Metaphysical Would-Be(s). Drunken Risibility.

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If one were to look at Quine’s commitment to similarity, natural kinds, dispositions, causal statements, etc., it is evident, that it takes him close to Peirce’s conception of Thirdness – even if Quine in an utopian vision imagines that all such concepts in a remote future will dissolve and vanish in favor of purely microstructural descriptions.

A crucial difference remains, however, which becomes evident when one looks at Quine’s brief formula for ontological commitment, the famous idea that ‘to be is to be the value of a bound variable’. For even if this motto is stated exactly to avoid commitment to several different types of being, it immediately prompts the question: the equation, in which the variable is presumably bound, which status does it have? Governing the behavior of existing variable values, is that not in some sense being real?

This will be Peirce’s realist idea – that regularities, tendencies, dispositions, patterns, may possess real existence, independent of any observer. In Peirce, this description of Thirdness is concentrated in the expression ‘real possibility’, and even it may sound exceedingly metaphysical at a first glance, it amounts, at a closer look, to regularities charted by science that are not mere shorthands for collections of single events but do possess reality status. In Peirce, the idea of real possibilities thus springs from his philosophy of science – he observes that science, contrary to philosophy, is spontaneously realist, and is right in being so. Real possibilities are thus counterposed to mere subjective possibilities due to lack of knowledge on the part of the subject speaking: the possibility of ‘not known not to be true’.

In a famous piece of self-critique from his late, realist period, Peirce attacks his earlier arguments (from ‘How to Make Our Ideas Clear’, in the late 1890s considered by himself the birth certificate of pragmatism after James’s reference to Peirce as pragmatism’s inventor). Then, he wrote

let us ask what we mean by calling a thing hard. Evidently that it will not be scratched by many other substances. The whole conception of this quality, as of every other, lies in its conceived effects. There is absolutely no difference between a hard thing and a soft thing so long as they are not brought to the test. Suppose, then, that a diamond could be crystallized in the midst of a cushion of soft cotton, and should remain there until it was finally burned up. Would it be false to say that that diamond was soft? […] Reflection will show that the reply is this: there would be no falsity in such modes of speech.

More than twenty-five years later, however, he attacks this argument as bearing witness to the nominalism of his youth. Now instead he supports the

scholastic doctrine of realism. This is usually defined as the opinion that there are real objects that are general, among the number being the modes of determination of existent singulars, if, indeed, these be not the only such objects. But the belief in this can hardly escape being accompanied by the acknowledgment that there are, besides, real vagues, and especially real possibilities. For possibility being the denial of a necessity, which is a kind of generality, is vague like any other contradiction of a general. Indeed, it is the reality of some possibilities that pragmaticism is most concerned to insist upon. The article of January 1878 endeavored to gloze over this point as unsuited to the exoteric public addressed; or perhaps the writer wavered in his own mind. He said that if a diamond were to be formed in a bed of cotton-wool, and were to be consumed there without ever having been pressed upon by any hard edge or point, it would be merely a question of nomenclature whether that diamond should be said to have been hard or not. No doubt this is true, except for the abominable falsehood in the word MERELY, implying that symbols are unreal. Nomenclature involves classification; and classification is true or false, and the generals to which it refers are either reals in the one case, or figments in the other. For if the reader will turn to the original maxim of pragmaticism at the beginning of this article, he will see that the question is, not what did happen, but whether it would have been well to engage in any line of conduct whose successful issue depended upon whether that diamond would resist an attempt to scratch it, or whether all other logical means of determining how it ought to be classed would lead to the conclusion which, to quote the very words of that article, would be ‘the belief which alone could be the result of investigation carried sufficiently far.’ Pragmaticism makes the ultimate intellectual purport of what you please to consist in conceived conditional resolutions, or their substance; and therefore, the conditional propositions, with their hypothetical antecedents, in which such resolutions consist, being of the ultimate nature of meaning, must be capable of being true, that is, of expressing whatever there be which is such as the proposition expresses, independently of being thought to be so in any judgment, or being represented to be so in any other symbol of any man or men. But that amounts to saying that possibility is sometimes of a real kind. (The Essential Peirce Selected Philosophical Writings, Volume 2)

