Abstract
We discuss the notion of truth in Mathematics as relative to certain structures, very much in line with Bernays’s conception of “bezogene Existenz”. Looking to some concrete examples, we argue that even so-called non-standard structures may have their own rationale. As a result, and in accordance with Bourbaki , structures turn out to be tools and have to be judged with respect to their usefulness rather than with respect to a concept of mathematical truth simpliciter.
Work partially supported by the Portuguese Science Foundation, FCT, through the projects The Notion of Mathematical Proof, PTDC/MHC-FIL/5363/2012, Hilbert’s 24th Problem, PTDC/MHC-FIL/2583/2014, and the Centro de Matemática e Aplicações, UID/MAT/00297/2013.
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Notes
- 1.
Toth (1980); our translation of the German original, as given in Mehrtens (1990, p. 51):
Dennoch hat er [seine nicht-euklidische Geometrie] abgelehnt. Seine Ablehnung beruht im wesentlichen auf seinem Glauben an die herrschende Auffassung, daß es <nur eine wissenschaftliche Methode, ein wissenschaftliches System der Geometrie geben> kann. Aus dieser Konzeption der Unizität folgt unmittelbar, daß sich das euklidische mit dem ihm entgegengesetzten System in der Relation einer ausschließenden Disjunktion, in einer Alternative befindet; ist das eine (als wahr) akzeptiert, so muß das andere (als unmöglich) zurückgewiesen werden: <wäre das dritte System (sc. das der hyperbolischen Geometrie) das wahre, so gäbe es überhaupt keine Euklidische Geometrie> — schrieb er.
- 2.
Frege (1983, p. 183); in German:
Wenn die euklidische Geometrie wahr ist, so ist die nichteuklidische Geometrie falsch, und wenn die nichteuklidische wahr ist, so ist die euklidische Geometrie falsch.
- 3.
See Volkert (2010, p. 118): “ ‘true’ is to understand here in the sense of the classical theory of adequacy” (“ ‘wahr’ ist hier zu verstehen im Sinne der klassischen Adäquatheitstheorie”).
- 4.
See, for instance, Volkert (2010, §2).
- 5.
- 6.
It would not be hard to find many other examples in the mathematical literature; we took this one, as its (uncritical) appearence in the first paragraph of a survey paper shows that it is assumed to be common knowledge.
- 7.
Our translation; the German original reads:
Die durch Euklid überlieferte Geometrie galt zwei Jahrtausende lang als Musterbeispiel einer logisch aufgebauten Wissenschaft. Ihre Axiome wurden als selbstverständlich angesehen, alles andere wurde aus ihnen durch logisch Schlüsse abgeleitet. Dieser axiomatische Standpunkt hat sich heute in der ganzen Mathematik durchgesetzt; nur sind Axiome keine selbstvertändlichen Wahrheiten mehr, sondern willkürliche Festsetzungen nach Zweckmäßigkeitsgesichtspunkten.
- 8.
Our translation; the German original reads:
Wenn von dem semantischen Wahrheitsbegriff gesprochen wird, so kann dies tatsächlich als eine unvollständige, abkürzende Formulierung angesehen werden. Es wird in der Semantik nicht vorausgesetzt, daß es einen ”Wahrheitsbegriff “gibt, vielmehr wird bloß das Prädikat ”wahr “in bezug auf ein bestimmtes System S als sinnvoll angenommen.
- 9.
Tarski’s “truth definition”, given in Tarski (1936), was a breakthrough for the mathematical notion of truth; in particular, it paved the way for a purely mathematical (set-theoretical) treatment of truth in formalized languages. Tarskian semantics may be subject to (philosophical) criticism: hardline formalists as well as intuitionists, for instance, will reject it as such. But, as Heyting pointed out in a short note with the distinctive title On Truth in Mathematics (Heyting 1958), in both cases, formalism and intuitionism, the very notion of truth makes no particular sense any longer. He actually continues by giving a sketch of the Tarskian notion of truth, albeit with the caveat “to me personally the assumption of an abstract reality of any sort seems meaningless”.
- 10.
To our knowledge, Hilbert never formulated the motto is this concise form, but it follows immediately from his argument in the letter to Frege of 29 December 1899, (Frege 1976, Letter XV/4, p. 66). This idea was later even backed by Poincaré (1914, p. 151f):
In mathematics the word exist can only have one meaning; it signifies exemption from contradiction.
French original (Poincaré 1908, p. 162): “en mathématiques le mot exister ne peut avoir qu’un sens, il signifie exempt de contradiction”.
- 11.
We use here the plural, as the negation of \(\textsf {CH}\) can be specified by concrete alternative cardinalities for the real numbers.
- 12.
Of course, any such intention has to be crossed-checked with the mathematical consequences one obtains when adding one or the other form as new axiom. Interestingly, nearly all properties, first of all, in connection with large cardinals, which were considered to decide \(\textsf {CH}\) turned out to be independent of it; and those few which relate to \(\textsf {CH}\), such as the proper forcing axiom or Martin’s maximum (both implying \(2^{\aleph _0} = \aleph _2\)), do not have the status of “evident truths” or intuitively clear axioms (but see also Woodin’s announcement for his Paul Bernays Lectures, Footnote 15).
