On Mon, March 23, 2009 12:17 pm, [email protected] wrote: > James, Good morning! Thanks - apparently it's -7. > On Mon, March 23, 2009 12:07 am, [email protected] wrote: >> Well spotted, James: but you are making an (entirely reasonable) >> assumption about the measure. >> Add |cos z| = 1 to the domain if you wish but then you will complain >> that it is (assuming z is a 'real variable', of course). > Which I did since $z<0$. I (subconsciously) played Axiom in my head and > looked for an ordered set with a cos functions. >>>> > ... maybe not you, but I would have started from the assumtion that z --> > 'complex variable' until, as you found, that becomes untenable and > starting to 'calculate the universal element in some subcategory'. > > The moral (or executive summary) being that this is exactly how many > mathematical formulas communicate their meaning: both by using assumptions > about deductions that the reader will make and/or by (more or less > implicitly) pointing out that the assumptions may be wrong. Indeed. In particular, though not exclusively, there is a strong element of what CS would call "type inference". AS you point out, we don't all do this the same way, and I suspect even mine is not amenable to full logical description. > This shows both the communicative (to humans) power of such very > abbreviated maths notations and the fact that much of it works only with > humans who make exactly the correct deductions about unstated assumptions > (particularly about 'variables'). > > The question is whether the 'mathematical semantics' can be said to be > captured without making explicit certain features that are often, but not > always, absent or very much implicit in the notation typically used. They > probably can in the computational and logical sense of semantics but is > this level of expressivity: > > -- all that OM3 is trying to achieve? > > -- sufficient for the 'semanticly rich encoding of mathematical material' > > For example, when encoding useful, concrete 'mathematical semantics' in > the 21C, is the concept of an: > > expression in a 'universal' or 'unconditional' bound variable > > sufficient to encode 'a function' or only a 'template for a collection of > functions'? > > Further if it is the only the latter, what has this got to do with > concrete mathematics as she is done and taught by many mathematicians? > > Thus it is the (apparently or actually) 'unconditioned variables' that > worry me: they are great for symbolic logic but that is not much of > 'mathematics as we know it'. Even 19C (and probably 17-18C) 'variables' > were not 'universal' but are (possibly by assumption) 'real' or 'complex' > variables or integers (or often more concretely 'quantities'). Quite so, and there is a (small) group of people who woryy about this in CA, and distinguish between 'variables ' and 'unknowns'. > As soon as the variables get conditioned then one gets some idea of the > 'domain over which they range': something that is essential if they are in > fact being used to 'describe a function'. And in anything I would > describe > as 'every day mathematics' such function-like things, with definite > domains > available when needed, are (and have been at least since Cauchy) > fundamental and omni-present. > > Hence I want all 'bindings' in OM to require a 'condition on the bound > variables'. Syntactically this can, perhaps, be omitted but with an > assumed default value. > > Manifesto: a pure, context-free lambda-expression neither describes a > mathematical function nor expresses any 'mathematical semantics'. This is a (doubtless intentionally) provocative statement, with which I have some sympathy. Of course, it is technically false, a "pure, context-free lambda-expression" expresses the mathematical semantics of such an object in the lambda-calculus. Whether this is relevant to much of the rest of the mathematics is the question.
James Davenport Visiting Full Professor, University of Waterloo Otherwise: Hebron & Medlock Professor of Information Technology and Chairman, Powerful Computing WP, University of Bath OpenMath Content Dictionary Editor and Programme Chair, OpenMath 2009 IMU Committee on Electronic Information and Communication _______________________________________________ Om3 mailing list [email protected] http://openmath.org/mailman/listinfo/om3
