Philosophers’ volume 19, no. 26 Pythagoreanism is the very surprising view that “all is number”. Imprint june 2019 If Pythagoreanism is true, then when Ernie asserted that a certain episode of Sesame Street was brought to you by the number three, his assertion’s bizarre implication that the episode in question was brought to you by some number or other is true. (Of course he may still have been wrong about which number.) Very surprising indeed. Could Pythagoreanism possibly be true? And why in the world would anyone believe it? Those are good ques- PYTHAGOREANISM: tions. But in §1 I first try to get clear on what the view is. As it turns out, there are actually several views that are all reasonable ways to precisify the basic Pythagorean idea. Then, I return to the good ques- A NUMBER OF tions. In §2 I try to understand why in the world anyone would believe at least some version of Pythagoreanism. And in §3 I try to determine whether any version of Pythagoreanism could possibly be true. Inter- THEORIES estingly, the best objections I uncover in §3 have no application to the versions that in §2 I argue we have some reason to believe. I am well aware that very few of my readers are Pythagoreans, and despite what I will argue hardly any more of them will be Pythagore- ans after reading this. (Perhaps even fewer of them will be Pythagore- ans after reading this.) I still don’t believe Pythagoreanism myself. What then is the point of engaging with such an archaic and exotic Aaron Segal metaphysical system?1 Well, for one thing, even if very few readers will come to believe Pythagoreanism outright, many more might be- come less confident in its denial, and maybe even become agnostic The Hebrew University of Jerusalem about it. Indeed, I am less confident in its denial myself than I was be- fore writing this. For another thing, even for those readers whose anti- Pythagorean beliefs and degrees of confidence will remain untouched by my discussion — or change in the anti-Pythagorean direction — © 2019, Aaron Segal the epistemic status of those beliefs and degrees of confidence might This work is licensed under a Creative Commons be improved. Pythagoreanism is no longer so archaic or exotic. It has Attribution-NonCommercial-NoDerivatives 3.0 License been endorsed in one version or another by an eminent philosopher

1See Hawthorne and Nolan[ 2006] and Turner[ 2011] for interesting answers to the same question about other exotic metaphysical views. aaron segal Pythagoreanism: A Number of Theories and an eminent physicist.2 And many of us know that. Their known conjunction with all definitional truths is a member of the theory. Then endorsement arguably diminishes or eliminates whatever justification we shall take Pythagoreanism to be the claim that there is a complete our unexamined denial of Pythagoreanism enjoys.3 Even if that’s not and closed theory that is in some respect entirely numerical.5 right, a careful and comprehensive examination of an otherwise unex- amined attitude can often serve to improve the epistemic status of that 1.1 Numerical or Mathematical attitude. “Know how to answer the Epicurean,” counsels the Mishna Well, maybe not entirely numerical. I have assumed until now that in the (2nd c.); we might add, “Know how to answer the Pythagorean”. For slogan ‘All is number’, ‘number’ means number, no more and no less. a final thing, engaging with a very surprising grand metaphysical sys- As an interpretation of ancient Pythagoreanism, such a reading has tem serves, as Bertrand Russell said, “to enrich our intellectual imagi- what to recommend it. Thus, Aristotle (Metaphysics 1.5 985b-986a26) nation”: it allows us to see the world in a radically different way.4 For says of the Pythagoreans that: me that is reason enough. ...they saw the attributes and ratios of musical scales in numbers, 1. What is Pythagoreanism? and other things seemed to be made in the likeness of numbers in their entire nature, and numbers seemed to be primary in I may have spoken too hastily at the beginning. Whether the bizarre all nature, they supposed the elements of numbers to be the implication of Ernie’s assertion is true if Pythagoreanism is true de- elements of all things that are. pends on a number of questions about what Pythagoreanism comes to. On their view, triangles are (or are made of) numbers as well. There are at least three axes along which versions of Pythagore- But as an interpretation of what all Pythagoreans have meant, it anism can differ. It is easiest to formulate the different options along falls short. Thus, while Kepler is often considered a Pythagorean, he is the three axes if we think of Pythagoreanism as the assertion of the said to differ from the ancient Pythagoreans in that for them, existence of a certain kind of theory. Let a theory be a (perhaps infinite) ...the first science is arithmetic, and the deepest level of expla- set of true sentences. Let a complete theory be a theory that is in some nation is the symbolical account furnished by numerology. Ke- robust sense a sufficient base from which to deduce every truth. (As I pler’s Pythagoreanism is of a different stripe. Kepler’s divine make clear in §1.3, a theory may in “some robust sense” be a sufficient archetype is geometrical in form, and his preference for geom- base from which to deduce a truth even if that truth can’t be logically etry is deeply motivated. He insists on an explanatory theory deduced from that theory, or even from that theory in conjunction with that can give results in terms of physical magnitudes...That is all definitional truths.) Let a closed theory be a theory such that any sen- why he insists on the importance of numeri numerati rather than tence that can be logically deduced from some subset of the theory in numeri numerantes, numbers as the measures of physical quanti-

2See Quine[ 1976] and Tegmark[ 1998, 2008, 2014]. On Pythagoreanism’s rela- tionship to the much more popular Ontic Structural Realism, see §1.5. 3See Frances[ 2005]. 5On the need for the theory to be closed, see nt. 11. 4See Russell[ 1912]. 6Cited in McKirahan[ 1994, p. 101].

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ties rather than ‘counting numbers’...7 inasmuch as we know which theories those are. Then let us say that a ‘mathematical predicate (functor)’ is a predi- Kepler’s view would be more accurately put by saying “All is number- cate (functor) in some mathematical theory or other (or one definable cum-shape”. wholly in terms of such predicates); a ‘numerical predicate (functor)’ Once we move beyond just the numerical, we might include still is a predicate (functor) in number theory (or one wholly definable in more than just the numerical-cum-geometrical. Modern mathemat- terms of such predicates); a ‘geometrical predicate (functor)’ is a pred- ics comprises fields some of which the ancient and early modern icate (functor) in geometry (or one wholly definable in terms of such Pythagoreans couldn’t even have dreamt of, such as set theory, group predicates); and so on. theory, and category theory. Then let us say that a ‘mathematical object’ is an object that can be This bring us to our first axis. Let Numerical Pythagoreanism be completely characterized — at least regarding how it is positively and the view that there is a complete and closed theory that is in some intrinsically — entirely in terms of mathematical predicates. More ex- respect entirely numerical, and let Mathematical Pythagoreanism be actly, let us say that a ‘mathematical object’ is an object such that there the view that there is a complete and closed theory that is in some is a closed theory, all of whose predicates and functors are mathemat- respect entirely mathematical. The former implies the latter, but not ical, that is a sufficient base from which to deduce every truth about vice versa. As the example of Kepler illustrates, there are of course the intrinsic positive features of that object;9 and a ‘numerical object’ intermediate views.8 is an object such that there is a closed theory, all of whose predicates and functors are numerical, that is a sufficient base from which to de- 1.2 Three Grades of Numerical Involvement duce every truth about the intrinsic positive features of that object; a Even after specifying a location along the first axis, the view is still ‘geometrical object’ is an object such that there is a closed theory, all of underspecified, since it’s not clear in what respect the complete and whose predicates and functors are geometrical, that is a sufficient base closed theory is supposed to be numerical, or geometrical, or mathe- from which to deduce every truth about the intrinsic positive features matical, or what have you. This brings us to our second axis. To clarify of that object; and so on.10 the different options, let’s first fix some terminology. I assume that the term ‘mathematical theory’ is unproblematic, at least inasmuch as we know which theories are mathematical and 9I am assuming a distinction between positive and negative properties, and I which aren’t. Likewise, I assume terms for specific mathematical theo- take the distinction to be primitive. I am also assuming a distinction between intrinsic and extrinsic properties, and I am understanding that distinction as ries, like ‘number theory’ and ‘geometry’, are unproblematic, at least Lewis[ 1986] does: An intrinsic property is one that never differs between pos- sible duplicates, and two things are duplicates if there is a one-to-one map- ping from the parts of one to the parts of the other that preserves all perfectly natural properties and relations. 7Kahn[ 2001, p. 160-161]. 10It is perhaps worth explaining why I have not employed alternative defini- 8It should be noted that while Mathematical Pythagoreanism is the weak- tions, especially since my definition might strike some as somewhat arbitrary. est form (along this axis) of Pythagoreanism worthy of the name, Numer- ical Pythagoreanism is not the strongest form: One might accept Logical First: Why not just take ‘number’/‘numerical object’ as primitive? Don’t we Pythagoreanism, which says that all is logic; it says that there is a complete have an independent grasp on those terms? Perhaps. But the same cannot be and closed theory that is in some respect entirely logical. Perhaps this is what said of ‘geometrical object’ or ‘mathematical object’, and so a uniform treat- Hegel thought. But, as far as I can tell, it is simply indefensible. ment of Ontological Pythagoreanism, independently of where we are along

