Convergence strategies for theory assessment

This paper addresses the issue of the import of convergence arguments in theory assessment. A ﬁrst part is devoted to making the point of the diﬀerent types of strategies based on convergence, providing new distinctions with respect to the existing literature. Speciﬁc attention is devoted to robustness vs consilience arguments and one representative example for each category is then discussed in some detail. These are: (a) Perrin’s famous robustness argument on behalf of the atomic hypothesis on the grounds of the concordance of thirteen diﬀerent procedures to the same result for the Avogadro number; (b) the consilience ar-gument motivating the trust in the viability of the extra-dimension conjecture in the context of early string theory. These two cases are expressly chosen in order to highlight possible diﬀerences, also including whether the convergence obtains in terms of empirical or theoretical procedures. Notwithstanding these various diﬀerences, in both cases the evaluation of the assessment strategy similarly depends, in a signiﬁcant way, on how the convergence argument is interpreted, as shown in the ﬁnal part of the paper.


Introduction
In current debates on the status of fundamental physics, a typical criticism to research fields at the frontiers of physics such as string theory or cosmic inflation is that they have more to do with pure mathematics (e.g., Hossenfelder,  2018), or even "fashion, faith and fantasy" (Penrose, 2016), than with traditional scientific methodology.The main reason for such criticism is the absence of empirical confirmation: the scenarios proposed in those theoretical frameworks are apparently so far away from the possibility of empirical testing that theory assessment grounded on empirical data does not seem a viable option.Thus, at least for the time being, one has to rely on assessment criteria that are not based on empirical testing: such as, for example, the "theoretical virtues" famously discussed by Kuhn (1970) for theory choice, or, more recently, the meta-empirical arguments individuated by Dawid (2013, 2021) for boosting the trust in the viability of a theory in the absence of empirical data.Such criteria, however, do not seem to be appropriate methodological tools for pursuing a line of research in investigating Nature, critics claim.Moreover, according to some of them, persevering under such motivations means "going astray", that is, abandoning "the" scientific method. 1 In fact, the issue of the legitimacy of the criteria employed for theory assessment in scientific practice is more nuanced than some of the contenders in the above debate seem to assume.As has been variously noted in recent literature,2 the debate (in its more mediatic form) has significantly suffered from not paying due attention to the subtleties of scientific methodology, as well as to the actual historical developments of the theories considered.Indeed, when examined under a more carefully detailed perspective, the effective deployment of theory building in fundamental physics shows a different story from what commonly depicted in the critical literature. 3 clear example is provided by considering those epistemic strategies which are common to both theoretical and empirical scientific practice.Precisely because they are shared, they provide an interesting perspective for discussing scientific methodology, especially when the debate is focused on contrasting empirical with non empirical cases.Particularly representative, in this respect, are the strategies for theory assessment which are based on convergence criteria.As we will see, notwithstanding the diversity of the convergence arguments one can envisage, the evaluation of the corresponding assessment strategy does not barely depends on such differences as, in particular, whether the convergence obtains in terms of empirical or theoretical procedures.The evaluation of the assessment process in these cases is a subtle task, and significantly depends on how the convergence argument is being interpreted, case by case.Now, turning to the actual subject matter of this paper, what is intended by a convergence argument?In fact, there is a rich variety of convergence reasoning acting at the level of scientific practice, informing both theory-building processes and arguments for theory assessment.More precisely, under the big umbrella of "convergence methodology" different strategies can be included, involving such notions as robustness, consilience, coherence and unification.Given the various uses of these notions in today's literature on theory assessment, section 2 is devoted to survey the main arguments based on the convergence (or "concordance", as some prefer to call it) of significant features or results.In particular, a new, historically oriented analysis of robustness and consilience arguments is provided.Section 3 discusses in some detail one representative example each for robustness and consilience.These are: (a) Perrin's famous argument on behalf of the atomic hypothesis on the grounds of the convergence of thirteen different procedures to the same result for the Avogadro number; (b) the convergence argument motivating the trust in the viability of the extra-dimension conjecture in the context of early string theory.The two cases are expressly chosen in order to highlight possible differences, also including whether the convergence obtains in terms of (theory-mediated) empirical procedures or of purely theoretical procedures (however physically motivated), besides the distinction between the types of the convergence arguments implied.In both cases, however, the trust in the theory or hypothesis involved is undoubtedly boosted on the basis of the convergence.In which way, exactly?The last section is devoted to address this point by examining the kind of epistemic strategy at work in each case, and the related interpretative issue.

Varieties of convergence
Convergence-based reasoning is widespread in scientific practice, both in empirical and theoretical cases.Various types of convergence can be singled out, depending on the context, the intended aim and the convergent feature one is dealing with.Broadly speaking, convergence in science indicates that some relevant elements in scientific activity -experimental or theoretical results, methods, models or even theories -turn out to be the same thing or to be strongly related to each other.On the grounds of the existence of such a convergence, conditions for a successful convergence-based argument are commonly held to be a) the existence of genuinely different starting points, and b) the variety and independence of the paths by means of which the convergence is obtained. 4How a convergence is obtained characterises the kind of reasoning which can be based on it.Commonly, convergence resulting from varied evidence is used to build arguments for boosting trust in the theoretical developments involved.This is the kind of convergence-based arguments -shortly, convergence arguments (CAs) -the paper focuses on.
