Abstract
This is the first of four papers prompted by a recent literature about a doctrine dubbed spacetime functionalism. This paper gives our general framework for discussing functionalism. Following Lewis, we take it as a species of reduction. We start by expounding reduction in a broadly Nagelian sense. Then we argue that Lewis’ functionalism is an improvement on Nagelian reduction.
This paper sets the scene for the other papers, which will apply our framework to theories of space and time. (So those papers address the space and time literature: both recent and older, and physical as well as philosophical literature. But the four papers can be read independently.)
Overall, we come to praise spacetime functionalism, not to bury it. But we criticize the recent philosophical literature for failing to stress:
-
(i)
functionalism’s being a species of reduction (in particular: reduction of chrono-geometry to the physics of matter and radiation);
-
(ii)
functionalism’s idea, not just of specifying a concept by its functional role, but of specifying several concepts simultaneously by their roles;
-
(iii)
functionalism’s providing bridge laws that are mandatory, not optional: they are statements of identity (or co-extension) that are conclusions of a deductive argument, rather than contingent guesses or verbal stipulations; and once we infer them, we have a reduction in a Nagelian sense.
On the other hand, some of the older philosophical literature, and the mathematical physics literature, is faithful to these ideas (i) to (iii)—as are Torretti’s writings. (But of course, the word ‘functionalism’ is not used; and themes like simultaneous unique definition are not articulated.) Thus in various papers, falling under various research programmes, the unique definability of a chrono-geometric concept (or concepts) in terms of matter and radiation, and a corresponding bridge law and reduction, is secured by a precise theorem. Hence our desire to celebrate these results as rigorous renditions of spacetime functionalism.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This programme (surveyed in Braddon-Mitchell and Nola, 2009) is familiar in metaphysics, philosophy of mind and ethics; but unfortunately little-known in philosophy of science. But our views are only ‘close to Canberra’: in particular, we will not require that the functional role of a concept, extracted from the given theory, gives a conceptual analysis of it.
- 2.
We will mostly use both ‘state’ and ‘concept’ throughout, although: (i) they are used more in philosophy of mind and metaphysics than in philosophy of physics (whose main analogues are ‘physical state’ and ‘quantity’ i.e. ‘physical magnitude’); and (ii) they are terms of art, without widely agreed criteria of individuation. But we shall not need to be precise about such criteria. Indeed, almost all our points are unaffected by what exactly these criteria are; and we will signal the exceptions.
It suffices for us to note that: (a) for most authors, a state is a localized fact or state of affairs; (b) some authors distinguish type-states, e.g. being in pain, or being in love, or seeing yellow in the top-left of the visual field, from token-states e.g. Fred’s being in pain at t, John’s loving Mary now, and these authors often go on to individuate token-states as n-tuples of the objects and times involved and the properties or relations attributed e.g. \(\langle {\mathrm {John}, \mathrm {Mary}, \mathrm {now}, \ldots \,\mathrm {loves}\ldots } \rangle \) (such n-tuples are often called ‘Russellian propositions’); (c) for most authors, a concept is the same, or pretty much the same, as a type-state. It is also the same, or pretty much the same, as a property or relation. For it is general, not localized, and is individuated more finely than by its set of instances; for example, it is individuated by a Fregean sense or Carnapian intension rather than by extension.
We can go along with (a) to (c), both in this paper and the others. We will simply note that some of the debates, e.g. in philosophy of mind about whether the mental state or concept of being in pain is reducible to material states or concepts, turn on controversies about the criteria of identity of properties—indeed a misty topic.
We can be sanguine in this way because, happily, for our examples for philosophy of physics, the state and concepts in question—the properties specified by their functional roles— are much less vague and controversial than e.g. pain. For they are physical properties: often, familiar physical quantities. Here are examples from our second paper: being freely mobile; being simultaneous with (as a relation between events); being congruent (as a relation between spatial intervals); and being isochronous, i.e. of equal duration (as a relation between temporal intervals).
- 3.
Beware: this strategy stumbles on Beth’s theorem. Namely: for first-order languages, supervenience i.e. determination is, surprisingly, equivalent to explicit definability, and so to reduction. In philosophy of science and mind, this was first pointed out by Hellman and Thompson (1975), and later stressed by various authors, e.g. Butterfield (2011a, Section 5.1, pp. 948–951). Cf. Dewar (2019) and footnotes 20 and 47.
- 4.
Our criticism is not meant to single out van Riel and van Gulick (2019): it has many merits. Our point is just that its emphases are part of a pattern: witness the fact that Lewis (1970), despite being definitive and general, has fewer citations than his (1972).
For both our first and second questions, we also recommend Lewis’ ‘Reduction of mind’ (1994). Its second half (pp. 421f.) is specialist: Lewis rebuts fashionable views, e.g. the ‘language of thought’ hypothesis, about Brentano’s problem of intentionality, i.e. the question what determines the contents of mental states like beliefs and desires. But the first half summarizes—and updates—the position we have reported in Sects. 7.1.1.1 and 7.1.1.2. We especially commend: Lewis’ advocacy of supervenience as reductive, his answer to the objection from qualia, his use of two-dimensional semantics to analyse the necessary a posteriori, and his answer to the idea that the term ‘pain’ is a rigid designator (pp. 412–421).