- 13.
This is one motivation behind Kreisel’s promotion of informal rigour. He writes (Kreisel 1967, p. 138f): “Informal rigour wants [...] not to leave undecided questions which can be decided by full use of evident properties of these intuitive notions.” And Kreisel has a formal argument: \(\textsf {CH}\) might be decidable as a second-order consequence from additional intuitive axioms. As second-order logic is not (recursively) axiomatizable, we may simply overlook it (Kreisel 1967, p. 152): “most people in the field are so accustomed to working with the restricted [first-order] language that they may simply not succeed in taking other properties seriously”. Thus, the independence of \(\textsf {CH}\) is, at this stage, related to the first-order nature of \(\textsf {ZFC}\); this is a quite different situation compared to the Parallel Postulate in Geometry which “is not even a second order consequence of this axiom [i.e., the second-order axiom of continuity].” (Kreisel 1967, p. 151). Still, up to today, nobody has come up with an intuitive second-order property deciding \(\textsf {CH}\) (but, again, see the reference to Woodin’s Paul Bernays Lectures in Footnote 15).
- 14.
- 15.
https://www.gess.ethz.ch/en/news-and-events/paul-bernays-lectures/bernays-2016.html. One may note his ostentatious use of “true” and “false” in the abstracts of his lectures.
- 16.
The quotation is taken from Franzen (2003, p. v).
- 17.
Of course, we presuppose here the consistency of Peano Arithmetic. Doubts about the consistency of \(\textsf {PA} \) cannot be taken mathematically seriously as long as nobody presents an explicit proof of an inconsistency. In particular, doubting the consistency of Peano Arithmetic puts any kind of Arithmetic in doubt, and clearly denies the very existence of the standard structure of the natural numbers (i.e., the first-order structure including addition and multiplication). Mathematically, it is pointless to even consider such doubts; and no philosophical debate about it provided so far any conceptual insight.
- 18.
We like to note in passing that, for Logic, Bernays observed that the difference of classical and intuitionistic logic can attributed to an overloading of the logical connectives, especially the negation (Bernays 1979, p. 4):
As one knows, the use of the “tertium-non-datur” in relation to infinite sets, in particular in Arithmetic, was disputed by L.E.J. Brouwer , namely in the form or an opposition of the traditional logical principle of the excluded middle. Against this opposition is to say that it is just based on a reinterpretation of the negation. Brouwer avoids the usual negation non-A, and takes instead “A is absurd”. It is then obvious that the general alternative “Every sentence A is true or absurd” is not justified.
German original: “Wie man weiß, ist die Verwendung des ‘tertium non datur’ in bezug auf unendliche Gesamtheiten, insbesondere schon in der Arithmetik, von L.E.J. Brouwer angefochten worden, und zwar in der Form einer Opposition gegen das traditionelle logische Prinzip vom ausgeschlossenen Dritten. Gegenüber dieser Opposition ist zu bemerken, daß sie ja auf einer Umdeutung der Negation beruht. Brouwer vermeidet die übliche Negation nicht-A, und nimmt stattdessen ‘A ist absurd’. Es ist dann klar, daß eine allgemeine Alternative ‘Jede Aussage A ist wahr oder ist absurd’ nicht berechtigt ist.”.
- 19.
To be a little bit more explicit: \(\lnot \textsf {Con}_{\textsf {PA} }\) is equivalent to \(\exists x.\textsf {Bew}(x,\ulcorner 0 = 1 \urcorner )\). While it is our intention that the existential quantifier should range over (standard) natural numbers, we have no formal means (in first-order logic) to prevent an interpretation where it could be instantiated by non-standard natural numbers. Just that happens in an insane model. But out of such an interpretation, of course, we cannot build an actual (finite) proof of the inconsistency of \(\textsf {PA} \), even in the insane model.
- 20.
From the Mengenlehrebericht of Schoenflies (1913) :
The development of set theory has its source in the endeavor to provide a clear analysis for two fundamental mathematical notions, namely the notions of argument and of function.
German original: “Die Entwickelung der Mengenlehre hat ihre Quelle in dem Bestreben, für zwei grundlegende mathematische Begriffe eine klärende Analyse zu schaffen, nämlich für die Begriffe des Arguments und der Funktion.”.
- 21.
In Moschovakis (2006) one can find a detailed account how such an encoding works.
- 22.
Beth (1959, p. 642) translates “bezogene Existenz” as “conditional existence” which we consider slightly misleading. It would correspond to “bedingte Existenz” and suggests just a condition for the existence; “bezogen” is stronger in presupposing a reference. Thus, “referring existence” would be a more literal translation.
- 23.
Cf. Corry (2004) for a critical evaluation of Bourbaki’s notion of structure.
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Kahle, R. (2017). Mathematical Truth Revisited: Mathematics as a Toolbox. In: Agazzi, E. (eds) Varieties of Scientific Realism. Springer, Cham. https://doi.org/10.1007/978-3-319-51608-0_22
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