In the same year, he states, in a letter to the Italian pragmatist Signor Calderoni:

I myself went too far in the direction of nominalism when I said that it was a mere question of the convenience of speech whether we say that a diamond is hard when it is not pressed upon, or whether we say that it is soft until it is pressed upon. I now say that experiment will prove that the diamond is hard, as a positive fact. That is, it is a real fact that it would resist pressure, which amounts to extreme scholastic realism. I deny that pragmaticism as originally defined by me made the intellectual purport of symbols to consist in our conduct. On the contrary, I was most careful to say that it consists in our concept of what our conduct would be upon conceivable occasions. For I had long before declared that absolute individuals were entia rationis, and not realities. A concept determinate in all respects is as fictitious as a concept definite in all respects. I do not think we can ever have a logical right to infer, even as probable, the existence of anything entirely contrary in its nature to all that we can experience or imagine. 

Here lies the core of Peirce’s metaphysical insistence on the reality of ‘would-be’s. Real possibilities, or would-bes, are vague to the extent that they describe certain tendential, conditional behaviors only, while they do not prescribe any other aspect of the single objects they subsume. They are, furthermore, represented in rationally interrelated clusters of concepts: the fact that the diamond is in fact hard, no matter if it scratches anything or not, lies in the fact that the diamond’s carbon structure displays a certain spatial arrangement – so it is an aspect of the very concept of diamond. And this is why the old pragmatic maxim may not work without real possibilities: it is they that the very maxim rests upon, because it is they that provide us with the ‘conceived consequences’ of accepting a concept. The maxim remains a test to weed out empty concepts with no conceived consequences – that is, empty a priori reasoning and superfluous metaphysical assumptions. But what remains after the maxim has been put to use, is real possibilities. Real possibilities thus connect epistemology, expressed in the pragmatic maxim, to ontology: real possibilities are what science may grasp in conditional hypotheses.

The question is whether Peirce’s revision of his old ‘nominalist’ beliefs form part of a more general development in Peirce from nominalism to realism. The locus classicus of this idea is Max Fisch (Peirce, Semeiotic and Pragmatism) where Fisch outlines a development from an initial nominalism (albeit of a strange kind, refusing, as always in Peirce, the existence of individuals determinate in all respects) via a series of steps towards realism, culminating after the turn of the century. Fisch’s first step is then Peirce’s theory of the real as that which reasoning would finally have as its result; the second step his Berkeley review with its anti-nominalism and the idea that the real is what is unaffected by what we may think of it; the third step is his pragmatist idea that beliefs are conceived habits of action, even if he here clings to the idea that the conditionals in which habits are expressed are material implications only – like the definition of ‘hard’; the fourth step his reading of Abbott’s realist Scientific Theism (which later influenced his conception of scientific universals) and his introduction of the index in his theory of signs; the fifth step his acceptance of the reality of continuity; the sixth the introduction of real possibilities, accompanied by the development of existential graphs, topology and Peirce’s changing view of Hegelianism; the seventh, the identification of pragmatism with realism; the eighth ‘his last stronghold, that of Philonian or material implication’. A further realist development exchanging Peirce’s early frequentist idea of probability for a dispositional theory of probability was, according to Fisch, never finished.

The issue of implication concerns the old discussion quoted by Cicero between the Hellenistic logicians Philo and Diodorus. The former formulated what we know today as material implication, while the latter objected on common-sense ground that material implication does not capture implication in everyday language and thought and another implication type should be sought. As is well known, material implication says that p ⇒ q is equivalent to the claim that either p is false or q is true – so that p ⇒ q is false only when p is true and q is false. The problems arise when p is false, for any false p makes the implication true, and this leads to strange possibilities of true inferences. The two parts of the implication have no connection with each other at all, such as would be the spontaneous idea in everyday thought. It is true that Peirce as a logician generally supports material (‘Philonian’) implication – but it is also true that he does express some second thoughts at around the same time as the afterthoughts on the diamond example.

