[FOM] Re. alleged quote by Hilbert (redirected from Michael Detlefsen)

[Martin Davis]

Dear Martin and fellow FOMers,

I know of nowhere where Hilbert says what Goldstein attributes to him in a 
"punkt und schluss" manner. Hilbert did, of course, say various things 
similar to what Goldstein quotes. In the end, though, any unqualified 
attribution of such a view to Hilbert must ignore important qualifications 
that Hilibert himself made. It must also trample on certain texts where 
Hilbert made claims contrary in spirit---and even letter---to Goldstein's 
"quote".

I would go farther and say that no serious formalist---not even 
Thomae---held the position Goldstein describes. In support of that I offer 
the following "classical" formulations of formalist theses.

 From a momentarily springlike South Bend,

Mic Detlefsen

P.S. I first sent this April 19th. There were formatting problems with that 
note, though, and I'm only now getting around to reformatting.

____________________________________

Hilbert ("Natur u. mathematisches Erkennen", Lectures Fall semester, 1919), 
p. 19. "Die Mathematik ist nicht wie ein Spiel, bei idem die Aufgaben durch 
willkŸrlich erdachte Regeln bestimmt werden, sondern ein begriffliches 
System von innerer Notwendigkeit, das nur so und nicht anders sein kann."

[Note: My translation: "Mathematics is not like a game in which the 
problems are determined by arbitrarily invented rules. Rather, it is a 
conceptual system of inner necessity that can only be what it is and not 
otherwise."]

Hilbert (1920, summer semester lectures in 'Probleme der mathematischen 
Logik', trans. in Ewald II, 943ff) said of Kronecker: "From the last 
paradox in particular we see that arbitrary definitions and inferences, 
made in the manner that hitherto was usual, are not allowed. And this leads 
to the expedient of prohibitions, of dictatorship.
The first, most far-reaching and most radical dictator in this area was 
Kronecker. ... he rejected set theory as a mere game of fantasy containing 
nothing but illegitimate combinations that are no longer mathematical 
concepts...."

[Note: Hilbert goes on to describe how arithmetic is indubitable and 
immediately comprehensible and that this owes to its 'enduring 
checkability'. And he says that Kronecker has adopted 'ostrich politics'. 
He also makes remarks about Kronecker's use of ideals and moduli (to take 
the place of irrational numbers) and gives examples of theorems that he 
rejected (e.g. Dirichlet's theorem that in every arithmetic progression 
where the difference is not divisible by the initial element) there are 
infinitely many primes, the logical law of excluded middle). Ultimately, he 
describes Kronecker's restrictions as restrictions on 'concept formation'.]

Hilbert 1927 (vH, 474-75): "Brouwer declares [just as Kronecker did in his 
day] that existence statements, one and all, are meaningless in themselves 
unless they also contain the construction of the objects asserted to exist; 
for him they are worthless scrip, and their use causes mathematics go 
degenerate into a game. The following may serve as an example showing that 
a mere existence proof carried out with the logical (epsilon)-function is 
by no means a piece of worthless scrip. ||(474|| ... What, now, is the real 
state of affairs with respect to the reproach that mathematics would 
degenerate into a game?

The source of pure existence theorems is the logical (epsilon)-axiom, upon 
which in turn the construction of all ideal propositions depends. And to 
what extent has the formula game thus made possible been successful? This 
formula game enables us to express the entire thought-content of the 
science of mathematics in a uniform manner and develop it in such a way 
that, at the same time, the interconnections between individual 
propositions and facts becomes clear. To make it a universal requirement 
that each individual formula then be interpretable by itself is by no means 
reasonable; on the contrary, a theory by its very nature is such that we do 
not need to fall back upon intuition or meaning in the midst of some 
argument. What the physicist demands precisely of a theory is that 
particular propositions be derived from laws of nature or hypotheses solely 
by inferences, hence on the basis of a pure formula game, without 
extraneous considerations being adduced. Only certain combinations and 
consequences of the physical laws can be checked by experiment-just as in 
my proof theory only the real propositions are directly capable of 
verification. The value of pure existence proofs consists precisely in that 
the individual construction is eliminated by them and that many different 
constructions are subsumed under one fundamental idea, so that only what is 
essential to the proof stands out clearly; brevity and economy of thought 
are the raison d'tre of existence proofs. In fact, pure existence theorems 
have been the most important landmarks in the historical development of our 
science. But such considerations do not trouble the devout intuitionist. ...

