Whenever I want to learn a new subject, I announce a graduate course in it, since the best way to learn is by teaching. But even better than teaching humans is teaching computers, i.e. program! Since computers will not let you wave your hands and wing it.
– Doron Zeilberger (1999)
Suppose that we are given an -dimensional first-order vector ordinary differential equation (ODE)
where is a vector field. A solution of this ODE is a function whose first derivative, , is equal to . Elementary, my dear Watson!
Despite the fact that the notation is rather pervasive, it is dangerous, in my humble opinion. One should keep in mind that such notation is an abbreviation. How could it not be an abbreviation? Note that is a function, and that the argument of is a vector in , not a function! In other words, writing results in a type error, and a compiler / interpreter with type checking would never accept the declaration . We, fallible humans, accept it because we know (do we?) that it is an abbreviation, and because it is tradition. Who are we to challenge a centuries-old tradition?
If is an abbreviation, what exactly is it an abbreviation for? It is an abbreviation for the following
where the symbol denotes equality of vectors. Note that the argument of is the vector (i.e., function evaluated at ), not the function . But, what does the equation above say? It says the following:
Proposition: Given a vector field , there exists a function , whose first derivative is , such that for every we have that .
This proposition may be true or false. Not all differential equations have solutions, after all, which is why the study of existence of solutions is (was?) an active area of research. Note the presence of the words “for every” in the proposition above. Do you see any universal quantifier, , anywhere? You do not, and that is due to the fact that is also an abbreviation! The universal quantifier is implicit. If we make it explicit, we obtain
where ranges over . The universally quantified formula above is the conjunction of infinitely many equations. However, do note that the words “there exists” also appear in the proposition, which suggests that an existential quantifier is missing. Therefore, the non-abbreviated notation would be as follows
where ranges over the set of all (continuous?) functions from to , and ranges over .
A better notation
So far, I have criticized the abbreviated notation for differential equations. I consider it harmful, as confusing a function with its evaluation is an atrocious crime against types. To make this post more “constructive”, I will take the liberty of proposing a better notation.
We start by noting that can be written in the form , where the symbol denotes function composition. We now introduce a differential operator , which maps functions to functions, so that we obtain the first derivative of function via . Hence, , which allows us to rewrite in the following form
where ranges over . The universally quantified formula above states that functions and , both of which from to evaluate to the same values for all possible choices of the input , i.e., functions and are equal. The formula above can thus be rewritten more compactly as .
Finally, we conclude that is equivalent to the following existentially quantified formula
where the symbol denotes equality of functions. This new notation may not be as compact as , but at least now the RHS does not result in type error. Criticism would be most welcome.