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Finally, let us say that a ‘broadly mathematical predicate’ is a pred- theory ranges over numerical objects.11 icate that is either (a) a mathematical predicate or (b) the predicate ‘x Thus, if as some Pythagoreans seemed to believe (Aristotle, Meta- is a mathematical object’ or (c) a predicate definable wholly in terms physics 986a17—2112), the statement, of predicates from (a) or (b) or in terms of which ‘x is a mathematical The elements of number are the even and the odd, and of these object’ is defined; a ‘broadly numerical predicate’ is a predicate that is the latter is limited and the former unlimited. The one is com- either (a’) a numerical predicate or (b’) the predicate ‘x is a numerical posed of both of these (for it is both even and odd) and number object’ or (c’) a predicate definable wholly in terms of predicates from springs from the one; and numbers, as I have said, constitute the (a’) or (b’) or in terms of which ‘x is a numerical object’ is defined; and whole universe, so on. The options along the second axis differ as to whether the respect in which the complete and closed theory is supposed to be numerical, 11 geometrical, mathematical, or what have you, is in its objects, its predi- It is to prevent Ontological Pythagoreanism from being trivial that I require the theory to be closed. It is trivial — or as trivial as there being numerical cates and functors, or in the theory itself. (Throughout the remainder of objects at all — that there is a complete theory whose ontology is entirely this section I will write as if we have specified the ‘numerical’ option numerical. Let ‘α’ be the set of all true sentences (in some suitably idealized language). Consider a predicate, ‘x alpharizes’, that is satisfied by an object along the first axis, so as to spare myself and the reader a constant just in case all the members of α are true. Then if the number two exists, then repetition of the various options along that axis.) the theory that consists in the sentence, ‘2 alpharizes’, is complete and its ontology is entirely numerical. But such a theory is not closed, for there is According to Ontological Pythagoreanism, there is a complete and much that can be logically deduced from the theory in conjunction with all closed theory whose ontology is entirely numerical. That is, any refer- definitional truths that isn’t part of the theory itself. More to the point, it is ring term that appears in any sentence in the theory refers to a nu- non-trivial that there is any theory whose ontology is numerical that is both complete and closed. (A referee for this journal noted that the requirement of merical object, and any quantifier that appears in any sentence in the closure may not completely solve the problem I am trying to solve — at least not for certain specifications along the third axis — since we can cook up even trickier predicates than ‘x alpharizes’. Consider the predicate, ‘x L-alpharizes’, that is satisfied by an object just in case if all the laws of metaphysics are true, the first axis, would elude us. Second: Why not just say that a ‘mathematical then all the members of α are true. The theory that consists in the sentence ‘2 object’ is an object that can be completely characterized entirely in terms of L-alpharizes’ is complete, at least given certain specifications along the third mathematical predicates (and analogously for ‘numerical object’ and so on)? axis (see §1.3). And one might be able to extend it so that it’s closed without Because that would collapse the distinction between Ontological Pythagore- quantifying over or referring to any non-numerical objects — since one can’t anism and Ideological Pythagoreanism. Indeed, it would be even worse: Pris- logically deduce from it (even in conjunction with all definitional truths) that tine Ideological Pythagoreanism (see §1.3) would follow from the mere fact all the members of α are true. If one could extend it in that way, that would that there is a single mathematical object! Clearly that notion of mathematical make (Nomic) Ontological Pythagoreanism as trivial as there being numeri- object is too demanding. cal objects at all. I’m not certain that one could so extend it — the negation of Third: Why not just say that a ‘mathematical object’ is an object to which the laws of metaphysics, which are themselves universal generalizations, are some mathematical predicate applies (and analogously for ‘numerical object’ existential assertions that might well end up quantifying over non-numerical and so on)? Because that would well-nigh trivialize Ontological Pythagore- objects. But I grant that more work needs to be done here, and there might anism; it would be as trivial as the fact that everything is a member of its not be any formal solution to the problem I’m addressing. In that case, I’d own singleton. Clearly that notion of mathematical object is not demanding reluctantly characterize Pythagoreanism as the view that there is a complete enough. and “non-tricksy” (or maximally metaphysically perspicuous) theory that is in some respect entirely numerical.) The way I have defined it is demanding enough, but not too demanding, for our purposes. 12Cited in McKirahan[ 1994, p. 101].

philosophers’ imprint - 4 - vol. 19, no. 26 (june 2019) aaron segal Pythagoreanism: A Number of Theories is a complete theory, then Ontological Pythagoreanism is true.13 first alternative, Ideological Pythagoreanism, there is a complete and But notice that it doesn’t follow from what those Pythagoreans be- closed theory whose ideology is numerical. That is, all the predicates lieve that the predicate ‘x springs from y’ is to be understood number- and functors that appear in any sentence in the theory are broadly theoretically. It need not be the case that necessarily x springs from y numerical. if and only if x < y, or anything else along those lines. Springing from, Thus, if as some Pythagoreans supposed (Alexander, Commentary along with pushing and pulling and hemming and hawing, might well on Aristotle’s Metaphysics 38.10—39.2015), have no number-theoretic analysis at all. There might well be no com- ...requital and equality were characteristic of justice and [they] plete theory (let alone complete and closed) whose predicates are all found these features in numbers, and so declared that justice numerical. The numbers of course live numerical lives, like being suc- was the first number that is equal-times-equal...[and] marriage cessors of one another and adding together, but they can be and do is the number five, because marriage is the union of male and other things as well. They can live a double life. female, and...the odd is male and the even is female, and this Quine came to embrace Pythagoreanism, but only in its Ontological number is the first which has its origin from two, the first even form. As he put it: number, and three, the first odd... We must note further that this triumph of hyper- and moreover, some natural extension of this theory is complete, then Pythagoreanism has to do with the values of the variables Ideological Pythagoreanism is true. of quantification, and not with what we say about them. It While Ontological Pythagoreanism fails to imply Ideological has to do with ontology and not with ideology. The things Pythagoreanism, the latter implies the former. For any theory that that a theory deems there to be are the values of the theory’s witnesses Ideological Pythagoreanism also witnesses Ontological variables, and it is these that have been resolving themselves Pythagoreanism. Consider any theory T that witnesses Ideological into numbers and kindred objects — ultimately into pure sets. Pythagoreanism. Anything to which T refers or over which it quan- The ontology of our system of the world reduces thus to the tifies has the following feature: There is some complete and closed ontology of set theory, but our system of the world does not theory all of whose predicates and functors are broadly numerical (i.e. reduce to set theory; for our lexicon of predicates and functors T ) that is a sufficient base from which to deduce every truth about still stands stubbornly apart.14 that thing. And so a fortiori there is some complete and closed the- Such is the first grade of numerical involvement. ory all of whose predicates and functors are broadly numerical (i.e. But as is clear from the above passage, there is an alternative. In- T ) that is a sufficient base from which to deduce every truth about deed, there are at least two alternatives, but they are conflated by the intrinsic positive features of that thing. But then there is also some Quine, and by others who actually embrace them. According to the complete and closed theory all of whose predicates and functors are numerical, period, that is a sufficient base from which to deduce ev- ery truth about the intrinsic positive features of that thing: that is, it 13At least so long as no sentence that can be logically deduced from that theory in conjunction with all definitional truths contains any term that refers to a non-numerical-object or quantifies over non-numerical-objects. 14Quine[ 1976, p. 503]. 15Cited in McKirahan[ 1994, p. 93].

