This remarkable book covers the history of the notion of formal proof and more recently the notion of computers dealing in proofs. There has been controversy since Aristotle over the rationale for thinking about what it means to be a proof, but when people began to program computers to recognize and find proofs, many thought that we had gone too far.
Very interesting perspectives are described around page 208 in the chapter ‘Social Processes and Category Mistakes’.
Dijkstra says early:
Computers can be said to understand things that we can’t. Computers know the sum of one million numbers in a snap and we generally take for granted whatever the computer says. There is yet controversy over whether computers can know propositions. They can certainly record them, using our words, but to know them is to deal with relationships between propositions, such as which may be derived from which. Should we believe a proposition because the computer says that it may be derived?
Only occasionally does the book observe that checking a proof may be tedious but can be done by a simple program which is an order of magnitude smaller than either the sorts of kernels or proof finders aspired to in the heyday of formal proofs. (He addresses the issue fully on about page 290.) Faith in such results relies on belief that:
I did not believe that simple definitions of proofs could capture the essence of proof until after a few courses in logic and foundational math at Berkeley. I do believe it now. Few have direct reason to believe that such simple rules capture logic faithfully. I am not sure that all convincing mathematical arguments can be put into this form but I suspect that the desired propositions about computer programs can be. We are left with a priesthood of artisans who can vouch for and attest to such validity.
There remains the significant problem of understanding what has been proven. The proof is not much good if the proposition cannot be understood or the proposition does not bear on our needs.
There remains the real practical problem: How many mistakes do computers make, and especially how can the enemy exploit them. This question has not yet been broached in the book, as of the end of chapter 6. In Keykos we attempted to introduce some random errors to see what fraction would impact system integrity. This was an uncertain process but it seemed that only a small fraction of random transient errors led to any external effects.
Page 290:
See text following “I would believe that” here.
At the end of page 292 MacKenzie describes Boyer’s project for his incomplete project for the verifier to verify itself. This would indeed be useful but then you are left with the significant problem of understanding the specifications of the verifier being proven.
I am amazed at MacKenzie’s report of Boyer’s attitude toward proof checkers. I seems a bit like a gold prospector returning to his grub stake provider and saying “Yes there is gold out there; I left it there; no I did’t keep exact notes as to where it was; maybe I can find it again.”. This is also a bit like the quote from Dijkstra suggesting that finding a proof was to convince one’s self that something was true, rather than convincing someone else that it was true. I think that we evolved a proof notion in order to convince others. Perhaps the proof finder is supposed to have so much credibility that others are convinced by mere claim of having found a proof. I would be less impressed than the original programer’s claim that the program was debugged.
I was not much impressed with the last chapter: Machines, Proofs and Cultures”. It examined the sociology around the problem of reliable software. I sense a duality in many of those that MacKenzie describes; bordering on dualism. If you are into dualism I would suggest that what goes on inside a computer may more properly of the world of the mind than of the physical world. It is not human; it is alien; but it is of the mind!
There is an important point that is made in several points in the book concerning the difficulty of understanding what has been proven. I fear that we will settle for proving what we can assert in some arcane language that only a few can understand. The web based research that I have done seeking the semantics of the languages used in proof technology has yielded nothing. The book is useful in providing grist for Google searches.
My biggest caution is the dfficulty of understanding what has been proven. Too often someone proves an insignificant fact about a program and announces that the program has been ‘proven’. How often do you hear the question “What was proven?”? How often do you hear an answer? How often do you understand the answer? If it was that the “Program matches specifications.” what language were the specs written in? How many people understand those specs?
Anticipating a phase of formal specifications for Keykos we contemplated some of these questions. We did not come up with a completely satisfactory answers but the effort led us to find a couple of high level bugs. Sometime threat of formality improves quality. It probably slows things down too.