A Functionalist Account of Epicurus’ Minima

Epicurus’ original version of atomism takes atoms to be physically indivisible but not completely unanalysable: each atom contains a finite number of minima. This paper explores the nature of the minima by focusing on a specific question: in which sense are the minima minimal ? I do so by investigating the notions of parthood and divisibility into parts that are at play in paragraphs 56–59 of the Letter to Herodotus , where the theory of minima is introduced. By focusing on the analogy (noticed by Francesco Verde) between Epicurean minima and Aristotelian limits, I argue that the minima should be understood as the minimal realiser of the atom’s physical functions. This allows me to keep together two very plausible but apparently incompatible claims: (i) the minima are supposed to block the paradoxes of theoretical divisibility, but (ii) their existence and their indivisibility can only be justified in physical (rather than geometrical) terms


Introduction
The core tenet of all versions of Ancient Atomism is that atoms are the smallest material constituents of physical reality: atoms are ingenerated, incorruptible and unbreakable, and all macroscopic bodies result from their composition.Epicurus' original version of Atomism adds a special twist to this core tenet.He argues that although atoms are the smallest components of physical reality and are unbreakable, they are not unanalysable.Each atom contains a finite number of smaller elements, that he calls elachista, minima.
The notion of elachiston is puzzling.How can atoms not be simple units, without being infinitely divisible either?What are the minima, and in which sense are they indivisible?Lucretius speaks of them as the 'minimae partes',1 but it is far from clear that it is adequate to refer to the minima as parts of the atom,2 at least if we think of parts as that into which a whole can be decomposed, and out of which it can be composed.The atom is unbreakable and ingenerated, so that its 'parts' cannot be compositional parts.On the other hand, if we do not require that the 'parts' of the atom are physically separable from the whole, but merely that they are identifiable by some mental procedure in the whole, it is unclear why the minima themselves should not contain smaller parts.
This paper examines the nature of the minima by focusing on a specific question: in which sense are the minima minimal?I aim to do so by investigating the notions of parthood and divisibility into parts that are at play in paragraphs 56-59 of the Letter to Herodotus, where the theory of minima is introduced.Identifying the peculiarities of the Epicurean notion of parthood and divisibility allows me to put forward an account of the minima which keeps together two very plausible but apparently incompatible claims: (i) the minima are supposed to block the paradoxes of theoretical divisibility, but (ii) their existence and their indivisibility can only be justified in physical (rather than geometrical) terms.
These two claims are central to the two alternative interpretations that polarise the scholarly debate, proposed respectively by David Furley and Gregory Vlastos.I submit that rather than opting for the one or the other interpretation, it is important to find a way to keep together the two core claims on which they are built.On the one hand, I argue that Furley is right in reading the theory of minima as a response to the Eleatic paradoxes of divisibility, and that the distinction between physical and theoretical divisibility does play a role in the theory.On the other hand, I believe that Vlastos is right in taking the theory of minima to be only justifiable in physical (and not geometrical) terms.
These two strands can be brought together by formulating a more nuanced, and unconventional, notion of divisibility into parts.A close reading of Ep.Hdt.56-59 shows that such a notion of parthood and divisibility into parts is in fact at play; this can be appreciated in particular from Epicurus' use of metabasis as a way of dividing into parts, and from his analogy between minima and limits.Building on Verde's suggestion that Epicurus is hinting at Aristotle's distinction between limits and parts, I argue that the basis of the analogy should be found in the way in which limits (as well as minima) are individuated and distinguished from the whole to which they belong, even if they cannot be physically separated from it.Limits are distinguishable from their whole because they fulfil a very specific set of physical functions, which are not fulfilled by the whole.The very same mechanism is at play in the individuation of Epicurus' minima: they can be distinguished from the whole because they have a very specific physical role to play, but are themselves indivisible in the sense that there is no further physical process which requires an internal differentiation of the minima.
Epicurus's original version of atomism is built on a core claim: each atom contains a finite number of minima.This claim raises a number of questions.
