I am relatively new to the world of academical mathematics, but I have noticed that most, if not all, mathematical textbooks that I’ve had the chance to come across, seem completely oblivious to the existence of lambda notation.
More specifically, in a linear algebra course I’m taking, I found it a lot easier to understand “higher order functionals” from the second dual space, by putting them in lambda expressions. It makes a lot more sense to me to put them in the neat, clear notation of lambda expressions, rather than in multiple variable functions where not all the arguments are of the same “class” as some are linear functionals and others are vectors. For example, consider the canonical isomorphism –
It would usually be expressed by Av(f)=f(v)
This was a notation I found particularly difficult to understand at first as there are several processes taking place “under the hood”, that can be put a lot more clearly, in my opinion, this way:
I agree that this notation may become tedious and over-explanatory over time, but as a first introduction of the concept I find it a lot easier as it makes it very clear what goes where.
My question is, basically, why isn’t this widespread, super popular notation in the world of computer science, not as popular in the field of mathematics? Or is it, and I’m just not aware?
As Derek already said, there is no essential difference between functions A×B→C and functions A→(B→C) via Currying (this is also more abstractly expressed by the universal property of an exponential which unifies the set-theoretical currying and currying in a typed lambda calculus).
On the notational side of things, I personally prefer x↦f(x) to λx.f(x) and I suspect many other mathematicians feel the same (especially since λ is such a commonly used letter).
EDIT: (now that my answer stopped being one, let me add some rambling that the 29 people so far have not upvoted for):
I’m guessing many mathematicians are less “comfortable” with nested expressions like v↦(f↦f(v)). That would be nothing extraordinary, since there are various concepts that some mathematicians feel less comfortable about. Here are two (unrelated) things that I have encountered:
- empty metric spaces: Some people deliberately require metric spaces to be non-empty which is a nuisance: given a metric space (X,d) and Y⊆X, (Y,d|Y2) is a metric space again… unless of course Y=∅; apparently it doesn’t feel “right” for metric spaces to be empty
- f(x) instead of f: Some people refer to a function f as f(x); this is (unfortunately) what I learned in high school and is (rein)forced by notation like df(x)dx and ∫f(x)dx
Although, your example:
Let A:V→V∗∗ such that Av(f)=f(v) for all v∈V and f∈V∗
is fine and not hard to understand, in my opinion. For every v∈V we have Av∈V∗∗, i.e. Av:V∗→K. Hence we can plug in an f∈V∗ to get f(v)∈K. If the author thinks it is easy to understand and is more used to it than v↦(f↦f(v)) then they would obviously have no reason to change the notation.
So the reason why v↦(f↦f(v)) (or a variant thereof) is not used as much is probably: “I’m not used to this notation and I’m perfectly happy with mine.”
By the way, my personal favourite is also not:
A:V→V∗∗,v↦_(v) where it is implied that _ is a placeholder, i.e. _(v):V∗→K,f→f(v).