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G. KREISEL
SECOND THOUGHTS AROUND SOME OF GDELS WRITINGS:A Non-Academic
Option?
TABLE OF CONTENTS
Preamble in the light of TENOS : : : : : : : : : : : : : : : : :
: : : : : : : : : : : : : : : : : : : : : : : : : : : : : :p.
99Miscellaneous reading around Gdels piece : : : : : : : : : : : :
: : : : : : : : : : : : : : : : : : : : : : p. 114Sundry items from
Vol. III : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
: : : : : : : : : : : : : : : : : : : : : : p. 121Diversity
remembered by whom it may concern : : : : : : : : : : : : : : : : :
: : : : : : : : : : : : : : :p. 133Appendix: Logical complements in
the light of TENOS : : : : : : : : : : : : : : : : : : : : : : : :
:p. 134Notes: Casual conversations; Views taken by many a (minds)
naked eye;
Non-academic aspects : : : : : : : : : : : : : : : : : : : : : :
: : : : : : : : : : : : : : : : : : : : : : : : : : : : : p.
149
Preamble in the light of TENOS, short for: tested experience,
not onlyspeculation (about possibilities). As has been stressed
before, for example,in [Nl(c)],1 Gdels work has been widely
publicized. This is repeated herein line with a property of
knowledge that will be prominent throughout: Itis one thing to have
knowledge, another to remember it when an occasionarises (tacitly,
in the part of the universe encountered and taken in). NBThe
erudites may remember this property more vividly by contrast
withFreges ideal(ization of another aspect of knowledge):
behauptende Kraft.In terms of a pun this is the power of a
proposition (Behauptung) to im-pose its(elf, in particular, its)
meaning; tacitly, on the attention of Fregesideal(ization of the
knowing) subject. That power is present, but may beweak in any
particular situation. More generally, the diversity of
properties(of knowledge) will be recalled below by contrast with
the idea(l) in thephilosophical literature of (its) mere truth.
As explained in the editorial note, particular attention will be
givento material in Vol. III (Gdel 1995), which was previously
unpublished,? Editorial Note. This article was originally
commissioned as a Review Essay (fo-
cussing on Volume III of Gdels Collected Works). But it became
clear that the broaderthemes discussed here have general interest
beyond that original telos. The article shouldthus be read as a
review in the sense described by the author, i.e. as second
thoughts on andaround foundational issues, with special reference
to some of Gdels writings (not only inVolume III).
Synthese 114: 99160, 1998. 1998 Kluwer Academic Publishers.
Printed in the Netherlands.
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100 G. KREISEL
but, broadly speaking, matches the publications reprinted in
Vols. I (Gdel1986) and II (Gdel 1990) in chronological order.
(Material written duringGdels final illness, which became evident
(to me) in the latish 60s, is hereleft aside.) The match is even
closer if published reports of conversationswith Gdel are added to
Vols. I and II. There are endless opportunitiesfor meticulous
comparisons, even of wordings, and as elsewhere in thebroad
academic tradition, subsidized academic trades that reward thosewho
seize those opportunities. For them the aspects stressed below are,
byTENOS, not rewarding.
1. One piece (pp. 376382 of Vol. III) stands out; it dates from
the early60s, and has the title: The Modern Development of
Foundations in the Lightof Philosophy. It is the last piece in Vol.
III written before Gdels finalillness. It is phrased in a way of
which Gdel was fond (and for which Iacquired a passive taste,
tacitly, when practiced by him in conversations).Details aside for
the moment, in plain English it is above all a reminder offamiliar
facts.
Logical Foundations had ambitious (cl)aims about mathematics
andmathematical knowledge, still enshrined in such terminology as
logical(in)dependence and (in)completeness; tacitly, as logical
idea(lization)s ofexperience meant by household words like proof;
with emphasis on prin-ciples, a.k.a. (mere)
understanding-in-principle. Contrary to those (cl)aimssuch logical
aspects, which are of course present in mathematical knowl-edge,
generally do not require (or reward) close attention. Both
internalsquabbles in so-called foundational debates in the 20s, and
such pastimesas belabouring paradoxes distract from this.
As an example of what is lacking in those foundations, Gdel
mentionscertain non-scientific meditation exercises for the
specific purpose of dis-covering suitable new axioms of infinity,
an idea to which he remainedattached since the 40s. Those
exercises, proposed in the early 60s, are tobe read in(to) Husserls
writings, a kind of Wesensschau; in contrast to thekind of
meditation popular by the end of the 60s such as the
transcendentalvariety or by better living through chemistry. (Gdel
explicitly warnedme that I should find Husserl boring, and, being
interested in other things,I had no occasion to test his
impression.) But, (these) specifics aside theexample is a reminder
that some aspects are lacking. It is a separate ques-tion which of
them, if any, lend themselves to rewarding study, let alone ofa
theoretical variety.
By the way, the only specific item in the modern development of
foun-dations used in Gdels piece (apart from those axioms of
infinity) is hisincompleteness theorem. This serves again, by
contrast and by TENOS,
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SECOND THOUGHTS AROUND GDEL 101
not only me as a salutary reminder; of such areas as real
algebra forwhich a complete set of principles has long been
familiar. But progressrequired giving principal attention to some
suitable aspects neglected inthose logical idea(lization)s of
mathematics; principal since as readerswith a little logical
education will know occasionally that completenessis used in
so-called transfer theorems; cf. 3(a) below for more.
2. Given the malaise about logical foundations sampled in (1),
one ideathat has struck many a minds eye, is to ignore the whole
enterprise (whilemany another minds not much less naked eye was
fascinated).2
(a) As so often with such ideas the option has to be tempered by
suit-able understandings; after all, as mentioned, knowledge has
some logicalaspects. Foundations for the Working Mathematician
(Bourbaki 1949) or rather what I read in(to) it now, not 35 years
ago provides one suchunderstanding. It selects suitable items of
logical knowledge, here meant incontrast to mere
understanding-in-principle, that is, of set-theoretic princi-ples.
It could be compared to (arithmetic) foundations for the working
car-penter, learnt before ones teens as basic arithmetic (in
contrast to Peanosaxioms). It will come up below, viewed as an
example of a general asym-metry in knowledge: By themselves such
foundations do not go far, butnot knowing them can be a disaster
(in a sense of this word suitable to theacademic situation
involved).
(b) Equally broadly, but at another extreme, that idea has, by
TENOS,not served well at least one reader of those who wanted to
pontificateabout their malaise. By TENOS many cases of prejudice
(by gifted peoplesteeped in the subject involved) are confirmed by
more thorough knowl-edge; but without those details it is harder to
say what is known. In the nextsection some earlier critical
literature on logical foundations will be takenup; complementary to
Gdels piece in Vol. III.
(c) Patently, the option of simply ignoring logical foundations
wouldbe hopeless for a review around Vol. III (or, by [N1], Vols. I
and II),where review is meant in the sense of second thoughts; in
contrast toa schoolboys prcis (required to satisfy the school
masters regulations).Though often perfunctory, references to
logical foundations and even tofoundational debates introduce or
conclude material in Vol. I (and Vol.II from before the mid 40s, or
appear in footnotes with afterthoughts).Sometimes the references
are implicit; famously in the words relativeconsistency in the
title of work which would otherwise come under theheading:
constructible sets. In Vol. II, in an essay on Russells
mathemati-cal logic, there is a fanfare about logic being a science
prior to all others,emphasis on paradoxes as related to an amazing
breakdown of various
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102 G. KREISEL
logical intuitions in the middle, and a peroration about
promises of logicfor mathematics. So unless Gdels piece from the
60s is to be ignored,too, a rereading in the sense of rethinking,
not of a prcis of Vols. IIIIrecommends itself.
This was actually done in [Nl], suitably for some, but, by
TENOS, notfor all (with a logical education). As a matter of
personal TENOS, with thereminders in Gdel more vividly before me, I
have spotted since then someitems that are not reviewed critically
enough in [N1]. (Some are taken upbelow in the section on Sundry
items from Vol. III under the heading hardcore foundations.)
(d) Given the intellectual vitality, both of Gdels own work and
ofsome (not all) familiar logical literature that continued it, it
was a foregoneconclusion that a review would not leave one
empty-handed. In eruditeterms, going back to Aristotle, the
enterprise of logical foundations pre-sented itself (to many a
minds naked eye) as a privileged telos for math-ematical logic. But
contrary to a familiar view of Aristotles sound bite,by TENOS, it
can be rewarding to shift the emphasis to different targets,among
infinitely many candidates in our infinitely diverse world.
Suitable TENOS for second thoughts on such shifts is available
to read-ers with a little experience in applied mathematics;
specifically, for re-viewing the application of mathematical logic
to that enterprise. It is easyenough to pay pious lip service to
the principle (of rewarding shifts); byTENOS, its practice is
demanding. Thus it is not merely the words asso-ciated with the
original telos that remain enshrined in the mathematicalscheme, but
the ideas or, if preferred, objects (categories in the
colloquialsense or whatever): Are they, let alone, their
descriptions, suitable for thenew purpose and, more generally, for
extended knowledge?
In short, there is no free lunch; (suitable) shifts of telos
have a price,which, depending on ones resources, may be a bargain
or a dead loss. Anecdotes. By a fluke, as an undergraduate I still
encountered those Tri-pos Exercises (at Cambridge, UK) in Applied
Mathematics before WorldWar II, notorious for elaborating
painstakingly primitive idea(lization)sof physical phenomena,
especially, in rational mechanics, mentioned al-ready in [N1], for
example, on p. 602 of (c). Some elementary items haveremained among
the foundations for the working (theoretical) physicist,others had
been developed with mathematical vitality in the theory of
func-tions (of a complex variable). But those elaborations were, by
and large,sterile; except for the fact that the dons concerned made
a, to them, by andlarge, satisfactory living. From what I remember
they had no illusions ofcontributing to physics or mathematics, but
were doing something differ-ent (tacitly, from such contributions).