If the atoms are indivisible, how can they have parts?And if they are somehow divisible into parts even if they are unbreakable, how can this divisibility have a lower limit?What grants that in each atom there is a finite number of minima?
In order to solve these questions, it is fundamental to understand why Epicurus posits the existence of internal differentiations in the atom in the first place, thus complicating the more straightforward account put forward by Leucippus and Democritus.Furley's (1967) seminal account starts from the hypothesis that Epicurus' introduction of the minima in the atom should be read on the background of the Eleatic paradoxes of divisibility, and of Aristotle's criticism of Leucippus and Democritus.Zeno's paradoxes aim to defend Parmenides' counterintuive rejection of motion and plurality, by showing that the (seemingly obvious) existence of motion and pluralism gives rise to equally (if not more) worrying paradoxes.In particular, his Argument from Complete Divisibility (reported by Aristotle in gc i.1, 316a15-34) claims that if a body is infinitely divisible (i.e.divisible in parts which are themselves further divisible), a paradoxical dilemma arises: what can be left, once the division has been carried out?Not magnitudes, for they would be further divisible; nor zero-dimensional items, for we should admit that a body was composed out of nothing.
According to a popular narrative (that goes back to Aristotle), the Classical Atomism of Leucippus and Democritus was born as a reaction to such paradoxes, and is heavily influenced by the Eleatic framework.In particular, Aristotle argues that the Atomists 'give in' to the Eleatics.3Although they oppose Parmenides' monism, which they deem contrary to the observable phenomena, they buy into the core idea that what is unitary and continuous is indivisible, and that infinite divisibility is paradoxical.Each indivisible atom resembles the Parmenidean One, in that it is continuous and indivisible.Macroscopic bodies are divisible (because they are composed by a plurality of atoms), but not infinitely divisible: their divisibility must stop when they reach the atom, which is continuous and extended but indivisible.4 Aristotle's famous criticism of Atomism develops on two levels: on the one hand he argues that Atomism is itself contradictory,5 while on the other he develops a theory of continuity and infinite divisibility that purports to solve the paradoxes.6Assuming that Epicurus is familiar at the very least with the content of Aristotle's scholarly works, if not with their letter,7 his theory of minima can be read as an attempt to strengthen the Atomistic response to the Eleatic paradoxes taking into account Aristotle's criticism.
In order to understand how the minima could achieve this result, Furley starts by distinguishing two kinds of division:8 -physical division: the division of something in such a way that formerly contiguous parts are separated from each other by a spatial interval; -theoretical division: parts are distinguished within an object by the mind, but they never end up being separated from one another by a spatial interval.It is famously unclear whether Democritus and Leucippus themselves conceived of their atoms as physically or theoretically indivisible, nor whether it is fruitful to use this contemporary distinction to make sense of their theory.Aristotle's criticism of the Argument from Complete Divisibility seems to take the atoms as both physically and theoretically indivisible: many of the problems that he points at arise from the assumption that atoms are theoretically indivisible.9In particular, Aristotle argues that the atoms could not be in contact without overlapping completely (not having parts, they could not be in contact part-to-part but only whole-to-whole), so that they could not even constitute macroscopic bodies;10 nor could they move (because they could not be part in one state and part in another).11In addition, note that the atoms have various geometrical shapes.It is impossible to have a shape without having theoretically distinguishable parts, at least in the sense laid out by Furley.This is because the possession of a definite geometrical shape and of an extended magnitude enables the mind to distinguish parts in the whole -for example, a rectangular object can be mentally divided in two halves by tracing the diagonal.On the contrary, physical indivisibility is perfectly compatible with the possession of a geometrical shape.
We have no proof that Epicurus was aware of, nor interested in, the distinction between physical and theoretical divisibility.The hypothesis that he was, however, is fruitful: it would become clear both why he introduces parts in the atom, and why these parts need to be indivisible.On the one hand, admitting the possibility of theoretically dividing a physically indivisible atom allows Epicurus to account for the atom's internal complexity while avoiding Aristotle's criticisms.On the other, as soon as the theoretical divisibility of the atom is admitted, the Eleatic paradoxes become a threat again.As Aristotle understands,12 the paradoxes do not require that the whole is physically divisible; theoretical divisibility is sufficient to trigger them.Consequently, they can only be blocked by positing a lower limit to theoretical divisibility, too (or to find a different way to disarm the paradoxes, as Aristotle does).The existence of theoretically indivisible minima in the atom would achieve the purpose.