The formula game that Brouwer so deprecates has, besides its mathematical 
value, an important general philosophical significance. For this formula 
game is carried out according to certain definite rules, in which the 
technique of our thinking is expressed. These rules form a closed system 
that can be discovered and definitively stated. The fundamental idea of my 
proof theory is none other than to describe the activity of our 
understanding, to make a protocol of the rules according to which our 
thinking actually proceeds. Thinking, it so happens, parallels speaking and 
writing: we form statements and place them one behind another. If any 
totality of observations and phenomena deserves to be made the object of a 
serious and thorough investigation, it is this one..."

Weyl on Hilbert's 'formalism' (Weyl [1944], p. 640): "Hilbert fully agrees 
with Brouwer in that the great majority of mathematical propositions are 
not "real" ones conveying a definite meaning verifiable in the light of 
evidence. But he insists that the non-real, the "ideal propositions" are 
indispensable in order to give our mathematical system "completeness". Thus 
he parries Brouwer, who had asked us to give up what is meaningless, by 
relinquishing the pretension of meaning altogether, and what he tries to 
establish is not truth of the mathematical proposition, but consistency of 
the system. The game of deduction when played according to rules, he 
maintains, will never lead to the formula 0­0. In this sense, and in this 
sense only, he promises to salvage our cherished classical mathematics in 
its entirety."

Weyl's own statement of formalism (from "The Mathematical Way of Thinking", 
an address given at the Bicentennial Conference at the University of 
Pennsylvania, in 1940. First published in Science 92 (1940): 437-446): "We 
now come to the decisive step of mathematical abstraction: we forget about 
what the symbols stand for. The mathematician is concerned with the 
catalogue alone; he is like the man in the catalogue room who does not care 
what books or pieces of an intuitively given manifold the symbols of his 
catalogue denote. He need not be idle; there are many operations which he 
may carry out with these symbols, without ever having to look at the things 
they stand for."

[Note: This is on p. 442 of the original version of the article in Science.]

von Neumann, 'The formalist foundations of mathematics', in B&P, 61-62: 
"The leading idea of Hilbert's theory of proof is that, even if the 
statements of classical mathematics should turn out to be false as to 
content, nevertheless, classical mathematics involves an internally closed 
procedure which operates according to fixed rules known to all 
mathematicians and which consists basically in constructing successively 
certain combinations of primitive symbols which are considered "correct" of 
"proved". This construction-procedure, moreover, is "finitary" and directly 
constructive. ... if we wish to prove the validity of classical mathematics 
... then we should investigate, not statements, but methods of proof. We 
must regard classical mathematics as a combinatorial game played with the 
primitive symbols, and we must determine in a finitary combinatorial way to 
which combinations of primitive symbols the construction methods or 
"proofs" lead."

Helmholtz ('Zahlen und Messen', Davis trans., 3-4): "In order to indicate 
briefly in advance the standpoint which leads to simple consequent 
deductions and to the solution of the contradictions mentioned, the 
following may serve: I consider Arithmetic or the theory of pure numbers 
(Zahlen) to be a method built upon purely pyschological facts by which is 
taught the consequent application of a system of symbols (namely of 
numbers) of unlimited extension and unlimited possibility of refinement. 
Arithmetic inquires especially what different methods of combination of 
these symbols (operations of calculation) lead to the same final result. It 
teaches also, among other things, how to replace remarkably complicated 
calculations, even such as could be completed in no finite time, by simpler 
ones. Apart from the test thereby made of the inner logicality of our 
thinking, such a procedure would indeed be primarily a pure play of 
ingenuity with imaginary objects, which P. du Bois-Reymond derisively 
compares to the knight-moves of the chess board, if it did not admit of 
such remarkably useful applications. For by means of this symbol system of 
numbers, we give descriptions of the relations of real objects, which, when 
they are applicable, can attain any required degree of exactness; and by 
means of the same, in a great number of cases where natural bodies meet or 
interact under the sovereignty of known laws of nature, the numerical 
values measuring the result are found in advance by calculation."