philosophers’ imprint - 5 - vol. 19, no. 26 (june 2019) aaron segal Pythagoreanism: A Number of Theories is a numerical object. For if there were no such theory all of whose other than the theory of numbers. predicates were numerical, then there would be no such theory all of whose predicates were broadly numerical: If using a certain vocabu- 1.3 Sufficiency lary I can’t say enough to characterize you positively intrinsically, then Pythagoreanism says that there is a complete theory, and hence in I still won’t have said enough if I merely add that I could not (using some robust sense a sufficient base for the deduction of every truth, that vocabulary) already have said enough to characterize you posi- that is entirely numerical. But what is this robust sense? In what does tively intrinsically. this sufficiency consist?17 This brings us to the third and final axis. Ideological Pythagoreanism, therefore, is the second grade of nu- Here again we confront a range of options. In its strongest form (along merical involvement. this axis) Pythagoreanism says there is a closed theory that is (in the Notice, however, that it doesn’t straightforwardly follow from Ideo- respect specified along the second axis) entirely numerical from which, logical Pythagoreanism that there is a complete and closed theory that all by itself, one can logically deduce every truth. This is equivalent, is as weak as number theory. Ideological Pythagoreanism says that given the meaning of ‘closed’, to the claim that there is a theory, which there is a complete and closed theory whose ideology and (as we’ve is (in the respect specified along the second axis) entirely numerical, just argued) ontology are both numerical. But any such theory might that has as a member every truth. Or even more simply, that every include sentences that are neither axioms nor theorems of number the- truth is (in the respect specified along the second axis) a numerical ory. For example, the following sentence is no axiom or theorem of truth. Call this ‘Pristine Pythagoreanism’. In its weakest form (along number theory: this axis) Pythagoreanism says that there is a theory that is (in the re- spect specified along the second axis) entirely numerical from which, Everything is a numerical object together with every metaphysically necessary truth, one can logically de- Number theory is the theory of numbers. It doesn’t say that there duce every truth.18 Call this ‘Metaphysical Pythagoreanism’.19 In be- is nothing other than numbers. But it’s consistent with Ideological Pythagoreanism that any complete theory includes that sentence (or 17The very same question needs to be asked about the sufficiency that plays a some sentence that implies it). This brings us to the second alternative role in defining ‘mathematical/numerical/geometrical object’. The term ‘suf- to Ontological Pythagoreanism: Theoretical Pythagoreanism. Accord- ficient’ plays a dual role then in defining Ontological Pythagoreanism, and I shall assume that it is to be used with a consistent meaning. Our specification ing to Theoretical Pythagoreanism, number theory itself is a complete of a location along this third axis is thus a specification of what sufficiency and closed theory.16 Since the ontology and ideology of number theory amounts to in both its roles. are numerical, Theoretical Pythagoreanism implies both Ontological 18Perhaps there is a still weaker form of Pythagoreanism, according to which Pythagoreanism and Ideological Pythagoreanism, but, as we’ve noted, there is a theory that is entirely numerical that metaphysically necessitates every truth. If there is some theory that is entirely numerical that contains not vice versa. an infinite number of true sentences and there is some truth that is meta- Theoretical Pythagoreanism, therefore, is the third and highest physically necessitated by an infinite subset of the theory but by no finite grade of numerical involvement. There is nothing to know, it says, subset, then there is a truth that is metaphysically necessitated by the theory but not logically deducible from it, even together with every metaphysically necessary truth. I set aside this complication. 16In order for it to be closed, number theory would of course have to consist in 19Note: if Necessitarianism is true — that is, if every truth is a necessary truth all the theorems of number theory, not just some set of axioms. — then Metaphysical Pythagoreanism is “automatically” true. Perhaps that

philosophers’ imprint - 6 - vol. 19, no. 26 (june 2019) aaron segal Pythagoreanism: A Number of Theories tween, we have still other options: Slightly stronger than Metaphysical implication of Ernie’s assertion is true, I was assuming that even if Pythagoreanism is the view, Essentialist Pythagoreanism, that there is Pythagoreanism is true, it’s still the case that television producers exist a theory that is entirely numerical from which, together with every essen- and bring shows to audiences. That assumption is false if any of these tialist truth, one can logically deduce every truth.20 Alternatively, but highly revisionary versions of Pythagoreanism is true. So I was assum- still slightly stronger than Metaphysical Pythagoreanism, there is the ing, and will continue to assume, the falsity of such versions. This view, Nomic Pythagoreanism, that there is a theory that is entirely nu- leaves non-Pristine versions of Ideological and Theoretical Pythagore- merical from which, together with every law of metaphysics, one can logi- anism and all versions of Ontological Pythagoreanism. cally deduce every truth.21 There are perhaps other options besides for the bridge principles that connect the numerical theory with the rest 1.5 Pythagoreanism and Structuralism of the truths. With these more specific versions on the table, we are ready to move to the arguments. But before doing so, one more clarification is in order. A 1.4 Specific Versions reader might think that Pythagoreanism logically implies Ontic Struc- Choices along all three axes give rise to specific versions of Pythagore- tural Realism (OSR), the view that ‘all is structure’.22 But it doesn’t. Its anism. Thus, for example, Pristine Numerical Ideological Pythagore- conjunction with the claim that ‘mathematics is just structure,’ i.e. Math- anism is the view that every truth has as predicates and functors only ematical Structuralism, indeed logically implies OSR. But Mathemati- broadly numerical ones. This, along with any other specific version of cal Structuralism is no part of Pythagoreanism, and I do not assume Pristine Ideological Pythagoreanism — and a fortiori any specific ver- the former’s truth (or its falsity) here. sion of Pristine Theoretical Pythagoreanism — is exceptionally strong. Conversely, a reader might think that OSR logically implies It is so strong as to demand a radical revision of our ordinary beliefs. If Pythagoreanism. But it doesn’t. At least not if OSR is standardly inter- it is true, then sentences that say that there are neutrons, or that there preted. It might well follow from any standard version of OSR that the are neurons, or that there are Neanderthals, are not true, for ‘x is a following sentence is a complete theory (in the sense specified above), neuron’, ‘x is a neutron’, and ‘x is a Neanderthal’ are not broadly nu- where S is some structure: merical predicates (or broadly mathematical predicates, period). And, S is instantiated for the same reason, neither are the negations of those sentences. They must not express any claims at all. But no version of Pythagoreanism is implied by the completeness of Any of these versions of Pythagoreanism is radically revisionary. that theory. For the predicate ‘x is instantiated’ is not a broadly mathe- Now, when I said that if Pythagoreanism is true, then the bizarre matical predicate. (And nor is any other predicate that would serve the same purpose, such as ‘x is actualized’ or ‘x is concretized’.) And so no version of Ideological or Theoretical Pythagoreanism is implied by the is reason to think Metaphysical Pythagoreanism too weak to be a version of completeness of that theory. And even if S is a mathematical object, no Pythagoreanism. version of Ontological Pythagoreanism follows from the completeness 20On essentialist truths — truths that specify the essences of things — see, inter alia, Fine[ 1994]. 21On laws of metaphysics, see, inter alia, Sider[ 2011] and Schaffer[ 2017]. 22On Ontic Structural Realism, see, inter alia, Ladyman and Ross[ 2007].

