Truth in Mathematics

Euclidean Geometry was for 2000 years the most elaborated mathematical theory. With Kant it was raised to an Anschauungsform a priori which gave it an allegedly untouchable status of eternal truth.

Nevertheless he rejected it [his non-Euclidean Geometry]. His rejection was essentially based on the belief in the dominant opinion that there could “be only one scientific method, one scientific system of Geometry”. From this conception of unicity it follows immediately, that the Euclidean system bears a relation of excluding disjunction to the opposed system; they are alternatives. If one is accepted (as true), then the other must be rejected (as impos­sible); he wrote: “would the third system (sc. hyperbolic geometry) be the true one, so there would be no Euclidean Geometry at all.”

And we still find this view in Frege who wrote^[162]:

If the Euclidean Geometry is true, then the non-Euclidean Geometry is false, and if the non-Euclidean Geometry is true, then the Euclidean Geometry is false.

Of course, one could consider the notion of truth here just as one referring to the physical world; this was the way, Gaufi used it when he spoke about true geometry.^[163] In fact, it took the mathematical community a long time to release Geometry from its empirical status as a theory of our physical space.^[164] But the result of this process was captured by Poincare in his famous dictum^[165]:

One geometry cannot be more true than another; it can only be more convenient.

And we put it as: One can be a better tool than another. And this, clearly, depends on the purpose. By analogy with our initial comparison, where a hammer is not “truer” than a screwdriver, but just a better tool to deal with nails, the Euclidean Parallel Postulate might be the more adequate to deal with certain spaces (physical or not) than its alternative (or vice versa).

Thus, is truth still an issue in Mathematics? Let us give here two citations which seem to deny this. The first one is from a mathematical survey paper on the founda­tions of geometry, published in 19686:

Geometry as passed on by Euclid was considered for two millennia as textbook example of a logically structured science. Its axioms were considered as evident, everything else was deduced from them by logical rules. This axiomatic viewpoint became today prevalent in all of Mathematics; just that the axioms are no longer evident truths, but arbitrary stipulations under aspects of convenience. (Lenz 1968, p. 64).^[166]

The second citation is from a philosopher in a book which bears the title Das Wahrheitsproblem und die Idee der Semantik (The truth problem and the idea of semantics):

If one speaks of a semantic notion of truth, then this can indeed be considered as an incom­plete, abbreviating formulation. In semantics, it is not supposed that there exist a “notion of truth”, but one assumes only that the predicate “true” as meaningful in relation to a concrete system S. (Stegmuller 1957, p. 216).^[167]

What Stegmuller called S is, of course, just a semantic structure in the Tarskian sense like, for instance, that of the natural numbers. There is no controversy about a notion of truth relative to such structures^[168]; what is at issue, is either the question of the very existence of such structures, or whether some structures are distinguished over others, often associated with the terms “standard model” and “non-standard models”.

As far as the existence of such structures is concerned, one may follow Hilbert’s motto^[169]:

Consistency implies Existence.

This claim is not unproblematic, and for a thorough discussion of it by Bernays, we refer to (Bernays, 1950). At least for first-order languages it can be taken for granted; but, in this case, one will be confronted with incompleteness phenomena which imply that the majority of our formal theories allow for non-standard models. This fact leads to the temptation to reserve the notion of mathematical truth for “truth in the standard model”. The problem is now how to distinguish standard models from the non-standard ones.

Apparently, we have no problem distinguishing the standard (first-order) struc­ture for the natural numbers from its competitors; thus, there is a generally accepted notion of truth for Arithmetic. For Geometry, however, a general notion of truth was abandoned, at least for the Parallel Postulate, in view of the success of non-Euclidean Geometries.

The crucial question today is the status of truth in set theory. Thus, in contempo­rary mathematics the primary controversy regarding truth concerns the status of the Continuum Hypothesis (CH). Knowing that it is independent of Zermelo-Fraenkel set theory with the axiom of choice (ZFC), we cannot yet see whether CH or one of its alternatives^[170] is actually intended to hold in our standard set-theoretic universe.^[171] At a first glance one could think that the situation should be analog to the parallel axiom. Godel’s proof of the consistency of CH with ZFC and Cohen’s proof of the consis­tency of its negation with ZFC show that we could consider different set-theoretic universes, one with CH and others with its alternatives—in the very same way that we can consider Euclidean Geometry and non-Euclidean Geometries. However, in sharp contrast to the situation in Geometry, neither Godel’s model of ZFC + CH, the constructible universe L, nor models of ZFC + -CH based on Cohen’s result could be considered as the natural or intended set-theoretic universe. This leaves space to suppose that in the intended universe CH turns out to be determined as true or false.^{[172] [173]} The debate concerning “new axioms” in set theory, and with it the question whether one speaks of one set-theoretic universe or admits a “multiverse”, is fierce.^[174] We may refer, for the latter, to Shelah (2003, p.

Whatever the ultimate outcome of the discussion might be, the status of mathe­matical truth as a relative one does not seem to be affected; only the possible dis­tinction of one particular set-theoretic universe is at issue—in L, CH will always be true; as much as it will be false in other forced universes.

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