[plt-scheme] Re: Why "lambda"?

On Sunday 31 May 2009, Todd O'Bryan wrote:
> The problem for many students is that they're not willing to make the
> jump from an intuitive understanding of a concept to its formal
> statement, and this is just as true in basic algebra as it is in the
> study of automata. It's always a balance to include enough formalism
> to make one's point clear, but not so much that people who could
> benefit from the concepts are turned off by the exposition. (Stephen
> Hawking, I think, mentioned in the preface to A Brief History of Time
> that his publisher warned him that every equation in the book would
> cut the readership in half.)
>
> The problem for many researchers seems to be that they're too willing
> to jump to a formalization, before deciding if the concept they're
> describing is important enough to deserve such treatment.

Researchers know the conceptual landscape of their area, something 
students, almost by definition don't do. Researchers have the largely 
intuitive background to understand why a concept is formalized in a 
particular way. Indeed, it is an entirely valid to question whether a 
specific formal definition is an appropriate formalization of a pre-
theoretical concept at all. Purely formally, all definitions are equal. 
But only few of them are interesting and useful.

The aspect of being "interesting and useful" is hidden below the 
academic waterline and, in my experience(!), is rarely taught, either in 
lectures or books. Usually, the definitions and theorems sticking out of 
the water are presented as accomplished facts, without a word about the 
reasoning and history behind them.

Coming full circle to the lambda in the subject, has lambda calculus 
always been the formalization of choice for computations? There are 
other contenders, such as Turing machines and more restricted machines. 
Presumably, there are failed formalizations that are forgotten and never 
taught. All these formalizations lend themselves to different purposes 
because they emphasize different aspects.

To students, this backdrop is unknown. They see a formal definition 
without the benefit of knowing its history, wondering why it is supposed 
to be a good formalization of their pre-theoretical understanding and 
what it's purpose might be.

Michael

-- 
Michael Schuerig
mailto:michael at schuerig.de
http://www.schuerig.de/michael/