What evidence is there that \(2^{\land}65536\) is a natural number?
At the outset the author warns that the title will appear strange to most mathematicians. By traditions that have evolved, actually, in most professions, in such cases the article will be treated with more or less benevolent neglect (as being sterile). Granted the statistical soundness of this rule of thumb, it turns out that in the present case closer attention leads, at least, to some salutary reminders. First, taken literally (but tempered by note 6), -- what is called on p. 472 -- the central assertion of the paper (a) (mis)represents Tarski's semantics: as interpreting derivations to refer to particular structures. In fact, Tarski interpreted (sets of) formulae, and thus all -- the infinitely many -- derivations from those formulae. (b) The assertion neglects -- not just a, but for an alert reader -- the principal (cl)aim of Hilbert's \(\varepsilon\)-notation, and especially his ideas on substitution methods. Here any (privileged) ranges of the bound variables in \(\varepsilon\)- terms are disregarded, and only the necessarily finitely many relations between such terms (in the finite proof considered) are respected. After more than 70 years of research on Hilbert's bright idea too much is known about blind spots in his expectations to dwell on the author's tinkering with his examples on pp. 467-478. (They do not even mention Hilbert's idea, let alone, the discovered blind spots.) A suitable salutary reminder here is two-fold. As far as logical (cl)aims are concerned the greater flexibility of considering models for single derivations -- than for all deductions from a (single) formula in (a) above -- has not found an effective use. But (later) experience in number theory provides a successful parallel; tacitly, for those with a suitable education: Hasse's local/global principle exploits the greater manageability of the finite fields \(\mathbb{Z}_ p\) (than of the ring \(\mathbb{Z}\) itself). Secondly, as to the -- to the author, dramatic, to those with (scientific) experience of large numbers, hysterical -- question in the title, it is a matter of course that effective knowledge requires -- possibly, the discovery of new -- suitable numerical notation. Certainly, it is not a matter of mere existence; least of all, of such physical situations (involving large numbers) as in the familiar inanities on p. 466. At one extreme the experience meant comes up in theories of (biological and other) chemical evolution; one large (dimensionless) number is the ratio of the age of the (living) universe to the duration of a single chemical reaction. Popular expositors of such theories have had to learn that evidence by visualization is either not available or, when it is, futile; in contrast to a suitable arithmetic education. At another extreme, experience in number theory removes malaise -- not about mere existence, but -- about the occurrence of situations in which knowledge of large numbers contributes to effective progress; cf. \textit{A. Baker} and \textit{H. Davenport}, `` The equations \(3x^ 2- 2= y^ 2\) and \(8x^ 2- 7= z^ 2\)'', Q. J. Math. Ox. II. Ser, 20,129-137 (1969; Zbl 0177.068), where the bound \(\exp_ 3\exp_{10} 487\) was good enough to (help) settle completely a problem of Diophantus, while earlier finiteness theorems (by exceptionally gifted number theorists) were not. For the record the reviewer finds such examples particularly memorable by contrast with suitable inanities. Perhaps the most salutary reminders concern spectacular differences between the traditions of exact philosophy (expecially, of the would-be scientific variety) on the one hand, and of mathematics and the ordinary exact sciences on the other, above all, w.r.t. assumptions made and examined. Thus, albeit generally more discretely (than in this paper), throughout exact philosophy there is (i) blithe talk (cf. p. 473) of some `onus of proof', `question-begging', and `circularity'; tacitly, as being necessarily vicious. Without exaggeration it is assumed that the acquisition -- or at least exposition -- of knowledge is (to be) like a tree in the mathematical sense. But also there is (ii) agony -- for example, in the author's tinkering (cf. (b) above) -- around `subtle' assumptions of this or that familiar numerical term having a value. By scientific experience there are many rewarding questions about such terms, but not this one. -- Note 1 (on p. 478) reports a delightful story illustrating the sweet reasonableness of many of those unable or unwilling to put 2 and 2 together; but without the reminder that most so- called relevance logics -- staples of exact philosophy -- prohibit putting 2 and two (or, specifically, putting the \(\varepsilon\)-notation and the author's central assertion) together. Disclaimer. The kind of preoccupations in exact philosophy are by no means unnatural (even if the degree found on its wilder shores may be). On the contrary, the kind matches the view(s) of many a mind's naked eye; usually exact (or easily made precise) enough to see that the naked eye tends to focus on aspects that are sterile for study (without technology and shifts of emphasis). As in (a) and (b) such a focus tends to distract from already available, but usually more demanding knowledge.
scientific article; zbMATH DE number 440194