For the readers above my experience
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SECOND THOUGHTS AROUND GDEL 103
in applied mathematics may be good as a salutary reminder: to be
preparedfor similar elaborations in the case of primitive logical
idea(lization)s ofknowledge introduced in logical foundations.
As to TENOS on the extent to which the likes of Tripos
exercisesactually occur in the logical literature, this has a
(social) price as follows.Over the last few years Zbl. Math. has
sent me a stream of articles, whichcontain, sometimes extreme,
examples; including occasionally the kind oflip service mentioned
earlier and thus providing TENOS that the prac-tice of the
principle is more demanding than appears to the authors; cf.Zbl.
815.03036 together with its editorial note for a convenient record.
Itincludes reports of (the authors) and speculations about (the
reviewers)intentions.3 The authors indignation aside, some malaise
about the reviewis fitting3 inasmuch as it is in conflict with
academic conventions. It isunconventional to use (academic)
publications for the kind of TENOSabove, which has been compared to
the use of bacterial cultures in bac-teriology. Also it is
unconventional to use Zbl. as a convenient record ofsuch TENOS, in
particular, in the case above concerning the diversityof knowledge
perceived as understanding among those of us with alogical
education; cf. 3(c) and 3(d) below for more under the heading:
anon-academic option.
So much for parochial rewards for attention to logical
foundations;parochial in contrast to the following:
3. A broader view looks at the likes of logical foundations and
at alternatives to formal (mathematical) refutations such as Gdels
incom-pleteness theorem. A familiar look alike is >2500 years
old; with arith-metic replacing logic: Number is the measure of all
things. The refutationuses the irrationality of
p2, the measure, a.k.a. length, of the diagonal of a
unit square. It can also be reworded in terms of Dedekind
(in)completeness:Q is incomplete, and incompletable by any
countable set (of real adjunc-tions). Remark A specific parallel
will be used below: all finite decimalsand all finite sequences of
integers are rational and recursive respectively.
A common gut feeling focuses on the lack of proportion between
themind-bogglingly crass (cl)aims of the look alikes above and the
elegantformal refutations. (NB. This feeling does not seem to be
shared by thetrade of so-called exact philosophy, with scholastic
logic chopping in for-mal dress as its principal commodity.) On the
other hand, while I haveshared the feeling since my teens,
especially then I also fancied it, and,by TENOS, so have others;
famously Andr Gide: Les extrmes me (notonly, se) touchent. Other
attractions of the formal refutations are familiar,so need not be
dwelt on, but must not be forgotten here either. For those
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104 G. KREISEL
in the market for entertainment or uplift (&), they are
jolly orbeautiful. The lasting market for the arithmetic item is
known, for Gdelsexpected. Given the intellectual vitality of the
ideas used (for refutations),the potential for shifts of telos has
been present from the start; at the kind ofprice documented in 2(d)
above. Readers with a suitable background willfind more in [A2(c)],
short for item (c) in the second part of the appendix;in
particular, about the rocky evolution of basins of attraction
within thearea of diophantine problems (evolved from the
insolubility of n2 = 2m2 inpositive integers).
Such alternatives as (a) and (b) below are reservations
regarding thosecrass (cl)aims rather than formal refutations, and
not nearly as flashy. Bya refrain of this article, it is a separate
matter where either alternative issuitable; more so than, for
example, the option of ignoring those claims.This is taken up in
(c) and (d) below; as mentioned, as a non-academicoption.
(a) In contrast to the irrationality ofp2, the alternative meant
empha-sizes diversity: generally, of the world, more specifically,
of numbers, of aspects of any one thing and of suitable measures.
This is by no meansad hoc (tacitly, after footnote 3), inasmuch as
it is in contrast with therepository of (not only logical)
foundational ideals. They all demand theessence, a.k.a. nature, of
things, and unity of, say, a body of knowledgearound them. When
taken in-the-raw, these ideals are not satisfied by thealternative
of relatively few measures suitable for relatively many situa-tions
encountered. (As always, they may be suitably tempered by
suitable,possibly tacit, understandings).
Anecdote. Already in my teens I was taken by the views and
personalityof J.E. Littlewood whose introduction describes A
mathematicans mis-cellany as suitable except for those
irreconcilable persons who demand anappearance of unity and a
uniform level. I dont remember if, in my teens,I associated this
demand with ill will. Since then I have learnt by TENOSthat some
simply either do not take in knowledge presented in a miscellanyor,
if they do, do not remember it suitably. Some of [N1] has the
characterof a miscellany, but less so than the world at large.4
(b) In the case of logical foundations, one alternative to Gdels
formalrefutation combines the last paragraph in (1) on real algebra
with the re-minder at the outset: (tacitly, true) knowledge has
other properties besides(mere) truth. In the background is the
master reminder (in [N2]): rewardingknowledge, here, around real
algebra, may not long continue to concernaspects that strike many a
minds naked eye. The choice of real algebra fits
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SECOND THOUGHTS AROUND GDEL 105
the idea earlier on that a critique (here, of logical
foundations) will besnappier if it uses knowledge of the
subject.
Logical foundations emphasize the fact that exactly the true
formulaein the (elementary) logical theory of R are (logical)
consequences of afamiliar axiom schema, say, RCF. As corollaries,
RCF is complete and(recursively) decidable; with some refinements
of the latter, going back>60 years. Now, by TENOS, within
mathematics another view among,by (a), infinitely many! has
evolved, which shares in the first place onlya literally
superficial, a.k.a. syntactic, element: the formal axioms RCF.
RCF is interpreted as describing the abstract idea, to the
erudites a.k.a.structure, of real closed fields, on top of a few
particularly well knownbasic structures (or structures mres) such
as groups or fields. The unde-cidability of their elementary theory
has been known for nearly 50 years(and its incompleteness much
longer).
A disclaimer and a reminder of a conflict. I have no idea to
what extent,if any, spotting abstract structures is an exercise in
Wesensschau; at least,in effect, not generally by intention of most
people in abstract mathematics(inasmuch as they, too, have not read
Husserl; cf. (1) above and [Nl(e)]). A view at an opposite extreme
is not merely implicit in the logical tradition,but quite explicit
in the (recent) literature on the allegedly only way offinding
theorems of a structure; cf. 1.-5 to 1.-3 on p. 159 of (Pohlers
1996).In plain English, in familiar mathematical reasoning formal
deduction gen-erally isnt used even if it could be. Instead without
premature precision such logical consequences are seen in ways not
obviously different fromseeing (old or new) axioms; reducing the
likelihood of errors resulting, forexample, from boredom (with
logical deduction). This point will come uprepeatedly below, with
labels like Wesensschau or intuition serving aspegs strong enough
to hang ideas on, which are rooted in TENOS.
Those of us who have paid attention to abstract mathematics, as
it hasevolved in the last few decades, have by now plenty of TENOS
for secondthoughts on the matter brought up in 2 (d): Where are
logical categories, inparticular, those of completeness and
decidability, suitable for understand-ing knowledge about and
around the abstract structures above? Reminderof a different kind
of TENOS: Logical categories have appeal for many aminds naked eye;
in the simple sense that they are related on purposeand in effect
to epistemological notions, which are of venerable vintage,and thus
enshrine views taken by many such eyes. Logical categories
aregeneral, being defined across the board. But by a(nother)
refrain of thisarticle, this is a separate matter from being
suitable across the board.
The mathematical TENOS reported in the following sections is
elemen-tary, but in terms introduced earlier good enough for
reservations about
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106 G. KREISEL
those logical categories; with corrollaries for (the
suitability, not meretruth, of) Gdels incompleteness theorem, which
refutes precise, some-what extravagant formulations of certain
foundational (cl)aims. In contrast,reservations are suitable not
only w.r.t. to those logical formulations, butalso when the
admittedly crass (cl)aims are stated in terms not specificto any
academic education. This is an example of what was called a
non-academic option, here, for using ones (mathematical,
including)logical education. It is not offered as a free lunch; cf.
2(d). When the time isripe (at the end of the article), it may be
considered (by individual readers)whether the price is right (for
them). But enough is known already for afew words on the label
non-academic.
(c) As mentioned, a principal alternative to the option is
simply toignore those (and similar) crass (cl)aims together with
refutations andreservations. As a matter of personal TENOS, some of
us perceive speech-lessness here, in the face of those (cl)aims as
a kind of helplessness, andthus find the alternative
unsatisfactory. Evidently by academic traditions,this (cl)aim for
the option is a mere luxury. NB The defect is parochial,inasmuch as
not all effective thought is academic (let alone, scientific;
cf.(d) below).
But this is not all. Closer inspection allows one to specify
actual con-flicts between the option and diverse idea(l)s of
various academictraditions; first, from academic philosophy. The
examples below quote2 memorable items in the philosophical
literature, which implicitly oreven explicitly are in conflict with
TENOS used by the option. The firstquotation comes from C. S.
Peirces (Coll. Papers: 4, 237): It is . . . easyto be certain. One
has only to be sufficiently vague.
Implicitly, this is assumed to be too easy to be rewarding. But
abstractmathematics provides much TENOS around situations where,
what is suf-ficiently vague for certainty, is also sufficiently
precise for the matter inhand. NB Sufficient in the sense of a
suitable kind, not only degree,of precision; in plain English, what
one is precise about (in contrast toscholastic logic chopping
earlier on). Here abstraction is meant as in ab-stract mathematics
(in (2) of the next section) and in contrast to logic,where it
means higher types. For example, a problem which is not easy
toanswer for a particular number field, say Q, may be discovered to
be easywhen (made) suitably vague (in the sense of non-specific,
for example, forall number fields). It is a logical convention that
this ceases to be vaguewhen it is put in formal dress: the totality
of fields considered is splendidlyvague, and precise enough for
most situations actually encountered.
