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Philolaus & Infinity: A Response to the Infinite Regress and Intelligibility Issues in

Philolaus’ Fragments

Nathan D. Ward

PHIL 4030H

Professor: Jessica Berry

Georgia State University

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Philolaus was a prominent Pythagorean philosopher who lived from c. 470 to ca.

385 BC. He was probably the first (and perhaps only) Pythagorean to write a book, titled

On Nature (Huffman, p. 8, 12).1 Many existing fragments and testimonia have been

attributed to Philolaus. However, spurious fragments and testimonia plague the

Pythagorean tradition for several interrelated reasons, foremost of which is the fact that

Pythagoras wrote nothing of his own (Huffman, SEP).2 As a result, a monetary incentive

existed in ancient times to produce “genuine” Pythagorean fragments. Additionally, later

Neo-Platonic and Aristotelian thinkers produced spurious works to try and tie their own

philosophical projects back into Pythagorean doctrines, further muddying the waters with

unoriginal fragments and testimonia (Huffman, SEP). Hence, we must be cautious about

treating any fragment assigned to Philolaus (or any Pythagorean thinker for that matter)

as genuine without further examination.i However, recent scholarly work has

demonstrated that eleven fragments can be reliably attributed to Philolaus.3 Fifteen other

fragments and six testimonia are now believed to have come from later Neo-platonic and

Aristotelian sources and are not genuine.

1 Philolaus, and Carl A. Huffman. Philolaus of Croton: Pythagorean and Presocratic: A

Commentary on the Fragments and Testimonia with Interpretive Essays. Cambridge: Cambridge UP, 1993. Print.

2 Huffman, Carl, "Philolaus", The Stanford Encyclopedia of Philosophy (Summer 2012

Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/sum2012/entries/philolaus/>.

3 (DK) Frs. 1–6, 6a, 7, 13, 16 and 17, Huffman argues that these fragments alone derive from Philolaus' book On Nature (Huffman, SEP). 3 There are 15 fragments that are considered spurious (8, 8a, 9–12, 14, 15, 19, 20a, 20b, 20c, 21– 23) and six testimonia

(11–13, 16b, 17b, 30) that are based on spurious works (Huffman, p. 341–420), (Huffman, SEP).

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I will stick closely to the fragments and testimonia utilized by Huffman. I will

provide a brief discussion of Philolaus’ three archai: limiters, unlimiteds, and harmonia.

I will then examine some apparent epistemological problems in Philolaus’ work

highlighted by McKirahan and sketch my own possible solutions to those problems.

Philolaus’ work is quite abstract; Parmenides and Melissus are perhaps the only other

Pre-Socratic philosophers to match him in this regard (McKirahan, p. 355).4 Huffman

highlights that Philolaus’ method of inquiry proceeds by positing as few explanatory

principles (archai) as possible (Huffman, p. 78–92). Fragment (18.6-DK6-§1-6) refers to

the kosmos as a whole and Philolaus argues for the necessary existence of three archai

within the kosmos: limiters, unlimiteds, and harmonia. Limiters and unlimiteds are the

two basic types of “stuff” that comprise the kosmos according to Philolaus. Huffman

thinks that the unlimiteds are best defined as “continua undefined by any structure or

quantity,” and that limiters, unsurprisingly, “set limits in such unlimiteds and include

shape and other structural principles” (Huffman, SEP). Huffman highlights that the

limiters and unlimiteds “join” together according to a “harmonia” (Huffman, SEP).

Harmonia is not random and can be described, in principle, in mathematical terms.

Philolaus' sole example of a harmonia arising from unlimiteds and limiters is a diatonic

musical scale, where sounds are limited according to certain ratios between whole

numbers (Huffman, SEP). Philolaus thought that the entire kosmos was structured

according to harmonia between numbers and that the possibility of human knowledge

4 McKirahan, Richard D. "Philolaus of Croton." Philosophy before Socrates: An

Introduction with Texts and Commentary. 2nd ed. Indianapolis: Hackett Pub. 2010. 352-

64. Print.

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was closely tied to grasping these numbers and their harmonious relations (Huffman,

SEP). I turn now to a discussion of Philolaus’ unlimiteds and limiters.