What Lewis says about the last topic (pp. 420–421) is also relevant to our point that, unfortunately, ‘functionalism’ is often understood as non-reductive. Thus Lewis writes: ‘It is unfortunate that this superficial question [in effect, a question about the best nomenclature for the occupant or realizer of a role] has sometimes been taken to mark the boundary of ‘functionalism’. Sometimes so and sometimes not—and that’s why I have no idea whether I am a functionalist’ (p. 421). We will briefly return to this in Sect. 7.4.3.2.
We also suspect, perhaps cheekily, that people have not sufficiently taken up functionalist reduction simply because its exposition in Lewis (1970) occurs only in the last two Sections (p. 441 et seq.). For most of the paper is taken up with a rigorous logical (though beautifully clear!) exposition of simultaneous unique definition within the first, i.e. reduced, theory. In short: we suspect the last two Sections have been ignored.
- 5.
The ‘dynamical approach’ is especially associated with Harvey Brown (especially his 2006); indeed, it was a member of his school, Eleanor Knox, who first coined the phrase ‘spacetime functionalism’ as a label for a position she favoured, and saw as close to Brown’s.
Recalling from this Section’s preamble, and Sect. 7.1.1.1, that functionalist reduction proceeds in two stages, and uses a ternary contrast, not a binary one, you will ask: what will be our ternary contrast? That is: what will be our ‘second theory’, accepted independently of (or after) the first one? In short: our answer, in subsequent papers, will vary from case to case; but the text’s contrast between spacetime and matter-and-radiation will be the unifying theme.
- 6.
Agreed: here, our labels ‘problematic’ and ‘unproblematic’ become strained. For in the present state of knowledge, it is spacetime, i.e. our established theories positing a spacetime continuum, that should be called ‘unproblematic’, and the speculative quantum gravity programme that should be called ‘problematic’. But labels aside, the intended analogy between the mind and spacetime cases is as clear as for the first contrast. Namely: just as mind is best understood in terms of mental concepts’ webs of relations to each other, and to material concepts, so also spacetime is best understood in terms of spatiotemporal concepts’ webs of relations to each other, and to the concepts of a postulated ‘non-spacetime’ theory.
- 7.
Agreed: it might be worthwhile to assess some emergent-spacetime research programmes using the taxonomy of problems that we will develop for our versions of spacetime functionalism. In particular, some of these programmes’ efforts to obtain a spacetime continuum may face the problems we label as Faithlessness, Plenitude and Scarcity (cf Sect. 7.2.2). Thanks to Julius Doboszewski for this point.
- 8.
We should mention here, since we will be advocating Lewis’ account of functionalism and reduction, that he himself did not espouse spacetime functionalism in our sense. In fact, he did not work in detail on philosophy of physical geometry. But his main view was substantivalist: spacetime is an object, the mereological fusion of its regions, that have various spatiotemporal relations to each other. Besides, in his mature metaphysical system, these relations are, in his jargon, perfectly natural and external. But we will not need these doctrines in this paper, or in our others. We only need Lewis’ treatment of functionalism and reduction.
- 9.
Agreed: we will start our discussion of reduction, in Sect. 7.2.1.1, by mentioning some historically influential proposed reductions that were ambitious and philosophical, like the ‘unified science’ picture. But these are just by way of example. In the same vein, note that our main claims will not depend on the labels, ‘problematic’ and ‘unproblematic’. To be sure: in some cases, the discourse to be reduced is unproblematic, but a reduction remains of interest. Cf. footnote 6.
- 10.
Lewis proposes that the extent to which a property encodes objective similarity—which he dubs: how natural it is—is a feature the property has across all of modal reality, wholly independent of contingencies such as what are the laws of nature. So this is indeed ‘limning the true and ultimate structure of reality’ (Quine, 1960, p. 202). Among less gung-ho replies, Taylor’s proposal, which he calls a ‘vegetarian substitute’ to Lewis, is to relativize similarity and associated notions to a theory (1993, especially Section IVf., p. 88f.); and van Fraassen answers Newman and Putnam from an empiricist and pragmatic perspective (2008, pp 229–235).
- 11.
For our use of ‘concept’ and ‘state’, recall footnote 2. References for each example, out of countless that could be given, are: (i) Smith and Jones (1986); (ii) Harman (1977, Part I); (iii) Benacerraf (1973); (iv) van Fraassen (1980, Chap. 2). For variety, we have here chosen expository references that are not committed to a reduction. Section 7.2.1.1 will discuss reductions for these examples.
- 12.
Again, there are many examples of these responses. The ‘mathematical atheism’ of Field (1980) exemplifies the second, eliminativist, response for example (iii) above. The condemnation of the ‘naturalistic fallacy’ by Moore (1903, Chap. 2) exemplifies the third, ‘sui generis’, response for example (ii).
- 13.
Saying this shows that we will take a body of claims, a theory, to be a set of sentences or propositions, rather than a set of models, as in the semantic or structural conception of theories. In Sect. 7.3.1, we will briefly defend this: in short, it will not matter to anything we say. Note also that (as mentioned above): among the ingredients that augment the unproblematic so as to enable deduction, there might be tools of construction, such as set theory.
- 14.
- 15.
- 16.
- 17.