CAs cases are variously described and interpreted in the literature.An exemplary illustration of such a variety is provided by the philosophical discussion on the famous argument attributed to Jean Perrin for assessing the atomic hypothesis.The argument, apparently based on the convergence to the same empirical result of the thirteen procedures followed for obtaining Avogadro's number, is undoubtedly the most debated case of CA. 5 While Cartwright (1983, pp.84-86) interprets it as an inference to the most probable cause, Salmon (1984, p. 220) considers it as a type of common cause argument.Most often, especially in recent literature, it is discussed as 4 How to spell out the independence condition is not a simple issue and there is a lively debate in regard -see for example Stegenga (2012), Stegenga and Menon (2017), Shupbach  (2018), Coko (2020b).Recent discussions of Bayesian accounts of the independence condition are provided, for example, in Stegenga and Menon (2017) and Landes (2020).
5 For more details on this case, see section 3.
a paradigmatic instance of a robustness argument, with further specifications depending on how robustness analysis is intended.6All the mentioned descriptions correspond to the same "orthodox robustness interpretation" according to Hudson (2020a, p. 196), who proposes instead a different understanding of the argument in terms of calibration (2020a) or analogical reasoning (2020b).Finally, Coko (2020a) provides a detailed discussion of Perrin's argument as a case of multiple determination.
Another representative example from the history of physics is Newton's theory of universal gravitation unifying Galileo's terrestrial mechanics and Kepler's laws of planetary motion, the so-called "Newtonian Synthesis" (Salmon 1998,  p. 85).This case is typically discussed as a example of theoretical convergence in the literature, most often specified in terms of unification or consilience (e.g., Friedman 1983; Morrison 2000).The classic reference, in this respect, is Willliam Whewell's analysis of Newton's achievement as a paradigmatic example of his notion of Consilience of Inductions, the second of his three confirmation criteria (i.e., novel predictions, consilience and coherence).7 Whewell's consilience account of Newton's case has been variously interpreted, as we will see.To give some significant examples: Foster (1988) views it in terms of a common cause argument for realism, Harper (1989) as a "Natural Kind inference" providing evidential support, while Janssen (2002, pp.488-89) understands it as a combination of "common-origin inferences" (COIs), that is, in his terminology, as a case of meta-COI.
As already apparent in the examples mentioned so far, robustness, unification and consilience are the key notions at stake when addressing CA cases.They form the core of the conceptual toolbox for discussing convergence arguments. 8n the examples above, robustness was typically used in the first case, while the discussion of the second case was mostly conducted in terms of unification and consilience.In fact, as we will see, it is the kind of convergence argument that determines which notion is indeed significant and what role it actually plays in the argument.This will emerge more clearly by having a closer look at the tools available in the "convergence box".In particular, given the case studies considered in this paper, special attention will be devoted to robustness vs consilience arguments.

Robustness
The most discussed types of CAs are undoubtedly the so-called robustness arguments: that is, those assessment arguments that are based on the robustness of some scientific feature or result.In fact, as often noted in the literature, there are several notions of robustness, differing "both in their normative credentials and in the conditions that warrant their deployment" (Woodward, 2006, p.  219).
Historically, robustness was considered across models in its original version.This was in the celebrated article by Richard Levins on the strategy of model building in population biology (Levins, 1966).Levins notoriously made the following claims: first, that model building in the study of complex systems involves a necessary trade-off among generality, realism and precision; second, in order to solve this trade-off problem, "that the reliability of an inference is increased when it is the joint inference of multiple models".9This latter was the claim about robustness: more precisely, in Levins' own words, "if these models, despite their different assumptions, lead to similar results, we have what we can call a robust theorem", whence his famous conclusion that "our truth is the intersection of independent lies" (1966, p. 423).
