On Mon, 31 Jan 2005, Chung-chieh Shan wrote:
(Is Lemming the same person as Henning Thielemann?)
Yes. :-)
For the expression '1+x' I conclude by type inference that 'x' must be a variable for a scalar value, since '1' is, too. But the expression '1/O(n^2)' has the scalar value '1' on the left of '/' and a set of functions at the right side. Type inference fails, so my next try is to make the operands compatible in a somehow natural way.
It seems to me that your classification of certain notations as "wrong" and others as "right" assumes a certain type inference system that allows adding a number to a placeholder but disallows dividing a function by a set of functions.
Let's see if we share a common interpretation of notation before discussing who uses it consistently and who does not. :-) For me 1, x and + are identifiers. The strings "1", "x" and "+" are not mathematical objects. But '1' denotes a unique mathematical object. Only almost, because depending on the underlying explanation it may be identified with the natural number 1, with the rational number 1, with the real 1, with the complex 1, you will probably need a different representation. Further this representation depends on how you represent natural numbers, reals and so on, e.g. by sets. '+' also denotes a mathematical object, more precisely a function, and again we have ambiguities like that for '1', since '+' might be the natural addition, the rational one or even a mixed one. 'x' denotes no special mathematical object but it may be replaced by one, and the special thing is, that within a certain scope all occurences of 'x' must be replaced with the same mathematical object. Functions are special, in the sense that if I write "1+2" I don't want to consider this as the sequence of three objects denoted by "1", "+", "2" but I want that the function denoted by "+" is applied on the values denoted by "1" and "2". So I imagine a layer of notation and a layer of mathematical objects. Do you share this interpretation? What are consequences of this interpretation? "2+2" and "4" denote the same object. A function has only access to the value (the mathematical object) but not to the generating expression. So how can I define e.g. differentation with respect to a variable? I can't, but I don't miss it because I can define it for functions, and yes Jerzy :-), also for other objects. For me differentiation was introduced by limits. The definition of limits don't need expressions as mathematical objects, the differentation of functions don't need them, too. But later we got used to differentiate expression (e.g. x^2 + y^2 is turned into 2*x * dx + 2*y * dy), but no-one defined what that is. What are the consequences of treating expressions as mathematical objects, too? "2+2" and "4" are different expressions. I guess we would still ask for a value, thus we need a mathematical function which maps expressions to values. I think Mathematicas Evaluate is made for this purpose. If mathematical functions can transform expressions - is it possible that they change my writing? ;-) Indeed, I really like this separation: On the one side expression, on the other side mathematical objects. Simplifications of expressions are nothing I allow a mathematical function. But if I simplify an expression I must assert that the denoted mathematical object remains the same. Differentation of a function is possible if you only know its graph, but a computer algebra system can find an expression for the derivation if you have an expression for the function. There are many functions that can be integrated, but a computer algebra system cannot find an expression without the Integrate function for it. When a pupil says "the function can not be integrated" he means that he couldn't find an expression for the integrated function without the integral sign. When a professor teaching calculus says "the function can not be integrated" he means that there is some divergence. It seems to me that in this case the pupil considers expressions as mathematical objects and the professor does not.
Lambda notation also involves much ambiguity and many implicit conversions.
Following the interpretation above I like to see lambda as a notation instead of an mathematical operator. lambda expressions denote functions. I can replace every occurence of lambda expressions textually. If "\A -> B" occurs I can also write "f" and "where holds \forall A f(A)=B". So lambda notation involves no more ambiguities than the rest of the notation.
You use 'ask' twice in the second expression. Does this mean that there may be two different values for 'ask'? If this is the case your second interpretation differs from my second interpretation.
No -- when I use "ask" I mean the one defined in Control.Monad.Reader.
Sorry for my imprecise question :-) What I meant was that 'ask' somehow injects a value into the mathematical operation, I wondered if the values it introduces are equal or not.