Shapiro (2000, Chapter 10) is an introduction to the issues. But note that Benacerraf was re-discovering an old theme. Potter (2000, Sections 3.2–3.5) describes how already in 1888, Dedekind articulated Benacerraf’s “over-shooting” critique, and a version of structuralism about numbers, as immune to it. The similarity of the problems shows up in Potter’s Section headings, ‘Existence’ and ‘Uniqueness’ (of natural numbers). For compare how our problem of Plenitude makes existence/uniqueness of the definiendum easy/difficult, respectively; while our problem of Scarcity makes existence/uniqueness of the definiendum difficult/easy, respectively.
Besides, there are several other precursors—indeed, luminaries like Frege and Quine; (thanks to Alex Oliver for these cases). (1): Already in the Grundlagen (1884: paragraph 69, pp. 80–81) Frege himself puts the over-shooting objection to his own official definition of the number of a concept ( immediately after propounding the definition!). He asks: ‘do we not think of the extensions of concepts as something quite different from numbers?’; and goes on to say that we say ‘one extension of a concept is wider than another’ (i.e. in modern jargon: is a superset of another), while ‘certainly we do not say that one number is wider than another’. But after discussion, he concludes that, although ‘it is not usual to speak of a Number as wider or less wide than the extension of a concept, . . . neither is there anything to prevent us speaking this way’. That is, Frege bites the bullet, and allows the definition to mildly revise what we say. To put the point in Carnap’s jargon: Frege says it is enough to give an explication. Dummett (1991, p. 177–179) endorses Frege’s moves. First, he raises the objection of Faithlessness. But then he urges that since nothing Frege will prove, or argue for, turns on the arbitrarily chosen features of his definiens that go beyond the received sense of the definiendum, Frege’s choice of definiens is legitimate. He sums up: ‘Benacerraf’s problem simply does not arise for Frege’. (2): Quine rehearses the same considerations in Word and Object’s discussion of the various ways set-theorists and philosophers define an ordered pair. He admits that each proffered definition overshoots in the way we have discussed, but breezily says that he doesn’t care (1960: Section 33, p. 166, Section 53, p. 238). We shall return to this theme—whether to care that one is Faithless—in Sect. 7.6.2.
We also note that a cousin of Benacerraf’s structuralism, called ‘structural realism’, is prominent in recent philosophy of science. It relates to functionalism through, for example, the use of Ramsey sentences. But it does not bear closely on our main claims; so we postpone a proper discussion to another paper.
- 18.
We say ‘almost any understanding’ so as not to presuppose sets: we mean, roughly, a plurality of objects with properties and relations among them. And we say ‘almost any structure’, to signal that some vast structures may need proper classes rather than sets: in which case, one could talk of a ‘class-theoretic mock-up’.
- 19.
Agreed: advocates of conceptual analysis often do not stress that they must capture all or most of the claims about the analysandum once it is interpreted in terms of the analysans. At least they do not stress this, with ‘capture’ meaning ‘derive’. That is hardly surprising since, as we have seen: in most cases of philosophical interest, such a derivation is a very tall order. Nevertheless: reduction, as we understand the enterprise, is thus obliged. And as we saw in the quotes above, Russell himself accepted this obligation: as did Carnap (1963, p. 16).
Not that we wish to put Russell or his bon mot on a pedestal. Oliver and Smiley (2016, 272) call it ‘one of the shoddiest slogans in philosophy’; and their reasons echo one of our themes for our spacetime examples, viz. that writing down the right functional role, or analysis, of a concept, before any ‘construction’ begins, can take considerable ‘honest toil’. Thus they point out that : (i) Russell originally aimed it, unfairly, at Dedekind’s treatment on continuity, and (ii) ‘it assumes [wrongly] that we already know what we want … the examples from Dedekind show just how much honest toil it takes to discover—to formulate precisely—just what it is that we want’.
- 20.
For rigorous details about the notion of definitional extension, and about answers and choices for these questions, cf. for example, Boolos and Jeffery (1980, pp. 245–249), Button and Walsh (2018, Sections 5.1–5.5) and Hodges (1997, Sections 2.3, 2.6.2, and 5.5). Note that the idea of ‘implicit definition’ in (iii) above, originally due to Padoa (in 1900), is not the idea usually understood by this phrase, that was advocated by Hilbert, and denounced by Frege. We will return to clarify both ideas in Sect. 7.6.2.1. And in Sect. 7.3.4 we will briefly relate definitional extension to the question when two theories count as equivalent.
- 21.
- 22.
Or rather: that is so, in a suitably historically sensitive sense of ‘wave theory of light’, e.g. up till 1870; since the success of Maxwell’s theory, most expositions of the wave theory of light of course emphasise that light is electromagnetic waves. We will return to the issue of meanings shifting over time.
- 23.
As we mentioned at the end of Sect. 7.1.1.2: we in fact reject the denial, and associated views like epiphenomenalism, since we endorse a Lewis-Armstrong-style functionalism about mind. But nothing in this or our other papers will turn on this.
- 24.
The problem should be distinguished from what has come to be called ‘Newman’s objection’: which was originally made by Newman against Russell’s (1927), but is now seen as a problem for various forms of ‘structural realism’. For while Plenitude is a matter of there being too many definitions , Newman’s objection is a matter of it being logically guaranteed that there is a realizer, a definiendum. The idea then is that this guarantee is a problem since structural realism wants the existence of the definiens to be its main substantial assertion. As mentioned in footnote 17, we postpone discussion of structural realism to another paper.
