Chris Wilcox - Thesis - Ch 1
Quine-Putnam Indispensability Argument
The Quine-Putnam indispensability argument for Platonism in the philosophy of mathematics (hereafter referred to as “the indispensability argument”) is essentially an empiricist argument. It attempts to show that mathematical Platonism follows from our best empirical scientific practice (or at least from the body of knowledge resultant from that practice). In order to demonstrate this, the argument relies to a large extent on two theses central to Quine’s philosophical program (his program, not necessarily his own, sometimes contradictory, thought): confirmational holism (the thesis that scientific theories face the “tribunal of experience” as a whole rather than individually) and (Quinean) naturalism (briefly put, the thesis that philosophy is in no sense “prior” to science). In this chapter I will engage in a detailed discussion of the indispensability argument and its supporting theses. Accordingy, it will be useful to begin with a brief characterisation of the argument itself.
in a Nutshell
Following Colyvan,~\footnote I summarise the structure of the indispensability argument as follows: {[green NEED A LIST HERE ]} Premise 1: We ought to have ontological commitment to all and only those entities that are indispensable to our best scientific theories. Premise 2: Mathematical entities are indispensable to our best scientific theories. {[green THIS SHOULD BE A CONCLUSION LINE THINGAME ]}
[[lime that a claim of ontological commitment to an entity X can be read in at least four (distinct but interrelated) ways: {[green NEED TO MAKE THIS A LIST ]} 1. As a claim of semantic realism: i.e. that statements involving X always have a determinate truth-value, regardless of whether or not there is ever any possibility of obtaining evidence as to what that truth-value is (Dummett is a famous semantic antirealist~\citet{Dummett:1978 and :1993 for example.}). 2. As a claim about the “width of cosmological role” of the entity X: i.e. that it plays a role in explaining things far beyond its own area of discourse (mathematics provides a good example of this). 3. As a claim about the judgement-independence of the truth of statements involving X: i.e. that, under ideal conditions, parties familiar with the area(s) of discourse involving X would not disagree as to the truth-value of statements involving X. 4. As a claim about the role of cognitive command in determining the truth-values of statements involving X: i.e. that a disagreement over the truth-value of one of those statements must be due to a cognitive mistake by at least one of the parties.
However, I agree with Colyvan that these sorts of characterisations do not really capture just what the Platonist’s claim is meant to be’s discussion will prove useful in later chapters however, once the coherence of the Platonist’s ontological claim has been called into question. Wright’s characterisation focuses largely on the linguistic role of the ontological assertion “X exists,” through the examination of the notion of truth at work in the area of discourse under consideration and of the explanatory role(s) of the entities involved. What the Platonist likely has in mind however, is a more meaty, metaphysical kind of ontological claim, a claim that mathematical entities exist in some sense “out there,” independently of how we capture them through language and/or thought. This is not to say that this strong metaphysical existence may not also manifest itself in ways consistent with Wright’s characterisation, just that the “out thereness” is more fundamental. Thus, for the purposes of the indispensability argument, mathematical entities are claimed to exist in much the same way that scientific realists hold unobservable entities such as electrons to exist (although it should be said that many Platonists would dispute whether mathematical entities play any causal role, this will become important later). Alternatively, one can approach the conception of ontological commitment at work here by way of extending the basic view that physical entities (such as chairs, tables, etc.) exist to the more “abstract” class of mathematical entities. At this point then, it will prove useful to examine more closely just what the proponents of the indispensability argument mean by the terms “entity” and, more importantly, “mathematical entity.”
- Mathematical or Otherwise
Primarily, it seems that Colyvan uses the term “entity” to refer to something without implying any ontological commitment via past associations of the term used (such as might be the case for example if he used the more ontologically weighty terms “object,” or, moving in an anti-Platonist direction, “concept”). As intimated, on this account, an entity is just supposed to be a bare, ontologically agnostic bearer for any properties we generally hold the particular entity to have. For example, the entity “2” would bear properties such as its place on the real number line, it status as the sum of 1 and 1, etc.. :1953 provides a good taxonomy via which one can get a better grip on the important ways in which things grouped together under the entity banner may differ. He distinguishes between four ontological categories: concrete particulars (for example, Socrates), abstract universals (for example, wisdom), abstract particulars (for example, Socrates’ wisdom), and concrete universals (for example Socraeity — the property of being like Socrates).
