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Date: Thu, 15 Jun 95 14:31:10 EDT From: tynor (Steve Tynor) Subject: fwd: [eiffel@swissoft.h.provi.de] powers To: nice-esg-libs ------- start of forwarded message (RFC 934 encapsulation) ------- From: eiffel@swissoft.h.provi.de (Michael Schweitzer) To: tynor@atlanta.twr.com Subject: powers Date: 15 Jun 1995 19:09:00 +0200 Dear colleagues, Bertrand Meyer said that he was disappointed by my notes about the power operator, that my tone was 'contemptous' and that I seem to claim to have a monopoly on mathematical soundness. Maybe that the tone was contemptous - excuse me. But I cer- tainly don't claim to have a monopoly on mathematical sound- ness - I just have a profound mathematical education. If the Eiffel community tries to redefine mathematics, Eiffel will not succeed, because there are too many people out there who can judge the mathematical soundness of (math.-) Eiffel clas- ses themselves - they don't need my advice or the advice of whomever (fortunately). Bertrand wrote: > I must say in particular that I don't understand > the relevance of Michael's distinction between alge- > braic and analytic properties. Applying his reasoning > to multiplication rather than exponentiation, we could > write, paraphrasing him: The main difference is that algebraic concepts are more 'general' than analytic concepts because there are (in a sense) much more algebraic structures than analytic struc- tures. For most number fields one can neither define lo- garithms nor an exponential function, polynomial rings don't have them either, etc. But they all have a power operator which takes an integer argument. It was not by accident that the natural numbers were discovered by humans long before someone even thought about 'zero' or even negative numbers - not to mention irrational numbers. Bertrand's 'satire' is in fact a satire, because multi- plication is in general not an abbreviation for addition: a * b does not mean a + a + ... + a (b times) in general. In fact, this is only true if 'b' is a non-negative integer. Or how would you interpret A * B if A and B are square matrices (or polynomials) in terms of addition? What should it mean to add the matrix A 'B' times to itself if B is another matrix? Now you could say that we've restricted the meaning of the power operator and that ISE wants to have a more 'general' notion. But the point is that NUMERIC must capture the common features of all possible descendants. Therefore one must answer the question: what is the _meaning_ of class NUMERIC; if someone asked you whether a certain class X could be made a descendant of NUMERIC, how would you decide whether to answer 'yes' or 'no'. We, the Gustave team, take the view that NUMERIC should approximate the concept of a ring (for those who don't know the term: take the integers, reals, complex numbers, polynomials (with real coefficients, say) and matrices (real entries, say) as examples - and then think about the commonalities). Rings are a mathematically sound and well understood concept which can be approximated very well in Eiffel. With best regards, Michael SwisSoft, Michael Schweitzer Geismar Landstr. 16 D-37083 Goettingen Fax : +49 551 770 35 44 email : eiffel@swissoft.h.provi.de ------- end -------