- 25.
- 26.
For why multiple realizability threatens Scarcity of definitions, despite being a plenitude of realizations at the level of \(T_b\), cf. (3) at the end of Sect. 7.2.2.4.
- 27.
Some say that there might even be infinitely many ways, according to the taxonomy provided by the vocabulary (the concepts) of \(T_b\), to be an instance of (to realize) \(T_t\)’s predicate F. This is the idea of supervenience or determination: that the \(T_t\)-facts merely supervene on, are determined by, the \(T_b\)-facts, and a definiens would have to be an infinite disjunction. Cf (iii) in Sect. 7.3.1. But we doubt there are such truly infinite cases; and if they occur, we doubt their scientific importance (cf. Butterfield (2011a, Sections 4.1, 5.1, pp. 940–944, 948–951; 2011b, Sections 4.2.3, 5.2.3 and 6.3.4, at pp. 1070, 1089, 1100, 1127). Anyway, such cases will not occur in our other papers’ examples. So we will only consider finite disjunctions.
- 28.
This admission is like the point often made by mathematicians and logicians about axiom systems that are said to ‘implicitly define’ the (non-logical) words within them (or the concepts referred to be those words). Namely, that ‘implicit definition’ is a misnomer, since in general, one cannot extract genuine definitions of each term from the axiom system. As it is often put: the elementary analogy with solving n simultaneous linear equations for n unknowns is misleading. We shall return to this, especially in Sect. 7.6.2.
- 29.
Agreed: if we are undertaking a reduction with a definitional extension, we ask for more: the definition, taken together with others and with the claims of the reducing theory, must imply the claims of the reduced theory. Cf. Sect. 7.3.1.
- 30.
Recall the traditional objection to logical behaviourism, that it ruled out what seems possible: for example, the conjunct in our sketched pain-role, ‘typically causes aversive behaviour’, seems to rule out perfect-actor “super-Spartans” who never flinch when in pain.
- 31.
As we noted in (3) in Sect. 7.2.2.4 and footnote 26: the labels ‘Plenitude’ and ‘Scarcity’ can be confusing in the present context. For they were introduced in Sects. 7.2.2.2 and 7.2.2.3 as about, respectively, having too many, or no, definitions. But in this Section, the focus has been on having too many, or no, occupants (realizers) of a definition: a different topic.
- 32.
A wrinkle about terminology. Nowadays, some (e.g. Janssen-Lauret and MacBride (2020)) use ‘global descriptivism’ for the weaker doctrine that the reference of all terms is settled en bloc by total theory. This is weaker than our definition above, since it makes no claim that the only constraint on reference-assignment is making true whatever we assert. So it can be combined with the sort of ‘reference magnetism’ Lewis espoused; or with some other constraint that is not ‘just more theory’. So there is no disagreement here.
- 33.
Here, ‘true’ means ‘completely true’. This logical strength prompts a clarification. Shortly after the quoted passage, Lewis writes:
A complication: what if the theorizing detective has made one little mistake? He should have said that Y went to the attic at 11:37, not 11:17. The story as told is unrealized, true of no one. But another story is realized, indeed uniquely realized: the story we get by deleting or correcting the little mistake. We can say that the story as told is nearly realized, has a unique near-realization… In this case the T-terms ought to name the components of the near-realization … But let us set aside this complication for the sake of simplicity, though we know well that scientific theories are often nearly realized but rarely realized, and that theoretical reduction is usually blended with revision of the reduced theory.
Well said. Indeed in (1) of Sect. 7.3.2, we saw Nagel, Schaffner and Fletcher also say what ‘we know well’.
- 34.
In this Section and the next, we are very indebted to Adam Caulton: whose insightful comparison of Lewis’ and Carnap’s views we have regretfully suppressed, for the sake of space.
- 35.
So in our other papers, we will use this notation in examples of defining a spacetime structure as the unique realizer of some role. Broadly speaking: it will be vocabulary about the physics of matter and radiation that are the O-terms, and vocabulary about chrono-geometry that are the T-terms.
- 36.
This may seem nonchalant: Lewis (1970, p. 429) gives a longer defence of this choice. But agreed: questions remain, both in general (e.g. ‘How does this bear on the usual construal of Newman’s objection as depending on the Ramsey sentence using second-order quantifiers?’) and for our own advocacy of spacetime functionalism (e.g. ‘In our examples, can the defined spacetime structures, e.g. a metric, be named?’). We address these questions in our other papers.
- 37.
To be precise: its purely troublesome consequences are merely all the logical truths (logically valid formulas) containing only T-terms.
- 38.
Note that although ‘entity’ connotes ‘object’, and Lewis mentions some objects as his examples of theoretical entities, viz. living creatures too small to see and the dark companions of stars, Lewis’ taking all T-terms as names means that properties and relations, like physical magnitudes or a spacetime metric, will count as theoretical entities. Cf. the end of footnotes 2 and 36.
- 39.
A minor clarification. The equivalence requires an appropriate stipulation about the truth-values of identity statements using descriptions that are improper, i.e. not realized or multiply realized. But no worries: as Lewis says (1970, pp. 430, 438; 1972, footnote 11, p. 254), the stipulations in Scott’s system are appropriate.
