A Quote by Kenneth E. Iverson
Most programming languages are decidedly inferior to mathematical notation and are little used as tools of thought in ways that would be considered significant by, say, an applied mathematician.
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Although mathematical notation undoubtedly possesses parsing rules, they are rather loose, sometimes contradictory, and seldom clearly stated. [...] The proliferation of programming languages shows no more uniformity than mathematics. Nevertheless, programming languages do bring a different perspective. [...] Because of their application to a broad range of topics, their strict grammar, and their strict interpretation, programming languages can provide new insights into mathematical notation.
With the computer and programming languages, mathematics has newly-acquired tools, and its notation should be reviewed in the light of them. The computer may, in effect, be used as a patient, precise, and knowledgeable "native speaker" of mathematical notation.
The properties of executability and universality associated with programming languages can be combined, in a single language, with the well-known properties of mathematical notation which make it such an effective tool of thought.
The precision provided (or enforced) by programming languages and their execution can identify lacunas, ambiguities, and other areas of potential confusion in conventional [mathematical] notation.
My impression was and is that many programming languages and tools represent solutions looking for problems, and I was determined that my work should not fall into that category. Thus, I follow the literature on programming languages and the debates about programming languages primarily looking for ideas for solutions to problems my colleagues and I have encountered in real applications. Other programming languages constitute a mountain of ideas and inspiration-but it has to be mined carefully to avoid featurism and inconsistencies.
An old French mathematician said: "A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street." This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect; for what is clear and easily comprehended attracts, the complicated repels us.
One main reason why the separate nature of the science of operations has been little felt, and in general little dwelt on, is the shifting meaning of many of the symbols used in mathematical notation. First, the symbols of operation are frequently also the symbols of the results of operations.
If a mathematician wishes to disparage the work of one of his colleagues, say, A, the most effective method he finds for doing this is to ask where the results can be applied. The hard pressed man, with his back against the wall, finally unearths the researches of another mathematician B as the locus of the application of his own results. If next B is plagued with a similar question, he will refer to another mathematician C. After a few steps of this kind we find ourselves referred back to the researches of A, and in this way the chain closes.
I think my hight had the most significant single effect on my existence, aside from my brain. In fact, it's part of an inferior-superior syndrome. I think I have an inferior brain and an inferior stature, if you really want to get brutal about it.
Computer scientists have so far worked on developing powerful programming languages that make it possible to solve the technical problems of computation. Little effort has gone toward devising the languages of interaction.
If the system exhibits a structure which can be represented by a mathematical equivalent, called a mathematical model, and if the objective can be also so quantified, then some computational method may be evolved for choosing the best schedule of actions among alternatives. Such use of mathematical models is termed mathematical programming.
The constructs of the mathematical mind are at the same time free and necessary. The individual mathematician feels free to define his notions and set up his axioms as he pleases. But the question is will he get his fellow mathematician interested in the constructs of his imagination. We cannot help the feeling that certain mathematical structures which have evolved through the combined efforts of the mathematical community bear the stamp of a necessity not affected by the accidents of their historical birth.
Eiffel borrows quite openly from several earlier programming languages and I am sure that if we had found a good language construct in C we would have used it as well.
Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. "Immortality" may be a silly word, but probably a mathematician has the best chance of whatever it may mean.
To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. Mathematics is the ultimate in technology transfer.
I was appalled to find that the mathematical notation on which I had been raised failed to fill the needs of the courses I was assigned, and I began work on extensions to notation that might serve. In particular, I adopted the matrix algebra used in my thesis work, the systematic use of matrices and higher-dimensional arrays (almost) learned in a course in Tensor Analysis rashly taken in my third year at Queen's, and (eventually) the notion of Operators in the sense introduced by Heaviside in his treatment of Maxwell's equations.
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