A second quotation is Quines quip on an idea(l) of ideas that
sustainintensive analysis, stated (at the end of the piece on
creation), but not
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SECOND THOUGHTS AROUND GDEL 107
followed in his Quiddities (Quine 1989). Many ideas satisfy the
ideal, forexample, when (infinitely) many logical consequences are
derived (relent-lessly; by TENOS, for some in our diverse world
such intensive analysis isits own reward). TENOS shifts the
emphasis to other well tested propertiesof ideas besides their
being subjects for analysis.
Thus they may be (suitably) selected or combined; with due
regard tocompatibility, prominent in (b) above in TENOS on
combining logical cat-egories and knowledge around abstract
structures (in Zbl. 815.03036, andagain in 2(a) of the next
section). In [A 713] Kant mentions that they maybe constructed;
true to form, given his general preoccupation. Last, but notleast,
ideas may be good enough as they stand, provided they are
remem-bered suitably. In this case they are liable to be spoilt by
elaboration; asmentioned in connection with foundations for the
working mathematician(or carpenter), underlined by the warning
provided by Tripos Exercises.In terms of a hackneyed metaphor, here
a first step restricted to an inch,may approach a target by, say
half an inch, but, pursued without suitablechange of direction, may
miss any sensible target by a mile.
W.r.t. the label non-academic the broad conclusions above are
de-scribed in colloquial terms without academic erudition. The
terms ac-quire additional meaning in the senses of being memorable,
of othersocalled uses, and presumably of some neurological aspects
when re-lated to the academic education involved in the TENOS
adumbrated. Inshort, the option consolidates the (non-academic)
knowledge expressedcolloquially.
(d) As to conflicts between the option and traditions evolved
within the sciences and mathematics, there is above all the
following contrast inconcerns of the common-or-garden varieties
among those traditions. (Thesocalled fundamental sciences are
discussed separately in (4) below.) Interms used above, those
traditions respect, at least tacitly, such matters asthe diversity
of the world and of kinds and degrees of suitable
precision(prominent in uses of the option); foundational ideals,
e.g. of unity, aretempered accordingly.
By and large, the relative contribution of a scientific or
mathematicaleducation generally outweighs that of consolidated
non-academic knowl-edge. For example, one does not expect to get
far without a scientific edu-cation in a study of phenomena in
astronomy and spectroscopy measuredwith an accuracy to 9 or 11
significant digits. Of course not all thought isscientific. But
then not all objects of (rewarding) thought are paid attentionto in
those traditions either. By TENOS, within those disciplines,
whathave been called sound intellectual reflexes serve academics
quite wellto get on with their jobs (laid down by their
traditions). By a refrain of
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108 G. KREISEL
this article this is a separate matter from describing those
jobs (withoutleaving the rest of us speechless). By and large, and
not, for example,in enterprises of discovering suitable shifts of
emphasis as in 2(d) above;away from the likes of Tripos
exercises.
For the present non-academic option the article mentioned in
2(d) andreviewed in Zbl. 815.03036, is of particular interest just
because it is notsome kind of isolated, socalled personal
aberration. It enshrines conve-niently in one place several
idea(l)s that lend themselves to correction byuse of the
option.
To conclude this general orientation, a couple of observations
are worthnoting. First, the option is genuine, even w.r.t
presenting (memorably) thekind of non-academic knowledge in
question: it is not suitable for all. ByTENOS, for some, colloquial
formulations are not weighty enough (to besuitably remembered).
Such people strike the rest of us as close relativesof those
irreconcilable persons in endnote 4. They have a different
option,called so wahrhaft philosophisch by G. Lichtenberg; cf.
footnote on p.143 of [N1(b)] for a quotation. Roughly, his
colloquial paraphrase recallshow in practice thoughtful scientists
have always taken into account thatboth events and their
observation have a part in knowledge of nature; butthen Lichtenberg
stresses that Kants way of saying this was exceptionalin being
s.w.p.
Secondly, there is a vivid relation between the option and Gdels
(dif-ferent) words in his piece from the 60s to the effect that not
all thought isscientific. This is certain even if to be realistic
the word scientific isleft vague; even because Peirces Law in (c)
requires a little care in thecase of negative propositions. Gdel
takes such popular extensions of non-academic knowledge, a.k.a.
gesunder Menschenverstand, as meditation orreligion; cf. Einsteins
credo about science without religion being lame;cf. Nature 146
(1941) p. 605. Now, I personally find science with religiousfervour
a drag. So the (low-brow) non-academic option suits me better:
ithelps me remember (available) non-academic knowledge better
(withoutagony over the kind of knowledge provided by those other
extensions).This evidently leaves open the matter of TENOS where
the option is goodenough.
4. Throughout [N1] there are (brief) references to another side
of logi-cal foundations, but without mention of the label in (3) of
any non-academic option. Foundational (cl)aims are viewed as
parallel to those of(successful) fundamental or even, in
contemporary jargon, final theories;the parallel being available
for comparisons and contrasts. In particular,logical foundations
are viewed as a final theory of mathematics; cf. top
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SECOND THOUGHTS AROUND GDEL 109
of p. 5 in Russells Principles of Mathematics (in the first
decade of thiscentury): The fact that all Mathematics is Symbolic
Logic is one of thegreatest discoveries of our age. He adds the
ideal of an understanding-in-principle, and, in the introduction to
the second edition of Principia, theideal of unity (achieved by the
formal reduction to just one kind of objectand relation, namely,
sets and membership, one propositional operator andone
quantifier).
So there is nothing disturbingly original, in particular,
untested, letalone unheard-of, in those references. The references
were brief, becauseat the time I had not come across (or had not
recognized) enough TENOSon actual rewards for attention to this
side of logical foundations.
(a) By the restriction to successful fundamental theories it is
a foregoneconclusion that the parallel will turn out to be a
parody. The view takenhere is:
Even if it is a parody, glaring limitations of logical
foundations areliable to have counterparts, which are much less
visible (in successfultheories), but as always, in certain
situations present a comparableobstruction; above all, to
successful combinations with other knowledge.
Inasmuch as it is a parody (that is, presents a striking
relation) at all,glaring successes in other theories are liable to
draw attention to elements whose counterparts are lacking in
logical foundations; and thus help saywhat they are (not
necessarily supply any, and not only see that somethingis
lacking).Reminder (cf. [N2]) This view is for many a minds naked
eye no lessperverse than the heliocentric view for many an ordinary
eye. In our diverseworld the view is not advocated generally; no
more than the heliocentricview to those making sun dials. (By
TENOS, a vivid description of thisview or of any other option is
liable to be (mis)understood as advocacy.)
(b) As in 3(d) on the non-academic character of the option, in
par-ticular, on not expecting some direct contributions to academic
business the parallel is not presented for any scientific interest
to successful theoret-ical scientists. By way of a manifesto I
dissociate myself from popularin(s)anities about (such thoughts as)
the parallel providing clarification,and about taking ones
impressions of some socalled heuristic value (ofsuch thoughts, for
example, to oneself) at face value. Of course, the im-pressions may
be as clear as the geocentric view of the suns motion; cf.the
reminder at the end of (a). More generally, blithe talk about
heuristicvalues, in which the idea(l) of causes is prominent,
recalls TENOS in end-note 3 on items from the repository around
knowledge of causes, and ofvalues associated with (unsuitable)
orderings.
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110 G. KREISEL
(c) In the 90s, after the material in [N1] was published,
several semi-popular books around fundamental science appeared; by
authors who, bycommon consent, have been very successful in the
broad areas of thosebooks (here meant in contrast to the kind of
speculation on brain physi-ology in Gdels Gibbs Lecture touched
below under the heading: Brainsand faster computers). As in (b), at
least for me, there is no question ofhelping the authors to be
(even) more successful in their science or (even)more popular,
tacitly, in suitable sectors of the market (in the commerce
ofideas). A main concern here is decidedly non-academic: to derive
fromthose books some knowledge outside the areas of ones academic
educa-tion, without (hopeless) illusions of understanding, but also
without being(hopelessly) put off by passages on matters of the
kind touched in 3(c), andcalled general by one of the authors.5
Now, the normal course of nature provides opportunities for spot
checkson ones impressions of understanding by specialists with
(academic)knowledge around those books. It is not (academically)
conventional tohave ones knowledge tested by colleagues; but then
there is no free lunch.As to those passages, some of them recall
(my) teenage emotions when(I was) left speechless both by (cl)aims
and by objections in the caseof logical foundations. One option is
to ignore the passages in question;tacitly, for the concern above,
not for the authors agenda in (d) below. Thenon-academic option
provides the (non-academic) luxury of an alternative.
On pp. 4344 of Dreams of a Final Theory S. Weinberg (1993)
ago-nises over an objection by another successful scientist, P.
Anderson: Arethe discoveries of the (double) helical structure of
DNA and of Turingsidea(lization) of computability not comparably
fundamental (to the dreamt-of theory)? A moments attention to the
diversity of the world, prominentin (3) as a principal element of
the non-academic option, is enough as awarning; of the need for
remembering tacit understandings, which temperidea(l)s of some
fundamental ordering of knowledge, theoretical or not; cf.endnote 2
(on terrestrial mechanics) extended to such terrestrial affairs
asliving organisms or software engineering, and on (metaphorically)
celes-tial matters of high energy physics (with astronomical
costs). Remark forreference in (d) below. Inasmuch as practically,
universal monetaryorders, a.k.a. values, do apply to (the costs of)
acquiring a wide rangeof knowledge, those tacit understandings are
liable to be incompatiblewith monetary orders; as with orders (of
prominence) of the same itemof logical knowledge in foundations for
the working mathematician and inelaborations satirized above by
reminders of Tripos Exercises.