Since then, robustness has been the subject matter of a growing philosophical literature, especially after Wimsatt (1981)'s generalisation of Levin's ideas by developing a systematic account of "robustness analysis" for scientific reliability (e.g.Soler et al., 2012, and references therein).In Wimsatt's terms, "Things are robust if they are accessible (detectable, measurable, derivable definable, producible, or the like) in a variety of independent ways", and these "things" can be entities, properties, processes, results, or theorems (1994, p. 210).Thus, with respect to Levins' original formulation, Wimsatt extends robustness analysis to include a much larger variety of procedures, ranging from experimental manipulations, non-interventive observation or measurement to mathematical or logical derivation. 10owever different, all these variants and uses of robustness have a "common theme" in Wimsatt's view: that is, distinguishing "that which is regarded as ontologically and epistemologically trustworthy and valuable from that which is unreliable, ungeneralizable, worthless, and fleeting" (Wimsatt, [1981] 2012, p.  63).How this claim about robustness could be effectively substantiated is a controversial issue in the literature, especially because of the diversity of the notions implied.Woodward (2006), for example, distinguishes four notions: inferential robustness (robustness as insensitivity of the results of inference to alternative specifications), derivational robustness, measurement robustness (robustness in agreement of measurement results), and causal robustness (robustness as a mark of causal or structural relationships).Calcott (2011), instead, identifies three kinds of robustness in Wimsatt's approach: robust theorems (theorems whose derivation can be supported in multiple ways), robust phenomena (phenomena which are reliably present in many different contexts), and robust detection (triangulation, multiple lines of evidence).11According to Eronen (2015, p. 3962) most of the discussion has focused on derivational robustness, while, he claims, there is "a more general form of robustness that is potentially more relevant for justifying inferences to what is real", that is, robustness as multiple accessibility.12Finally, Coko (2020b, 2022) distinguishes multiple determinationthe epistemic strategy of using multiple, independent procedures to establish the same result -from variants of robustness analysis with which, as he argues, it is confused in the literature: while the first refers to the multiple, independent establishment of empirical claims about the world, the second refers to an analysis of some sort of invariance to change or perturbations. 13hether these distinctions are indeed substantial and whether they cover the whole space of possibilities are debated issues.14In fact, there are two levels to consider in this debate.Let us assume in general that something is robust to the extent that it is obtained by many, different and independent means (e.g.Shupbach, 2018, p. 278). 15Then, two levels of discussion can be distinguished in the literature.On the one side, the discussion regards the "operational level" of the concrete procedures (empirical or theoretical) employed for establishing the robustness.Examples are the analyses of robustness in terms of reliability (e.g.Basso, 2017), calibration (e.g.Bokulich, 2020), triangulation (e.g.Kuorikoski and Marchionni, 2016), and multiple determination (e.g., Coko, 2020b and 2022).On the other side, the discussion is focused on the "meta level" of the assessment strategies which are grounded on the robustness obtained at the first, operational level.Thus, to mention a concrete case, the issue at stake at this second level is not how Perrin succeeded in getting a robust result for Avogadro's number,16 but rather the nature and import of his argument -based on the multiple determination of the empirical result -on behalf of the atomic hypothesis.
Here, we are specifically concerned with this second, "meta-level" type of analysis.Therefore, leaving aside questions regarding the operational level of the practices for getting a robust result, let us focus on the modalities of the robustness-based arguments for strengthening the trust in the viability of a theory (model, hypothesis).In this respect, we can distinguish three main types of strategies in the literature, depending on the starting points and the end goals of the convergence considered.
(1) A first type of strategy is the so-called robustness analysis developed in the framework of scientific model building (Levins, 1966 and 1993; Weisberg, 2006), based on robustness across models. 17Starting with different idealised or approximate models, the aim is to arrive at a robust model or, possibly, at a true theoretical core.In Levins (1993, p. 554)'s terms, the strategy is as follows: given that, in science, "most of the models .. are partly true and partly false", we can "strengthen our confidence in the implications of some assumptions by using ensembles of models that share a common core of these assumptions but also differ as widely as possible in assumptions about other aspects." 18In substance, the issue is how to deal with inevitably highly idealized models of complex systems, in order to determine which parts of these models make trustworthy predictions about their targets or can reliably be used in explanations (cfr.Weisberg, 2013, chap.9). 19 (2) A second strategy consists in increasing the confirmatory status of a given theory (model, hypothesis) by making it as robust as possible on the basis of varied, independent evidence. 20In this perspective, robustness is usually (though not always) employed in a justification context, rather than in a discovery process.The rationale is that underlying the so-called "variety-of-evidence thesis" (varied evidence confirms more strongly than less varied evidence). 21In the words of Hempel (1966, p. 34), to quote a classic historical reference in regard, "The confirmation of a hypothesis depends not only on the quantity of the favorable evidence available, but also on its variety: the greater the variety, the stronger the resulting 17 See Coko (2022) for a detailed analysis of the accounts of Levins (1966, 1993), Wimsatt  (1981) and Weisberg (2006, 2013), their rationale and their differences.
18 More in detail, Levins's idea is that "the more the variable part spans the range of plausible assumptions, the more valid the claim that the conclusions shared by all of them depend on the constant part."Thus, "if we also have confidence that the constant part is true, then we have strong support for the claim that the conclusion is generally true.This gives robustness to the conclusions" (1993, p. 554).
19 This kind of strategy has been applied especially to modeling complex systems, from biology to social and economical sciences.In the last decade, robustness analysis has been discussed also with respect to climate modeling (e.g., Lloyd, 2010; 2015; Parker, 2011; Weisberg, 2013; Vezér, 2017).More recently, robustness analysis has been extended to simulations in particles physics (e.g., Boge, forthcoming) and to evaluating cosmological modeling (e.g., Gueguen, 2020).