- 40.
‘Transitivity of identity’ is usually understood as identity of objects; and this accords with Lewis’ taking all T-terms as names not predicates (1972, p. 253, quoted in Sect. 7.4.3.2). But we can also understand it as co-extensiveness of predicates; cf. footnotes 2 and 36.
- 41.
For brevity, we set aside: (i) Lewis’ discussion of the auxiliary reduction premise (1970, p. 443) and the strong reduction premise (1972, p. 255); (ii) Lewis’ examples (1970, p. 443–444; 1972, p. 256–258—which is mind and body); and (iii) Lewis’ discussion of adding definitionally expanded bridge laws to \(T^*\) if they are not theorems already (1970, p. 444–445).
- 42.
Certainly, it seems to fit the example of the interdependence of Newton’s laws and inertial frames. That is: we propose that one can reasonably take classical mechanics (in a point-particle formulation of the sort Sneed et al. consider) as introducing simultaneously terms for time and length (and so: frame), and for mass: terms that are held by the theory to be uniquely realized in such a way as to make the theory true—including Newton’s laws, using these terms, being true, i.e. true for motions described in inertial coordinates.
Presumably, the starting point for this project would be Chapter III of Sneed (1979) and Chapter II.3 of Balzer et al. (1987). But the gap in the literature is wide. Sad to say: so far as we know, neither of the two sides—Lewis and other Canberra Planners, and the Sneed-Stegmüller school—refers to the other.
- 43.
One might similarly summarise the dispute as about whether it is defect of an axiom system that it can be realized, i.e. made true, by very diverse interpretations (‘models’ as we nowadays say in logic and model theory)— with Frege saying ‘Yes’ and Hilbert saying ‘No’. (The dispute is often summarised as about whether axioms implicitly define their terms (with Frege saying ‘No’ and Hilbert saying ‘Yes’); but as we mentioned, ‘implicit definition’ is a misleading term.) We recommend Potter’s discussions (2000: 87–94; 2020, 124–132).
- 44.
In fact, Hilbert already took this view, some ten years before: witness the similar remark in 1891, made while waiting for a train in a station waiting-room: ‘one must be able to say at all times—instead of points, straight lines and planes—tables, chairs and beer mugs’: cf. Kennedy (1972, p. 133).
- 45.
Below, we will mention another example, Pieri. We also note three historical points. (1): Torretti (1999, p. 408–414) portrays this view as prompting the semantic or structural (as against syntactic) view of theories that, as we saw in Section 6.1, he favours. (2): Gray (2008, Section 4.1, p. 176f.) makes an interesting case that this broad development represented a rise of modernism, in a sense analogous to that in art and literature. (3): The Peano school’s endorsement of the Hilbertian structural view of axiom systems also surfaced in the philosophy of arithmetic: where it leads us back to Russell’s quip about ‘theft over honest toil’. Thus recall from Sect. 7.2.2.1’s discussion of Faithlessness, especially footnote 17, that Frege “already played the role of Benacerraf”. That is: he objected to his own definition of natural number; and then replied that the Faithlessness (specifically: over-shooting) was a price worth paying for the benefit of an otherwise successful definition. A propos of this, Alex Oliver points out to us (p.c.) that Peano also made this objection in 1901; and the reply that the price is right came from Russell, in his Principles of Mathematics. Writing with his usual verve (but lack of argument!), he in effect just outfaced Peano: ‘To regard a number as a class of classes must appear, at first sight, a wholly indefensible paradox. Thus Peano remarks that “… these objects have different properties”. He does not tell us what these properties are, and for my part I am unable to discover them’ (1903/2010: Section 111, p.115). Similarly in later writings: Russell had the ‘structuralism’ of Peano, Dedekind and Hilbert in mind when he condemned the ‘method of postulating’ as ‘theft’: cf. Sect. 7.2.2.3, especially footnote 19, and Linsky (2019, Section 3).
- 46.
In Sect. 7.4.2 we agreed that this is in general impossible, but argued that this is no objection to Lewisian functionalism.
- 47.
Two further points, which will be centre-stage in our second paper. (1): Beware: ‘Implicit definition’, as used nowadays by logicians, means something else. The idea is due to Padoa (Torretti, 1978, pp. 226–227). It is essentially equivalent to metaphysicians’ notion of supervenience/determination (cf. footnote 20). But it is utterly precise, and logically weaker than explicit definition (in general—though equivalent to it, for first-order languages, by Beth’s theorem: cf. Boolos and Jeffery, 1980, p. 245f., Hodges, 1997, p. 149). Thus proving explicit undefinability by showing implicit undefinability is called Padoa’s method (Hodges, 1997, p. 58). (2): For us, Pieri’s own axiomatisation of geometry has the further interest that, following the tradition of the Helmholtz-Lie Raumproblem, it is based on the idea of rigid motions.
- 48.
Indeed, the alleged contrast between logical form and ‘surface grammar’ is itself very questionable: cf. Oliver (1999).
- 49.
So this causes trouble for combining the doctrines of the Tractatus with the sort of phenomenalism Carnap espouses in the Aufbau. It is also, of course, an Achilles heel of the Tractatus: it prompted Wittgenstein to return to philosophy, and write his ‘Remarks on logical form’ in 1929; cf. e.g. Potter (2020, pp. 337–338, 409–410).