Peirce is a forerunner of the attempts to construct alternatives such as strict implication, and the reason why is, of course, that real possibilities are not adequately depicted by material implication. Peirce is in need of an implication which may somehow picture the causal dependency of q on p. The basic reason for the mature Peirce’s problems with the representation of real possibilities is not primarily logical, however. It is scientific. Peirce realizes that the scientific charting of anything but singular, actual events necessitates the real existence of tendencies and relations connecting singular events. Now, what kinds are those tendencies and relations? The hard diamond example seems to emphasize causality, but this probably depends on the point of view chosen. The ‘conceived consequences’ of the pragmatic maxim may be causal indeed: if we accept gravity as a real concept, then masses will attract one another – but they may all the same be structural: if we accept horse riders as a real concept, then we should expect horses, persons, the taming of horses, etc. to exist, or they may be teleological. In any case, the interpretation of the pragmatic maxim in terms of real possibilities paves the way for a distinction between empty a priori suppositions and real a priori structures.

Quantifier – Ontological Commitment: The Case for an Agnostic. Note Quote.

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What about the mathematical objects that, according to the platonist, exist independently of any description one may offer of them in terms of comprehension principles? Do these objects exist on the fictionalist view? Now, the fictionalist is not committed to the existence of such mathematical objects, although this doesn’t mean that the fictionalist is committed to the non-existence of these objects. The fictionalist is ultimately agnostic about the issue. Here is why.

There are two types of commitment: quantifier commitment and ontological commitment. We incur quantifier commitment to the objects that are in the range of our quantifiers. We incur ontological commitment when we are committed to the existence of certain objects. However, despite Quine’s view, quantifier commitment doesn’t entail ontological commitment. Fictional discourse (e.g. in literature) and mathematical discourse illustrate that. Suppose that there’s no way of making sense of our practice with fiction but to quantify over fictional objects. Still, people would strongly resist the claim that they are therefore committed to the existence of these objects. The same point applies to mathematical objects.

This move can also be made by invoking a distinction between partial quantifiers and the existence predicate. The idea here is to resist reading the existential quantifier as carrying any ontological commitment. Rather, the existential quantifier only indicates that the objects that fall under a concept (or have certain properties) are less than the whole domain of discourse. To indicate that the whole domain is invoked (e.g. that every object in the domain have a certain property), we use a universal quantifier. So, two different functions are clumped together in the traditional, Quinean reading of the existential quantifier: (i) to assert the existence of something, on the one hand, and (ii) to indicate that not the whole domain of quantification is considered, on the other. These functions are best kept apart. We should use a partial quantifier (that is, an existential quantifier free of ontological commitment) to convey that only some of the objects in the domain are referred to, and introduce an existence predicate in the language in order to express existence claims.

By distinguishing these two roles of the quantifier, we also gain expressive resources. Consider, for instance, the sentence:

(∗) Some fictional detectives don’t exist.

Can this expression be translated in the usual formalism of classical first-order logic with the Quinean interpretation of the existential quantifier? Prima facie, that doesn’t seem to be possible. The sentence would be contradictory! It would state that ∃ fictional detectives who don’t exist. The obvious consistent translation here would be: ¬∃x Fx, where F is the predicate is a fictional detective. But this states that fictional detectives don’t exist. Clearly, this is a different claim from the one expressed in (∗). By declaring that some fictional detectives don’t exist, (∗) is still compatible with the existence of some fictional detectives. The regimented sentence denies this possibility.