20 This seems to correspond to what Nederbraght (2012, p. 123) calls "multiple derivability", defined as "the strategy by which a theory is supported by the evidence obtained through two or more independent methods that differ in the background knowledge on which they are based".
support".In rough terms, the idea is that the chance of being simultaneous wrong in each of the different, independent evidential checks declines with increasing their numbers (e.g.Wimsatt, 1994, p. 210). 22This kind of epistemic strategy has been much discussed in recent literature, especially in the framework of Bayesian approaches to confirmation. 23 (3) Finally, a third type of strategy is that aiming at increasing the trust in a hypothesis or a theory on the grounds of the possibility of obtaining, on its basis, a robust derivation of a given result, which can be of empirical or theoretical nature.This third type of second-level modality, based on multiple determination (at the first, operational level), 24 is different from both the two second-level strategies mentioned above: on the one side, robustness is not considered across models (as in ( 1)); on the other side, it is a characteristic of the result obtained, not of the hypothesis/theory to be assessed (as in ( 2)).In other words, it is not just a case of varied evidence, since the evidence is the same one (the same result): what is varied, is the way of obtaining it, not the result itself. 25In this case of assessment strategy, especially, the underlying rationale is often taken to be a nocoincidence (or no-miracle) argument, motivating (via an inference to the best explanation) the trust in the viability of the theoretical framework used for arriving at the robust result. 26This view, however, has been criticised, either on the grounds of rejecting the epistemic import of this kind of no-coincidence argument, 27 or by proposing different accounts of robustness reasoning. 28

Consilience
Beside robustness, convergence arguments are often analysed in terms of consilience and unification.Surely, there are close connections among these three notions.In particular, it is not always easy to disentangle one from each other when considering the role they play in specific convergence arguments.Consilience and unification, for example, are even treated as interchangeable in some literature.29Moreover, it is not rare to find discussions of consilience as a case of robustness reasoning (e.g., Wimsatt, 1981, p. 124). 30Let us focus here on the notion of consilience and consider unification only insofar it is related to consilience in arguments for theory assessment. 31he term consilience is often used in today's philosophical and scientific literature in the loose sense of concordance or convergence simpliciter.When applied in more precise terms, the notion is taken to mean different things and with different epistemic import, depending on the context and case examined. 32n fact, even the original nineteenth century notion has not received an unanimous account in the scholarly literature.Whewell's own treatment of the notion has originated much discussion, giving rise to different interpretations (e.g.Hesse, 1968, 1971; Laudan, 1971; Forster, 1988; Harper, 1989; Snyder,  2006, 2008).Here, without entering into the detail of this interpretative issue, let us just focus on those relevant features of the original notion on which a convergence argument for theory assessment can be founded.
In his XIV aphorism among those "concerning science", Whewell (1840, p. 469) gives the following, famous characterization of the nature of consilience: "The Consilience of inductions takes place when an Induction, obtained from one class of facts, coincides with an Induction, obtained from another different class.This consilience is a test of the truth of the Theory in which it occurs." Beside the multiplicity and independence of the evidence ("classes of facts altogether different"), what makes this "coincidence" or "agreement" a test of truth for hypotheses is also its unexpectedness -in Whewell (1840)'s own terms, an agreement "unforeseen and uncontemplated" (p.65), "the unexpected coincidences of results drawn from distant parts of the subject" (p.67).Note, in this regard, that what is unexpected -and therefore surprising -is the coincidence, not a new fact or prediction.
The epistemic role of this kind of surprise is well evident in Whewell's discussion of his most known example of consilience, that is Newton's Theory of Universal Gravitation. 33The fact that Newton found that "the doctrine of the Attraction of the Sun varying according to the Inverse Square" of the distance, which explained Kepler's Third Law, explained also Kepler's First and Second Laws "although no connexion of these laws had been visible before", and that, again,"it appeared that the force of Universal Gravitation .. also accounted for the fact, apparently altogether dissimilar and remote, of the Precession of the equinoxes" is, for Whewell, "a most striking and surprising coincidence, which gave to the theory a stamp of truth beyond the power of ingenuity to counterfeit" (1840, pp.65-66).
Thus, the striking and surprising fact that the consilient theory can explain unrelated additional phenomena or laws is an essential part of the assessment argument (leading to an increase in the trust in the theory's truth).In other words, we can say that the coincidence or convergence must be surprising for consilience to function as an assessment argument.
Note that there are two levels at which the element of surprise is epistemically relevant, here: on the one side, the first level of the surprising fact of the convergence per se; on the other side, the meta level of the reasoning thatgiven the surprising convergence -it would be very surprising if the theory were false: in Whewell's words, "no accident could give rise to such an extraordinary coincidence" (1840, p. 65).It is only this second level of reasoning from surprise which is working in the no-miracle/no-coincidence argument often used for justifying the rationale of the robustness strategies (2) and (3) discussed in 2.1.There is no surprise from unexpectedness working at the first, operational level of robustness reasoning (cfr.2.1): the fact that a result is obtained in multiple, different ways can be unlikely and asks for justification (for instance, by means of a no-coincidence IBE argument), but it is not unexpected per se -quite the opposite.In many cases, robustness is searched for, by varying circumstances, parameters, and so on.In this sense, if we want to transpose consilience in today's terms, there is an additional feature (the unexpectedness of the convergence) to be considered with respect to the reasoning from variety of evidence or multiple determination seen above.34However, this is often under-estimated in current literature, where consilience is frequently identified with the convergence of multiple independent evidence "streams" or lines tout court (e.g.Forber and Griffith, 2011; Vezér, 2016; Currie, 2018a; Bokulich,  2020). 35n fact, beside the elements highlighted so far, understanding Whewell's consilience requires considering, in the framework of his particular theory of induction,36 his notion of natural kind and common cause (e.g.Snyder, 2006;  Coko, forthcoming).As Snyder (2008, p. 187) puts it, "Consilience occurs when a theory brings together members of different kinds, showing that they belong to a more general classification.In the case of event kinds, individual types of events are members of the same kind when they share the same cause."This feature of "causal unification of different event or process kinds into more general kinds, in virtue of sharing a common cause" (ibid.) is precisely what has given rise, in the scholarly literature, to viewing Whewell's consilience in terms of unification and common cause (e.g.Forster, 1988; Harper, 1989; Janssen,  2002), as we have seen in the introductory part of section 2 (p.4).