Reading Epicurus on the background of the Eleatic paradoxes, and assuming that he was sensitive to the distinction between physical and theoretical divisibility, allows Furley to formulate a first hypothesis as to what it means for the minima to be minimal: they are absolutely indivisible, both physically and theoretically.
The viability of this interpretative line crucially depends on the plausibility of its assumptions.It is thus not surprising that the controversy between Furley and Vlastos revolves in great part around the role that the paradoxes of divisibility play in motivating the theory of the minima, and on the relevance of the distinction between physical and theoretical divisibility.
As I have briefly illustrated, Furley's interpretation is able to account both for the existence of minima -recognising the theoretical divisibility of the atom is required to explain how atoms can have shapes and sizes, how they can move, and how they can come into contact without overlapping -and for the decision of positing a limited number of minima in each atom, so as not to incur in the paradoxes of divisibility.Unfortunately, it falls short of providing a satisfactory justification for the latter.Notice that for the minima to be able to block the paradoxes of divisibility, it is necessary that they are spatially extended and have positive size (if they did not, the atom would paradoxically be an extended body composed out of un-extended parts).13How is this possible?It seems that if something is spatially extended, it also needs to be theoretically divisible -at least according to the definition of theoretical divisibility provided by Furley.In particular, we saw that possessing a definite geometrical shape is a sufficient condition for theoretical divisibility (again, in the very broad sense individuated by Furley).Thus, it follows that in order to be theoretically indivisible, the minima should be extended chunks of matter without any shape,14 which seems unlikely (if not overtly absurd), especially because Epicurus never suggests that the minima lack shapes.Moreover, positing a spatially extended but theoretically indivisible magnitude blatantly contradicts one of the basic assumptions of Euclidean geometry, namely that it is always possible to individuate a point between any two given points.Because of this, it is not at all clear how the theoretical indivisibility of the minima could be justified, once the theoretical divisibility of the atom is accepted.Vlastos (1965) argues that to justify the imposition of a lower limit to theoretical divisibility, Epicurus should invoke some a priori, theoretical principle -notably, he believes, a geometrical one.Euclidean geometry cannot do the job, obviously: an alternative geometry would be needed.It is indeed possible that a finitist and discretist geometry is developed in the Garden,15 but it is more plausible that the need to amend the principles of geometry was caused by this revolutionary physical theory of the minima, rather than vice versa; in any case, no geometrical justification is given in the Letter to Herodotus.Absent a geometrical justification, the imposition of a lower limit to theoretical divisibility would be completely arbitrary.
According to Vlastos, this proves that the theory of minima should not be considered a reply to the divisibility paradoxes.If he is right, then focusing on the distinction between physical and theoretical divisibility only muddles the water, letting us lose sight of the importance of the theory of minima as a purely physical doctrine.To understand its value, Vlastos believes that we should rather focus on unpacking the Epicurean notion of 'part' .Vlastos' core intuition is that in Greek there at the very least two senses of 'part' that are commonly used.16In a weak sense, anything that results from a division (be it physical or theoretical) can count as a part.In a stronger sense, however, something counts as a part of an object X if and only if X is an integral multiple of it.In this second sense, a part is a unit of measurement of the whole.
Note that if we understand 'part' in this way, saying that the minima are the smallest parts of the atom does not imply that they are theoretically indivisible; it simply amounts to saying that they are the smallest atomic length de facto existing in the world, of which all other atoms are multiples.Thus, the claim that there are minimal parts in the atom can be reduced to the statement of a physical law, that Vlastos calls L(I): L(I): atoms are so constituted that variations in atomic lengths occur only in integral multiples of the smallest atomic length.17 According to this second interpretation, the reason why the minima are minimal is not that they are theoretically indivisible, but that they are the smallest divisor of all atomic lengths.Note that a positive length can be recognised as the common unit of measurement of two different magnitudes, all while being in itself theoretically (and even physically) divisible.
