This paper defends the Quine-Putnam mathematical indispensability argument against two objections raised by Penelope Maddy. The objections concern scientific practices regarding the development of the atomic theory and the role of applied mathematics in the continuum and infinity. I present two alternative accounts by Stephen Brush and Alan Chalmers on the atomic theory. I argue that these two theories are consistent with Quine’s theory of scientific confirmation. I advance some novel versions of the indispensability argument. I argue that these new versions accommodate Maddy’s history of the atomic theory. Counter-examples are provided regarding the role of the mathematical continuum and mathematical infinity in science.
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According to Maddy (1997, 134), Quine’s theory of confirmation includes the doctrine of confirmational holism, the criterion of ontological commitment and the theoretical benefits of scientific confirmation.
See also Bangu (2012, chapter 9) for a discussion of these objections.
As in the case of argument Q-P1, I assume that there is a hidden naturalist premise in argument Q-P2.
The atomic debates had two sides: the physical side essentially based on the kinetic theory and the chemical side essentially based on the theory of thermodynamics.
Nyhof (1988) argues in the same direction as Brush.
In this vein, Clark (1976) argues that there were two competing research programmes during the nineteenth century—the atomic-kinetic research programme and the thermodynamics research programme. Clark denies external explanations concerning scientific evidence and makes the opposite claim: ‘it was the degeneration of the kinetic programme compared with the empirical progress of thermodynamics which accounts for the rise of scientific positivism’ (Clark 1976, 44). Clark’s arguments are shaped by the philosophy of science developed by Lakatos. Here, I will not discuss Clark’s view.
Berzelius (1813, 359).
Nyhof (1988, 90).
Clark (1976, 74).
As quoted in Clark (1976, 42).
For a full discussion of this point, see Chalmers (2009, 227–246). Here, I am ignoring Chalmers’ epistemic views on how scientific theories are confirmed: ‘a theory is supported only if the evidence conforms to predictions following in a natural rather than a contrived way from the theory in conjunction with subsidiary assumptions that themselves have independent support’ Chalmers (2009, 238). For a detailed analysis of this aspect, see also Chalmers (2011).
Other tensions in Maddy’s history are noted by van Fraassen (2009, footnote 2).
See Psillos (2011) for a detailed analysis of five key participants (Duhem, Stallo, Ostwald, Poincaré and Boltzmann) in the philosophical controversy concerning atomic theory. Psillos identifies two lines of dispute concerning the atomic hypothesis. By 1900, it was either perceived to be instrumentally indispensable (Stallo-Duhem-Mach) or was perceived to be epistemically precarious (Ostwald-Poincaré-Boltzmann).
This is a dispute that I cannot address in this paper. Thus, I will consider MC’s hypothesis, as other historians of science can address this dispute.
Maddy (1997, 139). Here, the words ‘virtues’ and benefits’ are interchangeable.
‘Epistemic values’ is the contemporary jargon for ‘theoretical benefits’.
For example, mild belief is the attitude that we should take ‘to the entities of a theory that is doing well but is not thoroughly established’; agnosticism is the attitude that we should take ‘to the entities of untried speculations at the frontiers’, Devitt (1984, 121).
Prima facie, this argument is redundant, as we could have reasons to be agnostic about numerous things, even if they are not indispensable to our scientific theories (not even when these are part of our speculations). However, in this case, I should recommend outright disbelief.
It may be objected that Quine’s criterion of ontological commitment is inconsistent with the gradation suggested by the arguments. Our scientific theories are either true or false. Hence, either certain objects lie in the domain of the quantification or not. There is no room for a middle ground. To this, I reply that this objection only shifts the problem to the truth value of the scientific theories. As, in light of Quine’s theory of confirmation, truth values and ontological commitments are on par, the above refined versions may be adapted into arguments regarding our epistemic attitudes concerning the truth value of scientific theories. It may be also objected that the same scientific statements can be common to different types of scientific theories. Therefore, it is possible that we should simultaneously be strongly committed, mildly committed and ontologically agnostic regarding entities that are indispensable to different types of scientific theories. To this objection, I reply that if an entity is indispensable to different types of scientific theories, then our type of commitment is determined by the stronger commitment of all available information on the entity in question. Here, there is a sort of separatism, but it is not the ‘current’ separatist objection to confirmational holism between mathematical entities versus empirical entities. See Busch (2011, 2012), Colyvan (2006) and Dieveney (2007) for a discussion of the separatist objection.
The Hubble constant is proportional to the space–time curvature. ‘The most recent measurements seem to indicate that the spatial curvature is almost zero (…) and that the total energy density is very close to the critical value [which depends on the current value of the Hubble constant, the speed of light and Newton’s gravitational constant]’ (Gasperini 2008, 29).
This technical scientific account was based on Gasperini (2008).
These theorems do not take quantum gravity effects into account. Nonetheless, these theorems imply that quantum gravity must, in the limit, recover the classical geometry of space–time provided by general relativity. That is, singularities are also reproduced as limiting cases of quantum gravity. See Heller (2011, 223–224).
Heller (2011, 223).
See Ellis (2007, 1268) for arguments in the direction that claims about infinity are unverifiable.
‘[M]athematical methods have been elaborated to deal with infinities. Although we are unable to work directly with infinities, we apply to them various versions of the ‘strategy of limits.’ The standard ‘going to the limit’ in a convergent series is the simplest and best-known method of this kind.’ Heller (2011, 224). Note, however, that this is not an idealisation, as we are attempting to address quantities that are ‘known’ to be infinite. Moreover, to the best of my knowledge, cosmological models do not involve any prior explicit idealisation concerning these quantities. Singularities are a scientific consequence of our best cosmological theories.
Here, I will not address the contemporary discussion regarding the relationalist or substantivalist views of space–time. The philosophical issues regarding the continuum go back to Zeno’s paradoxes.
A space is compact if it is closed and bounded.
See Penrose (2004, 62, 574).
‘In the quantum theory of a particle, the attribute of position (and most other physical properties), when they exist, still take values in the classical model R’ (Jozsa 1986, 395).
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Thanks to the audiences and individuals who commented on previous versions of this paper, especially, Alexander Bird, James Ladyman, Øystein Linnebo, John Mayberry, Leon Horsten and José Díez. I would also like to thank two anonymous referees for this journal for their comments and suggestions. This work was supported by grant SFRH/BPD/46847/2008, the project PTDC/FIL-FCI/109991/2009, “Hilbert’s Legacy in the Philosophy of Mathematics”, and the project PTDC/FIL–FIL/121209/2010, “Online Companion to Problems of Analytical Philosophy”, from Fundação para a Ciência e Tecnologia.
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Castro, E. Defending the Indispensability Argument: Atoms, Infinity and the Continuum. J Gen Philos Sci 44, 41–61 (2013). https://doi.org/10.1007/s10838-013-9222-8
- Atomic theory
- Mathematical indispensability
- The continuum