A review of Aristotle’s claim regarding Pythagoreans fundamental Beliefs: All is number?

The question-statement, “All is number?”, the title of Zhmud’s famous 1989 article in Phronesis, opens a challenge to the extremely important Aristotelian testimony that “all is number” was the fundamental definition of Pythagorean philosophy. Such a challenge is anything but easy, especially when one considers that, so far, the histories of both ancient philosophy and ancient mathematics seem to have no doubts that this definition is correct. This paper aims to submit Aristotle’s claim that the Pythagoreans believed that “all is number” to critical review. Our analysis of the many ways in which Aristotle states the thesis that “all is number” will reveal, beyond merely semantic variations, a fundamental theoretical contradiction that Aristotle himself seems incapable of solving. Three different versions of the doctrine are in fact present in the Aristotelian doxography: (a) an identification of numbers with the sensible objects; (b) an identification of the principles of numbers with the principles of things that are; (c) an imitation of objects by numbers. While versions (a) and (c) seem to identify numbers with the material cause of reality, in terms (“imitation”) reminiscent of Plato, version (b), numbers as formal causes of reality, is an Aristotelian reconstruction of the Pythagorean theory. Aristotle would have been pushed to such a reconstruction by the difficulty he found in accepting the Pythagorean material notion of number, and by considering it closer to its sensitivity, strongly marked by the reception of that same theory in the Academic realm.

The Porphyrian summary and a significant absence A Porphyrian summary of Pythagoras' most famous doctrines immediately brings us to the heart of the pro lem: there is no escaping the fact that his account contains no reference to mathematics or astronomy, or even to cosmology or politics, despite the critical role these doctrines have played in the definition of Pythagoreanism in other strata of the tradition, in particular the Aristotelian texts.
Let us consider the passage: The absence of a reference to numerical theory is significant for understanding how to define the historiographical category of Pythagoreanism that otherwise seems to largely depend on a link to numbers.Such an absence su gests the need for a closer consideration of the history of the assignment of a mathematical theory to ancient Pythagoreanism.
Although Ze ler himself was confident that Philolaus' theories that number is the essence of a l things, along with the doctrines of harmony, the central fire, and the spheres, were core pi lars of Pythagoreanism, contemporary criticism cha lenges the a leged Aristotelian dógma that in Pythago-reanism, "a l is number" .The interpretative tradition, led in recent times by Frank (1923), has become accustomed to regarding a l Pythagorean mathematics as an Academic invention created after Philolaus' fragments, which must be considered spurious.The influence of Frank's skepticism is such that even Cherniss (1935), who disagrees with Frank about the value of Aristotle's testimony, agrees with Frank's interpretation of the connection between the Aristotelian dógma that "a l is number" and ancient Pythagoreanism.The consensus of such scholars is e ecia ly impressive when it comes to the value to be given to Philolaus' fragments, which we regard as one of the fundamental loci of this debate: The fragments attributed to Philolaus are surely spurious, since they contain elements that cannot be older than Plato.Erich Frank has gathered the evidence against the fragments; and, apart from his own theory as to their origin and his conclusion of certain very weak arguments […] his analysis makes it superfluous to restate the overwhelming case against them (Cherniss, 1935, p. 386).
2 For general agreement with Frank's skepticism, see, among others, Burnet (1908, p. 279-284) and Levy (1926, p. 70ff.).It is certainly not correct to agree, therefore, with Spinelli (2003, p. 145, n. 345), when he "dispatches" the question of the authenticity of the fragments in this way: "despite much that has been written for and against them, the whole argument is exposed, in an appropriate manner, only in the work of three writers: Bywater, Frank and Mondolfo." 3In truth, Frank himself, in his subsequent writings, retreating from a position which, in extreme and, in a sense, paralyzing skepticism, could not resist other scholars' criticisms.In fact, in 1955, he readily admitted that "it can hardly be doubted that Pythagoras was the originator of this entire scientific development: he was a rational thinker rather than an inspired mystic" (Frank, 1955, p. 82).Nevertheless, in his review of Von Fritz's book on Pythagorean politics, Frank's skeptical verve is still strongly present (Frank, 1943).