The two interpretative lines proposed by Furley and Vlastos yield two opposite and incompatible interpretations: according to the first, the minima are spatially extended but theoretically indivisible chunks of matter; according to the second, they simply are the smallest atomic length existing in nature, but can be theoretically divisible.The core of the disagreement revolves on what is the goal of the theory of minima, and in particular on whether it is meant to block the divisibility paradoxes and whether it appeals to a notion of theoretical indivisibility.
I believe that neither interpreter got the full picture, but each of them individuated a key claim that is essential to Epicurus' theory.In particular, I submit that that Vlastos is right that the theory of minima is a physical theory that needs to be justified in physical (and not geometrical) terms, but that he is wrong in dismissing the relevance of the divisibility paradoxes (and consequently, of theoretical indivisibility).It is surely true that Epicurus is committed to L(I), and that the minima are in fact the common measure of all atoms,18 but this is not the whole story.
In the next section I will take a closer look at Ep. Hdt.56-59 and show that Furley is right in thinking that the paradoxes of divisibility are in the background.This also provides an indirect argument in favour of the idea that the distinction between physical and theoretical divisibility is in fact important to a correct understanding of the nature of atomic minima.The analysis shows that the paradoxes should be understood as relative to some kind of theoretical divisibility: the minima are meant to posit a lower limit to this kind of divisibility, so as to block the paradoxes.However, I will ultimately argue that Furley's distinction between physical and theoretical divisibility is not fine-grained enough, and does not grasp the specificities of Epicurus' minima.In the last section I will advance an hypothesis as to what kind of theoretical (but not geometrical) divisibility is at stake, and how it can be justified in physical rather than geometrical terms.

Metabasis and the Divisibility Paradoxes
Epicurus presents the theory of atomic minima in a very succinct way in Ep.Hdt.56-59.The location of these paragraphs in the text is important.The theory is introduced after the need for physically indivisible atoms has already been established, and their properties -shape (and all which is related to shape), weight and size -have been examined.In particular, the discussion of the minima begins (at §56) as a follow-up or explanation of the claim that only a finite number of atomic sizes can be found in nature.
Epicurus introduces the discussion as follows: Furthermore, we must not consider that the finite body contains an infinite number of bits (ogkoi), not even parts with no [lower] limit to size.
Therefore, not only must we deny division (tomē) into smaller and smaller parts to infinity, so that we do not make everything weak and be compelled in our conception of complex entities to grind away existing things and waste them into non-existence, but also we must not consider that in finite bodies there is traversal (metabasis) to infinity, not even through smaller and smaller parts.19 These lines strongly suggest that Epicurus had the paradoxes of divisibility in mind.The train of thought is: if a whole contains an infinite number of parts, paradoxes will follow; thus, nothing can be infinitely divisible.Since the physical indivisibility of the atom has already been established, the idea seems to be that atoms cannot be even theoretically divisible ad infinitum.Interestingly, Epicurus considers two ways in which divisibility might be achieved.Both seem to be kinds of divisibility in thought, and so would be classified as 'theoretical' according to Furley's distinction.Epicurus specifies that it is not sufficient that tomē, the process of dividing a whole into parts, reaches a limit, but he also argues that 'traversing' an object cannot require an infinite number of steps.Although the term 'metabasis' does not appear in the reports of the Eleatic paradoxes, the situation envisaged in these lines is clearly recognisable as one of the traditional puzzles of divisibility.The most famous version is probably the Zenonian paradox of the Dichotomy, which argues that if a distance is infinitely divisible, it is impossible to traverse it.20Thus, metabasis seems to consist in examining the whole by passing over its parts one by one, in succession.Notice that if this is the case, metabasis can be infinite even if the parts are not physically separated the one from the other: it is sufficient that the whole is theoretically divisible.21 Simply saying that all atoms are commensurable, as Vlastos suggests, would not help with respect to these problems.The minima must really be indivisible, in some sense.In which sense, though?And how to justify such indivisibility?Recall that their existence and indivisibility cannot be justified in geometrical terms.
