not recognize that their programs were no longer faithfully calculating risks.
A similar analysis had been applied to accounting methods prior to the recession associated with the "dot com" collapse. In the past, accounting records had been relatively stable because of costs involved with maintaining records. With the advent of spreadsheet programs, it became a simple matter to compare the analyses of different bookkeeping policies. With exception for the bare minimums of GAAP governed by legal policies associated with taxation, companies could undermine assumptions involved with annual reports by adjusting bookkeeping and business practice at a moment's notice.
One could argue that there is *more* reason to learn the mathematics with the power computational tools bring to the table.