It is important to note that the resolution of the ontological status of an entity does not necessarily imply that the ontological status of its properties will be resolved in the same way. For example, it is quite a common for someone to hold the belief that some physical object (a table for example) exists (possibly in virtue of its status as a collection of smaller objects such as atoms) whilst denying that some of its properties (for example, its colour) do not. Thus, as will become important later, a successful argument for the existence of an entity need not necessarily be interpreted as an argument for the existence of all of its properties. {[green NEED TO ADD THE REVERSE EG HERE - LOOK IN NOTES ]}
Now, the key entities that we are concerned with here are mathematical ones. Accordingly then, it is important to get clear on just what sorts of properties Colyvan holds mathematical entities to have. On this point Colyvan (unusually) is less than explicit. I take it that he regards a mathematical entity to have all its usual “mathematical” properties (e.g. real numbers have their position on the real line, sets can have the properties of being measurable or not, functions can have the property of being continuous or not, etc.). However, there seem to be many extra-mathematical properties about which he believes it is permissible to be agnostic. These include: causal properties (i.e. whether or not mathematical entities are causally inert), spatio-temporal locative properties (or mathematical entities’ lack thereof), and metaphysical necessity/contingency. He argues for this limited conception of what it is to be a mathematical entity on the basis of disagreement within the Platonist community over whether or not mathematical objects should have all or some subset of these properties. Unfortunately, I believe that in failing to include these properties, Colyvan creates what is at best an ambiguity (at worst a fatal flaw) in the conclusion of his argument. This is because there a more general additional property that mathematical entities are widely held to possess in some form or another: abstractness. In cutting off the discussion of these properties, Colyvan at the very least fails to provide any basis on which mathematical entities should not be considered to be concrete particulars according to Williams’ taxonomy above — a classification that would be deemed highly controversial by many Platonists and scientific realists alike. The nature and consequences of this oversight will be treated in detail in the next chapter.
and Our “Best” Scientific Theories
When considering what it is for a mathematical entity to be indispensable to a scientific theory, the first important thing to note is that indispensability in this sense is not meant to be synonymous with ineliminability. This is largely a result of concerns regarding S Craig’s Theorem in mathematical logic, which states that any recursively enumerable theory is also recursively axiomatisable.~ The details of this procedure are not important for present purposes. What is important is that, using this theorem, one can reaxiomatise the theoretical parts of a scientific theory so as to leave only axioms, logical laws, and observation statements - effectively eliminating all theoretical terms. Thus, if Craig’s Theorem is applicable in the present context, no theoretical commitments are ineliminable. Many criticisms of the application of Craig’s result to scientific theories have been made. Criticisms include: that the resultant reaxiomatised theories violate the principle of parsimony in scientific theory choice; that the reaxiomatised theory should not be considered as a truly new, independent theory, as it is parasitic upon its predecessor; and that the elimination approach relies upon a mistaken, syntactic view of theories.more discussion of some of these issues see \citet{Putnam:1962, :1976 and :1980.} However, given that Colyvan himself raises the possibility of theoretical terms being eliminated in this way, it will be useful to examine the way in which he claims that indispensability can be interpreted in a stronger sense.
In a move likely motivated by the parsimony-based criticisms (and by Field’s attempt to argue that mathematics is indeed dispensable\citet{Field:1980 and :1989 for example.}), Colyvan proposes the following, strengthened interpretation of indispensability: a theoretical entity should be held to be indispensable to a scientific theory if its elimination causes that theory to become less preferable (or, for an even stronger requirement, if its elimination does not cause an increase in the preferability of the theory). Of course, the strength of this new interpretation is largely dependent on what criteria are used to evaluate the preferability of a theory. According to Colyvan, the preferability decison should be based on the standard desiderata for good scientific theories. These include: empirical success, unificatory and/or explanatory power, parsimony, fertility (i.e. the propensity of a theory to generate further theories and/or research), and formal elegance (a desideratum meant to capture some sense of a theory’s “aesthetic” value). However, given that Colyvan does not give a detailed account of exactly how this is meant to work (a key question is just how one is to provide uncontroversial weightings for the various desiderata) and that the fine detail of his views on this matter is not of relevance to my concerns in later chapters, I will move on to other matters. Suffice to say that a lot remains to be clarified regarding the details of this interpretation of indispensabilty. Furthermore, the ongoing debate over explanation in the philosophy of science~more on this see ????. indicates that it is by no means clear that Colyvan will be able to do this in a way that is consistent and doesn’t raise further problems.