To sum up, how can we characterise a consilience argument for theory assessment in today's terms?As we have seen, depending on the context, interest and focus of the analysis, consilience is assimilated to different things in the literature: variety-of-evidence reasoning, multiple determination, natural kind inference, causal explanatory unification, a combination of "common-origin inferences".With respect to the convergence arguments seen in 2.1, however, there are two distinguishing features of consilience which emerge in the light of the genesis and development of the notion: i) the number of the different, independent evidence lines is not especially influential (already a small number of them are enough for boosting the trust in the theory's viability); ii) a distinct, fundamental role of surprise (corresponding to the unexpectedness of the convergence).In what follows, therefore, we will rely on these two elements for distinguishing consilience as a form of convergence argument.

CA arguments: Two representative cases
As history and scientific practice clearly show, convergence arguments for theory assessment are employed in a variety of cases of theory building.The question of interest, here, is how to understand the specificity of the epistemic import of these arguments -in particular, robustness and consilience arguments -from the viewpoint of theory assessment.
Let us address the question by considering two cases of CAs, the first one representative of a robustness argument, the second of a consilience argument: namely, a) the already mentioned case of Perrin's argument on behalf of the molecular hypothesis (hereafter, Perrin's case); b) the case of the convergence argument for boosting the trust in the "extra-dimension hypothesis" in the framework of early string theory (hereafter, the extra-dimension case).It is worth noting that Perrin's case, in addition to represent a robustness argument (as anticipated in section 2), is also commonly considered a typical example of empirical theory assessment, since the convergence is obtained in terms of measurements (however theory-mediated). 37The extra-dimension case, on the contrary, will be shown to represent an instance of a consilience argument, as well as a case for non-empirical theory assessment: the argument is grounded on the convergence to a theoretical result (22 extra space dimensions), which is obtained in terms of theoretical procedures (although based on physical assumptions).Indeed, as we will see, this latter case is a clear example of how a non-empirical CA can be effective in motivating the acceptance of a very surprising hypothesis. 38) Perrin's case.The CA in question, in this case, is the argument attributed to Perrin on behalf of the atomic hypothesis on the grounds of the convergence of thirteen different procedures to the same result for the Avogadro number.As already mentioned in section 2, this argument has been seen in different ways in the scholarly literature: as an inference to the most probable cause (Cartwright, 1983), as a common cause argument (Salmon, 1984), as an instance of no-miracle argument (e.g.Chalmers, 2009; Psillos, 2011b), as a variety of robustness argument (e.g.Shupbach, 2018; Landes, 2020), as an example of calibration reasoning (Hudson, 2020a), and as a paradigmatic instance of multiple determination (Coko, 2020a: Coko, forthcoming), to recall a number of stances.
Generally, these views are based on analyses of the concrete procedures followed by Perrin as well as on his own reflections.Here, since the paper's focus is on the "meta level" of convergence strategies for theory assessment, I will not be concerned with the details of Perrin's measurement procedures as rather with the following question: whether, in Perrin's case, there is a distinctive, epistemic import due to the fact of the convergence with respect to mere empirical confirmation (however strong or "robust"), and, if this is the case, in what this additional epistemic feature actually consists.
The CA attributed to Perrin is basically grounded on a number of Perrin's famous statements.A most quoted one is the following conclusion of his review of the various phenomena yielding concordant values for Avogadro's constant in his book Les Atomes ([1913]1916, pp.206-7): Our wonder is aroused at the very remarkable agreement found between values derived from the consideration of such widely different phenomena.Seeing that not only is the same magnitude obtained by 37 There are many analyses of the procedures followed by Perrin in the literature, starting with his own writings.Historical reconstructions are, first of all, Brush (1968) and Nye (1972).For more recent, detailed analyses see, in particular, Bigg (2008), Chalmers (2009), Psillos (2011b), Hudson(2020a), Coko (2020a), Smith and Seth (2020), Demopoulos (2022).
