Computability, Notation, and de re Knowledge of Numbers

Round 1

Reviewer 1 Report

Report on "Computability, notation, and de re knowledge of numbers" for the special issue Turing the Philosopher, for Philosophies

This is a brief but very well-written article that makes its point clearly. I recommend that it be published in your issue. The notion of de re knowledge has been challenged in general, and specifically it has been doubted, following Benacerraf, that there could be any de re knowledge of abstract entities such as numbers. The other difficulty about de re knowledge is that the standard for having such knowledge, as expressed by answers to the question "Which thing is that which..."?  seems hopelessly context sensitive and so not really an objective issue.

The author shows that attention to the familiar "stroke notation'' for numerals used in explaining Turing machines can lead to answers to these difficulties for the case of de re knowledge of numbers. The starting point is a remark of Kripke that computations can give us de re knowledge of numbers. This paper expands on this suggestion with convincing details. Kripke's  suggestion is that one can gain de re knowledge of the value of a function by computing its value. An algorithm for computing the greatest common divisor of two numbers, say, twelve and sixty-four gives us de re knowledge of which number it is (four) that is the value of the function. The use of stroke notation in Turing machines, say |||| for the number 4, and the operation of adding another stroke to indicate successor, so that ||||| is the name for the successor of four (namely five), gives a good sense in which we can be said to have de re knowledge of numbers. This solution is defended with a discussion about the idealization required of giving this representation of numbers and the Dedekind-Peano axioms a special status. While a brief 15 pages, this argument is backed up with a survey of a literature of twenty-five recent and classic articles on this topic. I recommend this for publication without requesting any changes.

Author Response

Please convey our thanks to this referee.  He or she recommended acceptance of the paper as it is (or was), and so we have nothing to report.

Reviewer 2 Report

The article "Computability, notation, and de re knowledge of numbers" takes on an interesting question and does an admirable job making a provocative argument that there can be de re knowledge of numbers. There is one central point that needs to be responded to in order for this paper to be publishable, an explanation as to why the central move is not a category mistake.

The argument has eleven steps:

1. Quine argues that there can be no de re propositional attitudes in general.
2. Kripke argues that Quine is wrong, that he overstates the problem by which can be answered by analyzing natural language use focusing on semantic and pragmatic elements of language.
3. This still leaves the problem of de re knowledge of numbers, specifically the result of computations which are seemingly restricted to the syntactic and therefore do not have the requisite semantic and pragmatic bits used by Kripke in the other cases.
4. One possible answer is that we can consider notational conventions to constitute the semantics and contextualization since there are cases where notation choice determines if the function is computable.
5. But there is a problem with the choice of notation here, so we need another route to de re propositional attitudes about the numerical results of computation.
6. A sufficient condition would be the finding of a Kripkean "buck-stopper" for numbers.
7. For normal people, unary notation, the sort used by Turning machines, is not a buck-stopper, indeed, some numbers have no buck-stopper at all. Hence, it seems like either the approach of the claim of de re propositional attitudes toward numbers is flawed.
8. HOWEVER (and this is the BIG move), we should not be thinking here about real people, ordinary people, actual living people, but rather idealized human-type calculators "not encumbered with limits in attention span, lifetime, materials, and the like."
9. By Turing's Notational Thesis, this makes such an idealized person equivalent to a Turing machine.
10. For such a machine, unary notation for the standard usages of numbers would be a buck-stopper.
11. Therefore, it gives de re propositional attitudes for numbers.

It is a clever argument and a nifty move. My major concern is with 8-10, the heart of the move. It seems odd to even speak of de re propositional attitudes of idealized entities. The very concept of de re propositional attitudes seems to require a real cognitive being in a context with an intent and understanding of propositions. And that, at least at first glance, is exactly what is eliminated with the move to the idealization. Is there really a de re/de dicto distinction for idealizations or isn't it a result of real people being less than ideal that is a precondition for the possibility of the distinction? It seems to follow that we could attribute propositional attitudes to Turing machines. A fascinating and provocative claim, but one that needs a bit of justification in particular places in the discussion. Are idealized calculators the sorts of things that CAN possess de re propositional attitudes? If the argument is bolstered around that point, I think it would be publishable.

Also, a two minor typos: "numbers" is misspelled in the abstract, on page 9, "whatever" is spelled "wahtever."

Author Response

Please convey our thanks to this referee.  He or she raised a very interesting point concerning de re attitudes toward numbers with respect to the idealizations.  We addressed this concern near the start of Section 5.  We also corrected the typos that this reviewer found.  We do not know how to track changes in LaTeX files.  Besides addressing the interesting concern of this referee, the other changes consist of either correcting a few typos, making very minor stylistic tweaks, and adding a couple of items to the references (some noted in footnotes).