I believe that to find a way out of this dead-end it is necessary to re-think Furley's distinction between physical and theoretical divisibility.This distinction is a very useful tool, and a good first step in the right direction, but it is not fine-grained enough.As Steven Makin (1989) pointed out, each time someone wishes to speak about 'indivisibility' , they should specify at least two variables: (i) what counts as a part; (ii) what kind of modality is in play.The problem with Furley's distinction is that the two parameters are confused and conflated, so that the notion of 'theoretical (in)divisibility' remains ambiguous and ends up being too generic.
The most common example of theoretical division (which, it is useful to remind, is characterised by Furley simply as the distinction of parts into a whole operated by the mind) is geometrical division.As Vlastos argues, this does not seem to be what Epicurus has in mind.Epicurus, however, could be using a different kind of theoretical division, one in which the parts are indeed identified by the mind rather than being physically separated from the whole, but where further (and possibly physical) conditions need to be met for something to count as a part.
To understand what counts as a part for Epicurus, we need to establish which sense of indivisibility is needed to make it the case that the metabasis of an atom only involves a finite number of steps.Thus, understanding precisely what metabasis is, and which parts it takes into account, is of key importance.
Ep. Hdt.57 is instructive in this respect, for it presents some arguments in support of the §56 claim that the division of the atom needs to have a lower limit.In particular, the last argument purports to show that metabasis cannot be allowed to go on ad infinitum by pointing out that, if it were, it would be possible to reach infinity in thought: And third, since the finite body has an extremity which is distinguishable, even if not imaginable as existing per se, one must inevitably think of what is in sequence to it as being of the same kind, and by thus proceeding forward in sequence it must be possible, to that extent, to reach infinity in thought.22 These lines provide useful information as to how metabasis works.In particular, we can infer that: a) The metabasis starts from one of the extremities (ἄκρον) of a body and should finish at the opposite one, passing through all the intermediate parts (κατὰ τὸ ἑξῆς εἰς τοὔμπροσθεν βαδίζοντα); b) The extremity which constitutes the starting point of the metabasis is the model on the basis of which all the other parts are thought (οὐκ ἔστι μὴ οὐ καὶ τὸ ἑξῆς τούτου τοιοῦτον νοεῖν); c) It is possible to go over these parts in succession, passing from the one to the next (τὸ ἑξῆς; κατὰ τὸ ἑξῆς εἰς τοὔμπροσθεν); d) The extremity (and consequently all the parts) is distinguishable but not separable from the body: it cannot be thought as existing per se (ἄκρον τε ἔχοντος τοῦ πεπερασμένου διαληπτόν, εἰ μὴ καὶ καθ' ἑαυτὸ θεωρητόν).It is not immediately clear what kind of part can fulfil these criteria.More information comes from Ep. Hdt.58-59, where Epicurus describes the nature and arrangement of the minima in the atom by using an analogy with the minima in sensation: visible bodies: minima in sensation = atoms: minima in the atom This is how he introduces the minima in sensation: As for the minimum in sensation, we must grasp that it is neither of the same kind as that which admits of traversal, nor entirely unlike it; but that while having a certain resemblance to traversable things it has no distinction of parts.Whenever because of the closeness of the resemblance we think we are going to make a distinction in it -one part on this side, the other on that -it must be the same magnitude that confronts us.We view these minima in sequence, starting from the first, neither all in the same place nor touching parts with parts, but merely in their own peculiar way providing the measure of magnitudes -more for a larger magnitude, fewer for a smaller one.23Visible bodies are described as 'that which admits of metabasis' ,24 while the minima in sensation do not admit of metabasis because they are indivisible (and metabasis consists in passing over the object part after part, in succession: if there are no parts, there can be no metabasis).