Colyvan’s proposal of a preferability-based interpretation of indispensability leads nicely into the next topic for discussion: which theories are meant to constitute our best scientific theories. Quine puts forwards several criteria upon which scientific theories may be judged: mathematical conservatism, modesty (i.e. logical weakness), simplicity, generality, and refutability. Note here the distinct similarity between the criteria Quine proposese and those put forward by Colyvan in relation to indispensability. It appears that that what Colyvan has effectively done in responding to the potential problems for indispensability posed by Craig’s theorem, is build the idea of what it is to be the best scientific theory into his interpretation of indispensability. Accordingly, these terms have fundamentally intertwined for the purposes of the indispensability argument. Unless of course, one wishes to argue against the applicable of Craig’s Theorem, then reducing indispensability to ineliminability. However, this does not appear to be a route that Colyvan wishes to traverse (indeed, it may not be a viabe one for him given the key role preferability plays in his criticism of Field~).
{[green MAYBE A SUMMING UP PARAGRAPH HERE? ]} {[lime Yes, very briefly indeed. ]}
Naturalism
It will be useful to consider the indispensability argument’s two supporting Quinean theses (naturalism and confirmational holism) in terms of their contribution to what I shall call the “Quinean Ontological Machine” (hereafter referred to as the QOM). The QOM is a machine linking language and metaphysics for the purpose of determine the components of the world. It takes scientific theories (in this case our best ones) as its input and produces real existing (in the sense of a genuinely metaphysical ontological commitment) entities as its output.
At its most general, naturalism is the attempt to delineate a philosophical methodology that is respectful of the great successes of the natural sciences. However, many (if not most) contemporary, analytic approaches to philosophy (and I suspect many continental approaches too) would wish to lay claim to the label naturalism so construed. Accordingly, more work needs to be done in order to track down the particular Quinean variant of naturalism we are concerned with here. As mentioned briefly above, Quinean naturalism is the doctrine that there is no “first” philosophy, i.e. that philosophy is not in any sense prior to science. But what exactly does that mean? The details of Quinean naturalism can be cashed out in terms of its three main components: what Colyvan refers to as its normative and descriptive strands, and the Quinean ontic thesis. I will now discuss each of these in turn.
As intimated previously, the first strand of Quine’s naturalism takes on a normative form. It is normative in the sense that it concerns how one to investigate the nature of the world (i.e. it is concerned with what one’s method of ontological investigation should be). For Quine, science is the primary method via which one investigates what the world. Accordingly, if one has questions about the world, one should always look first to science for answers. In relation to the QOM, this strand of thought tells the machine where to look for its input — namely, to our best scientific theories.
The second, descriptive strand of Quine’s naturalism is primarily constituted by the claim that philosophy is continuous with science. However, as Colyvan points out, what this entails is a little unclear — the claim of continuity obviously doesn’t rule out that philosophy and science differ in some way. However, the exact nature of this difference is ambiguous in Quine’s writings. Given that this strand does not play a significant role in the QOM (at best it may play a supporting role in limit the machine’s domain of operation), I will set aside this ambiguity for now (although it will reemerge below in the discussion of the various ways in which one can satisfy the naturalist urge).
In his influential paper On What There Is~\footnote Quine puts forward what is here referred to as his “ontic thesis.” This thesis specifies exactly what the ontological commitments of a given scientific theory are. Quine does this via an examination of the variables necessary for a logical respresentation of a given theory. Once that theory is represented in a particular canonical logical form it is held to be committed to the existence of whatever the bound variables used in that logical representation range over. To put it in the form of a well-worn slogan: for Quine, to be is to be the value of a bound variable. Thus, the ontic thesis further narrows down the QOM’s domain of input. The thesis implies that the QOM should look to the bound variables of our best scientific theories. At this stage I should note that this Quinean claim will subjected to serious criticism in Chapter 3. Specifically, I will claim that the arguments of Stephen Yablo and Huw Price raise significant doubts as to the viability of this interpretation of ontological commitment.