However, it’s perfectly straightforward to express (∗) using the resources of partial quantification and the existence predicate. Suppose that “∃” stands for the partial quantifier and “E” stands for the existence predicate. In this case, we have: ∃x (Fx ∧¬Ex), which expresses precisely what we need to state.

Now, under what conditions is the fictionalist entitled to conclude that certain objects exist? In order to avoid begging the question against the platonist, the fictionalist cannot insist that only objects that we can causally interact with exist. So, the fictionalist only offers sufficient conditions for us to be entitled to conclude that certain objects exist. Conditions such as the following seem to be uncontroversial. Suppose we have access to certain objects that is such that (i) it’s robust (e.g. we blink, we move away, and the objects are still there); (ii) the access to these objects can be refined (e.g. we can get closer for a better look); (iii) the access allows us to track the objects in space and time; and (iv) the access is such that if the objects weren’t there, we wouldn’t believe that they were. In this case, having this form of access to these objects gives us good grounds to claim that these objects exist. In fact, it’s in virtue of conditions of this sort that we believe that tables, chairs, and so many observable entities exist.

But recall that these are only sufficient, and not necessary, conditions. Thus, the resulting view turns out to be agnostic about the existence of the mathematical entities the platonist takes to exist – independently of any description. The fact that mathematical objects fail to satisfy some of these conditions doesn’t entail that these objects don’t exist. Perhaps these entities do exist after all; perhaps they don’t. What matters for the fictionalist is that it’s possible to make sense of significant features of mathematics without settling this issue.

Now what would happen if the agnostic fictionalist used the partial quantifier in the context of comprehension principles? Suppose that a vector space is introduced via suitable principles, and that we establish that there are vectors satisfying certain conditions. Would this entail that we are now committed to the existence of these vectors? It would if the vectors in question satisfied the existence predicate. Otherwise, the issue would remain open, given that the existence predicate only provides sufficient, but not necessary, conditions for us to believe that the vectors in question exist. As a result, the fictionalist would then remain agnostic about the existence of even the objects introduced via comprehension principles!

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?

Conjuncted: Demise of Ontology

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The demise of ontology in string theory opens new perspectives on the positions of Quine and Larry Laudan. Laudan stressed the discontinuity of ontological claims throughout the history of scientific theories. String theory’s comment on this observation is very clear: The ontological claim is no appropriate element of highly developed physical theories. External ontological objects are reduced to the status of an approximative concept that only makes sense as long as one does not look too closely into the theory’s mathematical fine-structure. While one may consider the electron to be an object like a table, just smaller, the same verdict on, let’s say, a type IIB superstring is not justifiable. In this light it is evident that an ontological understanding of scientific objects cannot have any realist quality and must always be preliminary. Its specific form naturally depends on the type of approximation. Eventually all ontological claims are bound to evaporate in the complex structures of advanced physics. String theory thus confirms Laudan’s assertion and integrates it into a solid physical background picture.

In a remarkable way string theory awards new topicality to Quine’s notion of underdeterminism. The string theoretical scale-limit to new phenomenology that makes Quine’s concept of a theoretical scheme fits all possible phenomenological data. In a sense string theory moves Quine’s concept from the regime of abstract and shadowy philosophical definitions to the regime of the physically meaningful. Quine’s notion of underdeterminism also remains unaffected by the emerging principle of theoretical uniqueness, which so seriously undermines the position of modest underdeterminism. Since theoretical uniqueness reveals itself in the context of new so far undetected phenomenology, Quine’s purely ontological approach remains safely beyond its grasp. But the best is still to come: The various equivalent superstring theories appear as empirically equivalent but ‘logically incompatible’ theories of the very type implied by Quine’s underdeterminism hypothesis. The different string theories are not theoretically incompatible and unrelated concepts. On the contrary they are merely different representations of one overall theoretical structure. Incompatible are the ontological claims which can be imputed to the various representations. It is only at this level that Quine’s conjecture applies to string theory. And it is only at this level that it can be meaningful at all. Quine is no adherent of external realism and thus can afford a very wide interpretation of the notion ‘ontological object’. For him a world view’s ontology can well comprise oddities like spacetime points or mathematical sets. In this light the duality phenomenon could be taken to imply a shift of ontology away from an external ‘corporal’ regime towards a purely mathematical one. 