philosophers’ imprint - 7 - vol. 19, no. 26 (june 2019) aaron segal Pythagoreanism: A Number of Theories of that theory, since, even if complete, it might well not be closed. In- 2.1 Tegmark’s Arguments deed, given the definition of ‘x is instantiated’, the conjunction of that Max Tegmark [1998, 2008, 2014] puts forward deductive, abductive, theory and all definitional truths quite plausibly logically implies that and probabilistic arguments (without labelling them as such).25 His there is something that is an instance of S . And we have no reason to deductive argument goes as follows. Realism (as opposed to good believe that any instance of S is itself a mathematical object. (Whether old-fashioned 19th century Idealism) is true. If Realism is true, then it is will depend on which particular structure S is.23) then there is a true fundamental theory none of whose predicates or Perhaps there are non-standard versions of OSR according to which functors is anthropocentric. But all predicates and functors other than the sentence mathematical ones are, at the end of the day, anthropocentric. So there is a true fundamental theory, all of whose predicates and functors are S exists broadly mathematical. And from there, it’s but a small step to what I is a complete theory. But I’m not aware of any argument for structural- have called Ideological Pythagoreanism (albeit not in its Pristine form). ism that would establish a conclusion so strong, and such a version A thorough examination of this argument would require us to get seems to be nothing but the conjunction of OSR and Pythagoreanism. clear on what it is for a predicate or functor to be anthropocentric. Of course, that view logically implies Pythagoreanism. But it’s not OSR But even without doing so, it’s clear enough that, given any sense of — or the reasons to accept it — that’s doing the implying, then. ‘anthropocentric predicate (functor)’ that makes the second premise The idea that the world is thoroughly mathematical is at least logi- true, there is no reason at all to think that the third premise, given cally independent of the idea that the world is thoroughly structural, that sense, is true. That is, there is no reason at all to think that only and needs a treatment of its own.24 mathematical predicates and functors manage not to be objectionably anthropocentric. What reason is there to think that the predicate ‘x has 2. Arguments mass’, for example, is anthropocentric in a way that would run afoul One might think that, even setting aside any objections one might have of Realism? There is none. Or at least Tegmark hasn’t begun to give to Pythagoreanism, there simply is no good reason at all to accept it. In- us one.26 Let us move on to his slightly more promising abductive deed, the extant arguments of which I am aware range in quality from argument. moderately underwhelming to very underwhelming. We can survey Tegmark points to what Eugene Wigner [1960, p. 1] famously called those arguments rather briskly.

25It’s not just the labels that are missing, but a good number of the premises. 23See Saunders[ 2003, p. 129]: “But I share French and Ladyman’s puzzlement What follows is therefore my best attempt to fill in Tegmark’s frequently as to why Cao assumes that I, and they, are wedded to the belief that struc- laconic suggestions. See also Butterfield[ 2014]. merely tures are mathematical. I have never said so, and nor to my knowledge 26 have they. And my position is entirely neutral with respect to Platonism, as I See Zimmerman[February 2017]: “Tegmark’s reason for taking the latter ap- believe is theirs.” proach is his conviction that physics must be purged of anything but math- ematical terms. Non-mathematical concepts, he says, are “anthropocentric 24Likewise, the idea that the world is thoroughly mathematical is logically in- baggage,” and must be eliminated for objectivity’s sake. But why think that dependent of the idea that the world is fundamentally abstract — the latter the only objective descriptions that can truly apply to things as they are in of which has occasionally been identified with Pythagoreanism (see Cowling themselves are mathematical descriptions? So far as I can see, he never justi- [2017, p. 91]) — and each deserves its own treatment. fies this assumption.”

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“the unreasonable effectiveness of mathematics”. Let us assume that with the denial of Tegmark’s. And both explanations would seem to our current physical theories are (at least) roughly correct. Such the- explain our datum in exactly the same way. Tegmark has given us no ories are chock full of mathematics (and increasingly so). While they reason to prefer his explanation to its rival.29 (apparently) still quantify over and refer to non-numbers, and use non- Second, being the best explanation is a holistic matter. It takes ac- mathematical predicates (like ‘x is massive’), they also quantify over count not only of explanatory power, but of other virtues, including fit and refer to numbers (like ‘the mass in kg of x’), and use numerical with pre-theoretical belief. It’s hard to think of another candidate ex- predicates (in order to to express the numerical relations between quan- planation for the effectiveness of mathematics that scores so low in this tities, like mass). The upshot: Given our assumption of the correctness regard. Let us move on to Tegmark’s slightly more promising proba- of current physical theories, mathematics effectively describes physical bilistic argument. reality. But what, if anything, explains the fact that mathematics is ef- His probabilistic argument is an argument for Tegmark’s own ver- fective in doing so? This question can be put more incredulously: How sion of Pythagoreanism, or ‘Tegmarkian Pythagoreanism’ for short. could mathematics be effective — and so very effective — in describing Tegmarkian Pythagoreanism says (at least) that (a) everything is a cate- physical reality if mathematical reality and physical reality (the mathe- gory (in the sense relevant to category theory), (b) there is some relation matical objects and the physical objects) are as “distant” as the abstract R expressible in category theory that is (or into which we can analyze) and concrete realms of which each takes a respective part? What could the “universe”-mate relation, i.e. the relation that x and y bear to one explain this mystery? Tegmark suggests that the best explanation is another iff they are parts of the same “universe”, and (c) there are cate- that they are not so distant at all. Indeed, they are one and the same: gories that do not stand in R to any part of our “universe”. Tegmarkian All physical objects just are mathematical objects. And from there it’s Pythagoreanism is a version of Pythagoreanism since it entails Pristine a but a small step to what I have called Ontological Pythagoreanism.27 Mathematical Ontological Pythagoreanism: Categories are mathemati- There are at least two problems with this argument. (Even setting cal objects, and everything is a category, so everything is a mathemat- aside problems with abductive arguments in general.) First, being the ical object. But then every truth is such that all its quantifiers range best explanation is a comparative matter. So one would have to consider only over mathematical objects and all its referring terms refer only to other proposals of which we are aware, and there are a number of mathematical objects. those.28 Closest to home, perhaps, would be a rival explanation that Now to the argument30: Tegmark notes that it is a live hypothesis agrees that mathematical reality and physical reality are one and the among contemporary physicists that we inhabit a Multiverse, that our same. But rather than say that all physical objects just are mathematical objects, it says that all mathematical objects just are physical objects. The rival explanation is consistent with Tegmark’s, but also consistent 29Butterfield[ 2014] focuses on the abductive argument I’ve just sketched, and he also replies by putting forward an alternative explanation. Butterfield’s alternative explanation is that the physical universe is an “applied mathemat- ical structure” rather than a “pure mathematical stucture”. We can put this in 27A step in which we assume that physical reality exhausts all of reality — that terms of the point I made in §1.5: The datum can be explained just as well by there is nothing else in addition to the physical things (no ghosts or spirits, the claim that physical reality is an instance of some mathematical structure. for example). If there is a complete theory at all, then it will be one which 30 quantifies over and refers to no non-mathematicals. I should say that this argument, if it’s Tegmark’s, was even more laconically expressed than the other two. Consequently, I’m not absolutely certain that 28See Steiner[ 1998] and Yablo[ 2005]. Tegmark subscribes to it.

