Platonist Assertory Mathematics. Thought of the Day 88.0

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Traditional Platonism, according to which our mathematical theories are bodies of truths about a realm of mathematical objects, assumes that only some amongst consistent theory candidates succeed in correctly describing the mathematical realm. For platonists, while mathematicians may contemplate alternative consistent extensions of the axioms for ZF (Zermelo–Fraenkel) set theory, for example, at most one such extension can correctly describe how things really are with the universe of sets. Thus, according to Platonists such as Kurt Gödel, intuition together with quasi-empirical methods (such as the justification of axioms by appeal to their intuitively acceptable consequences) can guide us in discovering which amongst alternative axiom candidates for set theory has things right about set theoretic reality. Alternatively, according to empiricists such as Quine, who hold that our belief in the truth of mathematical theories is justified by their role in empirical science, empirical evidence can choose between alternative consistent set theories. In Quine’s view, we are justified in believing the truth of the minimal amount of set theory required by our most attractive scientific account of the world.

Despite their differences at the level of detail, both of these versions of Platonism share the assumption that mere consistency is not enough for a mathematical theory: For such a theory to be true, it must correctly describe a realm of objects, where the existence of these objects is not guaranteed by consistency alone. Such a view of mathematical theories requires that we must have some grasp of the intended interpretation of an axiomatic theory that is independent of our axiomatization – otherwise inquiry into whether our axioms “get things right” about this intended interpretation would be futile. Hence, it is natural to see these Platonist views of mathematics as following Frege in holding that axioms

. . . must not contain a word or sign whose sense and meaning, or whose contribution to the expression of a thought, was not already completely laid down, so that there is no doubt about the sense of the proposition and the thought it expresses. The only question can be whether this thought is true and what its truth rests on. (Frege to Hilbert Gottlob Frege The Philosophical and Mathematical Correspondence)

On such an account, our mathematical axioms express genuine assertions (thoughts), which may or may not succeed in asserting truths about their subject matter. These Platonist views are “assertory” views of mathematics. Assertory views of mathematics make room for a gap between our mathematical theories and their intended subject matter, and the possibility of such a gap leads to at least two difficulties for traditional Platonism. These difficulties are articulated by Paul Benacerraf (here and here) in his aforementioned papers. The first difficulty comes from the realization that our mathematical theories, even when axioms are supplemented with less formal characterizations of their subject matter, may be insufficient to choose between alternative interpretations. For example, assertory views hold that the Peano axioms for arithmetic aim to assert truths about the natural numbers. But there are many candidate interpretations of these axioms, and nothing in the axioms, or in our wider mathematical practices, seems to suffice to pin down one interpretation over any other as the correct one. The view of mathematical theories as assertions about a specific realm of objects seems to force there to be facts about the correct interpretation of our theories even if, so far as our mathematical practice goes (for example, in the case of arithmetic), any ω-sequence would do.

Benacerraf’s second worry is perhaps even more pressing for assertory views. The possibility of a gap between our mathematical theories and their intended subject matter raises the question, “How do we know that our mathematical theories have things right about their subject matter?”. To answer this, we need to consider the nature of the purported objects about which our theories are supposed to assert truths. It seems that our best characterization of mathematical objects is negative: to account for the extent of our mathematical theories, and the timelessness of mathematical truths, it seems reasonable to suppose that mathematical objects are non-physical, non- spatiotemporal (and, it is sometimes added, mind- and language-independent) objects – in short, mathematical objects are abstract. But this negative characterization makes it difficult to say anything positive about how we could know anything about how things are with these objects. Assertory, Platonist views of mathematics are thus challenged to explain just how we are meant to evaluate our mathematical assertions – just how do the kinds of evidence these Platonists present in support of their theories succeed in ensuring that these theories track the truth?

