What is the Nature of Mathematical–Logical Objects?

Abstract

This article deals with a question of a most general, comprehensive and profound content as it is the nature of mathematical–logical objects insofar as these are considered objects of knowledge and more specifically objects of formal mathematical theories. As objects of formal theories they are dealt with in the sense they have acquired primarily from the beginnings of the systematic study of mathematical foundations in connection with logic dating from the works of G. Cantor and G. Frege in the last decades of the nineteenth century. Largely motivated by a phenomenologically founded view of mathematical objects/states-of-affairs, I try to consistently argue for their character as objects shaped to a certain extent by intentional forms exhibited by consciousness and the modes of constitution of inner temporality and at the same time as constrained in the form of immanent ‘appearances’ by what stands as their non-eliminable reference, that is, the world as primitive soil of experience and the mathematical intuitions developed in relations of reciprocity within-the-world. In this perspective and relative to my intentions I enter, in the last section, into a brief review of certain positions of G. Sher’s foundational holism and R. Tieszen’s constituted platonism, among others, respectively presented in Sher (Bull Symb Log 19:145–198, 2013) and Tieszen (After Gödel: platonism and rationalism in mathematics and logic, Oxford University Press, Oxford, 2011).

Appendices

Appendix 1

A convenient reference to the possibility of introduction of nonstandard elements by means of the notion of an ‘internal horizon’ of (standard) objects is E. Nelson’s axiomatical foundation of Internal Set Theory (IST) (Nelson 1986, pp. 4–14). More specifically, a key part of the syntax of the theory stands the undefined predicate standard which is the formal equivalent, so to say, of the notion of a fixed object in informal mathematical discourse. Though this new predicate has a syntactical rather than a semantical content it acquires by three ad hoc postulated axioms (the transfer (T), the idealization (I) and standardization axioms (S)) a metatheoretical sense of the ‘fixedness’ of individuals, these latter possibly conceived as either corresponding to real-world counterparts or as pure products of imagination acting, e.g., as substrates of objectivities of understanding. In the present case these objectivities of understanding may be represented by predicate formulas of classical mathematics whose variables are standard (classical) ones. The following two axioms may be taken as corroborating the claim that the introduction of nonstandard individuals can be associated on the constitutional level with the notion of an ‘internal horizon’ of standard (‘fixed’) objects.

The Transfer Principle (T):

$$\begin{aligned} \forall ^{st}t_{1}...\forall ^{st}t_{n}\;[\forall ^{st} x\;A\leftrightarrow \forall x\;A] \end{aligned}$$

where A is an internal formula (i.e., one of classical mathematics which does not include the new predicate standard) whose only free variables are $x, t_{1}, t_{2},...., t_{n}$

The intuition behind (T) is that if something is true for a fixed, but arbitrary x, then it is true for all x.

The Idealization Principle (I):

$$\begin{aligned} \forall ^{stfin}x^{'}\;\exists y\;\forall x\in x^{'}\;A\leftrightarrow \exists y\;\forall ^{st}x\;A \end{aligned}$$

where A is again an internal formula.

In loose terms, to say that there is a y for all fixed x such that we have A, is the same as saying that there is a y for any fixed finite set of x’s such that A holds for all of them. Put more naively, we can only fix a finite number of objects at a time.

Now we can prove by means of these principles that any infinite set includes a nonstandard element, in other words it includes an element that might be larger or smaller than any standard one. The formal proof is very easy and straightforward (ibid. pp. 7–8). Let the classical mathematical formula (internal formula) be $x\ne y$. Then by the (I) principle for every standard finite set $x^{'}$ there is an element y such that for all $x\in x^{'}$, $x\ne y$ and this is equivalent to saying that for every standard x we have $x\ne y$. Further, if we take x and y to range over infinite sets we deduce that in every infinite set there is at least a nonstandard element. In particular, there exists at least a nonstandard natural number.