philosophers’ imprint - 9 - vol. 19, no. 26 (june 2019) aaron segal Pythagoreanism: A Number of Theories universe is just one among many such universes. Let us grant the truth hypothesis does in fact confirm Pythagoreanism tout court: It should of the Multiverse hypothesis. But the Multiverse hypothesis is itself en- be evident from the division of Tegmarkian Pythagoreanism into three tailed by Tegmarkian Pythagoreanism. For according to Tegmarkian theses that it is really (a) that makes it Pythagorean while it is the Pythagoreanism, there are categories that are not “universe”-mates conjunction of (b) and (c) that is being confirmed by the Multiverse with anything in our universe. But presumably, every such category, hypothesis. And so it’s far from clear that the Multiverse hypothesis like the ones that are parts of our “universe”, is “universe”-mates with confirms Pythagoreanism. at least itself. So consider a maximal set of such categories that are mutual “universe”-mates: Any two members in the set are “universe”- 2.2 Quine’s Argument mates, and they are “universe”-mates with no category not in the set. As noted above, Quine[ 1976] came to embrace Ontological Pythagore- Then, there is a universe whose parts are all and only the members anism.32 The general contours of his reason will be unsurprising to of that set, and it is not our universe (since no member of the set anyone familiar with Quine: a taste for desert landscapes. But the spe- is “universe”-mates with us). Thus, since the prior probability of the cific shape of that reason is still of interest. On the one hand, he argues Multiverse hypothesis is less than one, Tegmarkian Pythagoreanism for the position that came to be known as ‘supersubstantivalism’, that raises the probability of the Multiverse hypothesis; and so the Multi- spacetime exists along with regions of spacetime, and further, that ev- verse hypothesis raises the probability of Tegmarkian Pythagoreanism. ery concrete object just is a region of spacetime.33 On the other hand, So the Multiverse hypothesis, whose truth we have granted, confirms he also famously and begrudgingly argues for platonism: There are ab- Tegmarkian Pythagoreanism. stracta, like numbers and pure sets.34 With supersubstantivalism and At last we have an argument that seems right, as far as it goes. But platonism in hand, he then wields Occam’s Razor: If the numbers are the problem is that it does not go very far. For one thing, it’s all well there anyway, and they or sets of them can do double duty as space- and good that a hypothesis whose truth we have granted confirms time regions, then we should let them. That way our theory is more Tegmarkian Pythagoreanism, but it still might be downright irrational parsimonious. And Quine claims they can do such double duty. Sim- of us to assign Tegmarkian Pythagoreanism any credence greater than ply identify the spacetime points with the quadruples of real numbers, 0.5, say. For the argument tells us nothing about what our prior proba- bility in Tegmarkian Pythagoreanism is nor about the degree to which 32I should note that according to Peter Hylton [2007, 307–10], even Quine did the Multiverse hypothesis confirms Tegmarkian Pythagoreanism. And not really mean to embrace Pythagoreanism, as much as stress that “what while I suspect that the latter is quite high, I suspect that the former is, particular objects fill the theoretical roles in our theory may be a matter of for most of us, extraordinarily low, and rationally so. Second, it doesn’t indifference”. (Thanks to Sam Cowling for this reference.) Be that as it may, Quine’s argument is an argument for Pythagoreanism. follow from his conclusion that the Multiverse hypothesis confirms 33Given that he is going to go on and endorse Pythagoreanism, according to Pythagoreanism tout court, or even the more specific Pristine Mathe- which these regions are just sets of sequences of numbers — and so it’s not matical Ontological Pythagoreanism.31 As to whether the Multiverse clear if they are concrete — it might be best to put the second part of the view as: “Every object we ordinarily take to be concrete just is a region of spacetime”. 31It would follow if the so-called Special Consequence Condition (SCC) held, 34For reasons similar to those adduced in nt. 33, it might best to put the view but SCC famously does not hold for the relation of probability-raising, the as: “There are things we ordinarily take to be abstract, like numbers and pure relation at work in the argument. sets.”

philosophers’ imprint - 10 - vol. 19, no. 26 (june 2019) aaron segal Pythagoreanism: A Number of Theories and regions in spacetime with sets of such quadruples. And since, ac- might reply that the argument trades on a confusion between two pred- cording to supersubstantivalism, every concrete object is just a region icates: ‘x is numerically-original’ and ‘x is spatiotemporally-original’. of spacetime, every concrete object turns out to be just a set of se- Quine might say that no quadruple of reals, and hence no spatiotem- quences of numbers. And from there it seems to be but a short step poral point, is the spatiotemporal origin, while some quadruple of re- to Ontological Pythagoreanism.35 Put simply: Assuming the truth of als, and hence some spatiotemporal point, is the numerical origin. In a supersubstantivalism and platonism, there is an argument from parsi- word, we save the identification in ontology by granting a distinction mony for Ontological Pythagoreanism. in ideology. In reply to this reply: While such a position is coherent, This argument is more promising than any we have seen so far. But it is hard to see how one could reasonably come to believe it on the there are still at least three problems with it. First, whatever arguments grounds of parsimony. Anyone who came to believe (Quinean) On- there are in favor of supersubstantivalism, there are serious, perhaps tological Pythagoreanism on Quine’s grounds should maintain, with insuperable, objections to it.36 Quine himself, that spatiotemporal structure just is numerical struc- Second, it’s not true that quadruples of real numbers can do dou- ture: It would be absurdly profligate for the quadruples of reals to ble duty as spacetime points. For there is less structure to the set of have both numerical structure and a distinct spatiotemporal structure. spacetime points than there is to the set of quadruples.37 There is, for But then there aren’t really two predicates here (except in a morpholog- instance, a quadruple that enjoys the privilege of being the origin, but ical sense): ‘x is spatiotemporally-original’ just means ‘x is numerically- no spacetime point enjoys such a privilege. So (by Leibniz’s Law) there original’. Quine could just say that there is in fact no ontological or is some quadruple of real numbers that isn’t a spacetime point, and the ideological distinction, and so there is a spatiotemporal origin after all. spacetime points are not identical with the quadruples of reals.38 One But that is a highly revisionary bullet. Third, and perhaps most importantly, it’s not true that from Quine’s conclusion that every concrete object is a set of quadruples of real num- 35Even if there are other non-concrete things besides numbers and sequences of numbers and sets of such sequences, presumably the whole truth about bers, it is but a short step to Ontological Pythagoreanism. For Quine how those are is a matter of necessity, and so at the very least it would seem nowhere denies that mass and charge and other such features (or quan- Metaphysical Mathematical Ontological Pythagoreanism is true. tities) are intrinsic. And if they are intrinsic, and their instantiation by a 36Schaffer[ 2009] catalogues no fewer than seven arguments in favor of super- given point/region is not entailed by the facts about that point/region substantivalism. Quine’s own reasons varied. In his [1976], his reason is that supersubstantivalism is the only way to make sense of the non-particle-like wholly expressible using mathematical predicates, then it turns out behavior that contemporary physics attributes to particles. In his [1981], his that quadruples of reals and sets of such quadruples are not mathe- reason is parsimony. For some of the serious objections to supersubstantival- matical objects after all. They don’t just lead a double life; they live a ism, see Hawthorne[ 2008]. divided life. 37See Field[ 1980, p. 32]: “No platonist would identify the real numbers with the points on any physical line: for one thing, it would be arbitrary which point on the line to identify with 0 and which with 1..." But rather than providing a real resolution, this would only demand a com- 38One might reply that Quine didn’t need to say that every spacetime point is plication in the formulation of the problem. We could point to the fact that a quadruple of reals and that every quadruple of reals is a spacetime point, every quadruple of reals has some property of the form, being such-and-such just that every spacetime point is a quadruple of reals. And that is of course distance from a privileged origin, while no spacetime point has such a prop- consistent with there being a quadruple — one that is privileged enough to erty. And that would suffice to show that no spacetime point is a quadruple be the origin — that is not one of the spacetime points. of reals, and a fortiori that not every spacetime point is a quadruple of reals.