Three versions of the Pythagorean doctrine of numbers
The question-statement "A l is number?", the title of Zhmud's famous 1989 article in Phronesis, opens a challenge to the extremely important Aristotelian testimony that "a l is number" was the fundamental definition of Pythagorean philosophy. 4Such a cha lenge is anything but easy, especia ly when one considers that, so far, the histories of both ancient philosophy and ancient mathematics seem to have had no doubts that this definition is cor ect (see, Heath, 1921, p. 67;Guthrie, 1962, p. 229 ff.;Huffman, 1988Huffman, , p. 5, 1993, p. 57), p. 57).
Reasons for this confidence are not absent.Indeed, in Aristotle, the assignment of the doctrine of "a l is number" to the Pythagoreans is recur ing and ultimately summarizes his interpretation of Pythagoreanism.
Aristotle states a number of times that: (1) "They thought the elements of numbers to be the elements of a l things (2) and the whole heaven to be a harmony and a number" (Met.986a 3).( 3) "Numbers, as we said, are the whole heaven" (Met.986a 21).( 4) "They say numbers are the things themselves" (Met.987b 28).( 5) "Those [philosophers] say that things are numbers" (Met.1083b 17).( 6) "They've made the numbers to be things that be" (Met.1090b 23).
In contrast, seven other times, Aristotle seems to su gest that the Pythagoreans say something slightly different: (1) "There is no other number than the number by which the world is constituted" (Met.990a 21).( 2) "For the Pythagoreans there is only the mathematical number, but they say that it is not separate and that, but that sensi le substances are composed of it (3) because they build the entire heaven with numbers" (Met.1080b 16-19).(4) "It is impossi le to say that […] the bodies are made of numbers" (Met.1083b 11).( 5) "They assumed that real things are numbers, but not in a separate way, rather, that real things are composed of numbers" (Met.1090a 23-24).
(6) "They derived the physical bodies from the numbers" (Met.1090a 32).( 7) "Those who believe that heaven is made of numbers reached the same result as them [the Pythagoreans]" (De caelo 300a 16).
In the above quotes, Aristotle makes the Pythagoreans claim more precisely that the foundation of the world is ex a ithmôn, that is, that numbers are constitutive of and therefore immanent in the world.
This variability of the Aristotelian lectio marks his whole ap roach to Pythagoreanism (Burkert, 1972, p. 45).This is not the only case where Aristotle shows some difficulties in expressing Pythagorean doctrines in the terms of his philosophy.Here the presentation of the doctrine of "a l is number" by Aristotle is, at worst, contradictory, and at best presents three different versions. 5In a dition to the first version, which identifies numbers with sensitive objects, two other versions are provided by Aristotle.
The second identifies the principles of numbers with the principles of the real things: The so-called Pythagoreans are contemporary and even prior to these philosophers [Leucippus and Democritus].They have applied first in mathematics, making them grow, and nurtured by them, believed that their principles were the principles of all beings .
This claim is closely related to the above quote from Met. 986a 3, which is stated in terms of stoic eîa instead of a c aí.
The third version is that real objects imitate numbers, as su ge ed by a famous passage in which a para lel is drawn with the Platonic conception of participation: The Pythagoreans say that beings exist by imitating the numbers.Plato, on the contrary, says it is by participation, changing only the name.In any case, either one or the other neglected equally to indicate what participation and imitation of ideas mean .
The first claim, that "things are numbers" , is clearly inconsistent with the other two.Cherniss (1935, p. 387) rightly notes that Aristotle seeks to reconcile the first claim with the second, that numbers are principles of a l things.His attempt depends on his claim that the Pythagoreans derived a l of reality from the number one, a theory that is not present in the sources, and ap arently confuses Pythagorean cosmology with their theory of numbers (Cherniss, 1935, p. 39).Aristotle himself seems to recognize that this ap roach is bankrupt: These philosophers also did not explain how the numbers are causes of substances and being.Are they causes as limits of greatness, and just as Eurytus established the number of each thing?(For example, a number for man, one for the horse, reproducing with pebbles the shape of the living beings, similar to the numbers that refer to the figures of the triangle and the square […] (Met.1092b8-13).
