Vague Definitions

The Capabilities community has not succeeded in defining ‘capability’. There are many attempts, a few by me. They are all relative to some perspective of what it is to compute. They frequently include negative elements so as to distinguish them from more conventional systems. There are other problems too.

Recently it occurred to me that I have seen such failure elsewhere and it is not always a terminal condition. This 2017 October page on Hilbert space has many virtues. It introduces Hilbert space to that category of mathematician with roughly an undergraduate math degree from a good college. The history section there gives facts that I knew but had not put together. It was von Neumann that named “Hilbert Space”. Hilbert was the main generator of these ideas. Others, before and after Hilbert, expanded such ideas and applied them in other areas. Von Neumann grabbed an exemplar from this mass of ideas and named it ‘Hilbert space’. In his constructions there were, up to isomorphism, just two: over the reals or over the complex numbers. He showed them to be simply and rigorously definable. He collected and proved many theorems which usually simplified previous work. This inaugurated a new flood of research. I took a “Hilbert Space” course at Berkeley in 1954 and the professor said “That physics book is the best math book on Hilbert Space.”. Later some stretched the term to include larger and smaller spaces and that causes occasional confusion that you cannot blame on Hilbert or von Neumann.

There is today (2017) no Hilbert for capabilities. I see no path yet to such an outcome for capabilities. I see no impediment either. Keykos makes no such claims. When pushed computer people point to the Turing machine when asked for theoretical basis for what they are doing. That is an insufficient foundation upon which to define what it means to be secure. Modern machines, post 1970, go meta to support programs designed to control other programs, usually with privileged mode.

The “Talk” page associated with the Hilbert space page has some interesting remarks as well. Hilbert space has proven useful enough to science and engineering that many non mathematicians need to understand it as well. The introduction mentions “Cauchy sequences” as not helpful to that class. The math courses that mention Cauchy sequences generally are taken only by aspiring mathematicians. Another introduction would indeed serve this expanded class as well. Hilbert space is famous for providing a class of theorems that are true and useful in three dimensions. This intuition bridge make precise deductions by non mathematicians possible. The objection to the introduction is thus justified. The same issues abound with capabilities.

I propose that Wikipedia establish a tradition of sections with headings such as “Introduction for Mathematicians” or even “Introduction for Undergraduate Mathematicians”. This would be in addition to the general introduction. There is another mathematics strain: Category Theory. There seems to be a breed of mathematician who believes that all foundational maters should be in terms of Category Theory.

I recall some advanced math text book that included in a chapter 0, a list of mathematical notations adopted in the book. This chapter also served as a filter for readers, warning some that they might not be ready for the book.