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Pythagoreanism Is the Very Surprising View That “All Is Number”

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Pythagoreanism Is the Very Surprising View That “All Is Number” 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 <www.philosophersimprint.org/019026/> 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]. philosophers’ imprint - 2 - vol. 19, no. 26 (june 2019) aaron segal Pythagoreanism: A Number of Theories 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.
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