Philolaus argues disjunctively in fragment (18.2-DK2) that the kosmos must be

either all unlimiteds or all limiters or it must be comprised of both of these (McKirahan,

p. 352). Philolaus discounts the possibility of everything being comprised of unlimiteds

because it is clearly the case that our world contains at least some limiters, some

unlimiteds, and others that are composed of both unlimiteds and limiters (McKirahan, p.

352). In a bold skeptical sweep, Philolaus also argues in fragment (18.2-DK6-§2) that

“the being of things” is beyond the realm of human understanding; we cannot know

things as they are in and of themselves (McKirahan, p. 352). Huffman says, “We can

only go so far as to say that they must have included limiters and unlimiteds in order for

the world we see around us to have arisen. Beyond this description the basic principles of

reality admit only of divine and not of human knowledge” (Huffman, SEP). This raises

an interesting epistemological issue as well, but I will bracket that issue for now.

Philolaus thinks we can still apprehend limiters and unlimiteds because the kosmos is

composed of limiters and unlimiteds (Huffman, SEP). I will now turn to a discussion of

how exactly Philolaus thinks limiters and unlimiteds remain intelligible.

McKirahan provides a succinct interpretive reconstruction of Philolaus’

arguments concerning limiters and unlimiteds. McKirahan takes (18.2-DK2-§1) to mean

that all possible entities are “either a limiter or an unlimited or an unlimited limiter”

(McKirahan, p. 355). I agree with him here, except that I find it more accurate to refer to

“unlimited limiters,” which McKirahan defines as “something joined together from one

or more unlimiteds and one or more limiters,” as composite entities joined via harmonia.

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McKirahan takes (18.2-DK2-§2) to imply that entities of all three kinds exist and he

argues that this is verified by (18.2-DK2-§6). McKirahan argues that (18.2-DK2-§2) “but

not in all cases only unlimited,” follows immediately from (18.3-DK3), “There will not

be anything that is going to know at all, if all things are unlimited” (McKirahan, p. 355).

In essence, fragment (18.3-DK3) assumes as an a priori fact that entities capable of

knowing exist and that if everything were unlimited those entities would not exist.

Philolaus apparently regards knowing as a form of limitation that would be

impossible if the kosmos was entirely comprised of unlimiteds (Huffman, p. 119–120).

Hence, it cannot be the case that every existing thing is of an unlimited sort, and (18.2-

DK2-§2) supports this (McKirahan, p. 355). (18.2-DK2-§3) begins another argument, the

conclusion of which is (18.2-DK2-§4), which follows immediately from (18.2-DK2-§3),

which follows from (18.2-DK2-§6) the purpose of which is stated in (18.2-DK2§5)

(McKirahan, p. 355). The conclusion that results is that everything within the kosmos is

ultimately “joined together from unlimiteds and limiters” (McKirahan, p. 355). This

conclusion is supported by evidence of how things behave in the kosmos, “things derived

from limiters limit, things derived from both limiters and unlimiteds both limit and do not

limit, and things derived from unlimiteds are unlimited” (McKirahan, p. 355). How this is

supposed to support Philolaus’ argument is not immediately clear, but McKirahan makes

a helpful distinction between “things” and what those things are derived from, which he

calls “products” and “principles” respectively (McKirahan, p. 355). Philolaus holds in

(18.2-DK2-§5) that the principles underlying a given product can be inferred based upon

the “behavior” of that product (McKirahan, p. 355). (18.2-DK2-§6) clearly posits that

there are products that limit, are unlimited, and that both limit and do not limit, hence it

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follows in (18.2-DK2-§3) that corresponding principles exist and that it cannot be the

case that those underlying principles are all either a limiter or unlimited (18.2-DK2-§4)

(McKirahan, p. 355). Unlimiteds and limiters are said to join together, but how this

occurs is not yet clear. I will no turn to a discussion of Philolaus’ third key concept,

harmonia.

Fragment (18.6-DK6-§5-6) explicitly introduces the concept of harmonia into

Philolaus’ methodological framework. McKirahan says, “The basic idea is clear enough:

limiters and unlimiteds do not form products simply by being thrown together. They must

be joined together in a way appropriate to form the product in question” (McKirahan, p.