38 A discussion of the case of the extra-dimension conjecture in early string theory is provided in Castellani (2012).Castellani (2019) focuses again on this story, reconstructing it as an example of scientific methodology based on a convergence argument in non-empirical theory assessment.Linnemann (2020) uses Castellani (2019)'s account of the different paths to arrive at the extra-dimension conjecture as an example of non-empirical robustness argument.In what follows, we will analyse in more detail the kind of convergence argument represented by this case, showing that it is more appropriate to see it in terms of consilience rather than in terms of robustness.
each method when the conditions under which it is applied are varied as much as possible, but that the numbers thus established also agree among themselves, without discrepancy, for all the methods employed, the real existence of the molecule is given a probability bordering on certainty.
This passage is representative of many similar reflections to be found in Perrin's writings. 39In the literature, these remarks are usually taken to indicate that, according to Perrin, the trust in the truth of the molecular (atomic) hypothesis is boosted on the grounds of obtaining, on its basis, a robust result for the number of molecules in a mole (whatever other assumptions are used in the different procedures for arriving at the value of Avogadro's number). 40In other words, in terms of the distinctions introduced in section 2.1, the argument attributed to Perrin can be seen as a case of robustness CA corresponding to the third type of convergence strategy discussed.Now, whether this sort of argument has effectively played a significant role in viewing Perrin's contribution as conclusive for establishing the existence of atoms is a debated issue, from both a historical and an epistemic point of view.41From this latter point of view, in particular, much of the discussion has focused on the presumed rationale of the argument.The key question regards the distinctive epistemic role to be attributed to the convergence of the many, independent ways to obtain Avogadro's number in assessing the atomic hypothesis, to which we will turn in some detail in the next section.
(b) The extra-dimension case.The context for discussing this case is the so-called Early String Theory (EST): that is, the first developments of string theory from the 1968 formulation by Gabriele Veneziano of his famous scattering amplitude to the first string revolution in 1984. 42In the framework of this "founding era" of string theory, the extra-dimension conjecture emerged in the first phase, characterised by the developments of the dual theory of strong interactions in the years 1968-1973. 43 This EST initial phase was originally aimed at finding a viable theory of hadrons in the framework of the so-called analytic S-matrix (or S-matrix theory) developed in the early Sixties. 44Its programme was to determine the relevant observable physical quantities, i.e. the scattering amplitudes, only on the basis of some general principles such as unitarity, analiticity and crossing symmetry and a minimal number of additional assumptions, among which the so-called duality principle. 45n this framework, the problem of finding a scattering amplitude obeying also the duality principle was brilliantly solved by Veneziano for the case of four mesons.This ground-breaking result, universally recognised as the starting point of string theory, immediately gave rise to a period of intense theoretical activity aimed at extending Veneziano's amplitude: from the first two models for the scattering of N particles -the generalised Veneziano model, known as the Dual Resonance Model (DRM), and the Shapiro-Virasoro Model46 -to all the subsequent endeavours to extend, complete and refine the theoretical framework, including its string interpretation and the addition of fermions (see Cappelli et al., 2012, Part III).
Two particularly significant conjectures were introduced in this process.First, the string conjecture in 1969: in independent attempts to gain a deeper understanding of the physics described by dual amplitudes, Nambu, Nielsen and Susskind each arrived at the conjecture that the underlying dynamics of the dual resonance model was that of a quantum-relativistic oscillating string.47Second, the conjecture or "discovery" of extra spacetime dimensions: independent developments of the dual theory led to the critical value d = 26 for the spacetime dimension (the critical dimension), reducing to the value d = 10 when including fermions.
In what follows, we briefly illustrate the three independent theoretical processes leading -by surprisingly converging to the same surprising result (d = 26) -the research community to accept the critical-dimension conjecture, however bold and apparently unphysical. 48Three ways to the critical dimension In the framework of the theoretical endeavours to extend the original dual theory in order to overcome its initial limitations and problems, the critical dimension conjecture first emerged in the context of two independent programmes: 1) the "unitarization programme", in the context of which Claud Lovelace arrived at the conjecture d = 26 while addressing a problematic singularity case arising in the construction of the nonplanar one-loop amplitude; 2) the "ghost elimination programme", where the critical value d = 26 for the spacetime dimension issued from studying the spectrum of states of the Dual Resonance Model.In some more details: 1. Lovelace's result.The original dual amplitudes didn't respect the S-matrix unitarity condition.To go beyond the initial narrow-resonance approximation, the "unitarization programme" substantiated in generalising the initial amplitudes, considered as the lowest order or tree diagrams of a perturbative expansion, to include loops.As a first step for restoring unitarity, one-loop diagrams were constructed, and in this building process the calculation of a nonplanar loop diagram led Lovelace, in order to solve a singularity problem emerged in the process, to the 1971 conjecture of the value d = 26 for the spacetime dimension.492. The "no ghost" result.In the endeavours for generalising Veneziano's amplitude to the scattering of an arbitrary number N of scalar particles, a serious problem was represented by the presence of negative-norm states ("ghosts") in the state spectrum of the model.50These states, leading to unphysical negative probabilities, had to be eliminated from the theory.In this "ghost elimination" programme, a decisive step was the 1971 construction by Del Giudice, Di Vecchia and Fubini of an infinite set of positive-norm states (the so-called DDF states), which were found to span the whole space of physical states if the spacetime dimension d was equal to 26.Soon after, the proof of the so-called No-Ghost Theorem, establishing that the Dual Resonance Model has no ghosts if d ≤ 26, was achieved by Brower, and independently by Goddard and Thorn. 51 spacetime of 26 dimensions was not easy to accept. 52While initially almost nobody had taken Lovelace's conjecture seriously, after the proof of the No-Ghost Theorem the attitude changed and the extra dimensions started to be gradually accepted in the dual model community. 53A further decisive support to the conjecture came from the third theoretical process leading, independently from the previous two ways, to the same "critical" value d = 26 for the spacetime dimension: the 1973 work of Goddard, Goldstone, Rebbi and Thorn (GGRT) on the quantisation of the string action.