At this stage it may look like the proponent of the indispensability argument has all the theoretical framework they require in order to argue for the existence of mathematical entities. However, as Colyvan points out,~\footnote this is not quite the case. In the discussion above we have seen that there are ways in which entities may be removed from theories, i.e. there are ways in which one bound variables in a logial representation may be eliminated or replaced by another. This seems to introduce a problematic degree of “unsettledness” into any resultant ontology. Accordingly, a little more work needs to be done before the QOM can be put to work determining the ontological commitments of our best scientific theories. For Quine and his followers, that work is done by confirmational holism. This willbe the subject of discussion in the next section.
In light of the discussion that is to follow (in Chapter 3 in particular) it will be useful here to briefly examine some of the other ways in which one may incorporate the naturalist urge into a philosophical methodology before moving on to discussion of confirmational holism. As Papineau points out~\footnote naturalist urges can broadly be separated into the methodological and the ontological (which is not to say that many naturalists are not subject to both). Those who indulge the naturalist methodological urge claim that there is (or at least should be) no distinction between the methods of science and those of philosophy.~ Those subject to the ontological urge on the other hand, claim that there is nothing unscientific “out there” in the world. They argue that what exists does not exceed the domain of science.~the Quinean position is good example of this. In what follows I will focus primarily on the ontological naturalist urge. In doing so there is one major distinction that I wish to consider: the distinction between representationalist and antirepresentationalist views. Representationalism,' likenaturalism,’ is a term in wide and diverse use in current philosophical discourse. As such it requires further definition here. The type of representationalism I have in mind is that famously criticised by Richard Rorty: the view that the philosophical mediums with which which we are in direct contact (i.e. thought and language) somehow, if interpreted correctly, represent a metaphysically independent reality. Representationalist naturalism could almost be referred to as the received view in current analytic philosophy. Major representational naturalists include Quine (as earlier discussion should have made clear) and Armstrong (who argues for a naturalist ontology via the eleatic principle: that what exists is what plays a physical causal role). In response, antirepresentationalists such as Rorty and Huw Price argue that positing some sort of semantic ladder, between thought and/or language on the one hand, and metaphysical reality on the other, as representationalists must do, is not a valid move. Furthermore, they claim, one can satisfy the naturalist urge without recourse to such problematic claims. In Chapter 3 I will argue, following Price and Yablo, that there is a fundamental disconnect between the entities invlved in a scientific theory and the strong metaphysical commitment that the Platonist requires.
Holism
The last major component of the QOM takes the form of Quine’s doctrine of confirmational holism, sometimes referred to as epistemological holism or the Quine-Duhem thesis.it is questionable how much of Duhem remains in Quine’s formulation — see Quine’s doctrine consists of two (fundamentally intertwined) claims: i) that empirical statements are fundamentally interconnected, they cannot be confirmed or disconfirmed (in any sort of logical, principled way at least) independently; and ii) that in the face of recalcitrant experience one can always hold any particular statement to be true, provided that one is willing to give up the truth of certain other statements. For example, suppose that one has a physical theory that predicts the occurrence of some event E given a specific set of circumstances C. Now suppose that an experiment has been set up to replicate C but that E has not been observed. Quine’s point is that in these circumstances there is any number of ways in which one can interpret the results of the experiment without giving up on the truth of one’s theory. For example, one could cite some problem with the observational apparatus used, deny that the circumstances of the experiment were actually those specified by the theory (possibly due to the interference of some unobservable background conditions), etc..