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2.3 (What Could Have Been) Lewis’s Argument property: let’s take mass, as an example. We can offer the following Alas, Quine’s argument suffers from serious problems. But there is a argument.41 much more promising argument in the vicinity. Let us assume David 1. We know what mass is, and we (can) know when something Lewis’s thesis of Humean Supervenience: The distribution of all prop- changes its mass, and we (can) know when something keeps its erties and relations globally supervenes on the distribution of perfectly mass (Shoemaker[ 1980], Schaffer[ 2005]) natural local properties and perfectly natural geometrical relations — 2. If we know what mass is, and we (can) know when something where all perfectly natural properties and relations are freely recombin- changes its mass, and we (can) know when something keeps its able (subject, perhaps, to certain logical constraints). That implies that mass, then mass is essentially dispositional42 (Shoemaker[ 1980], there is a complete and closed theory all of whose predicates express Langton[ 1998], Lewis[ 2009]) either perfectly natural local properties (or their negations) or broadly geometrical relations.39 Therefore, If we are already assuming Humean Supervenience and it implies Mass is essentially dispositional that, then it seems to me that there is an interesting case to be made that we ought to accept something stronger, viz. that there is a complete But, and closed theory all of whose predicates express broadly geometric 3. If mass is essentially dispositional, then it is not freely recombin- relations, period. Or, in other words, that there is a complete and closed able, and so it is not perfectly natural (Schaffer[ 2008], Lewis[ 2009]) theory all of whose predicates are broadly geometrical. And that of course implies Ideological Pythagoreanism. Therefore, Here is the case.40 Consider a candidate for a perfectly natural local Mass is not perfectly natural

And then we can offer analogous arguments for analogous conclusions 39The theory will specify for every object which perfectly natural properties it has and which it doesn’t; and it will specify which geometrical relations those regarding charge, spin, and all the other candidate perfectly natural objects stand in; and it will specify that there isn’t anything other than those objects. (If there is an infinite number of objects, then saying that there isn’t anything else will require some ingenuity and a substantive commitment — for instance, the theory could say that everything is at some specified dis- 41After writing this, I discovered that elements of this argument can be found tance from a chosen “origin” and that no two things are co-located — or in Esfeld[ 2014] and Esfeld and Deckert[ 2017]. They argue for a similar con- an infinitary language. On this, and the relationship between supervenience clusion, which they term ‘Super-Humeanism’, regarding mass and other pu- and entailment more generally, see Sider[ 1999, §1] and Bricker[ 2006].) Given tatively perfectly natural local properties. However, they employ these ele- Humean Supervenience, such a theory will be complete; and it’s easy to make ments only as part of a larger argument that relies on the thought that, ac- it closed. cording to Humeanism, such candidate perfectly natural local properties are explanatorily idle. I am much less confident than they are that it follows from 40The case is an epistemological argument that relies essentially on an anti- Humeanism that such properties are explanatorily idle, and it seems to me skeptical premise, i.e. the claim that we know what mass is and when it comes th that the anti-skeptical argument can stand on its own. In any case, they don’t and goes. It is noteworthy, perhaps, that Philolaus’ (5 c. BCE) argument discuss the implications of Super-Humeanism for Pythagoreanism. for Pythagoreanism is also epistemological and relies essentially on an anti- skeptical premise, viz. that every truth is (in principle) knowable. He then 42By ‘mass is essentially dispositional’, I mean that for every mass value there goes on to claim that “all things which are known have number; for nothing is some disposition or cluster of dispositions such that necessarily anything can be known or understood without number” (fr. 4, cited in Kahn[ 2001]). with that mass has that disposition or cluster of dispositions.

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(non-geometrical) local properties. The premises of the analogous argu- logical or definitional consequence of that sentence).44 ments will be just as plausible as the premises of the “mass argument". Now, none of Shoemaker, Lewis, Langton, and Schaffer accepts all From the arguments taken together, we can conclude that nothing in- three premises of the above argument. Shoemaker accepts the first two, stantiates any perfectly natural (non-geometrical) local property.43 But and so denies the third: He thinks that perfectly natural, or fundamen- then, if in addition there is a complete and closed theory T all of tal, properties might fail to be recombinable. Schaffer accepts the first whose predicates express either perfectly natural local properties (or and the third, but denies the second: He thinks we can know what their negations) or broadly geometrical relations — as Humean Super- mass is, when something changes its mass, and when something keeps venience implies — then there is a complete and closed theory T ’ all its mass, even if mass is not essentially dispositional — and the same of whose predicates express either the negations of perfectly natural lo- goes for other perfectly natural properties — much the same way that cal properties or broadly geometrical relations. But if there is a theory we can know there is an external world even if there is no absolutely like that, then there is a complete and closed theory T ” all of whose necessary connection between our sensory evidence and the external predicates express broadly geometrical relations, and nothing else: Let goings-on. And Lewis and Langton accept the second and third,45 and T ” be just like T ’, except that in place of all the sentences that con- so deny the first: They think we don’t really know what mass is — and tain predicates for the negations of perfectly natural local properties, the same goes for a host of other perfectly natural properties. But that it contains the sentence “Everything is a geometrical object” (and any doesn’t mean the argument is no good. Each premise is independently plausible; there is no contradiction involved in accepting all of them; and it’s far from obvious that the denial of our argument’s conclusion 43Note well: This implies that the property, being matter-filled, is either unin- stantiated or is not a perfectly natural (non-geometrical) local property. But is more compelling than their conjunction. it is presumably a perfectly natural (non-geometrical) local property, so the Due to space constraints, I can hardly dwell on the plausibility of conclusion of the argument implies that that property is uninstantiated. Thus, the conclusion of the argument is inconsistent with substantivalism (let alone each premise or on the alleged implausibility of the argument’s conclu- supersubstantivalism), or at least any empirically adequate version thereof. Thus, I want to make clear that the version of Ideological Pythagoreanism for which I think there is a Humean argument is not one according to which 44That sentence is true if T ’ is. If T ’ is true, then the only perfectly natural fea- there is a complete and closed theory that just specifies the geometry of space- tures anything instantiates are geometrical. Given the definitions of ‘intrinsic’ time, since on the view under consideration there is no spacetime. Rather, it in nt. 9 and ‘geometrical object’ in §1.2, everything then qualifies as a geo- is one according to which there is a complete and closed theory that just metrical object (at least given a liberal enough construal of ‘sufficiency’). And specifies the geometrical relations (topological and metrical) between pieces that sentence in turn implies every true sentence in T ’ that says of an object of matter. Thanks to a referee for this journal for helping me see the need for that it doesn’t have some perfectly natural (non-geometrical) local property, this clarification. since if it did have some such property, it wouldn’t be a geometrical object. Significantly, there is no analogous (and equally compelling) argument for an (Here I assume, contra Weatherson[ 2006], that every perfectly natural prop- analogous conclusion regarding geometrical relations — or the geometrical erty is intrinsic — an assumption that follows from the definition of ‘intrinsic’ local properties, being point-sized and being topologically pointy — since in nt. 9 — and that necessarily for anything that has a non-geometrical per- the analogue of premise 2 has much less to recommend it: For a given geo- fectly natural property, there is no closed theory all of whose predicates and metrical relation or property, we know exactly what it is and what its essence functors are geometrical that is a sufficient base from which to deduce that is by purely a priori reflection. It need not be dispositional for its quiddity the object has that perfectly natural property.) to be transparent to us. (I should say that this might depend on whether So T ” is a complete and closed theory if T ’ is. our justification for premise 2 is Lewis’s Ramseyan one or Langton’s Kantian one. On the contrast see Langton[ 2004]. Making an exception of geometrical 45Or at the very least, Langton accepts an analogue of the third premise in features might not be feasible according to the Kantian justification.) which ‘intrinsic’ replaces ‘perfectly natural’. See Langton and Lewis[ 1998].