Weinbergs kind of objection to the comparison with Turings
ideawould also apply to the present parallel between logical
foundations and
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SECOND THOUGHTS AROUND GDEL 111
fundamental scientific theories. In effect, though not in these
words, Wein-berg appeals to the doctrine of category mistakes;
here, the alleged mistakeof mixing the categories of knowledge
about natural science and aboutmathematics (or engineering). As a
general doctrine it is hardly compellinginasmuch as, by TENOS, it
is often a good idea to count objects in differentcategories (which
would count as a category mistake).
The irony is that, with better knowledge, Anderson(s case) would
belaughed out of court, at least by readers with suitable
background. As tocontributions of Turings idea to contemporary
mathematics or to down-to-earth computation, the items on hard core
foundations provide secondthoughts; cf. the section (below): Sundry
items from Vol. III. Briefly,those contributions are relatively
limited; compared to the prominence ofTurings idea in connection
with the idea(l) of absolute definitions for epis-temological
notions (on p. 150 of Vol. II). But then these second
thoughtstemper the ideal itself.
As to broad knowledge about and around DNA here meant in
con-trast to the specific geometric form of the molecule there is,
realistically,no chance of competition for general interest by any
dreamt-of final the-ory. (I dont know the money expended by
pharmaceutical companies onknowledge around even single genes.)
W.r.t. that specific geometric as-pect of DNA, the matter is more
delicate: It is good enough to refutecertain precise formulations
of traditional vitalism, with claims which areno more coarse-minded
than those refuted by the irrationality of
p2 or
by Gdels incompleteness theorem. (Realistically, the knowledge
of bio-chemistry used is more demanding than that of arithmetic and
logic inthe other cases.) However, neither Weinbergs book loc. cit.
nor presum-ably Andersons case mentions what even I (as an
outsider) know about limitations on the understanding-in-principle
provided by that helicalstructure w.r.t. not just Life Itself, but
specifically self-replication;cf. Is DNA really a helix? (Crick et
al. 1979). For the present option it isworth noting that the
background is typical. A mathematician, Pohl, madea song and dance
about a (perfectly valid) point many (of us) had noticed inthe 50s:
the advertisement about self-replication by just un-zipping DNAdid
not bear (even my) second thoughts; (it was known) there were just
toomany turns of the helix to be unwound in the time available
during mitosis.But apparently Herr Pohl was not satisfied with this
observation, but wanted to have an opinion on what does happen. He
collected some chumsto propose a different (false) geometrical
structure to which the title of thepaper above refers. (More
interestingly (to me), the paper also shows, byuse of technology
evolved since the 50s, that a battery of enzymes cut andglue
fragments of DNA and their replicates respectively, and thus adds
to
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112 G. KREISEL
those elementary second thoughts more demanding knowledge.) So
muchfor those pages in Weinbergs book.
Cricks words (1994, cf. endnote 5) about all our thoughts and
feelingsbeing a matter of neurons etc. are reminiscent of Russells
at the begin-ning of this subsection; just add a matter of after is
in: mathematics issymbolic logic. The hypothesis in question, that
the scientific search (byCrick) for the soul might be unsuccessful,
does not astonish me; in fact,w.r.t. this matter of astonishment,
there is enough of it (for my taste) inTENOS around
common-or-garden varieties of psychological and psychi-atric
phenomena not to seek it in the mere (non-)existence of the
soul.But also, by reference to the parallel with Russells sound
bite I am notleft speechless nor put off from reading the material
offered in Cricksbook (and having my understanding tested whenever
an occasion presentsitself). Perhaps, it would have been fitting
(for my taste in view of the soundbite on neurons) to have more on
brain-damaged patients; not instead, butin addition to straight
experimental psychology. Fittingly (for applicationsof the
non-academic option), time and again the text recalls (endnote 3
witha sound bite of) Aristotle on the business of understanding and
knowledgeof causes; here, neurological causes of psychological
phenomena.
Even now enough is known about the parallel between logical and
neu-rological (would-be) fundamental theories for TENOS on the idea
of aparody adumbrated above by comparison between logical
foundations forthe working mathematician and neurological
foundations for the workingpsychologist and especially
psychiatrist. The little I know of neurology hasrelatively more
weight for the common-or-garden varieties of psychologi-cal
phenomena than all of logic in mathematics. Reminder. This
conclusionis already discounted (in the stockbrokers sense) by
shifts to discoveredaims for a logical education.
On the other hand the parallel above, between those
scientifically verydifferent foundations, highlights similarities
if the emphasis is shifted backto such foundational ideals as the
essence or nature of this or that X; sat-isfied by definitions of X
in answer to the nursery question: What is X?Reminder (of Platos
enthusiasm about definitions in Euclids geometry,for example, of
the circle, an idea that has struck many a naked eye). Aselsewhere
by general TENOS, with suitable tacit understandings on the X(and
on the kind of definitions) considered, the ideal becomes
impecca-ble. Now, in conversations with Gdel, some 40 years ago,
about diverseepistemological notions there was like-mindedness (cf.
[N1(e)]) regardingthe view that such notions have struck many (a
minds naked eye; in plainEnglish) before or without scientific
experience. In terms of Gdels piecein (1) such notions are defined
by a kind of Wesensschau, which I pursued
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SECOND THOUGHTS AROUND GDEL 113
as what I called at the time a calculated risk (without his
conviction);now (cf. (4) of the next section) I prefer the term
milksop foundations.These are rewarding to me, in retrospect, not
foreseen for informationon points of diminishing returns for
attention to those notions.
Viewed this way some claims, popular at the time (in the 50s),
aboutDNA and life, another idea that strikes many a minds eye, and
about DNAbeing the secret or at least the molecule of life, no
longer leave me speech-less. For one thing, in the light of TENOS
around epistemological notions,this is (now) simply no big deal for
me, if my what else? idea of life(at that time) was meant. That
idea did not include much about viruses,nothing about retroviruses
etc. (nor such matters as protein synthesis). Bynow those popular
claims are among the less interesting sides of what isknown (even
by me) around DNA: I want a better idea (of life). Anec-dote. In my
teens the people whom I heard pontificate about these
matters,struck me as pretentious. Now, with a little more TENOS,
they are seen tobe clueless; in particular, about the general
diversity (even) in the (small)world of human affairs. Reminder.
This diversity includes of course alsoareas of uniformity, as it
were basins of attraction, especially within thoseacademic trades
that have evolved traditions aptly called disciplines. Herethe
non-academic option adumbrated is relatively rarely suitable.
(d) The emphasis throughout this Preamble has been on
personalTENOS; evidently in the last paragraph, but also in the
fanfare about theproperty of knowledge being suitably remembered,
tacitly, by oneself.Plainly, there are other uses of knowledge,
including communication, per-suasion or even inspiration. This
applies to Weinbergs and Cricks differ-ent agenda (reflected) in
their books. The former set out to communicatehis enthusiasm for a
final theory to a bunch of people, and to persuadethem to spend $12
billion dollars (over 5 years) on that theory (rather than,perhaps,
on metaphorical black holes such as national or social
security).6
Cricks message (of scientific salvation) was to inspire with
luck,gifted people to work on (tacitly, neurological aspects of)
conscious-ness now. Without exaggeration, once the soul(ful
tradition) is discarded,inspiration is seen as a matter of
statistical TENOS, which I do not have(and so, by temperament, I do
not have any opinion on it; cf. endnote 5for contrast). Remark on
personal TENOS, in particular, reported in (c).For me the broad
idea of the message is not at all controversial (no morethan the
broad idea of determining the molecular structure of DNA,
tacitlywithin the tradition of biochemistry in, say, the 40s). This
leaves openthe chance of success with the resources available (to
an individual). Thewords of the message do not suit me at all; I am
no more interested inmy idea of consciousness than of life above.
There is no word on the (to
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114 G. KREISEL
me) most encouraging, if not inspiring, aspect of TENOS around
DNA:when progress did not quite live up to expectations (and/or
rhetoric) aroundself-replication, and the geometric (helical)
structure remained a mem-orable, but isolated bit of information,
(also Cricks) attention turned toother, almost immediately
rewarding aspects such as protein synthesis andthe linear order of
bases. (Flexibility of mind and energy can be usefulresources,
compensating a taste for flat generalities.)
Reminder of relations between this Preamble and Gdels piece on
pp.376382 of Vol. III (at the end of (3) above): Both have qualms
about log-ical foundations, but differ w.r.t. the kind of attention
given to them and tooptions for using the mathematical logic
originally developed for them. The Appendix is for readers with a
taste for the jolly side of (mathematical)logic.
Miscellaneous reading around Gdels piece. The words in its title
arehere understood broadly, as in the Preamble. Thus attention is
given tophilosophy as a repository of memorable reminders; both of
views taken bymany a minds naked eye, of feelings about them and of
thoughts generallypractised, but not elsewhere described so
wahrhaft philosophisch. Morespecifically, w.r.t. (logical)
foundations attention is given to the idea(l)s ofknowledge which
were embraced by the pioneers (and certainly continueto strike many
a minds naked eye).
1. One such idea(l), on understanding-in-principle, was related
in 2(a) ofthe last section to a pun on principles given in advance.
Another suchideal, aiming at mere truth, was highlighted at the
outset by contrast withTENOS on knowledge having other properties
(besides truth). In the lightof those ideals the possibility of a
constant intervention by (Kants) intu-ition in reasoning appears as
a spectre; in the first case, at least as long asno principles
determining such intervention are available. But, to me, muchmore
strikingly there is plenty of TENOS on capacities actually
available(to many) for the exercise of such intuition: they differ
(strikingly) amongdifferent subjects so that mere truth soon
becomes relatively marginal (forunderstanding). Digression. This
aspect of (Kants) subjectivity is a bitat odds with his idea(l) of
universal laws. But so is his sound bite onmankind being made of
crooked timber, which brings to (my) mind suchunsuitable laws as
all timber being inflammable if dry. Like many of Kants(splendidly)
flashy ideals those above are tempered by tacit
understandingsprovided by practical reason. End of digression.