Aristotle's reference to Eurytus introduces a theory known as "numerical atomism" , according to which numbers are the real things because numbers (thought of as psê hoi, pe bles) are the material of which a l real things are made.With good reason, indeed, Cherniss (1951) notes that in this way, numbers can identify any kind of phenomenal object: Numbers are held to be groups of units, the units being material points between which there is 'breath' or a material 'void'; and they quite literally all identified with phenomenal objects as aggregations of points, without, of course, considering whether these material points were themselves divisible or not.This was rather a materialization of number than a mathematization of nature, but it undoubtedly seemed to the Pythagoreans to be the only way of explaining the physical world in terms of those genuinely mathematical propositions which they had proved to be independently valid (Cherniss, 1951, p. 336).Tannery (1887b, p. 258ff.),Cornford (1923, p. 7ff.) and even Cherniss (1935, p. 387), fascinated by Eurytus's primitive atomistic-numerical method, found it to be quite old.They a l essentia ly fo low Frank's hypothesis (1923, p. 50) that the theory was bor owed by Archytas from Democritus.Not coincidenta ly, the citation from Met. 985b 23-26 refers to the atomists Leucip us and Democritus.Moreover, it has been su ge ed that some of Zeno's ar uments against plurality presup ose a Pythagorean theory of numerical atomism.However, Burkert (1972, p. 285-288) and Kirk et al. (1983, p. 277-278) have raised serious doubts about this assignment, and there are many ar uments for both views.However, we cannot mention a l of them here. 6n any case, it is not hard to imagine that the material nature of Pythagorean numbers has an archaic sense, without needing to postulate a theory of numerical atomism.This sense is summed up quite we l by Nussbaum's now classic definition: [T]he notion of arithmós is always very closely connected with the operation of counting.To be an arithmós, something must be such as to be counted-which usually means that it must either have discrete and ordered parts or be a discrete part of a larger whole.To give the arithmós of something in the world is to answer the question 'how many' about it.And when the Greek answers 'two' or 'three' he does not think of himself as introducing an extra entity, but as dividing or measuring the entities already in question (Nussbaum, 1979, p. 90).
On this interpretation, the number is sti l "itself a thing" (Burkert, 1972, p. 265).Burkert rightly notes, in the same context, that it should not be forgotten that the a ithmos has a certain "aristocratic sound" , which refers to what "counts" in the sense of being important, "worthwhile" to count.The term can be so ap roximated to the pre-Socratic a c e.
Thus, the second sense of "a l is number" , by which the principles of numbers were the principles of a l things, cor esponds more readily to what Cherniss (1935, p. 390) defines as an "Aristotelian construction of the Pythagorean theory." Aristotle would have been led to this synthesis, on the one hand, by his difficulty in accepting the overly simplistic material notion of number as analogous with Eurytus' pe bles, and on the other hand, by considering it more logical to understand the existence of Pythagorean numbers in the same way as the Platonists treated them, that is, by considering the a ithmoí as a c aí.But with this move, Aristotle shifts the pro lem of a Pythagorean theory of numbers into an Academic sphere.In fact, Frank (1923, p. 255) su gests that the source of this "misunderstanding" in Aristotle is in fact Speusip us; therefore, part of the Academy was deeply connected to the Pythagorean traditions.Speusippus is directly quoted by Aristotle in the Metaphysics (1085a 33) when he mentions those "according to whom the point is not one, but similar to one" that is, hoíon tò hén.The point, in fact, plays a central role in Speusip us' work; Speusip us was both a scholar of Philolaus and had openly declared that he based his writings on the latter.This statement is located in fragment 4 (Lang), preserved by Nicomachus as part of his book On the Pythagorean numbers.This fragment is clear evidence of the Academic origin of the principles of the Pythagorean theory of numbers.In this vein, Speusip us asserts that "when considering the generation: the first principle from which greatness generates is the one, the second the line, the third the surface, the fourth the solid" (44 A 13 DK = Fr.4 Lang).7 The first sense in which "a l is number" also contradicts the third sense, that is, the idea of a mím sis of the numbers by real objects.In fact, this thesis is mentioned by Aristotle only once (Met.987b 11), in a passage in which the Pythagorean conception is identified with the Platonic one of paticipation.This makes Cherniss (1935, p. 392) and Zhmud (1989, p. 186) consider it quite likely that Aristotle was trying to diminish the originality of the Platonic idea of méthexis by pointing to Aristoxenus, whose antagonism towards Plato is we l atte ed.Indeed, Aristoxenus' testimony reproduces the same idea of imitation: Pythagoras "likens a l things to numbers" (fr.23, 4 Wehrli, 1967).
In fact, other passages in Aristotle refer to something very similar to the concept of mím sis by using words that involve a conception of similitude: Since just in the numbers, precisely, more than in fire, earth and water, they thought they saw many similarities of what is and comes into being; for example, they believed that a certain property of numbers was justice, another soul and intellect, yet another the moment and opportunity and, in a few words, similarly with all other things (Met.985b 27-32).