- 50.
Of course, Davidson, like Quine, advocates predicate logic as the preferred language for logical forms or ‘regimentations’. So he sees this situation as an argument for an ontology of events, in this example a human action. For then one has regimentations like, for our first two sentences: \((\exists x)((Butter(x) \wedge By(x,John) \wedge Quick(x) \wedge Of(x,toast) \wedge With(x,knife))\), and \((\exists x)((Butter(x) \wedge By(x,John) \wedge Quick(x) \wedge Of(x,toast))\). So our first inference becomes formally valid in predicate logic, i.e. an instance of dropping a conjunct in the scope of an existential quantifier.
- 51.
Cf. Sects. 7.2.2.1 and 7.3.3. But our rationale for expounding this example is not just to illustrate this paper’s themes. It also sets up another analogy, that we explore in our second paper: between Beltrami’s construal of hyperbolic geometry, and how a modern relativist who advocates a conformally flat spacetime would construe that paper’s second example of spacetime functionalism: viz. Robb’s axiom system for causal connectability. There is also another analogy hereabouts, which is both recent and mathematically deep. Beltrami’s embedding of 2-dimensional hyperbolic geometry in Euclidean space is a ‘baby version’ of the famous Nash embedding theorem, that any compact Riemannian manifold can be isometrically embedded in Euclidean space; cf. e.g. Tao (2016).
- 52.
We take up Helmholtz in our second paper. Agreed, Gauss in his unpublished work envisaged that a non-Euclidean geometry could be the real geometry of space: so, cheekily, ‘H’ could also stand for Gauss.
- 53.
This is not to say the objection is right: recall footnotes 17 and 45. Note that H might also accuse E of what Sect. 7.2.2.2 called ‘Plenitude’. Thus the variety of ways to identify numbers with sets, which Benacerraf emphasised—and we called ‘Plenitude’—corresponds to e.g. different placements of the Beltrami pseudosphere within \(\mathbb {R}^3\). So H might accuse E of having many equally good, and therefore equally bad, placements of the hyperbolic space, within \(\mathbb {R}^3\). (We write \(\mathbb {R}^3\), but the affine Euclidean space would be more accurate: no matter—nothing turns on this.)
References
Austin, J. (1962). Sense and sensibilia. G.J. Warnock (Ed.). Oxford: Oxford University Press.
Balzer, W., Moulines, C., & Sneed, J. (1987). An architectonic for science. Dordrecht: Reidel.
Beaney, M. (2004). Carnap’s conception of explication: From frege to husserl. In Carnap brought home; the view from jena (pp.117–150). Chicago: Open Court.
Beltrami, E. (1868). Saggio di interpretazione della geometria non-euclidea. Giomale di matematiche, 6, 284–312. (Reprinted in his collected works, Opere matematiche (1902, Milan) Volume I, 374–405.)
Benacerraf, P. (1965). What numbers could not be. The Philosophical Review, 74, 47–73.
Benacerraf, P. (1973). Mathematical truth. Journal of Philosophy, 70, 661–679.
Blanchette, P. (2018). The Frege-Hilbert controversy. Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/frege-hilbert/
Boolos, G., & Jeffery, R. (1980). Computability and logic, 2nd edn. Cambridge: Cambridge University Press.
Braddon-Mitchell, D., & Nola, R. (2009). Conceptual analysis and philosophical naturalism. MIT Press: Bradford Books.
Braithwaite, R. (1953). Scientific explanation. Cambridge: Cambridge University Press.
Brown, H. (2006), Physical relativity. Oxford: Oxford University Press.
Butterfield, J. (2011a). Emergence, reduction and supervenience: A varied landscape. Foundations of Physics, 41, 920–959.
Butterfield, J. (2011b). Less is different: Emergence and reduction reconciled. Foundations of Physics, 41, 1065–1135.
Butterfield, J. (2014). Reduction, emergence and renormalization. Journal of Philosophy, 111, 5–49.
Butterfield, J. (2018). On Dualities and equivalences between physical theories. In N. Huggett, B. Le Bihan, & C. Wüthrich (Eds.), Philosophy beyond spacetime. Oxford: Oxford University Press. http://philsci-archive.pitt.edu/14736/. Forthcoming (abridged).
Button, T. (2013). The limits of realism. Oxford: Oxford University Press.
Button, T., & Walsh S. (2018). Philosophy and model theory. Oxford: Oxford University Press.
Carnap, R. (1936). Testability and meaning. Philosophy of Science, 3, 419–471.
Carnap, R. (1963). Intellectual autobiography. In P. A. Schilpp (Ed.), The philosophy of rudolf carnap: The library of living philosophers (vol. XI). Open Court.
Coffa, J. A. (1983). Review of Torretti (1978). Nous, 17, 683–689.
Coffa, J. (1991). The semantic tradition from kant to carnap: To the Vienna station. L. Wessels (Ed.) Cambridge: Cambridge University Press.
Coffey, K. (2014). Theoretical equivalence as interpretative equivalence. British Journal of Philosophy of Science, 65, 821–844.
Davidson, D. (1967). The logical form of action sentences. In N. Rescher (Ed.), The logic of decision and action. Pittsburgh: University of Pittsburgh Press.