Russell himself described Principia memorably as a parenthesis
in therefutation of Kant. Perhaps it is or can be turned into one
with a lit-
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SECOND THOUGHTS AROUND GDEL 115
tle formal care if that staple of Kants business, preoccupation
withpossibilities-in-principle, is taken at all literally.
Otherwise 3(b) of the lastsection is good enough to be prepared for
reservations about the refuta-tion in turn (by TENOS on a Pyrrhic
victory), for example, by remindersaround the logical (categorical)
definition (in Peanos axioms) of the suc-cession of the natural
numbers. Now, the axioms state properties (of naturalnumbers as
previously understood) used in number theory. One is
thecontrapositive of the principle of induction known as Kstners
principleat the time of Kant. But also, first, the validity of
those axioms was notrecognized only by logical deduction, and,
secondly, there is the moredemanding matter of TENOS: in more than
a century of familiarity withthose axioms and of a great deal of
progress in number theory there havebeen few occasions for
referring to the axioms as a definition, sometimesaka explanation;
in contrast to TENOS in the case of, say, groups (in placeof
natural numbers); cf. 2(b) below and the disclaimer and reminder in
3(b)of the last section.
2. Around 1950 Bourbaki published 2 articles already mentioned
in thePreamble about logical foundations (with and without this
word in thetitle), in(to) which I now read much more than I did
(even) at the timeof Gdels piece. By TENOS, for example, in reviews
(at the time) bylogicians, also some others had, let us say,
similar blindspots to mine. Theywere (for me) removed as
follows.
(a) In Larchitecture des mathmatiques (Bourbaki 1950), aka
Bour-bakis manifesto, some strong language is used, but adequate if
PeircesLaw in 3(c) of the last section is remembered. In
particular, the founda-tional ideal of unity is embraced (w.r.t.
mathematical knowledge), onlythe logical variety proclaimed by
Russell, cf. (4) in the last section is declared to be the wrong
kind. Logic, tacitly, its foundational side,7 isdubbed useful, but
also the least interesting side of mathematics. Bour-bakis as it
were competing offer was an abstract side of mathematics, witha
different kind of unity; (to be) achieved by suitable combinations
of so-called structures mres, aka basic structures, including
groups, topologicalspaces and the like. On a tacit understanding
the offer was to be effectiveacross the board of contemporary
elementary, aka basic, pure mathematics,and correspondingly
marginal in more demanding situations.
At the time I did not read that understanding into the manifesto
nordid an open-ended list of basic structures fit the idea(l) of
unity formed bymy minds (pretty) naked eye. Furthermore the
structures themselves weredefined in logical terms, in particular,
of sets; with symbols for operationson sets being ubiquitous. This
seemed interesting (not only to the eye in
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116 G. KREISEL
question). NB By the early 60s, there was no doubt about the
successof abstract mathematics, especially, in comparison with the
enterprise ofinfinitistic mathematics, popular in the 20s (around
Fund. Mathematicae,genitive singular). The question was how to say
what was known; cf. also(4) below.
To say it in terms of the Preamble, its reminder about the
diversityof the world is good enough to be prepared for tempering
the idea(l)of unity, by the alternative of relatively few basic
structures suitable forrelatively many situations (actually
encountered, not merely, possible-in-principle, when a suitable
understanding of relatively would be required).As to orders of
interest of logical knowledge, by prominence, again oneis prepared
for (relatively few) distinctions. But within available TENOSvery
few occasions have been encountered in abstract mathematics
whereall the knowledge of sets available (in familiar axioms) is
rewarding. NBThis is not interpreted here as reducing dubious
doubts over ontologicalcommitments, but to specify the kind of
interest that Bourbaki found lack-ing: rewards in abstract
mathematics for intensive analysis of principlesfor sets (at an
opposite extreme to Peirces Law). Again, it is a matter ofTENOS
where the evolution of suitable combinations, aka compatibilityof
basic structures proceeds smoothly. It does, for example, in
combi-nation of (the logical structure) order and group (in
algebra) inasmuch asin (suitably) ordered groups the operation is
merely required to preservethe order, and this has been good enough
in many (not all) situations inalgebra. Suitable orders of interest
are a bit more demanding in broaderareas of knowledge, even within
(very parochial) proof theory (cf. Zbl.815.03036 cited early on in
the Preamble). More memorably, there is theformal conflict between
the logical ideal of a universal well-ordering andTENOS (from
algebra) of suitable orderings, many of which, includingR,are not
well-founded.
(b) Foundations for the Working Mathematician (1949) also by
Bour-baki, is here viewed as a footnote to the manifesto in (a),
incidentally,with second thoughts in endnote 7 (with a so to speak
less sterile idea(l)for using logical knowledge). The emphasis is
not on principles (for sets),but as in 2(a) of the Preamble on a
selection of particular properties of,and notation for, sets
(decidedly at odds with Quines quip in 3(c) of thelast section
about the ideal of, tacitly, analytical philosophy). The
selectionis (cl)aimed to be suitable across the board, evidently, a
matter of TENOS.
As in the case of basic structures in (a), the list of (logical)
items insuch foundations suitable for the working mathematician
remains open-ended. Candidates from a broader logical education
(than mere set theory)have crossed not only my mind. But my
knowledge of broad math-
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SECOND THOUGHTS AROUND GDEL 117
ematics is not good enough for TENOS, and thus (by endnote 5)
not for(my being) interest(ed in my impressions). Digression on
some familiarliterature about not only this kind of uncertainty.
This had a dramaticside for Keats (cf. [N1(d)] on negative
capability) or for Russell (cf. p. 11of History of Western
Philosophy, Allen and Unwin, 1946). In contrast, for some of us,
who remember the diversity of the world, other things areto be
found in it for which ones knowledge at hand or within onesreach is
good enough; as always, depending on ones resources andappetites.
[N3] (re)views these last two paragraphs as an exercise of
thenon-academic option for using ones logical education.
3. Wittgenstein, like Bourbaki, was given to strong language
about logicalfoundations; at least by the time I met him in the
40s. (By that time hecertainly paid attention to the diversity of
the world, but was not one of those of us mentioned for contrast at
the end of (2) above; almost everyside he looked at seemed to have
a tragic or at least dramatic elementfor him.) By TENOS, of which
more in (d) below, I know too little forscholarly (cl)aims
regarding details of his ideas. But what I know is goodenough for
the following:
(a) In Tractatus logico-philosophicus, aka as an ode to
propositionallogic, Wittgenstein had presented a logical
ideal(ization) of the world andof knowledge; tacitly, with finitely
many simples (in the 20s, the decade fa-mous for its craving for
crisis). In our first conversation, but also elsewhere,he
emphasized that his later ideas were in sharp contrast to that
logi-cal idea(lization); without specifying any particular aspects
that presentparticularly striking differences.
Now, in Cantors terms (used for explaining his idea of sets), a
logicalview is a subject, a set as it were, that can be grasped as
a unity, but not itscomplement, where Wittgensteins contrasts are
to be found. This is satire,and easy; in Juvenals words difficile
est saturam non scribere. But it isgood enough as a reminder of an
alternative: to discover relatively few keywords adequate for
labelling relatively many contrasts as they come up; asalways, at a
price, for example, in quality of literary form (of this
articlecompared to Wittgensteins writings).
(b) Wittgensteins oft-quoted key words our language(s?), never
cameup in our conversations. If they had they would have reminded
me ofMephisto (in Goethes Faust), who had enchanted me in my
mid-teens;especially, his response to Fausts grand-standing about
his troubles with words: Willst Du was in Worten sagen, musst Du
Worten nachjagen.
One to me, particularly unsatisfactory element in
Wittgensteinswritings is his very frequent reliance on imagined
situations. They are OK,
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118 G. KREISEL
and occasionally singularly attractive, if used by such
singularly giftedpeople as Einstein, especially w.r.t. celestial
and other inaccessible matters(cf. [N2]). Popular and
correspondingly primitive speculations on influ-ences (and similar
causes, cf. endnote 3) aside, in retrospect I view mypresent
emphasis on TENOS not only in this article as a contrast to
thatelement; again, at a (here, social) price, described already in
the Preambleat the end of (2), for the sake of TENOS recorded in
some of my reviews.
(c) If Wittgensteins writings are interpreted as a philosophers
non-philosophical miscellanies (cf. endnote 4 and its
surroundings), a greatdeal fits (a) above; sometimes more memorably
if background is added(provided one has it, deconstructionist
doctrine being more suitable forthose lacking suitable background).
Thus somewhere Wittgenstein writeseloquently about phenomena of
reading. Here suitable background is inthe introduction to: Was
sind und was sollen die Zahlen (Dedekind 1888),which states a
counterpart to the logical idea(l) of rigour in the case ofreading:
spelling out words (letter by letter); not haphazardly, but as
aparallel to the rigour provided by logical deduction from his now
calledPeanos axioms for arithmetic. NB This quotation is, by TENOS,
oftentaken as ridicule. For the non-academic option of the
Preamble, such mis-cellanies are comparable to (miscellanies of)
contrasts between dynamicidea(lization)s of liquids (in motion) and
other aspects of liquids; dynamicidea(l)s corresponding to logical
idea(l)s and liquids to the world or toknowledge at large.
Relatively little need be known about properties ofso-called ideal
liquids to see that they are not suitable for understandingsuch
striking phenomena as drag or turbulence in hydrodynamics,
wetness,chemical composition, etc. Such a miscellany is not a
theory (to which thenon-academic option adds the reminder that not
all effective thought, letalone, all understanding of a diverse
world, is theoretical). But, at leastfor me, who does not happen to
be one of Littlewoods irreconcilablepersons, a miscellany by an
acute observer can be a sight more rewardingthan Tripos Exercises
in mechanics or scholastic exercises in formal dress,aka
philosophical logic.