3. The GGRT result.After the 1969 string conjecture and the immediately successive studies of a Lagrangian action for the string, 54 the quantisation of the string action by Goddard, Goldstone, Rebbi and Thorn was a decisive step for the string interpretation of the dual resonance model to be fully accepted.In the resulting quantized theory, all what had been previously obtained by proceeding according to a bottom-up approach and following different paths could now be derived in a more clear and unitary way.In particular, the critical dimension was obtained as a condition for the Lorentz invariance of the canonical quantisation of the string in the light-cone gauge: only for d = 26 the quantisation procedure was Lorentz invariant. 55 course, the story of the critical dimension goes further, and other decisive support to this conjecture came from successive developments of string theory, especially after it was re-interpreted as a unified quantum theory of all fundamental interactions including gravity. 56But let's stop at this point and turn to consider the rationale of the convergence arguments operating in the two representative cases described so far.

Conclusion: The interpretative issue
The two cases of CAs considered in the previous section are surely very different.They represent distinct types of convergence arguments (robustness vs consilience) and, in addition, different cases of theory assessment (empirical vs non empirical).In both cases, however, the trust in the theory or hypothesis involved is undoubtedly boosted on the basis of the convergence.In which way, exactly?This section is devoted to address this point by examining the kind of epistemic strategy at work in each case, and the related interpretative issue. 53A good example is given in the following quote by Goddard (Cappelli et al. (2012, Chapter  20, p. 285): "The validity of the No-Ghost Theorem had a profound effect on me.It seemed clear that this result was quite a deep mathematical statement ..., but also that no pure mathematician would have written it down.It had been conjectured by theoretical physicists because it was a necessary condition for a mathematical model of particle physics not to be inconsistent with physical principles.... I could not help thinking that, in some sense, there would be no reason for this striking result to exist unless the dual model had something to do with physics, though not necessarily in the physical context in which it had been born." 54 Nambu (and then Goto) proposed the Lagrangian action for the string formulated in terms of the area of the surface swept out by a one-dimensional extended object moving in spacetime, in analogy with the formulation of the action of a point particle in terms of the length of its trajectory.
55 Details on this point, and in general on the quantisation of the hadronic string, are provided by Di Vecchia and Goddard in their contributions to Cappelli et al. (2012, Chapter  11, 11.8 and Chapter 20, 20.7), respectively.
(a) Perrin's case.As already said, different interpretations of the rationale behind Perrin's reasoning have been proposed and discussed in the literature. 57onetheless, there is a substantial agreement on the fact that the "miracle of concordances" (Psillos, 2011b, p. 360) has played a significant role in boosting the trust in the atomic hypothesis.Salmon (1998) notoriously comments on Perrin's multiple determination of Avogadro by noting that "such agreement would be miraculous il matter were not composed of molecules and atoms" (p.82).In a similar vein, Chalmers (2009, p. 243) remarks: "The concordance of a variety of indisputable evidence with the predictions of the kinetic theory amounted to a powerful argument from coincidence.How could the theory get things so right if it were not at least roughly true?".
In fact, the epistemic relevance of an "argument from coincidence" in this case has been understood in a number of different ways over the years.Cartwright  (1983, p. 82), for example, argues that, while for many it is "a paradigm of inference to the best explanation", what Perrin really makes is "a more restricted inference -an inference to the most probable cause".58Coming to the current stage of this long-standing debate, the details of which have been thoroughly analysed in recent literature, 59 a new, more sophisticated way of seeing Perrin's reasoning as a no-coincidence argument is offered by Coko (2020a, 2020b)  in terms of his "multi-dimensional approach": the epistemic force of the argument, according to Coko, depends on the modality of the concurrence of the several elements ("dimensions") of multiple determination, such as the independence, reliability and number of the converging procedures.On this view, the epistemic import of a CA has to be analysed case by case, by looking at how well the different dimensions are instantiated. 60n alternative point of view is provided by Dawid (2021), who argues that, beside the implausibility of the coincidence scenario, two meta-empirical criteria are needed "for making a convincing case for atomism based on Perrin's results": namely, the absence of no non-atomist explanation other than mere coincidence, and the unlikeness of the existence of unconceived alternative explanations.Also Smith and Seth (2020) do not endorse a no-coincidence account based on IBE reasoning, although from a different perspective. 61More precisely, in reflecting on the evidential significance (for the reality of molecules) of Perrin's "converging theory-mediated measurements", they propose to under-stand the force of the evidence provided by the convergence by construing it "as a form of same-effect-same-cause reasoning -specifically as same-magnitudesame-quantity-being-measured reasoning" (p.310). 62inally, a further, different way of intending the import of the convergence in Perrin's case is offered by Demopoulos (2022, chap.2).Demopoulos rejects "an account of Perrin's success that is based on the hypothetico-deductive method or the method of inference to the best explanation", while, at the same time, maintaining a realist understanding of the argument.63Perrin's argument, for Demopoulos, is indeed an argument for molecular reality, but "it has a subtlety that is easily missed" (p.78).In order to show how it concretely works, Demopoulos provides a careful reconstruction of Perrin's argument in five stages.As regards specifically the concordance, its epistemic role enters at the fourth stage: that is, the stage which "consists in recounting the support that the connecting link [for the empirical determination of a host of molecular parameters] receives from the remarkable uniformity and concordance of the determination of parameter values to which it leads with various other determinations of these parameter values" (p.92).Then, the final (fifth) stage "infers from what the earlier stages have revealed the explanation of Brownian motion in terms of the molecular hypothesis" (ibid.).