Whilst the example in the preceding paragraph was primarily meant to illustrate claim ii) of Quine’s confirmational holism (although it can certainly be viewed as illustrative of claim i) as well), it is primarily claim i) that is of most use when discussing confirmational holism’s role in the QOM. Confirmational holism’s role here is to eliminate the possibility that anything might get “left out” when the ontological commitments of a particular theory are determined. For example, someone opposed to the indispensability argument may wish to claim that, due to the fact that some theoretical entities differ in regard to certain properties, not the theoretical entities involved in a theory should be held to exist. Such a view may well be employed by a scientific realist unwilling to accept the existence of mathematical entities. Quine’s confirmational holism allows the proponent of the indispensability argument to respond by arguing that the theoretical entities involved in a scientific theory cannot be separated out in this way — ontological commitment is something that must take place at the level of the entities involved in a scientific theory as a whole.
However, as I shall argue in detail in Chapter 2, it may well be the case that, in the case of the indispensability argument at least, the QOM achieves this holistic ontological result only at the expense of stripping away anything that is distinctively mathematical about the mathematical entities used in scientific theories.
It is important to distinguish here between Quine’s, relatively uncontroversial, confirmational holism from his, far more controversial, doctrine of semantic holism, the claim that the unit of meaning is not the sentence (as is commonly thought), but rather a system of sentences, possibly the whole of a language. Quine used this doctrine as the basis for two of his most famous claims: that translation is indeterminate (i.e. that one can never be sure of having grasped the meaning of a word in another language); and that there is no analytic/synthetic distinction (i.e. that there are no sentences that are true in virtue of meaning alone)., Quine’s denial of the analytic/synthetic distinction is not completely contained, as many believe, in his famous paper Two Dogmas of Empiricism~\citep(Quine:1961) (which just involves a denial of Fregean analyticity). Rather, it is spread over this and two other papers — \citet{Quine:1976 and :1960. See :1997 for more on this.} Quine also used semantic holism in his formulation of the indispensability argument for Platonism. However, the indispensability argument as depicted here is only reliant on the less controversial doctrine of confirmational holism and thus should strictly be seen as a descendent of Quine’s view.
Indispensability to IBE
As discussed earlier, the IBE-based scientific realist is a major target for Colyvan’s version of the indispensability argument. This is largely because Colyvan believes there to be an important connection between indispensability arguments in general and the argument given for (scientific) realism based on IBE. He claims that the principle underlying IBE-based realism just indispensability (i.e. that the scientific realist employs a particular kind of indispensability argument). On this view, the IBE-based scientific realist argues for the existence of scientific entities on the grounds that they are indispensable for the purpose of explaining how the world works.is by no means an uncontroversial claim. For example see Thus, given that the scientific realist has already endorsed one indispensability argument involving scientific theories, Colyvan claims that it will be difficult to avoid accepting his argument too.
However, I believe that the IBE-based scientific realist can indeed plausibly deny Colyvan’s indispensibility argument without falling into inconsistency (indeed, the discussion of Chapter 2 should provide them with the means via which to do so). To see how, one first needs o explicitly state the purpose for which mathematical entities are claimed to be indispensable in Colyvan’s argument: they are claimed to be indispensable for the purpose of formulating our best scientific theories. Now, assuming that “purpose” here is in some sense transitive (if it is not, then the scientific realist is already able to avoid Colyvan’s claim), and that one of the purposes of our scientific theories is to explain (which seems an uncontroversial claim in this context). Given these assumptions, the fate of the scientific realist hinges upon exactly how the transfer of purpose (via its transitive property) takes place. But now that we have a clear idea of what is needed, it seems that the scientific realist can, whilst holding all of the principles expounded above to be true, easily give an account of how they come to believe in the existence of scientific entities but not mathematical ones: namely, that they are not holists — that they do not hold that mathematical and scientific entities both play the same explanatory role. Furthermore, this claim can likely be expounded in a clear and detailed way in terms of the different properties of mathematical and scientific entities. These different properties will be the focus of Chapter 2.
it All Together - A Case Study
Now all that all the major components of the indispensability argument have been discussed in detail, it will be useful to looke at an example of the Quinean Ontological Machine in action. Accordingly, I will now examine some of the ontological commitments of the theory of special relativity.~have chosen special relativity as a relatively uncontroversial member of the set of our current best scientific theories. {[lime omit ]} Examples drawn from it are used throughout Colyvan’s book when he seeks to refute criticism (see in particular his discussions of Armstrong’s eleatic principle and Field’s nominalism).[[lime Note that in order to be one of our “current best theories” SR has to be considered as a limit case within GR. In particular, I wish to focus on the role of Minkowski spaces and the Lorentz transformations in the theory. I believe looking at these entities will serve to highlight important ambiguities in the indispensability argument (many of which have been discussed above).