Given Wittgensteins limited knowledge of both logic and
mathematicsthere is (fittingly) nothing (I know of) in his writings
that corresponds to2(b) above, on foundations for the working
mathematician. Admittedly,at the time in the mid 40s, I knew some
snippets in proof theory which Ithought of as candidates for what I
now, not then, call such foundations;in contrast to what, by above,
I have realized since then. Be that as it may Istressed
particularly the shift of telos involved in (new) uses of
consistencyproofs; Wittgenstein took all this in (his stride), and
seemed to find it allcongenial; but cf. (d) below.
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SECOND THOUGHTS AROUND GDEL 119
(d) By a fluke I dragged some comments by Wittgenstein into
footnote4 on p. 281 of Fund. Math. 37 (1950) concerning formal
incompleteness,including an anticipation of what later became known
as Henkinsproblem. The paper was a write-up of something I had done
in the early40s, but published only after the war. The background
to this footnotewas a relatively short conversation with
Wittgenstein, in which he wantedme to tell him Gdels proof. (He
told me he had never read it, havingbeen put off by the
introduction.) NB As I see it now at length in (3) ofthe Preamble,
certainly not then a formal refutation of logical founda-tions,
which he had already recognized as crassly defective, would
fittinglyarouse his passions. At the time, by reflex I described
the proof in termsof diagonalizing sequences (of partial functions)
without (ab)use of self-reference; in short, as a jolly piece of
mathematical logic. (I have describedhis enthusiasm on various
occasions.)
Until recently, I lacked TENOS striking enough (for me) to
consider theextent to which the ideas that interested him in our
conversations wereactually taken up by him or at least fitted those
he pursued when on hisown. Recently, in the socalled Wiener Edition
some pages with marginalcomments by Wittgenstein to Hardys Pure
Mathematics are reproduced.By a fluke his copy was published in
1941 (though this date was not men-tioned loc. cit.), and my
extended conversations with Wittgenstein aboutthis very book took
place in the first 5 months of 1942. So those commentswere
certainly not written long before the conversations, which I have
oftenhad occasion to relate to matters of (to me) continuing
interest. In contrastId not merely have been bored by the published
marginalia, but havefound them barmy (as I still do).
Added in proof. Only in autumn 1997 did I come across a
typescript (ofa MS in Wittgensteins Nachla) with the title:
Mathematik und Logik. Itstarts on 6.I.1943, almost exactly a year
after the conversations just men-tioned started, and continues
beyond 3.III.1944. In para. 197 near the endhe refers to
transformations of proofs, which had been prominent in
ourconversations (and seemed to me, at the time, to satisfy him).
But loc. cit.he recognised in such transformations a strong
temptation to (mis)under-stand them as contributing to his idea(l)
of philosophy, adumbratedimmediately before in para. 196; cf. a
fuller statement of this ideal (inanother typescript) on
30.XII.1942. Quite simply, philosophy (of mathe-matics) should not
add new (mathematical) knowledge, but linger, albeitcritically,
over what are called in this article views taken by many aminds
naked eye.
Patently, my disclaimers in the opening paragraph of the present
sub-section (3) were sound; the new knowledge of Wittgensteins
feelings about
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120 G. KREISEL
the matter is needed as a threshold for informed views on it.
Correspond-ing disclaimers apply to conversations with Gdel, for
example, in [Nl(e)]or (4) below under the heading milksop
foundations.
4. In terms of the Preamble the (cl)aims of logical foundations
are viewstaken by many a minds naked eye, and so Wittgensteins
observationsin (3) are viewed as reservations about those (cl)aims;
in effect, noton purpose, they paraphrase Bourbakis strong language
(about the leastinteresting side). As other such language it
becomes rewarding when tem-pered by a suitable reminder; it leaves
open where which kind and degreeof elaboration of those foundations
is suitable or at least good enough; cf.[N2] on situations where a
geocentric view is suitable (with due regard forthe temptation of
Tripos Exercises on epicycles). Anecdote. As describedon top of p.
613 of [N1(d)], 40 years ago I was blind to any interest ofany
foundational idea(l)s, and delighted by my interpretation of
pointsmade by Gdel in conversations on the potential of some of
them. The timeseems ripe for the following comments around that
interpretation.
(a) A milksop view of the foundational debates about different
ismsis a reminder of the truism that most knowledge has aspects
involvingobjects, ideas, subjects and even formal notation
(particularly prominent insymbolic data processing). Einstein may
have compared those debates tooo (cf. Aesops Fable CLXVIII), but
his own debateswith Bohr over the foundations of quantum mechanics
had a distinctlysimilar general flavour. (They differed from most
foundational debates bythe intellectual vitality of the specific
ideas for areas of interest to Bohrand Einstein, and missing in the
scholastic logical tradition).
(b) A milksop view of debates on logical laws regards them as an
op-portunity for a tour de force on the theme of diversity; even
within themodest domain of minding ones ps and qs (in classical and
intuitionisticlogic). Later, in the 50s, in line with the medieval
tradition of lumping logicand rhetoric together (among the trivia),
one had (Lorenzens) dialogues;where scoring Debating Points
replaced the search for Truth.
(c) A milksop counterpart to Gdels specific (cl)aim for
Wesensschau(to discover axioms of infinity suitable for settling
CH) was broader; itwent back to various (elementary, relatively
neglected) epistemologicalnotions regarding properties of proofs
and to somewhat related notionsin the area of choice sequences. As
expected this was good enough forseeing similar flaws of course,
not mere lack of precision in popularepistemological notions (cf.
1(c) of the next section), and saying what isseen.
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SECOND THOUGHTS AROUND GDEL 121
Remark for readers interested in details how this is largely
superseded byevents outside logic. J.R. Moschovakis (1994) returned
recently to socalledlawless sequences providing an occasion for a
review that goes into suchdetails; cf. Zbl. 795.03083. It should be
added that a little later, on p.829, the author corrected an
oversight (about free variables in schemata),which A.S. Troelstra
has pointed out repeatedly and conscientiously, butapparently not
memorably enough.
Sundry items from Vol. III, preceded by 3 loose ends from Vol.
II, re-viewed (too) briefly in [N1]. Wherever available, trade
jargon and refer-ences will be used; allowing readers to test if
they have a suitable back-ground and interest: they have the former
when the jargon is familiar,and the latter, when they follow up the
references. (Here the market forillusions of understanding is
neglected since, once again, I do not knowenough to be interested
in any ideas I might have about it.)
1. Hardcore foundations follow the idea in Quines quip about
sustainedanalysis; they are at an opposite extreme to milksop
foundations in (4) ofthe last section.
(a) On p. 124 of Vol. II Gdel reports (his) amazement at . . .
logicalintuitions concerning . . . being . . . being
self-contradictory. At the time,in the mid 40s, he related this to
the familiar paradoxes, later (on p. 376 ofVol. III) relegated to
epistemology; cf. also (b) and (c) below. A milksopview replaces
speechlessness for example, Weyls (cf. p. 632 of [N1(d)])at (t)his
amazement; it is enough to remember suspicions (rather than
intu-itions) in the 19th century, even of such modest objects as
Cantors abstractsets, where here and below, object is used rather
than being, let alone,beable.
W.r.t. objects, specifically, in abstract mathematics, Hilbert
establisheda suitable milksop view in his bestseller on the
Foundations of Geometry:their nature does not matter. It turned out
by TENOS that contrary to, forexample, Freges (usual) convictions
by and large enough is known: it isenough that there be infinitely
many objects (in the situation) considered.But also and this
elementary reminder, too, is part of a milksop view in any model of
an abstract geometry the objects (such as points andlines) are
specified as part of the data. This loss of drama or, according
totemperament, feeling of enlightenment is tempered as follows (at
least,for those with a suitable education).
The milksop view above is not good enough in some parts of the
algebraof free uncountable abelian groups; by TENOS, it is not
always easy torecognize such parts on sight. NB. A milksop view is
good enough to be
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122 G. KREISEL
prepared for this possibility; of needing a closer look at the
objects (thegroups and the free bases). Samples Such a look is
needed in the caseof Whiteheads problem; cf. P. Eklof (1976). It is
not needed in the caseof another uncountable abelian group, which
has (all) bounded sequencesof integers as elements and the
operation of pointwise addition (shownin the 50s by Specker, to be
free, when the objects are elements from acollection satisfying
CH); cf. G. Nbling (1968), and (with a twist) G.Bergman (1972).
(The twist consists in a shift to matters of discoveredinterest
away from the particular group; including transfinite induction
onsuitable pseudo-well-orderings.)
(b) In footnote h on p. 275 of Vol. II Gdel touches another
familiarfoundational idea(l), although he does not give it a
familiar name; as inthe case of ramified sets, which he called
constructible (for his particularvariant). The material below is
complementary to p. 615 at the end of 5in [Nl(d)] on the idea(l) of
reductive proof.
In familiar terms, purity of method has struck many a minds
naked eyeas an ideal, at least since those Greeks who objected to
the use Archimedesmade of (his knowledge of) 3 dimensional cones
for knowledge of ellipsesetc., which are conic sections. The
peroration of Hilberts Foundations ofGeometry propounds the ideal
as a principle, which Hilbert rarely followedin practice. (In the
Foundations he has an entertaining exercise regardingDesargues
theorem in 2 and 3 dimensional projective geometry.) In thiscentury
it was popular in number theory under the heading of elementary,aka
direct, proofs of the prime number theorem, and so forth; cf.
(Kreisel1951, 248). It may also be viewed as a local version of
[A476/B504], citedon p. 294 of Vol. III, on self-contained bodies
of knowledge, once againexpounded so wahrhaft philosophisch (in
Kants manner).