To sum up, we can say that there is a shared agreement, in these representative positions, that the concordance of the various determinations of Avogadro's constant plays a distinctive, additional epistemic role in Perrin's reasoning besides mere empirical confirmation.How, then, this role is precisely specifiedwhether in terms of an inference to the best explanation, in terms of an inference to the most probable cause, in terms of a same-effect-same-cause inference or in terms of meta-empirical reasoning -significantly depends on the interpretative stance adopted, as we have seen.
(b) The extra-dimension case.As described in the previous section, the critical-dimension conjecture emerged from endeavours to extend the original dual theory and thus overcome its initial limitations and problems.These endeavours were mostly of theoretical nature, but justified or motivated on the grounds of the physics studied -that is, let us stress, on the grounds of assumptions and constraints of both phenomenological and theoretical nature. 64In this theory-building process, characterised by a close interplay of mathematically driven creativity and physical constraints, the fact that the value d = 26 for the spacetime dimension was obtained in three different, independent ways surely was an influential reason for taking it seriously.In fact, already after the second result (i.e., the no-ghost one), the initial skeptical attitude started to change.Now, a first question is whether this fact can be considered a sufficient basis for a convergence argument on behalf of the viability of the extra-dimension hypothesis.The real independence of the three ways leading to the critical dimension could be questioned, for example.But, analogously, one could question the independence of the different lines of evidence in many other (empirical) convergence cases, including Perrin's one, as has been done by some authors. 65ssuming, to the contrary, that the convergence of the different paths to the same surprising numerical result d = 26 provides a legitimate convergence argument for boosting the trust in the extra-dimension conjecture, the question becomes: what are the distinctive features of the argument doing the epistemic work in this case?
First of all, an analysis in terms of robustness does not seem to be appropriate, here.To start with, the number of independent ways of arriving at the result is very low (when compared to the thirteen ways of Perrin's case, for example).Beside, the convergence is not searched for.Quite the opposite: it is completely unexpected.Moreover, the result itself is very surprising.
As underlined in section 2.2, these features -a (small) number of concordant procedures, the unexpectedness of the convergence and the role of surprise -can be taken as the distinguishing characteristics of consilience as a convergence argument for theory assessment.The fact that we are dealing with a nonempirical case -in the sense that the convergence is to a theoretical result, obtained on the grounds of theoretical procedures (though physically motivated, it is worth recalling) -is not relevant from the viewpoint of the consilience structure of the argument.
The extra-dimension case can thus be seen as a particular instance of consilience.As highlighted in the previous section, the element of surprise plays a distinctive epistemic role in boosting the trust in an hypothesis in consilience cases.Actually, there are two kinds of surprising facts in this specific case: a) the convergence of the different, independent paths to the same numerical result d = 26 for the spacetime dimension; b) the result itself, which is undoubtedly very surprising.Correspondingly, the surprise factor has a double role here: first, by motivating a no-coincidence argument on behalf of the extra-dimension conjecture; second, by providing further support to the force of the argument.
Of course, this is a very particular case of theory assessment, where no empirical evidence is available.In this sense, it is naturally different from other examples of consilience, such as the case of Newton's unification discussed by Whewell (see section 2).However, it is worth underlining, the specific epistemic role of surprise in the consilience argument is independent of the empirical or non-empirical nature of the case considered.In other words, the assessment strategy based on consilience can be successful or defective in both cases, depending on how the epistemic import of consilience is interpreted.In current literature, there is a growing attention to the epistemic role of surprise in scientific pratice (e.g., Currie, 2018b; French and Murphy, 2023).Without entering in the details of this discussion, 66 what is of interest to underline, here, is that we can draw a similar conclusion for this kind of consilience argument as in the previous robustness case: the evaluation of the assessment strategy in such cases significantly depends on how the convergence argument is interpreted.And this is independent of whether we are dealing with an empirical case of convergence argument, as in the Perrin's case, or with a non-empirical case, as for the extra-dimension conjecture.