357). Unfortunately, how harmonia is supposed to work is less than clear. McKirahan

notes “harmonia is not a force that binds limiters and unlimiteds together,” but that it

“comes upon” limiters and unlimiteds when they are in a certain arrangement

(McKirahan, p. 357). How harmonia “comes upon” limiters and unlimiteds is unclear,

McKirahan claims that limiters and unlimiteds “do not lose their identity when they form

products.” Rather, “the harmonia is a harmonia of the still existing unlimiteds and

limiters” and that “the harmonia supervenes upon them” (McKirahan, p. 357). I think that

this interpretation is correct, harmonia appears to be something that supervenes upon

limiters and unlimiteds, but it does so only after two objects are joined together in the

appropriate way. However, we are given no criterion to determine the “appropriate” way

for unlimiteds and limiters to join together.

Philolaus clearly thought that harmonia was necessary in the kosmos and this

claim is supported by (18.6-DK6-§6). I also think that Philolaus implicitly utilizes the

Principle of Sufficient Reason in (18.2-DK2) and in (18.6-DK6). Hence, it seems

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unlikely that he would not have also utilized it, or at least recognized the need for its use,

regarding his concept of harmonia. There must be a reason why harmonia is needed and

Philolaus explains at least that much in (18.6-DK6-§6) when he says that “similar”

products do not need harmonia whereas “dissimilar” products do need harmonia

(McKirahan, p. 353). However, my contention is that Philolaus does not provide

sufficient reasons for how harmonia is supposed to work. I think that harmonia might

work like this, when a limiter and an unlimited come together in a way that is appropriate

for the relevant products, and a properly equipped knower (such as a human being) is

present to perceive those two products together, that a harmonia results. Hence, I think

that harmonia is a sort of aesthetic epiphenomenon that exists only relative to a properly

equipped knower i.e. a human being. In other words, harmonia is not something that

exists on its own, because if it were we could not know of it and it would not exist (18.6-

DK2-§2). Hence, it must be something that arises only when humans are around to

perceive it as a harmonia. Time is an excellent example. Time is an unlimited product

that is limited by human beings (I am assuming here that limiters do not need to be

limited to number alone, I think that any abstract constraint counts as a limiter). Once we

place abstract limiters on time, such as hours, minutes, seconds, etc., we perceive it as an

ordered harmonia, rather than an unlimited. In other words, human perception recognizes

the order it has imposed upon an unlimited by limiting it and deems it aesthetically

pleasing. Hence, harmonia does not “come upon” products as an external, independently

existing entity, it comes upon them only relative to humans and it cannot exist otherwise.

I will now turn to a discussion of Philolaus’ epistemology and the role number plays

within his concept of harmonia.

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Philolaus was clearly interested in ratios, his sole example of harmonia, “the

harmonia,” in fact, involves musical ratios (18.7-DK6a) (McKirahan, p. 353). Limiters

and unlimiteds combine according to certain ratios of numbers, just as everything else in

the kosmos does; nothing can arise solely due to the chance combinations of unlimiteds

and limiters (although it still seems as if chance must play some role, but I digress)

(Huffman, SEP). Rather, the limiters and unlimiteds form a harmonia in a way that is

(somehow) in accordance with number. Hence, it appears that Philolaus was not only

interested in numbers, but in the sorts of relationships between numbers that result in

ordered harmoniai. The question, at this point, is an epistemological one. What is the

relationship between numbers and the three archai (unlimiteds, limiters, and harmonia)

posited by Philolaus?

It seems that Philolaus introduces numbers to respond to epistemological issues,

he thought that nothing could be known or comprehended without number (18.4-DK4 &

18.5-DK5); otherwise he only speaks in terms of his concepts of limiters and unlimiteds.5

Clearly, number plays an important role in Philolaus’ kosmos because harmonia is a

“fitting together of limiters and unlimiteds in accordance with number” (Huffman, SEP).

But there is nothing in the genuine fragments that would appear to indicate that numbers

are literally what comprise the things in the kosmos (Huffman, SEP). The issue appears to

lie with how “have number” is best interpreted in (18.4-DK4). Kirk, Raven & Schofield

appear to think that number is a necessary condition of perception, if X cannot be counted

5 Huffman posits that Philolaus might have been responding to Parmenides' metaphysical constraints. He says, “Mathematical relationships certainly fit a number of the

characteristics of a proper object of thought as set out by Parmenides in Fr. 8, e.g., they are ungenerated, imperishable, and unchangeable” (Huffman, p. 67–68).