To come up to this result we have first to assume a notion of ‘fixedness’ for those objects termed as standard, expressed as such by syntactical means, which cannot be meant otherwise than by taking these objects in a lowest-level sense as concrete individuals of intentionality possibly presentified any time at will in reflection. Next, they can be thought to possess a horizon of ‘fixedness’ ideally extensible to indefinite bounds by means of universal or universal-existential quantifiers respectively in the principles (T) and (I). Referring to the principle (I) in particular, one may ‘keep track of the fixedness’ of standard elements as concrete individuals and each time verify the existence of an element distinct from those fixed ones apprehended thus far. To the extent that we can project this kind of intuition from the finitary level to any ‘fixed’ level extending it indefinitely, we can say that it may exist a nonstandard element whose existence is conditioned on the possibility of applying ideally ad infinitum intentional acts performed within the ‘internal horizon’ of standard objects in a non-arbitrary way; that is, in accordance with the syntactical–logical structure of the corresponding predicate formula. In this view, T and I principles are irreal objectivities as intended senses which are always the same (being intersubjectively identical) and whose substrates are formal individuals, i.e., individuals not necessarily corresponding to ‘real-world’ counterparts, having an ‘internal horizon’. A further elaboration of this horizon by means of the intentional faculties of a subject’s ego may lead, and indeed does on the formal level in the case at hand, to new mathematical entities/states-of-affairs.

Appendix 2

Here is an easy example from forcing theory of how a particularization of a formal individual from a universal sentence of a general form to fulfill another predicative sentence is subject to the requirement of ‘confirmation’ by a continuous connection of actual and possible intuitions. I will skip some subtleties of relevant definitions which are too technical for a philosophical reader (or even for a mathematician not knowledgeable of forcing theory) and moreover do not have a special significance for my arguments. Let’s start with the definition of a set A that is dense below an element p:

If P is a partial order and $p\in P$ then the set A is dense below p iff:

$$\begin{aligned} \forall q\le p\; \exists r\le q\;(r\in A)\;\;(\mathbf{I}) \end{aligned}$$

Let $p\in P$ and $\varphi (\tau _{1},..\tau _{n})$ a formula whose free variables $x_{1},..,x_{1}$ have been replaced by P-names $\tau _{1},..,\tau _{n}$ which may be roughly thought of as individuals of forcing theory (for further details see: Kunen 1982, pp. 186–204).

The forcing relation $p\Vdash _{P, M} \varphi (\tau _{1},..,\tau _{n})$ relative to a base model M of the standard set theory ZFC is defined by a certain logical equivalence which notably involves a second order quantification over generic sets G (Kunen 1982, p. 194). Then it holds that the following relation (1) implies (2):

$$\begin{aligned}&\mathbf{(1)}\;\text{ Given } \text{ that }\; p\Vdash \varphi (\tau _{1},..,\tau _{n})\;\text{ holds, } \text{ then }\; \;\forall r\le p\;\;[r\Vdash \;\varphi (\tau _{1},..\tau _{n})]\\&\mathbf{(2)}\;\text{ The } \text{ set }\;\{r;\;r\Vdash \;\varphi (\tau _{1},..\tau _{n})\}\;\;\text{ is } \text{ dense } \text{ below }\;p \end{aligned}$$

The almost trivial proof, given the definition (I) and and relation (1), is as follows: Let $A=\{r;\;r\Vdash \;\varphi (\tau _{1},..\tau _{n})\}$. By definition (I) we must prove that

$$\begin{aligned} \forall r\le p\;\; \exists w\le r\;(w\in A)\;\; \end{aligned}$$

Then by relation (1) given any r there is always a $w\le r$ such that obviously $w\in A$ $\diamond$.

The point here is that the element w satisfying the density below p property of A is confirmed as such after first being identified as the one that fulfills the universal quantification formula $\forall r\le p\;\;[r\Vdash \;\varphi (\tau _{1},..\tau _{n})]$ of (1) which further means that we must have previously formed in actual unity the formula $\forall r\le p\;\;[r\Vdash \;\varphi (\tau _{1},..\tau _{n})]$ as an objectivity of understanding implying the continuous connection of all possible intuitions. All the more so as here is involved a universal quantification formula over bounded variable r which presupposes a notion of a completed infinite whole, the scope of the bounded variable r, in presentational immediacy.