Note the very peculiar interaction between the 'traversability' of visible objects and the indivisibility of the minima in sensation: macroscopic objects admit of metabasis precisely because the minima do not.As Aristotle points out, if a magnitude is infinitely divisible there cannot be a relation of succession between the parts.25It is only because the minima in sensation are not further divisible that it is possible to pass from one minimum to the successive one, and to complete the metabasis of the visible object.
Note also that the two operations of tomē and metabasis, that are introduced together in §56, are closely connected as well: what has no parts does not 'admit of metabasis' either; while if the tomē could go on ad infinitum, completing the metabasis would require an infinite number of steps (thus being impossible or paradoxical).
Indeed, Epicurus specifies that the minima in sensation 'have no distinction of parts' and every time that 'we think we are going to make a distinction in it -one part on this side, the other on that -it must be the same magnitude that confronts us' .Note that this cannot mean that the minima in sensation are theoretically indivisible in an absolute way, since not even atoms (which are invisible because smaller than the minima in sensation) are.Epicurus does not provide enough details to precisely understand in which way the minima in sensation are indivisible, but it is clear that their indivisibility is linked to a specific physical mechanism, vision.What I take to be the core intuition (which, as I will explain in the next section, constitutes the basis for the analogy between the minima in sensation and the minima in the atom) is that there is a physical process that has a lower limit of accuracy, something beyond which it cannot, and need not, make discriminations.And as a consequence, we cannot discriminate any further with respect to that one physical process.
24 'That which admits of traversal' could in principle refer both to macroscopic bodies and to atoms.However, the context makes it clear that Epicurus is here thinking of macroscopic bodies: 'that which admits of traversal' is immediately put into relation with the minimum in sensation, showing that it must be a sensible body.In what follows, moreover, Epicurus develops the analogy with the 'things before our eyes' , and focuses on vision as the exemplar case of sensible perception.His analysis of vision respects the features of metabasis that I have briefly spelled out above: we see the minima in succession, starting from the first.This raises a question that has puzzled the commentators: do we really see the visual minima, when we look at a visible object?Unfortunately, this question goes beyond the scope of this paper: see Vlastos (1965); Purinton (1994) The last few lines of the paragraph provide further details on the arrangement of the minima in the whole.Epicurus argues that we see the minima 'in sequence' (ἑξῆς), but they are ordered 'in their own peculiar way' (ἐν τῇ ἰδιότητι): they are not in contact part to part (since they do not have any parts), but they do not even coincide.This is interesting, because as part of his criticism of Atomism, Aristotle argues that a continuous magnitude cannot be composed out of indivisibles because indivisibles cannot be in contact without overlapping completely.26By claiming that the minima do not overlap, but are arranged 'in their own peculiar way' , Epicurus shows that he is aware of the problem pointed out by Aristotle and thinks that the existence of minima in the atom is sufficient to provide a solution.Atoms can touch without because they do have internal differentiation: they are in contact minimum to minimum and not whole to whole.And the minima themselves are arranged 'in their own peculiar way' , which apparently is such that the problem of arrangement should not arise.
What is this 'peculiar way'?Without further details, the theory is bound to fail: Epicurus cannot simply postulate that the minima escape the problem of arrangement.But there might be something in his description of the minima that can help us understand why he thinks he can avoid the problem, and that we can use to fill the gaps in the exposition of the theory, casting light also on the very specific kind of indivisibility that Epicurus attributes to the minima.