Here a milksop view recalls, first, realizations of the ideal,
and, sec-ondly, TENOS that has accumulated around them; for
example, about theircontributions (on second thoughts) compared to
(original) expectations; cf.the case of real algebra in 3(b) at the
Preamble. But also one will recallthe diversity of the world,
perhaps by now effectively (at least, here), interms of the
following elementary, decidedly milksop reminder.
Every relation between an object and any (other) thing is a
property ofthat object, where, as in (a), a milksop view of objects
is taken.
NB. This is certainly elementary, and seems, by TENOS, to many a
mindsnaked eye innocent; deceptively, inasmuch as it is quite good
enough forreservations about the broad ideal of completeness; cf.
(3) of the Pream-ble. It thus prepares one for limitations of
formal (in) completeness results,but does not replace what has been
said above in detail. In a related way,
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SECOND THOUGHTS AROUND GDEL 123
the milksop view is good enough for correcting totally
unrealistic expec-tations of the ideal of purity of method, and
especially of Tripos Exercisesaround it. Reminder. As usual, once
the limitations of the ideal itself havebeen grasped, material
(with intellectual vitality) evolved in the course ofpursuing the
ideal, may be discovered to serve a more suitable discoveredtelos
(which, by TENOS, will generally not be as vivid to many a
mindsnaked eye as the ideal itself).
Samples. First, in the case of the prime number theorem
mentioned a mo-ment ago, direct proofs were given in the 40s (by
Selberg and Erdos); real-istically speaking, helping to put the
ideal in its place. Less well known isa later interpretation, aka
restructuring, of those proofs on pp. 102103 ofW.J. Ellisons Les
nombres premiers, Paris, 1975. This exhibits vividly atleast one
element lacking in direct proofs, which for some of us is mademore
memorable by contrast with the familiar idea(l) of
Beziehungsreich-tum. Secondly, at another extreme, Gentzens
cut-free rules for elementary,aka first-order, logic satisfy the
ideal with knobs on; not merely in thecrude form that logical
theorems ideally have (ordinary) logical proofs.Only (logical)
parts (in the technical sense of subformulae) of the formulaproved
appear in a cut-free proof, which is thus reductive in a
colloquialsense, too.8 Here, a (heavy) price is well known: the use
of lemmas andof proving specific cases from general theorems is
excluded by the ideal.Shifts of telos have been known (but not
always so seen, let alone, suitablypursued); for reading off
numerical bounds from general theorems at oneextreme, and general
theorems from computations at another (for some 60and 10 years
respectively; cf. also [A2 a(iii)]).
Remarks. Hilberts programme requiring finitist proofs for
finitist(icallystated) theorems fits the ideal, and thus his
peroration loc. cit. Now the(epistemological) notion of finitist
proof, which like other such notionsstrikes many a minds naked eye,
has long been known; enough (about it) torecognize that the
programme was carried out up to the hilt for real algebra.For a
milksop view this (TENOS from algebra) has, at least, some
epis-temological value; some compared to the immense value
proclaimedon p. 85 (or p. 112) of Vol. III: Falls das ursprngliche
Hilbertsche Pro-gramm durchfhrbar gewesen wre, so wre das
zweifellos von un-geheurem erkenntnistheoretischen Wert gewesen.
The milksop view recog-nizes but does not enter into the separate
question: Where, outside the(subsidized) trade of epistemology, are
the orders, aka values, which haveevolved in it, suitable for, aka
compatible with, other demands on knowl-edge, here, around proofs?
Readers with suitable background may look
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124 G. KREISEL
at [A2(d)] for a little more detail. I do not know enough to be
interestedin my ideas around the fact that Einstein himself, to
whose generaltheory of relativity the citation by the Nobel Prize
Committee attributed(great) epistemological value, did less well in
foundational debates aroundthe quantum theory, after having done
very well in establishing quantumeffects. (The behauptende Kraft of
his ideas may have been different forhim and for those
irreconcilable persons who assume that power to beindependent of
subjects, aka objective; for TENOS, cf. Zbl. 815.03036.)
(c) On p. 150 of Vol. II epistemology comes up (in a lecture in
the mid40s at Princeton University): . . . the great importance of
general recursive-ness (or Turing computability) . . . is largely
due to the fact that with thisconcept one has succeeded in giving
an absolute definition of an interestingepistemological notion.
This hard core view of recursiveness certainly fits Gdels ideas
inour conversations beginning some 10 years later. It also fits a
quip, re-ported on p. 128 of Quines Quiddities (as a bit odd, if
not cheeky), bya mathematician at (the Institute of Advanced Study
at) Princeton; if notin the 40s, certainly by the early 60s.
Roughly, Gdels interests wouldbe more suitable for the School of
Historical Studies than for the Schoolof Mathematics (to which Gdel
belonged); at least, on the then prettydominant parochial view of
(pure) mathematics, noted in Zbl. 795.03083(and cited in 4(a) of
the last section). Less flippantly, Gdels would-bebold formulation
of his view is below the threshold established by thenon-academic
option in the preceding pages.
Reminders. The assumption (in that view) of some universal order
of im-portance or some privileged place for orders evolved in
epistemology neglects the matter of compatibility with orders
evolved in other options.This neglect has already been adumbrated
at one extreme, in foundationsfor the working mathematician, and in
the view of philosophy as a repos-itory of ideas that strike many a
minds naked eye, at another. There isa patent parallel between
logical foundations of mathematics and socalledrecursion-theoretic
foundations, a.k.a. the logical idea(lization), of compu-tation,
and thus the parallel option of foundations for the working
(humanor electronic) computer. Further, it must be remembered that
experiencemeant by the household word computation does not consist
only of eval-uating expressions, aka execution of programmes; no
more than experi-ence meant by proof (in mathematics) consists only
of formal deduction:the discovery of suitable properties of
(suitable) objects for mathematicalstudy and of suitable
descriptions by programmes is, by TENOS, generallymore demanding
(and correspondingly rewarding). A familiar reminder
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SECOND THOUGHTS AROUND GDEL 125
stresses the balance between contributions and distraction (from
more re-warding aspects) by attention to those logical aspects of
computation. (By[N2] that TENOS is fitting inasmuch as the
computation meant here, incontrast to the epistemological variety,
is a terrestrial business.) Readersfamiliar with the subject of
finitely generated groups will remember that(in the jargon of 2(b)
of the last section) some simple facts about recursive-ness have
belonged to the foundations for the working group theorist
(whoworks on those groups) for more than 30 years; cf. G. Higman,
Subgroupsof Finitely Presented Groups, (1961). In other parts of
group theory otheridea(lization)s of computability have found a
place, for example, by finiteautomata, whence the name automatic
groups; cf. Word processing andgroups by Cannon, Epstein, Holt and
Paterson.
For any of those working foundations there is little difference
betweenGdels own and Turings descriptions of recursiveness:
GdelHerbrandequations and Turings universal machine, resp.
Unflinchingly, ever sincethe mid 30s, Gdel himself stressed the
advance made by Turing over hisown description (but also over
Churchs in terms of the -calculus). Milk-sop foundations dotted
some is and crossed some ts w.r.t. those equations,and established
additional relations between various descriptions; for ex-ample, by
matching steps in the corresponding evaluation procedures,
akaChurchs superthesis. But milksop foundations have nothing
memorableabout aspects of the kind involved in the advance, which
Gdel saw(without describing it). One candidate for such a
description focuses onideas that strike many a minds naked eye.
Thus Turings idealization iscertainly more vivid to such eyes (than
Gdels equations); cf. also Gdelsexpression finite mind used in his
Gibbs lecture quoted in (2) below. Asusual and this is very much a
matter of TENOS , the naked eye ishere, too, not dramatically in
error, but it is often a bit short-sighted; inthe present case,
without a clue of the often (more) laborious path from avivid
description to properties of discovered interest; here from
Turingsto such properties in the case of recursiveness, for
example, in Higman(1961) above.
In short, orders of importance (of such properties), evolved in
epis-temology and in the other trades considered, generally
conflict; with thepotential for the kind of distractions (from
other contributions) stressedthroughout. Those with suitable
intellectual reflexes will take this in stride.But, by TENOS, not
all; in ways related to the following asymmetry.
The epistemological tradition is so undemanding that Gdel
(tacitly,following it) could properly use titles containing the
phrases some basictheorems or, as already mentioned, the modern
development of the foun-dations of mathematics when talking about
incompleteness and recursive
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126 G. KREISEL
functions or recursively enumerable sets of axioms. This is in
sharp con-trast to the reservations above about the notion of
completeness itself in(b), as a corollary to diversity , and about
recursiveness as a logical, buthardly realistic, idea(lization) of
computation. This asymmetry is generallyreflected in the markets
for which these different traditions are suitable.
Reminder of the famous exception mentioned in (b) above:
Einsteins the-ories of relativity (tacitly, particularly his
introductions) and their (great)epistemological value; so to speak
in contrast to (marginal) relativistic ef-fects known at the time.
Now, an exception does not prove a rule, neitherin the sense of
establishing nor of testing it; often it draws attention to
therule. But the present exception is good enough not to
pontificate generallyagainst attention to the epistemological
tradition; in particular, attentionby those with Einsteins
particular gifts, who admittedly are not likely tobe discouraged by
any such pontification anyway.
2. Brains and faster computers. On pp. 304323 of Vol. III a
draft or thetext of the 1951 Gibbs Lecture of the American Math.
Soc. is printed forthe first time. The broad outline has long been
known (and more recentlybeen, let us say, popularized by critics of
socalled strong AI). In the late40s, Turing (often regarded as a
patron saint of that kind of AI) had pub-lished his exercise in the
cult of black boxes, enshrined in what has cometo be called Turings
test.