Therefore, it is in this sense of homoió mata that the reference to mím sis must be understood. 8 The analogy between numbers and Eurytus' pe bles (Met.1092b 8-13) also relates to conceptions of similitude and imitation.Alexander of Aphrodisias, in his commentary on Aristotle's Metaphysics, explains the reasoning that would have led to the imitative connection between justice and the number four: Assuming that the specific nature of justice be proportionality and equality, and realizing that this property is present in numbers, for this reason the Pythagoreans used to say that justice is the first square number; […] This figure some used to say it was four, as it is the first square, and also because it is divided into equal parts and is equal to the product of these (indeed, it is two times two) (In Metaph.38, 10 Hayduck).Burkert (1972, p. 44-45) notes that this conception of mím sis, even if the terminology is Aristotle' s, must cor espond to a pre-Platonic theory.The fundamental idea of magic or of Hip ocratic medicine is that of a "two-way" match between two entities (body and cosmos, art and nature).In this ecific case, there is a two-way match between the cosmos and numbers-the cosmos imitates numbers, and vice versa.Cornford (1922) considered this idea of imitation rather ancient, precisely because of its mystical nature; he uses etymology (mîmos = actor) to connect the term to Dionysian cults and the fact that the protagonists of the cults play the role of god himself: At that stage 'likeness to God' amounts to temporary identification.Induced by orgiastic means, by Bacchic ecstasy or Orphic sacramental feast, it is a foretaste of the final reunion.In Pythagoreanism the conception is toned down, Apollinized.The means is no longer ecstasy or sacrament, but theoría, intellectual contemplation of the universal order (Cornford, 1922, p. 143). 9 Against these hypotheses, however, the fact that Aristotle does not actua ly indicate the imitation of prágmata, but of abstract realities such as justice, time, etc., plays an important role. 10In any case, even though one may concede that Aristotle is here refer ing to a proto-Pythagorean, acousmatic, doctrine, in the fo lowing page (Met.987b 29), he ar ues forcefu ly that the Pythagorean and Platonic notions of méthexis assigned to numbers differ. 11This would su gest, in this case, that a controversial anti-Academic intention would perhaps be the most ap ropriate explanation of the reference to mím sis. 12 We can conclude that the three versions of the doctrine "a l is number" (that of identification, of numbers as principles, and of imitation) are imperfectly articulated and ultimately contradictory within Aristotle's work.
However, it is significant that Aristotle never mentions that the three different lectiones of "a l is number" belong to different groups of Pythagoreans.He seems to consider them, if not coherent among themselves, at least reconcila le, and refers to them a l without distinction as defining the "soca led Pythagoreans" .
Recognition of this fact has led several authors to adopt conciliatory solutions to the pro lem.First of a l, Ze ler himself.Although Ze ler felt that Aristotle' s testimony should be read 8 See for this approach Centrone (1996, p. 107-108). 9Casertano (2009, p. 67) also agrees with the possibility of this "mystic numbers" origin. 10Burnet (1908, p. 119), on the other hand, warns that one should not take these passages seriously: "They are mere sports of the analogical fancy." 11The term "proto-Pythagoreanism" is introduced here as a new term because it is necessary to distinguish between this first founding moment of Pythagoreanism and the development of Pythagoreanism during the fifth century BC, which is still "pre-Socratic," but which is in writing and corresponds to the era of the immediate sources of Plato and Aristotle.For the uses and meaning of the analogous term "proto-philosophy", see Boas (1948, p. 673-684).
12 This is also one of the reasons forcing one to reject Burnet's hypothesis (1908, p. 355) and Taylor (1911, p. 178ff.),taken up also by Delatte (1922, p. 108ff), whereby Pythagoras would be the inventor of the theory of the Platonic forms.Thus, Burnet (1908, p. 355): "the doctrine of 'forms' (eíde -, idéai) originally took shape in Pythagorean circles, perhaps under Sokratic influence." with a l due care, its historical proximity to the Pythagorean doctrines should sup ort its authenticity.Thus, for Ze ler: No doubt that in Aristotle's exposure we must seek first of all and only his own way of seeing, and not an actual and immediate testimony of reality, however even in this case [that of the numerical theory], everything speaks in favor of a recognition of the fact that his way of seeing was based on a direct knowledge of the actual connection of the very ideas of Pythagoreanism (Zeller and Mondolfo, 1938, p. 486, my translation).Frank (1923, p. 77, n. 196) and Rey (1933, p. 116), seeking to show the possibility of the compatibility of the three versions of "a l is number" , imagine that Aristotle understood the different versions to be logica ly derived from one another.Rey draws up a proposed compromise between the version of numbers being the things and that of numbers imitating things: numbers would be things when considering their nature and would imitate things when one considers their properties (Rey, 1933, p. 356ff.). 13More elaborate is Raven's conciliatory ar ument (1966 [1948], p. 43-65), whereby: To suppose, as so many scholars appear to suppose, that Aristotle was hopelessly confused about it, is not only to lay a very serious charge at his door, but also, incidentally, to demolish the main basis upon which any reliable reconstruction of Pythagoreanism must be erected (Raven, 1966(Raven, [1948], p. 63)], p. 63).