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then X cannot be distinguished from any other X, hence in order for things to have their

own identity they must be countable, otherwise they are indistinguishable and we end up

with an impossible kosmos comprised entirely of unlimiteds (Kirk, Raven & Schofield, p.

327).6 McKirahan appears to endorse a similar position; he says, “According to (18.4-

DK4), a necessary condition for anything to be known is that it ‘have number.’ Thus, we

have two criteria for intelligibility: in order to be intelligible a thing must not be purely

unlimited (18.3-DK3) and it must have number (18.4-DK4)” (McKirahan, p. 361). I think

that that number is best thought of as a limiter for Philolaus. But whether or not he

restricted the scope of number to the whole number integers or included relations among

numbers such as ratio is unclear. Hence, the epistemological question of intelligibility

remains because we do not know quite how products must “have number” (McKirahan,

p. 361). Moreover, McKirahan thinks that the problem of an infinite regress is also

present (18.2-DK6-§2) because “the being of things” is unknowable to us, despite our

ability to analyze it into constituent unlimiteds and limiters. I will now discuss the

particulars of the infinite regress and intelligibility issues and provide my own tentative

solutions

McKirahan thinks it is possible that Philolaus “simply identified limits with

numbers, but it is not clear that all possible limits have anything to do with number or

that all limiters operate in a way that can be reasonably described as imparting number to

their products” (McKirahan, p. 361). I think that Philolaus probably included numbers as

well as their relationships. McKirahan appears to notice this as well, he says, “This calls

6 Kirk, G. S., Raven, J. E., and Schofield, M., 1983, The Presocratic Philosophers, 2nd

ed., Cambridge: Cambridge University Press.

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for a more general notion of number than simply the positive integers. (Conceivably, the

reference to the “many forms” of even and odd is an indication that Philolaus attempted

to make a suitable generalization or at least saw the need to do so)” (McKirahan, p. 362).

I think that the intelligibility problem (how objects must “have number”) dissolves if

Philolaus’ scope is expanded to include abstract relationships between numbers, such as

ratio, in addition to the whole number integers.

McKirahan says, “Philolaus holds that a thing can be analyzed into limiters and

unlimiteds that can themselves be analyzed into other limiters and unlimiteds. He refers

to the limiters and unlimiteds that constitute a thing as its principles (archai), (18.6-

DK6§5) The ‘being’ of things, then, cannot be known, because knowing it involves

knowing the limiters and unlimiteds of which it and its constituent limiters and

unlimiteds are composed, and there is no telling how far back such an analysis goes”

(McKirahan, p. 356 emphasis my own). McKirahan thinks this analysis ends if and when

we reach “limiters and unlimiteds that are ultimate in that they cannot be analyzed

further” (McKirahan, p. 356-357). McKirahan reasons that such an “ultimate” unlimited

would be entirely indeterminate, lacking all shape, size, and structure, and that it “would

not even be any kind of thing” (McKirahan, p. 356-357).

Clearly, McKirahan realizes the potential for an infinite regress here. However, I

do not think this is problematic for Philolaus. The issue appears to be that there is no

criterion for when to stop the analysis of unlimiteds and limiters, but I do not think

Philolaus needs any such criterion. I think that the “ultimate unlimited” that McKirahan

has in mind is best thought of as a sort of number, albeit an abstract one; I think it would

have to be either zero or infinity. Zero is more akin to a placeholder than a number and

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infinity is essentially a description of a recursive feature of enumeration. At the risk of

being blatantly anachronistic, I think that infinity fits McKirahan’s bill here nicely.