Gdels and Turings views are formally at opposite extremes, and
so at least according to what is said about extremes would be
expected totouch. They do; by neglecting points particularly
prominent in this article,including the warnings (in [N2]) about
views taken by many a mindsnaked eye; here, views of the mind
concerned or of other minds, and corre-sponding practical
consequences. Thus expanded TENOS but not only from scientific
studies, say, of brain damaged subjects would be expectedto provide
idea(lization)s around phenomena meant by such householdwords as
intelligence or, more generally, mind, which are more suitable for
such expanded TENOS than (and therefore different from)
thoseimagined; at least, imagined by those of us without Einsteins
particulargifts. This expectation is here viewed as parallel to the
emphasis atthe end of the Preamble w.r.t. Life Itself, but also in
(1) above w.r.t.computation and proof. Readers with a little
background in number theorywill know examples of the evolution of
more suitable ideas in the lightof TENOS also in the humble area of
diophantine problems; cf. Mazursquestion in [A2 c (ii)].
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SECOND THOUGHTS AROUND GDEL 127
It would be unrewarding (for me) to review the vast literature
aroundGdels and Turings views in the last 45 years; let alone,
belabour thesound bites on diverse aspects of those views in [N1];
on mechanics being(non-)mechanical (in [N1(a)]), on ideal(izations
of) mathematicians (in[N1(b)]), on Pyrrhic victories of strong Al
(in [N1(c)]), or on Turingsphilosophical error in (in [N1(d)]).
Instead, some items in Gdels GibbsLecture will be used to underline
some of those neglected points, includingof course the potential of
those items for additions to the repository (inendnote 3). One of
Gdels conclusions, prominent on p. 290 of Vol. III,gives fair
warning; not to look in the Gibbs Lecture for contributions
toaspects (emphasized in this article as) outside that
repository:either . . . the human mind . . . infinitely surpasses
the powers of any finite mind, or elsethere exist absolutely
unsolvable diophantine problems.
(a) As to a difference between Turings and Gdels descriptions of
re-cursiveness adumbrated in 1(c) above, the epistemological
idea(l) of afinite mind is more vivid at least, to my mind when one
thinksof Turings machine than of Gdels equations. As to the
business of infinitely, or at least demonstrably surpassing the
powers of this orthat, a memorable example is provided by familiar
relations betweenTurings machine and any finite automaton; cf.
1(c), too. Readers mayremember here (and in (b) below) Juvenal
quoted in 3(a) of the lastsection.
(b) Peirces Law, applied to the words absolutely unsolvable, is
goodenough to be certain of Gdels conclusion; at least, for those
of usfor whom these words are sufficiently vague. But once again
theyprovide fair warning against assuming as some of those
popularizersmentioned earlier do that the first alternative in that
conclusion, aboutminds and machines, is established outright in the
piece (by the incom-pleteness theorem). NB It is a matter of formal
routine to refute thatassumption formally (for those with suitable
formal resources); cf. [A2d (i)] for some items in the Gibbs
Lecture that have been adapted inthe literature for logical
exercises on diverse epistemological notions. The next point is
less parochial.
(c) In footnote 13 on p. 309 of Vol. III Gdel envisages future
advances inbrain physiology which would establish with empirical
certainty thatthe brain is a machine in the sense of Turing.
(Fortunately, I heard this let us say, innocent vision from him
some 40 years ago, leaving meplenty of time for second thoughts.)
Now, given Gdels preoccupa-tion with non-scientific meditation
exercises it would be unrealistic toexpect much of his idea(l)s on
empirical certainty. So one option here
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128 G. KREISEL
is to apply again Peirces Law; in particular, (in my case) given
(my)ignorance of any specific aspects of the brain that are even
candidatesfor such advances.
But too much is known (also, by me) about the infinitistic
characterof recursiveness to be satisfied with this option. In
terms of (3) in thePreamble, Gdels footnote is comparable to
envisaging future advancesin, say, scientific carpentry to
establish with empirical certainty that thelength of a diagonal of
a unit square is rational. NB. In contrast to Gdelsfootnote, here
specific (geometric) aspects are stated.Reminder: Every finite
sequence of (pedantically, hereditarity finite) datais recursive,
just as every finite (initial segment of a) decimal or
binaryexpansion is a rational real number. In this connection Gdel
liked toquote (in conversation) the hoary canard about all theory
involving ideal-izations, but was impatient with elementary second
thoughts about someidealizations being less suitable than
others.
For milksop foundations the emphasis shifts to TENOS on where,
if any-where, such infinitistic distinctions do have empirical
interpretations, forexample, as follows:
(i) Interpretations aside, the mathematical apparatus used for
some sci-entific theory may be better understood by attention to
infinitistic(including infinitesimal) distinctions. A recent
spectacular exampleuses algebraic (among analytical) solutions of
the differential equa-tions of Yang and Mills; as usual, by a
suitable shift to (previouslyneglected) discovered aspects of those
equations. It is a separate mat-ter in which way(s), if any, such
mathematical understanding findsan empirical counterpart; for
example, in an interpretation of a new(discovered) parameter such
as the degree of those algebraic solutions. The next point is more
general, and hence demands less background.
(ii) In my teens there was a wide-spread superstition that
infinitistic dis-tinctions, including discontinuities (in
continuous parameters), werein-principle empirically meaningless.
(A distinction by Hadamardwas so (mis)interpreted.) Various atomic
and more generally particletheories are counter examples. Here the
phenomena considered are assumed to be described in integral terms,
and so approximateknowledge of empirical effects such as a
potential is good enough tobe interpreted by an integral result (on
the number of particles). Lesselementary uses of infinitesimal
properties, by M. V. Berry, are quotedon p. 640 of [N1(d)]; quite
apart from discontinuities in continuummechanics, when, for
example, discrete drops drip from a tap.
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SECOND THOUGHTS AROUND GDEL 129
In short, the weakness of Gdels footnote 13 loc. cit. is here
seen notas a mere matter of principle, but in not being rooted in
(enough) knowl-edge around brain physiology. Reminder. For many of
us there is an air ofunreality in attempts to use Turings
idealization of a computer for strongAI, which is engaged in barely
doing what people and other animals dowell. The alternative is to
use a few properties of that idea in foundationsfor the working
software engineer, using sensible, aka robust, AI for robotsthat do
their jobs better than people etc. As in other engineering, this
oftenrequires very different (material) structures.
(iii) Some of the literature around (i) and (ii) above is
rewarding; tacitly(as always), for those with suitable resources.
At one extreme there iswork by Pour El and Richards (cf. [N1(a) and
(d)]), on equations oc-curring in (well established) physics. Some
of their solutions turn outto be formally non-recursive; only
formally inasmuch as the aspects,aka parameters, considered are
found, on closer inspection, not to besuitable for the empirical
interpretation envisaged (cf. pp. 900902 ofJSL 47, 1982, on an
abuse of socalled initial values in the case ofhyperbolic
equations). At another extreme there are aberrations by oth-erwise
realistically minded applied mathematicians (like V. I. Arnold)who
ask teratological questions around recursiveness; roughly,
aboutsubsets of Q , where in fact subsets of R are suitable, and
appropriatedefinitions (of recursiveness) are available; cf. also
da Costa and Doria(1993).
(iv) Brain physiology aside, the broad contrast between minds
and ma-chines is also familiar from the foundational debates in the
20s be-tween Brouwer and Hilbert: formal operations (on formal
objects)are mechanical while not only those of higher mathematics,
but even logical operations are interpreted intuitionistically to
have mentalconstructions (including proofs) as arguments and
values. Gdel putthis into formal dress in the 30s in terms of
(formal and intuitionistic)provability, cf. [Al b (iii)]; with
second thoughts on using proofs aspart of the mental data (but
leaving open specifics; cf. [A2 (b)]).
NB. Though Kleene presented recursive realizations as formal
exercisesthese can be (and were) interpreted in the terms above;
for example, thefamiliar closure condition (of many systems, with
the usual notation): If .8x 2 !/ .9y 2 !/ A.x; y/, then, for some
recursive V ! ! ! andeach n!,
AT Nn; .n/U; where Nn is the numeral of n:(This is in sharp
contrast to classical rules for suitable501-formulae A.)
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130 G. KREISEL
In plain English, some thought is needed to check that the
properties(enshrined in intuitionistic logic) of those mental
operations are notblatantly non-recursive (despite striking
differences in general laws forformal and other provability loc.
cit.). In retrospect, this closure propertyof familiar, not of all
intuitionistically valid, formal systems was related toanother
(closure) property:If .9y 2 !/B.y/, and .9y 2 !/B.y/ is closed
then, for some n!, B. Nn/;cf. JSL 37 (1972), p. 327, where NIE
should be replaced by NID in l.-20and -19. (For validity, B. Nn/
need not be proved in the same, necessarilyincomplete, formal
system.)
Now, academic traditions have evolved in the last 25 years so as
to makethe (academic) option of continuing such exercises viable
(for making anacademic living). But for the non-academic option
adopted in this article,second thoughts have priority. At one
extreme, (even) the old exercisespresent an alternative to a cult
of black boxes. Thus enough is knownaround those mental operations
(put in black boxes) for, let us say, modesttheory; cf. [A 2 d(i)]
for more, comparable to exercises on neural nets. Atanother
striking extreme, there is the topic of suitable areas for
dif-ferences between (not necessarily conscious) mental operations
and thoseof (Turing) machines. Unless the primary interest is in
those exercises onintuitionistic logic, one would not start by
assuming that differences in theareas of mathematics, let alone, of
logic are particularly rewarding. It maywell be that differences
between humans and other animals are striking inthese areas. But
some elements common to humans and (suitable) animalsseem to differ
enough from Turing machines at least without fancy sen-sors for
establishing empirical differences (if one wants to do this);
withmore scope for experimentation.
Reminders of the non-academic option for using ones logical
education;cf. 3(c) of the Preamble. Above, it was used for co