philosophers’ imprint - 13 - vol. 19, no. 26 (june 2019) aaron segal Pythagoreanism: A Number of Theories sion. But let me say the following. First, as to the argument’s premises: understood, that’s not to say that those masses and charges are per- One would have to be a pretty serious skeptic to think that we don’t fectly natural; no, they are highly extrinsic but extremely useful ways know what mass is; and something close to the second premise is to summarize all the geometric facts. No more and no less. Thus, any widely endorsed, not only by Lewis and Langton, and not only by two worlds that agree on there being that number of pieces of matter Shoemaker and many of his dispositional essentialist allies, but also that stand at exactly those distances from one another and instantiate by the neutral monist likes of Russell[ 1927], Chalmers[ 2015], and oth- no perfectly natural local properties will also agree on everything else, ers;46 and the third premise is simply part and parcel of the Humean and in particular on the distribution of mass, charge, and so on. Supervenience package that we are assuming. Of course, much more can be said to develop this account.47 But Second, one can argue for the interim conclusion — that mass is hopefully enough has been said so that the argument is not most char- essentially dispositional — on other grounds. itably interpreted as a reductio of one of the premises.48 As to the alleged implausibility of the conclusion: There are several Finally, and as a natural followup, if the conclusion is thought to possible reasons it might be thought implausible. If it is thought to be implausible because it implies that the properties that figure into be implausible because it implies the allegedly implausible Ideological the laws are not perfectly natural, it’s worth noting the following. As- Pythagoreanism, I will address the alleged implausibility of the latter suming, as we are, Lewis’s thesis of Humean Supervenience, no law in section §3. is fundamentally a law, and some or all laws might not be fundamen- If the conclusion is thought to be implausible because it has been tal at all. But then there would seem to be no obstacle in principle to accompanied by no positive sketch of how such a slender theory might non-fundamental (or, in Lewisian ideology, less-than-perfectly-natural) be complete, let me very briefly sketch one natural way — though by properties figuring into them. It is true, as I have just said, that on no means the only way — that this might go according to an adherent Lewis’s own view, lawhood is a matter of striking the best balance be- of Humean Supervenience. At bottom, there are just a huge number of tween simplicity, strength, and fit. And it is true that on Lewis’s own pieces of matter at various distances from one another. As in Lewis’s view, simplicity is a matter of brevity when expressed in a language best-system analysis of lawhood, the laws are just the best systematiza- whose predicates express perfectly natural properties and relations. tions of these facts about where things are relative to one another; that But even the conjunction of both of those views — neither of which is, they are axioms of the system that strikes the best balance between is strictly entailed by Humean Supervenience — fails to entail that no simplicity, strength, and fit with those geometric facts. As it turns out, less-than-perfectly-natural properties can figure into the laws. It could the best way to systematize these facts is by assigning masses and be that whatever our laws sacrifice in simplicity, in virtue of “mention- charges (and the like) to the pieces of matter. (This much seems clear ing” properties like mass and charge, they more than make up for in enough from the practice of actual physicists.) But lest that be mis- strength.49

47 46They accept the following (or something that implies it): If we know what Indeed, for a fuller development of an account along these lines, see Esfeld 2017 mass is, then mass is either essentially dispositional or (proto-)phenomenal. and Deckert[ ]. So if we wish to cast a wider net in our second premise, we can use that and 48Thanks to a referee for this journal for raising this issue. then add the widely accepted anti-pan(proto)psychist premise that mass is not (proto-)phenomenal. 49Indeed, Hicks and Schaffer[ 2017] argue that less-than-perfectly-natural (or

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So it seems to me that barring any serious objection to Ideological is a priori, there are truths about the structure of physical space that Pythagoreanism, the view deserves to be taken seriously. aren’t a priori. Thus, there are truths about the structure of physical space that aren’t members of any mathematical theory. And thus there 3. Objections are truths, period, that aren’t members of any mathematical theory. So But aren’t there serious, even devastating, objections to any ver- Pristine Theoretical Pythagoreanism is false. sion of Pythagoreanism, including Ideological Pythagoreanism? Well, Here’s one reply. According to Pristine Theoretical Pythagoreanism, Pythagoreanism certainly sounds crazy, and many of us would stare every truth is a member of some mathematical theory. But there’s more incredulously at a flesh-and-blood Pythagorean. But it’s harder to say to know than just all the truths. Even once you know all the truths, you just what’s wrong with Pythagoreanism. still might want to know, among other things, which things we are (i.e. which mathematical objects we are) and the nature of the space which 3.1 A Posteriori we inhabit (i.e. in which mathematical space we are to be found). But Hartry Field offers what might be taken to be an argument against ‘physical space’ just means (at least as it’s used by Field in the passage Pythagoreanism: above) the space which we inhabit. (There might be other spaces, dis- connected spatiotemporally from ours, that are physical; each of these But however this may be, space-time points are not abstract enti- would be a physical space, but not the physical space of which Field tites in any normal sense. After all, from a typical platonist per- speaks.) Thus, the Pristine Theoretical Pythagorean can say as follows: spective, our knowledge of mathematical structures of abstract There aren’t, strictly speaking, any truths about the structure of physi- entities (e.g. the mathematical structure of real numbers) is a pri- cal space that aren’t a priori. Every truth is a mathematical truth and ori; but the structure of physical space is an empirical matter. all of those mathematical truths can be known a priori. But there is That is, most platonists who believe current physical theory be- what to know about the structure of physical space that is not a priori. lieve that it is a priori true that there are real numbers obeying the For knowing which mathematical objects we are and in which math- usual laws, and that it is a high-level empirical hypothesis (not ematical space we are to be found is something to know, and not the easily disconfirmed, but subject to revision by the development sort of thing we can know a priori. of an alternative physical theory) that there are lines in space This reply is promising. But it relies on an assumption that, while which (locally anyway) are isomorphic to the real numbers.50 orthodox, is contentious, viz. that there is “irreducibly de se knowl- We can put the argument this way (setting aside the particular pu- edge” or “essentially indexical attitudes". That assumption has come tative identification with sequences of reals): If Pristine Theoretical under attack recently.51 For those who don’t accept that assumption — Pythagoreanism is true, then every truth is a member of some math- and even for those who do — here’s another reply: Field’s epistemolog- ematical theory. But while our knowledge of mathematical theories ical argument establishes nothing more than the falsity of Pristine The- oretical Pythagoreanism. We have already noted that Pristine Theoreti- cal Pythagoreanism — along with Pristine Ideological Pythagoreanism ‘non-fundamental’) properties do figure into the laws, and, moreover, that this supports a Humean view of laws. 50Field[ 1980, p. 32] 51See, inter alia, Cappelen and Dever[ 2013].