A Minkowski space is four-dimensional (one dimension for time, three for space) real vector space with certain distinctive mathematical properties. For example, it is equipped with an inner product that is bilinear, symmetric, nondegenerate, and has signature (-,+,+,+) or (+,-,-,-).is an alternative definition of the Minkowski space as an affine space. However, this does not affect my argument here. A detailed discussion of these mathematical properties is unnecessary for present purposes. What important is that there are numerous different mathematical properties that are possessed by a Minkowski space. Furthermore, a Minkowski space is used to represent space-time in most formulations of the theory of special relativity.
In Colyvan’s words the equations that constitute the Lorentz transformations “are an integral part of special relativity.”~\citep[p. ???]Colyvan:2001 The transformations are used to specify the relation between spatiotemporal positions of two observers position ]]lime typo* and their relative velocity under special relativity. The equations of the transformations may be derived from certain group theoretic postulates and the fact that a Minkowski space is isotropic. Again, the majority of the details of this derivation are unimportant. I will simply note here that the derivation involves the exploitation of certain properties of linear transformations and the real numbers, as well as the aformentioned isotropy. What is more importance for my purposes is such a derivation is possible at all.
Now that the basic nuts and bolts of the example have been specified, how does it {[lime typo ]} in the context of the indispensability argument? Firstly, special relativity is assumed to be one of our best scientific theories. Hence, it is in the domain of operation of the QOM and has been selected it as something from which ontological commitments need to be extracted. Naturally, my focus here will be on its ontological commitments. To this end, one must look for the mathematical components that are indispensable to the canonical logical representation of the theory of special relativity. As discussed above, Minkowski spaces and the Lorentz transformations seem like good candidates for indispensability here. However, as expected given the discussion above, when one looks more closely as just what this supposed ontological commitment to Minkowski spaces and Lorentz transformations entails, things become a little more unclear.
There seem to be three main areas of ambiguity in the ontological commitments entailed by the QOM in relation to the specified aspects of special relativity. Firstly, it is unclear just how many of the properties of a particular mathematical entity one is committed to. For example, in the case of a Minkowski space is one just committed to those properties that remain indispensable to special relativity or is one committed to all of the properties of a Minkowski space as presently conceived (while these two sets of properties are likely the same in the case of the Minkowski space {[lime I think this is too ghenerous: for example, the affine space definition has dispensible properties, as can be seen by the fact that the vector space definition, which lacks some properties, is adequate (and vice versa) ]}, it is by no means certain that this will always remain so and this is certainly not the case for other mathematical entities (consider the various properties of groups for example — many of these are not involved in the role of groups in our best scientific theories) maybe omit group example here — Minkowski example is better)? Secondly, is one also ontologically committed to those entities used to derive indispensable mathematical entities? For example, in the case of the Lorentz transformations, if one is ontologically committed to them, is one also ontologically committed to linear transformation, the real numbers, etc.? If so, then how far “back” does this commitment go? Is one committed to the axioms of a set theory and the real number system simply on the basis of the indispensability of one mathematical entity? {[lime Note that set theory and the real number system themselves have multiple definitions. ]} Finally (and most importantly), just what sort of mathematical entity (in the sense of the extra-mathematical properties discussed above) is one ontologically committed to? Is one committed to entities that are abstract in some sense? Are they types or tokens? For example, does the if I {[lime Heideggerian ]} accept the indispensability argument and that the Minkowski space is indispensable to special relativity (which I hold to be one of our best scientific theories), does this commit me to the existence of just that Minkowski space used to represent space-time or to the existence of Minkowski spacemore generally? All of these questions don’t seemed {[lime typo ]} to be addressed by the indispensability argument as it stands.
The ambiguities highlighted in the preceding paragraphs will be the primary subject of discussion in the following chapter. In particular, I will focus on problems arising from the ambiguity in the sort of entity resultant from the Quinean Ontological Machine. These problems, I will argue, have the potential to call the whole Platonist indispensability program into question.
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