To put external and mathematical ontologies into the same category blurs the central message the new physical developments have in store for philosophy of science. This message emerges much clearer if formulated within the conceptual framework of scientific realism: An extrapolation of the notion ‘external ontological object’ from the visible to the invisible regime remains possible up to quantum field theory if one wants to have it. It fails fundamentally at the stage of string theory. String theory simply is no theory about invisible external objects.

Frege-Russell and Mathematical Identity

Frege considered it a principal task of his logical reform of arithmetic to provide absolutely determinate identity conditions for the objects of that science, i.e. for numbers. Referring to the contemporary situation in this discipline he writes:

How I propose to improve upon it can be no more than indicated in the present work. With numbers … it is a matter of fixing the sense of an identity.

Frege makes the following critically important assumption : identity is a general logical concept, which is not specific to mathematics. Frege says:

It is not only among numbers that the relationship of identity is found. From which it seems to follow that we ought not to define it specially for the case of numbers. We should expect the concept of identity to have been fixed first, and that then from it together with the concept of number it must be possible to deduce when numbers are identical with one another, without there being need for this purpose of a special definition of numerical identity as well.

In a different place Frege says clearly that this concept of identity is absolutely stable across all possible domains and contexts:

Identity is a relation given to us in such a specific form that it is inconceivable that various forms of it should occur.

Frege’s definition of natural number, as modified in Russell (Bertrand Russell – Principles of Mathematics) later became standard. Intuitively the number 3 is what all collections consisting of three members (trios) share in common. Now instead of looking for a common form, essence or type of trios let us simply consider all such things together. According to Frege and Russell the collection (class, set) of all trios just is the number 3. Similarly for other numbers. Isn’t this construction circular? Frege and Russell provide the following argument which they claim allows us to avoid circularity here: given two different collections we may learn whether or not they have the same number of members without knowing this number and even without the notion of number itself. It is sufficient to find a one-one correspondence between members of two given collections. If there is such a correspondence, the two collections comprise the same number of members, or to avoid any reference to numbers we can say that the two collections are equivalent. This equivalence is Humean. Let us define natural numbers as equivalence classes under this relation. This definition reduces the question of identity of numbers to that of identity of classes. This latter question is settled through the axiomatization of set theory in a logical calculus with identity. Thus Frege’s project is realized: it has been seen how the logical concept of identity applies to numbers. In an axiomatic setting “identities” in Quine’s sense (that is, identity conditions) of mathematical objects are provided by an axiom schema of the form

∀x ∀y (x=y ↔ ___ )

called the Identity Schema (IS). This does not resolve the identity problem though because any given system of axioms, generally speaking, has multiple models. The case of isomorphic models is similar to that of equal numbers or coincident points (naively construed): there are good reasons to think of isomorphic models as one and there is also good reason to think of them as many. So the paradox of mathematical “doubles” reappears. It is a highly non-trivial fact that different models of Peano arithmetic, ZF, and other important axiomatic systems are not necessarily isomorphic. Thus logical analysis à la Frege-Russell certainly clarifies the mathematical concepts involved but it does not settle the identity issue as Frege believed it did. In the recent philosophy of mathematics literature the problem of the identity of mathematical objects is usually considered in the logical setting just mentioned: either as the problem of the non-uniqueness of the models of a given axiomatic system or as the problem of how to fill in the Identity Schema. At the first glance the Frege-Russell proposal concerning the identity issue in mathematics seems judicious and innocent (and it certainly does not depend upon the rest of their logicist project): to stick to a certain logical discipline in speaking about identity (everywhere and in particular in mathematics).