De Haro, S. (2020). Theoretical equivalence and duality. Synthese, 198, 5139–5177. https://doi.org/10.1007/s11229-019-02394-4
Dewar, N. (2019). Supervenience, reduction and translation. Philosophy of Science, 86, 942–954.
Dizadji-Bahmani, F., Frigg R., & Hartmann S. (2010). Who’s afraid of Nagelian reduction? Erkenntnis, 73, 393–412.
Dummett, M. (1991). Frege: Philosophy of mathematics. London: Duckworth.
Field, H. (1980). Science without numbers. Hoboken: Blackwell.
Fletcher, S. (2019). Counterfactual reasoning within physical theories. Synthese, 198, pp. 3877–3898. https://doi.org/10.1007/s11229-019-02085-0
Frege, G. (1884). The foundations of arithmetic (Translated and edited by J. Austin 1974). Hoboken: Blackwell.
Gray, J. (2008). Plato’s ghost. Princeton: Princeton University Press.
Halvorson, H. (2019). Logic in the philosophy of science. Cambridge: Cambridge University Press.
Harman, G. (1977). The nature of morality: An introduction to ethics. Oxford: Oxford University Press.
Hellman, G., & Thompson, F. (1975). Physicalism: Ontology, determination and reduction. Journal of Philosophy, 72, 551–564.
Hempel, C. (1965). Aspects of scientific explanation. New York: Free Press.
Hempel, C. (1966). Philosophy of natural science. New York: Prentice-Hall.
Hodges, W. (1997). A Shorter Model Theory. Cambridge: Cambridge University Press.
Hudetz, L. (2019a). The semantic view of theories and higher-order languages. Synthese, 196, 1131–1149.
Hudetz, L. (2019b). Definable categorical equivalence. Philosophy of Science, 86, 47–75.
Hurley, S. (1989). Natural reasons: Personality and polity. Oxford: Oxford University Press.
Jackson, F. (1998). From metaphysics to ethics: A defence of conceptual analysis. Oxford: Oxford University Press.
Janssen-Lauret, F., & MacBride, F. (2020). Lewis’s global descriptivism and reference magnetism. Australasian Journal of Philosophy, 98, 192–198. https://doi.org/10.1080/00048402.2019.1619792
Kennedy, H. (1972). The origins of modern axiomatics: Pasch to peano. The American Mathematical Monthly, 79, 133–136
Kroon, F. (1987). Causal descriptivism. Australasian Journal of Philosophy, 65, 1–17.
Lewis, D. (1966). An argument for the identity theory. Journal of Philosophy, 63, 17–25.
Lewis, D. (1969). Review of ‘art, mind, and religion’. Journal of Philosophy, 66, 23–25.
Lewis, D. (1970). How to define theoretical terms. Journal of Philosophy, 67, 427–446.
Lewis, D. (1972). Psychophysical and theoretical identifications. Australasian Journal of Philosophy, 50(3), 249–258.
Lewis, D. (1983). New work for a theory of universals. Australasian Journal of Philosophy, 61, 343–77.
Lewis D. (1984). Putnam’s paradox. Australasian Journal of Philosophy, 62, 221–236.
Lewis, D. (1989). Dispositional theories of value. Proceedings of the Aristotelian Society, LXIll, 113–137.
Lewis, D. (1993). Many, but almost one. In K. Campbell, J. Bacon, & L. Reinhardt (Eds.), Ontology, causality and mind: Essays on the philosophy of D. M. Armstrong (pp. 23–38). Cambridge: Cambridge University Press.
Lewis, D. (1994). Reduction of mind. In S. Guttenplan (Ed.), A companion to the philosophy of mind (pp. 412–431). Hoboken: Blackwell.
Linsky, B. (2019). Logical construction. Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/logical-construction
Lutz, S. (2017a). What was the syntax-semantics debate in the philosophy of science about? Philosophy and Phenomenological Research, 95, 319–352.
Lutz, S. (2017b). Newman’s objection is dead. Long live Newman’s objection! https://philsci-archive.pitt.edu/13018/
Moore, G. (1903). Principia ethica. Cambridge: Cambridge University Press.
Nagel, E. (1961). The structure of science: Problems in the logic of scientific explanation. San Diego: Harcourt.
Nagel, E. (1979). Issues in the logic of reductive explanations. In Teleology revisited and other essays in the philosophy and history of science. New York: Columbia University Press; reprinted in Bedau and Humphreys (2008); page reference to the reprint.
Niebergall, K.-G. (2000). On the logic of reducibility: Axioms and examples. Erkenntnis, 53, 27–61.
Niebergall, K.-G. (2002). Structuralism, model theory and reduction. Synthese, 130, 135–162.
Oliver, A. (1996). The metaphysics of properties. Mind, 105, 1–80.
Oliver, A. (1999). A few more remarks on logical form. Proceedings of the Aristotelian Society, 99, 247–272.
Oliver, A., & Smiley, T. (2013). Zilch. Analysis, 73, 601–613.
Oliver, A., & Smiley, T. (2016). Plural logic. Oxford: Oxford University Press.
Potter M. (2000). Reason’s nearest kin. Oxford: Oxford University Press.
Potter, M. (2020). The rise of analytic philosophy, 1879–1930. Oxfordshire: Routledge.
Psillos, S. (1999). Scientific realism: How science tracks truth. Oxfordshire: Routledge.