Thomae's statement of formalism from, Elementare Theorie der analytischen 
Functionen einer complexen Veränderlichen, 2nd ed. (1898), p. 3 (my 
translation): "The formalist conception of number stretches to the same 
reasonable (bescheidenere) borders as the logical. It does not ask what are 
the numbers and what do they do. Rather, it asks what one requires of the 
numbers in arithmetic. For the formalist, arithmetic is a game with signs, 
which are called empty. That means they have no other content (in the 
calculating game) than they are assigned by their behavior with respect to 
certain rules of combination (rules of the game). The chess player makes 
similar use of his pieces; he assigns them certain properties determining 
their behavior in the game, and the pieces are only the external signs of 
this behavior. To be sure, there is an important difference between 
arithmetic and chess. The rules of chess are arbitrary while the rules of 
arithmetic are such that by means of simple axioms numbers can be referred 
to perceptual manifolds and thus make an important contribution to our 
knowledge of nature. ...

The formal standpoint rids us of all metaphysical difficulties; this is the 
advantage it affords us. Metaphysical questions, for whose disposal 
(Erledigung) axioms (principles) (Axiome) are necessary (wanting?) (nötig), 
first surface (auftauchen) with the connection of the numbers with given 
multiplicities (gegebenen Mannigfaltigkeiten). Certainly there are also 
cases in arithmetic in which the numbers do not merely take on a formal 
meaning (formale Bedeutung); e.g. in the statement "this equation of degree 
three", thus if the numbers appear as demonimations (benannte). In such 
cases we must remember the reckoning stones (Rechensteine), and allow the 
meaning of the predicate three to be seen (ansehen) just as certainly as 
the meaning of the predicate white in the proposition "Snow is white". It 
is a popular saying (Redensart) that via the formal framework calculation 
(das Rechnen) is degraded (herabsinken) to a worthless (unwürdigen) game. 
Because of the special advantages (Vorteile), however, which the formal 
standpoint offers (bietet), I am quite happy to forego (verzichte) the 
search for the properties of the numbers along with the search for the 
property of worthiness (Würde)."

Thomae, Elementare Theorie der analytischen Functionen einer complexen 
Veränderlichen, 1st ed.(1880), ¤4, p. 4 (speaking of the 'creation of 0, 
the negative and rational numbers, my translation): "From these newly 
introduced numbers we require (verlangen) that they shall perform according 
to the calculation rules (Verknüpfungsregeln) for whole numbers set out 
(aufgefunden) in paragraphs 2 and 3, and even then we will only permit them 
as useable (brauchbare) creations (Gebilde) if application of the 
calculation rules to the new numbers does not lead to logical inconsistency 
(logischen Widerspruchen), regardless of whether or not a reflection 
(Spiegelbild) of them is to be found in the real world (realen Welt). 
(Whether or not an analog to the numbers can be found in the real world 
does not, of course, have an influence on their meaning (Bedeutung). For in 
the former case (i.e. when they DO have an analog, MD), what we can find is 
a useful application (nützlicher Verwendung), while in the latter case 
(i.e. where there is not an analog, MD) they would afford only the interest 
of an ingenious (geistreichen) logical (logischen) game (amusement) 
(Spielerei)."

Michael Detlefsen <detlefsen.1 at nd.edu>




                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at eipye.com
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                        http://www.eipye.com