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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.