Robert Harper > Quotes

“I used to do machine language programming in the lights on the front panel of a computer; now I do higher-dimensional type theory. It's a little bit crazy.”
― Robert Harper

“The central dogma of computational trinitarianism holds that Logic, Languages, and Categories are but three manifestations of one divine notion of computation. There is no preferred route to enlightenment: each aspect provides insights that comprise the experience of computation in our lives.
Computational trinitarianism entails that any concept arising in one aspect should have meaning from the perspective of the other two. If you arrive at an insight that has importance for logic, languages, and categories, then you may feel sure that you have elucidated an essential concept of computation—you have made an enduring scientific discovery.”
― Robert Harper

“The whole beauty of constructive mathematics lies in the fact that it is just mathematics, free of any self-conscious recognition that we are writing programs when proving theorems constructively. We never have to reason about machine indices or any such nonsense, we just do mathematics under the discipline of not assuming that every proposition is decidable. One benefit is that the same mathematics admits interpretation not only in terms of computability, but also in terms of continuity in topological spaces, establishing a deep connection between two seemingly disparate topics.”
― Robert Harper

“Mathematics is itself a branch of computer science, since according to Brouwer’s Dictum all of mathematics is to be based on the concept of computation.”
― Robert Harper

“The force of Church’s Law is that it postulates that all future notions of computation will be equivalent in expressive power (measured by definability of functions on the natural numbers) to the λ-calculus. Church’s Law is therefore a scientific law in the same sense as, say, Newton’s Law of Universal Gravitation, which makes a prediction about all future measurements of the acceleration in a gravitational field.”
― Robert Harper

“The doctrine of computational trinitarianism holds that computation manifests itself in three forms: proofs of propositions, programs of a type, and mappings between structures. These three aspects give rise to three sects of worship: Logic, which gives primacy to proofs and propositions; Languages, which gives primacy to programs and types; Categories, which gives primacy to mappings and structures. The central dogma of computational trinitarianism holds that Logic, Languages, and Categories are but three manifestations of one divine notion of computation. There is no preferred route to enlightenment: each aspect provides insights that comprise the experience of computation in our lives.”
― Robert Harper

“The interesting conception of the propositions-as-types principle is what I call Brouwer’s Dictum, which states that all of mathematics, including the concept of a proof, is to be derived from the concept of a construction, a computation classified by a type. In intuitionistic mathematics proofs are themselves “first-class” mathematical objects that inhabit types that may as well be identified with the proposition that they prove. Proving a proposition is no different than constructing a program of a type. In this sense logic is a branch of mathematics, the branch concerned with those constructions that are proofs. And mathematics is itself a branch of computer science, since according to Brouwer’s Dictum all of mathematics is to be based on the concept of computation. But notice as well that there are many more constructions than those that correspond to proofs. Numbers, for example, are perhaps the most basic ones, as would be any inductive or coinductive types, or even more exotic objects such as Brouwer’s own choice sequences. From this point of view the judgement t ∈ A stating that t is a construction of type A is of fundamental importance, since it encompasses not only the formation of “ordinary” mathematical constructions, but also those that are distinctively intuitionistic, namely mathematical proofs.

An often misunderstood point that must be clarified before we continue is that the concept of proof in intuitionism is not to be identified with the concept of a formal proof in a fixed formal system. What constitutes a proof of a proposition is a judgement, and there is no reason to suppose a priori that this judgement ought to be decidable. It should be possible to recognize a proof when we see one, but it is not required that we be able to rule out what is a proof in all cases. In contrast formal proofs are inductively defined and hence fully circumscribed, and we expect it to be decidable whether or not a purported formal proof is in fact a formal proof, that is whether it is well-formed according to the given inductively defined rules. But the upshot of Gödel's theorem is that as soon as we fix the concept of formal proof, it is immediate that it is not an adequate conception of proof simpliciter, because there are propositions that are true, which is to say have a proof, but have no formal proof according to the given rules. The concept of truth, even in the intuitionistic setting, eludes formalization, and it will ever be thus. Putting all this another way, according to the intuitionistic viewpoint (and the mathematical practices that it codifies), there is no truth other than that given by proof. Yet the rules of proof cannot be given in decidable form without missing the point.”
― Robert Harper