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— has radically revisionary ontological and semantic implications. It is but go on to offer a substitute for contingency.53 Here, we can take no wonder that it also has radically revisionary epistemological impli- our cue from the reply we first suggested to the previous argument. cations. If the former weren’t reason enough to set the view aside, as Compare: According to Lewis’s Modal Realism, the (non-perspectival) we have indeed done, the latter might well be. Let us now set it aside truths about the pluriverse, which are all necessary, suffice to settle even more resolutely. everything. Well, not everything, for they don’t settle which world in the pluriverse is actual (nor anything that follows from the correct an- 3.2 Contingency swer to that question). But according to Modal Realism, when I say One might object to Theoretical Pythagoreanism, whether Pristine “world W is actual”, I mean no more and no less than the indexical or not, on the grounds that it eliminates contingency. If Theoretical statement “world W is the world of which I am a part". Thus, there Pythagoreanism is true, then Necessitarianism is true, i.e. every truth is room for contingency, but all the contingency there is room for is is a necessary truth. For the truths of mathematics are themselves nec- what we might call “de se contingency". So we can say the same for essary. And so any truth that can be logically deduced from the truths the Theoretical Pythagorean; indeed she can just piggyback on Modal of mathematics together with other necessary truths is itself necessary. Realism. There are a host of mathematical structures, some of them — According to Theoretical Pythagoreanism, every truth can be logically the ones that are maximal and integrated, according to some way of deduced from the truths of mathematics together with other necessary making that precise — are possible worlds, and every possible thing truths. So if Theoretical Pythagoreanism is true, Necessitarianism is is part of some such structure. The (non-perspectival) truths about the true. But, goes the objection, the world could have been different in so collection of these structures, which are all necessary, suffice to settle many ways. So Theoretical Pythagoreanism is false. nearly everything. What they do not settle is which of these mathemat- One reply is to just accept Necessitarianism and say nothing fur- ical structures is actual, i.e. which mathematical structure we are part ther. But that is costly.52 A second reply is to accept Necessitarianism, of. Thus, there is room for contingency, but all the contingency there is room for is de se contingency. Now, if this reply is going to be consistent with Theoretical 52Just how costly depends on how obvious it is that there is contingency. I will note that compelling and non-question-begging arguments against Ne- Pythagoreanism, it had better be the case that this sort of contingency cessitarianism are in fact hard to come by. Conceivability arguments for the isn’t contingency in what’s true. Nothing in this reply, after all, casts possibility of a certain (non-actual) state of affairs (see, inter alia, Yablo[ 1993] and Chalmers[ 2002]) rely either on a conceivability-possibility link that is any doubt on the premises of the objection; mathematics is still nec- far too generous or on an assumption that the state of affairs is “strongly” essary, and what follows of necessity from what’s necessary is still or “deeply” conceivable, the latter of which a Necessitarian ought not grant her opponent regarding any non-actual state of affairs. Intuition-based argu- ments for the possibility of a certain (non-actual) state of affairs (see Bealer [2002]) rely on an “intution of possibility,” which may be some evidence or alleged routes to modal knowledge. See Strohminger and Yli-Vakkuri[ 2017]. reason for believing that state of affairs to be possible, but it is evidence or a And in any case, I am not myself advocating Necessitarianism or ultimately reason that can be easily outweighed by arguments on the other side, such relying on it in my defense of the plausibility of Pythagoreanism. My defense as those of the Theoretical Pythagorean. Perception-based arguments for the is as I say in the text: The versions of Pythagoreanism for which we have good possibility of a certain (non-actual) state of affairs (see Strohminger[ 2015]) arguments are untouched by the objection from contingency. Thanks here to rely on an extremely controversial claim about our perceptual capacities. a referee for this journal. But this extraordinarly brief discussion does not do justice to the intricate 53I am deeply indebted here to an ingenious suggestion from a referee for this and sophisticated literature on the epistemology of modality, including other journal.

philosophers’ imprint - 16 - vol. 19, no. 26 (june 2019) aaron segal Pythagoreanism: A Number of Theories necessary, and according to Theoretical Pythagoreanism, every truth to be perfectly natural after all — surely there could have been some follows from the truths of mathematics. It follows ineluctably that ac- perfectly natural local properties, alien to the actual world, that were cording to Theoretical Pythagoreanism, every truth is necessary. The instantiated. What would prevent such a thing? reply under consideration works if and only if it offers a substitute Indeed, nothing prevents such a thing. What the objector says for garden-variety contingency. But as we noted before, the assump- is right. But what she says is entirely consistent with Ideological tion that the perspectival runs deep — and now we might add, deep Pythagoreanism and so is no objection. The first premise of our ar- enough to provide this substitute for contingency — is contentious. gument — the anti-skeptical premise — is contingent. We could have And on top of that, Lewis of course doesn’t think the contingency for failed to know what mass is. More importantly, the fact that every in- which Modal Realism allows is a mere substitute for garden-variety stantiated property is such that if it is a perfectly natural local prop- contingency. As far as he’s concerned, that is contingency, contingency erty, then we know what it is and when it comes and goes is itself even with respect to what’s true. contingent. Things could have been bad epistemically — some categori- This brings us to the third reply, which is to concede the case. The cal and unknowable perfectly natural local property could have been objection, even if succcessful, establishes nothing more than the falsity instantiated. All the argument assumes is that things aren’t in fact bad of Theoretical Pythagoreanism. For Ideological Pythagoreanism, and epistemically. a fortiori Ontological Pythagoreanism, is consistent with there being The objector might press further: If that is indeed a contingent mat- truths that don’t follow of necessity from the truths of mathematics; ter, what justifies us in believing it? That’s a fair question, but it is not all that’s needed is that every truth follows of necessity from some the Pythagorean’s alone to answer. Anyone who thinks that physics truths that are expressible in the language of mathematics. Some of does give us, or is at least capable of giving us, the complete story those truths might well be contingent. Perhaps they’re de se truths about physical reality will need to confront the same question. about which mathematical structures we are part of. Perhaps not. In any case, the objection from contingency leaves Ideological and Onto- 4. Conclusion logical Pythagoreanism unscathed. And the argument we offered was Pythagoreanism is not a single theory, but a number of them. The argu- in any case for Ideological Pythagoreanism. ments for Pythagoreanism that we have considered don’t conflict with the compelling objections to Pythagoreanism of which I am aware: The 3.3 Contingency, Second Pass arguments are for versions to which the compelling objections don’t ap- Falling back on Ideological Pythagoreanism has other modal advan- ply. Of course, there might be other compelling objections of which I am tages. Ideological Pythagoreanism is not only consistent with there be- not aware, which do apply more broadly. In the meantime, however, it ing contingency but is itself — and presumably unlike its Theoretical seems to me that Pythagoreanism ought to be a serious metaphysical cousin — a contingent thesis. It is possibly true and possibly false. That should hopefully dispel a confusion at the heart of the next and final objection. One might object that even if no perfectly natural local property is in fact instantiated — if all the plausible candidates turn out not

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