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Supervenience

Past decades witness an increasing interest in the concept of supervenience, which has traditionally been used as a relation between sets of properties. A set A of properties (called ‘supervenient properties’) is said to supervene on another set B (called ‘subveinent properties’), just in case if B-properties are indistinguishable, then so are A-properties; in other words, agreement in respect of B-properties implies agreement in respect of A-properties. In slogan form, “there cannot be an A-different without a B-difference”. The core idea of supervenience is that fixing subvenient properties fixes its supervenient ones; or equivalently, subvenient properties determine supervenient properties.

The notion of supervenience dates back at least to G. E. Moore’s classical work, where he described some certain dependency relationship between moral and non-moral properties. However, Moore did not use the term ‘supervenience’ explicitly; it was R. M. Hare that introduced the term into the philosophical literature, to characterize a relationship between moral properties and natural properties. Hare stated

First, let us take that characteristic of ‘good’ which has been called its supervenience. Suppose that we say ‘St. Francis was a good man’. It is logically impossible to say this and to maintain at the same time that there might have been another man placed in precisely the same circumstances as St. Francis, and who behaved in them in exactly the same way, but who differed from St. Francis in this respect only, that he was not a good man.

Thanks to Donald Davidson, the term ‘supervenience’ was first introduced into contemporary philosophy of mind, which opened up a new research direction in this area and other branches of philosophy. Donald Davidson used psychophysical supervenience to defend a position of anomalous monism that although the mental supervenes on the physical, the former cannot be reduced to the latter, as he said:

Although the position I describe denies there are psychophysical laws, it is consistent with the view that mental characteristics are in some sense dependent, or supervenient, on physical characteristics. Such supervenience might be taken to mean that there cannot be two events alike in all physical respects but differing in some mental respect, or that an object cannot alter in some mental respect without altering in some physical respect. Dependence or supervenience of this kind does not entail reducibility through law or definition.

It is alleged that every major figure in the history of western philosophy has been at least implicitly committed to some supervenience thesis. For example, Leibniz used the Latin word ‘supervenire’, to state the thesis that relations are supervenient on properties; G. E. Moore stated that “one of the most important facts about qualitative difference · · · [is that] two things cannot differ in quality without differing in intrinsic nature”; David Lewis used a thesis of Humean supervenience to express that the whole truth about a world like ours supervenes on the spatiotemporal distribution of local qualities.

The notion of supervenience is ubiquitous in our daily life. For instance, the aesthetic properties of a work of art supervene on its physical properties, the price of a commodity supervenes on its supply and demand, effects supervene on causes, and the mental supervenes on the physical. According to the chart of levels of existence, atoms supervene on elementary particles, molecules supervene on atoms, cells supervene on molecules, and so on.

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Moreover, a number of interesting doctrines and problems can be formulated in terms of supervenience. A paradigmatic example is physicalism, which may be construed as a thesis that “everything supervenes on the physical”. Mereology may be explained as mereological supervenience, i.e., the whole supervenes on its parts. Determinism can be roughly construed as a thesis that everything to the future supervenes on the present, and perhaps past, facts. All of the distinction between internalism and externalism can be characterized by means of supervenience theses. Mind-body problem may be rephrased as to whether the psychophysical supervenience thesis holds, i.e., are psychological properties supervenient upon physical properties?

There are so many distinct formulations for this concept, e.g., individual supervenience, local supervenience, global supervenience, weak supervenience, strong supervenience, similarity-based supervenience, regional supervenience, local-local supervenience and strong-local-local supervenience, multiple domain supervenience, that David Lewis thought of it as an ‘unlovely proliferation’. No matter how different the formulations are, they all conform to the aforementioned core idea of supervenience – that is, fixing the subvenient properties fixes the supervenient properties.

Supervenience has many applications, among which a central use is so-called ‘argument by a false implied supervenience thesis’. It is well known that the reduction of A to B implies the supervenience of A on B; in short, reduction implies supervenience. Thus for one to argue against a reduction thesis, it suffices to falsify the corresponding supervenience thesis. Other applications include characterizing the distinctions between Internalism and Externalism, characterizing physicalism, characterizing haecceitism, and so on.