In an open controversy with Cornford (1923, p. 10) and his idea that Aristotle failed to distin uish two moments of Pythagoreanism (a first one on the idea of the materiality of numbers, and a second one where the Pythagoreans would be more concerned with the numerical make-up of reality), Raven instead proposes a radical inseparability of the dual use of these senses within ancient Pythagoreanism. 14Aristotle would thus simply be getting a conception of nature as "equal to numbers" from Pythagoreanism, that is, constituted by an a gregation of spatia ly extended units (Raven, 1966(Raven, [1948]], p. 62).However, numbers would not constitute only the matte of reality, but would also be the origin of the qualitative differences that distin uish each material object from others.This is the only way one might think either version of the imitation and of the number of the p inciples as articulated with the first version. 15t the very least we can say that the idea of mím sis that Aristotle attributed to the Pythagoreans shares little with the Platonic conception of mím sis according to which phenomenal realities mimic the forms, in the sense of being "similar to" supra-sensi le realities of a higher ontological level.If this observation is cor ect, what Aristotle must attribute to the Pythagoreans when eaking of mím sis cannot be anything other than a generic cor espondence between things and the numerical relationships that explain them and make them inte ligi le.Casertano summarizes the matter very we l: Immanent intelligibility, therefore, and not transcendent to the same things.This is why the Pythagorean formula, 'things are numbers' and 'things are similar to numbers', are not contrasted, but rather are expressions of the same basic intuition, which is one of homogeneity between reality and thought, between the laws of reality and the laws of thought: to comprehend things is essentially to mirror them, to reproduce at the mental level that fully intelligible structure, which is characteristic of material reality (Casertano, 2009, p. 65, my translation).
Although the fundamental insight of the Pythagoreans, an attempt to understand the nature of numbers by analogy with the nature of the world, is clear, the fact is that the Aristotelian attempt to reconcile the different versions of the theory does not seem at a l successful.
If, moreover, we think that the main version of the Pythagorean doctrine, that of the identity of number with realities, pays obeyance directly to the controversial intention of Aristotle with regards to Platonism, making him consider the Pythagorean a ithmós a material cause, in op osition to the Platonic militancy in favor of its being a formal cause (Cherniss, 1935, p. 360), this makes it difficult to ap eal to the Aristotelian "a l is number" as a genuine and pure piece of historiographical evidence for the foundations of Pythagoreanism. 16ifficult, but not impossi le, I would say.

First solution:
The Aristotelian reduction In fact, two solutions have been proposed to the problem of whether Aristotle's claim that "a l is number" accurately describes Pythagorean philosophy.
The first engages in a radical cha lenge of the validity of the Aristotelian testimony, even coming to deny that a doc-trine of number belongs tout court in proto-Pythagoreanism.There is no lack of reasons for this cha lenge, and they center on the fact that no testimonies earlier than Aristotle attest to this doctrine.Zhmud's article (1989), quoted above, begins with this ar ument, and we wi l fo low it step-by-step.
Zhmud's article operates in the context of determining the criteria for identification as a Pythagorean.His fundamental concern is to consider the impression that the Aristotelian text seems to give, that is, that "someone who eaks of numbers" would be the best definition of a Pythagorean.The use of the criterion of numbers to identify a Pythagorean (Zhmud, 1989, p. 272) would be either circular or question-be ging.Indeed, despite several attempts in this regard, no historian-says Zhmud-has succeeded in finding any doctrine about numbers in the pre-Aristotelian sources on Pythagoreanism (Zhmud, 1989, p. 272).