Psillos, S. (2012). Causal descriptivism and the reference of theoretical terms. In A. Raftopoulos & P. Machamer (Eds.) Perception, realism, and the problem of reference. Cambridge: Cambridge University Press.
Quine, W. (1960). Word and object. Cambridge: MIT Press (new edition: 2013).
Quine, W. (1964). Implicit definition sustained. Journal of Philosophy, 61, 71–74.
Russell, B.(1903/2010). The principles of mathematics. Crows Nest: Allen and Unwin (2010 reprint by Routledge).
Russell, B. (1918). The philosophy of logical atomism. In The monist, 28, 495–527; 29 (Jan., April, July 1919): 32–63, 190–222, 345–80. Page references to The Philosophy of Logical Atomism, D.F. Pears (ed.), La Salle: Open Court, 1985, 35–155.
Russell, B. (1924). Logical atomism. In D. F. Pears (Ed.), The philosophy of logical atomism. La Salle: Open Court. 1985, 157–181: also in The Collected Papers of Bertrand Russell: vol. 9, Essays on Language, Mind and Matter: 1919–1926, J.G. Slater (ed.), pp. 160–179; 2001: London and New York.
Russell, B. (1927). The analysis of matter, London: Kegan Paul.
Russell, B. (1919). Introduction to mathematical philosophy. New York and London.
Ryle, G. (1949). The concept of mind. London: Hutchinson.
Schaffner, K. (1967). Approaches to reduction. Philosophy of Science, 34, 137–147.
Schaffner, K. (1976). Reductionism in biology: Prospects and problems. In R. Cohen, et al. (Eds.), PSA 1974 (pp. 613–632). Dordrecht: Reidel.
Schaffner, K. (2006). Reduction: The Cheshire cat problem and a return to roots. Synthese, 151, 377–402.
Schaffner, K. (2012). Ernest Nagel and reduction. Journal of Philosophy, 109, 534–565.
Shapiro, S. (2000). Thinking about mathematics. Oxford: Oxford University Press.
Sklar, L. (1982). Saving the noumena. Philosophical Topics, 13, 89–110.
Smith, P., & Jones, O. (1986). The philosophy of mind. Cambridge: Cambridge University Press.
Sneed, J. (1979). The logical structure of mathematical physics. Dordrecht: Reidel, Pallas Paperbacks.
Sober, E. (1999). The multiple realizability argument against reductionism. Philosophy of Science, 66, 542–564.
Stein, H. (1992). Was Carnap entirely wrong, after all? Synthese, 93, 275–295.
Tanney, J. (2015). Gilbert ryle. Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/ryle/
Tao, T. (2016). Notes on the Nash embedding theorem. https://terrytao.wordpress.com/2016/05/11/notes-on-the-nash-embedding-theorem/
Taylor, B. (1993). On natural properties in metaphysics. Mind, 102, 81–100.
Torretti, R. (1978). Philosophy of geometry from riemann to poincaré. Kufstein: Reidel.
Torretti, R. (1986). Observation. British Journal for the Philosophy of Science, 37, 1–23.
Torretti, R. (1990). Creative understanding: Philosophical reflections on physics. Chicago: University of Chicago Press.
Torretti, R. (1999). Philosophy of physics. Cambridge: Cambridge University Press.
Torretti, R. (2008). Objectivity; a Kantian perspective. Royal Institute of Philosophy: Supplement, 63, 81–94.
van Fraassen, B. (1980). The scientific image. Oxford: Oxford University Press.
van Fraassen, B. (1991). Quantum mechanics: An empiricist view. Oxford: Oxford University Press.
van Fraassen, B. (2008). Scientific representation. Oxford: Oxford University Press.
van Riel, R., & van Gulick, R. (2019). Scientific reduction. Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/scientific-reduction/
Weatherall, J. (2018a). Why not categorical equivalence? https://arxiv.org/abs/1812.00943
Weatherall, J. (2018b). Theoretical equivalence in physics. Philosophy Compass, 14, e12592–e12591. https://arxiv.org/abs/1810.08192.
Weyl, H. (1934). Mind and nature. In Mind and nature: Selected writings in philosophy, mathematics and physics, P. Pesic (Ed.) (2009). Princeton: Princeton University Press.
Acknowledgements
We are grateful to: audiences at talks in Cambridge UK, Harvard (Black Hole Initiative), MIT, Munich, New York (MAPS), Oxford, and the ‘Quantum information structure of spacetime’ Network. For conversations and comments on previous versions, we are very grateful: to David Chalmers, Grace Field, Sam Fletcher, Eleanor Knox, Dennis Lehmkuhl, James Read, Alex Roberts and Bobby Vos; especially to Erik Curiel, Sebastian De Haro, Josh Hunt, Alex Oliver and Bryan Roberts; and above all, to Adam Caulton for—as ever—such insight and generosity. We are also very grateful to Cristián Soto, not least for his patience.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Butterfield, J., Gomes, H. (2023). Functionalism as a Species of Reduction. In: Soto, C. (eds) Current Debates in Philosophy of Science. Synthese Library, vol 477. Springer, Cham. https://doi.org/10.1007/978-3-031-32375-1_7
Download citation
DOI: https://doi.org/10.1007/978-3-031-32375-1_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-32374-4
Online ISBN: 978-3-031-32375-1
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)