Except that rather than being potentially “operated” upon by any and all possible limiters

to produce any and all potential products, the reverse is actually the case; the ultimate

unlimited i.e. infinity, is actually the necessary precondition for the production (meaning

the possibility of infinite enumeration) of all possible limiters. I think that this would help

make some sense of fragment (18.10-DK16) that “some logoi are too strong for us,” this

fragment could be taken to refer to both the limits of human knowledge and to a

realization that some numerical relationships are perhaps too complex for humans to

understand.7

In conclusion, I think that number is best thought of as a composite harmonic

entity. It is both a limiter and an unlimited joined via harmonia. This is best captured by

the concept of infinity. Introducing infinity dissolves the infinite regress issue. In

principle numbers can be counted infinitely, hence there is an infinite number of potential

limiters. This appears to mean that there are an unlimited number of potential limiters.

My contention is that the risk of an epistemological regress dissolves, just as it did for the

intelligibility issue, if we expand Philolaus’ notion of number beyond the positive

integers and include more abstract mathematical relations such as ratios, formulae, and

numerical infinity. Expanding Philolaus’ scope of what counts as a number to include

infinity seems paradoxical because it appears to worsen the regress issue. However, I

7 McKirahan also cites this testimonia in support, although Huffman does not make

mention of it. 18.20 “The Pythagoreans declare that logos is the criterion of truth—not logos in general, but the logos that arises from the mathematical sciences, as Philolaus

used to say.” (Sextus Empiricus, Against the Mathematicians 7.92=DK 44A29

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think infinity actually helps dissolve the regress, especially since numerical infinity

supplies an infinite (“ultimately unlimited”) number of justified true beliefs. For example,

I am clearly justified in thinking that one is greater than zero that two is greater than one,

and so on. Hence, the regress dissolves because the “ultimate unlimited” i.e. infinity is

actually the necessary precondition for the production of all possible limiters, which is

what I hinted at previously when I claimed that infinity becomes the necessary

precondition for the production of all possible limiters; limiters are justified, not

regressively unjustifiable, because their relationships both hold infinitely and are infinite

in number.

To conclude, I discussed Philolaus’ three archai: limiters, unlimiteds, and

harmonia. I then examined some apparent epistemological problems in Philolaus’ work

highlighted by McKirahan and sketched my own possible solutions to those problems by

expanding the scope of Philolaus’ conception of number to include infinity.

i McKirahan breaks fragments (DK2) and (DK6) into six subsections each and I will attempt to keep his organization for the sake of clarity. I have listed the fragments most

relevant to my purposes in this paper for reference purposes, I have excluded (DK6a, 7, 17, 20, 13). I utilized the translations given by McKirahan, which are identical to those used by Huffman.

18.1 (DK1) Nature in the kosmos was joined from both unlimiteds and limiters, and the

entire kosmos and all the things in it. 18.2 (DK2) 1. It is necessary that the things that are be all either limiters or unlimited or both limiters

and unlimited; 2. but not in all cases only unlimited.

3. Now since it is evident that they are neither from things that are all limiters nor from things that are all unlimited, 4. it is therefore clear that both the kosmos and the things in it were joined together from

both limiters and unlimiteds. 5. The behavior of these things in turn makes it clear.

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6. For those of them that are from limiters limit, those that are from both limiters and unlimiteds both limit and do not limit, and those that are from unlimiteds will evidently

be unlimited. 18.3 (DK3) There will not be anything that is going to know at all, if all things are unlimited.

18.4 (DK4) And in fact all the things that are known have number. For it is not possible for anything at all either to be comprehended or known without this.

18.5 (DK5) In fact, number has two proper kinds, odd and even, and a third kind even-odd, from both mixed together. Of each of the two kinds there are many forms, of which each thing itself gives signs.

18.6 (DK6) 1. Concerning nature and harmonia this is how it is:

2. the being of things, which is eternal—that is, in fact, their very nature—admits knowledge that is divine and not human, 3. except that it was impossible for any of the things that are and are known by us to have

come to be 4. if there did not exist the being of the things from which the kosmos is constituted—

both the limiters and the unlimiteds. 5. But since the principles are not similar or of the same kind, it would be completely impossible for them to be brought into order [or, “for them to be kept in an orderly

arrangement (kosmos)”] if harmonia had not come upon them in whatever way it did. 6. Now things that are similar and of the same kind have no need of harmonia to boot, but

those that are dissimilar and not of the same kind or of the same speed must be connected together in harmoniai if they are going to be kept in an orderly arrangement (kosmos). 18.10 (DK16) Some logoi are too strong for us.