With these ar uments in mind, Zhmud admits only two possi le explanations for Aristotle's testimony: either the expression "a l is number" belongs to an ancient and secret teaching of the "divine" Pythagoras, which must have been directly revealed to Aristotle and first pu lished by him, or the expression "a l is number" was not actua ly a Pythagorean doctrine.17This second possibility cor esponds to the classic position of Burnet, in which "Pythagoras himself left no developed doctrine on the subject, while the Pythagoreans of the fifth century did not care to a d anything of the sort to the school tradition" (Burnet, 1908, p. 119). 18lthough not surprising, given the aforementioned studies of Cherniss su gesting that Aristotle's own "historiographic" method freely reformulated the doctrines of his predecessors in his own terms, it is important to ask what would make Aristotle falsely attribute such a doctrine of "a l is number" to the Pythagoreans.
Our ar uments so far have given a crude first response to this question.Aristotle was faced with a great diversity of Pythagorean sources, both ancient (Hip asus) and closer to him (Ecphantus, Philolaus, Archytas).However, for purposes internal to Aristotle's Metaphysics, this plethora of Pythagoreans needed to be brought back to a common denominator, under a sc ool that would somehow fit into the theoretical-historical course that Aristotle intended to draw on in his doxography.
Without reducing Pythagoreanism to a set of core theoretical doctrines, it would have been impossi le to find a place for it inside the agonic model by which Aristotle describes the history of his predecessors (Cherniss, 1935, p. 349).For example, only in this way could the Pythagorean a c é be an antagonist of the Ionic material cause.At the same time, terminological imprecision in the Pythagorean sources (which Aristotle himself complains about in Met.1092b 1-13) a lows the postulation of Pythagorean numbers as the precursor of the Platonic formal cause.Even if number did not already have this dual valence, Aristotle would proba ly have invented it, for it fits to perfection within his doxographic model.Thus, the postulation that "a l is number" would have been Aristotle's solution to a historiographical pro lem, and in some ways the beginning of a long tradition which, starting with Ze ler (Ze ler and Mondolfo, 1938, p. 435), reduced the category of Pythagoreanism to the nar ow limits of this metaphysical doctrine.

Second Solution: Philolaus
The first solution leaves us at a hermeneutic impasse: Aristotle himself invented a historiographic category ("the so-ca led Pythagoreans") and a doctrinal common denominator defining it ("a l is number").A second solution seeks to avoid tracing the category back to a mere invention by undertaking a reassessment of the Pythagorean sources of the fifth century BC for possi le historical references to Aristotle's term "so-ca led Pythagoreans" .
Starting from an important observation: the great number of references to Pythagoreanism and their theory of numbers in Aristotle reveals an indisputa le fact: Aristotle must have rea ly had several Pythagorean texts on his desk. 19The certainty with which Aristotle presents some statements about the Pythagoreans seems to presup ose his access to a sufficiently broad literature of their authorship.Consider the debate on whether the Pythagoreans considered the world to be generated or not.Aristotle says it is impossi le to doubt it: "There is no reason to doubt whether the Pythagoreans do or do not introduce generation of things which are eternal" (Met.1091a 13).Likewise, he ap ears to be absolutely certain that the Pythagoreans had not philosophized about sensi le bodies: "They did not say anything about fire nor earth, or on other bodies" .
Moreover, tradition informs us that Aristotle devoted at least two books to the Pythagoreans, not to mention the works devoted ecifica ly to Pythagoras or particular Pythagoreans such as Archytas. 20Any account of who the "soca led Pythagoreans" to whom Aristotle wants to assign the doctrine of numbers were depends, for the most part, on the possibility of identifying the subjects of these books.However, tradition only te ls us of books on Philolaus and Archytas.As Aristotle seems to deal with Archytas separately from the rest of pythagoreans, it is likely that the books of Philolaus constitute Aristotle's Pythagorean sources.
It is important to point out, again, that it is no wonder that this same methodological conclusion has not been reached before, that is, the pro lem to which the doctrine "a l is number" was intended as a solution was the study of Philolaus' fragments.Much of the tradition, beginning with Cherniss himself (1935, p. 386), could not pursue this direction because the texts of Philolaus were considered spurious in the wake of Frank 1923.Only after the "rediscovery" of the value of an essential part of Philolaus' fragments, first with Burkert (1972, p. 218ff.), and then with Huffman (1988Huffman ( , 1993)), did that path become possi le.
The recent reassessment of the historical value of Philolaus' fragments, therefore, a lows new, previously impossi le, hermeneutical steps.