4
Limits, Parts, and Minimal Realisers of Physical Functions I believe that the key element is identified by Francesco Verde (2010;2013;2020).Verde underlines that in §59, when Epicurus applies his analysis of the minima in sensation to the minima in the atom, he characterises the latter as limits: This analogy, we must consider, is followed also by the minimum in the atom: in its smallness, obviously, it differs from the one viewed through sensation, but it follows the same analogy.For even the claim that the atom has size is one which we made in accordance with the analogy of things before our eyes, merely projecting something small onto a large scale.We must also think of the minimum uncompounded limits (perata) as providing out of themselves in the first instance the measure of lengths for both greater and smaller magnitudes, using our reason to view that which is invisible.For the resemblance which they bear to changeable things is sufficient to establish this much; but a process of composition out of minima with their own movement is an impossibility.27 Verde suggests that by using the word 'limits' (πέρατα) Epicurus is hinting at the Aristotelian distinction between limit and part.According to Aristotle, limits are distinguishable, but not divisible, from the body to which they belong: they are both conceptually and ontologically dependent on their bearer.Thus the analogy with limits is particularly relevant for the issue at hand, because it provides a concrete example of a mental operation that allows us to distinguish internal differentiations in a whole, but that is not a physical nor geometrical division.Limits cannot be physically divided from their bearer, but it is undoubtable that we do identify them and distinguish them by the mind.I am going to argue that thinking of the minima along the lines of limits is instrumental to understanding not only how Epicurus thinks that the minima can be identified by the mind, but also why the very same procedure cannot be used to further divide the minima themselves.However, this analogy should be treated carefully, and its scope clearly delimited.
The distinction between limits and parts plays a major role in Aristotle's theory of continuity, and in his own solution to the paradoxes of divisibility.Aristotle argues that (infinitely many) points can be identified on a continuous line, but that we should not think of them as parts of the line.Aristotle works with the strong notion of parthood that Vlastos refers to, and that we saw in the first section.Such notion requires that two conditions are met for something to count as a part: (i) it needs to measure the whole, and (ii) it must be that into which a whole can be decomposed, and out of which it can be composed.
Aristotelian limits fail to fulfil these conditions because they are lowerdimensional objects.We would say that the limit of a n-dimensional object is 27 μικρότητι γὰρ ἐκεῖνο δῆλον ὡς διαφέρει τοῦ κατὰ τὴν αἴσθησιν θεωρουμένου, ἀναλογίᾳ δὲ τῇ αὐτῇ κέχρηται.ἐπεὶ περ καὶ ὅτι μέγεθος ἔχει ἡ ἄτομος, κατὰ τὴν ἐνταῦθα ἀναλογίαν κατηγορήσαμεν, μικρόν τι μόνον μακρὰν ἐκβαλόντες.ἔτι τε τὰ ἐλάχιστα καὶ ἀμερῆ πέρατα δεῖ νομίζειν τῶν μηκῶν τὸ καταμέτρημα ἐξ αὑτῶν πρῶτον τοῖς μείζοσι καὶ ἐλάττοσι παρασκευάζοντα τῇ διὰ λόγου θεωρίᾳ ἐπὶ τῶν ἀοράτων.ἡ γὰρ κοινότης ἡ ὑπάρχουσα αὐτοῖς πρὸς τὰ ἀμετάβατα ἱκανὴ τὸ μέχρι τούτου συντελέσαι, συμφόρησιν δὲ ἐκ τούτων κίνησιν ἐχόντων οὐχ οἷόν τε γίνεσθαι.Ep.Hdt.59, 708-18, Long & Sedley's transl.Notice that the Greek τὰ ἐλάχιστα καὶ ἀμερῆ πέρατα is ambiguous between two readings: Epicurus might mean that (a) the minima are limits, or that (b) the limits are minima.I think that Verde (2013: 69) and Purinton (1994: 122-3) are right in defending reading (a).However, note that what follows can be accepted also by those who favour (b).This is because Epicurus argues that the extremity of the atom is the model for all the other parts: if the limit of the atom is a minimum, so are all the other parts.It is still legitimate to use the notion of limit to enlighten the notion of atomic minimum.only extended along (n-1) dimensions: the limits of a three-dimensional body are two-dimensional planes, the limits of a plane are lines, and the limits of a line are points.Since they have measure 0 along (at least) one dimension, limits cannot be used to measure the whole -Aristotle believes that there is no ratio between zero and positive numbers.28It is also impossible to obtain an extended magnitude adding up limits, since Aristotle agrees with the Eleatics that adding up unextended objects will never result in a positive magnitude, however long we keep adding.
Conversely, according to Aristotle a continuous line cannot be divided into points.Aristotle maintains that whenever we divide a line in parts, these parts are themselves lines that are further divisible, so that it is impossible to complete the process of division.Points are generated as a by-product of each division, but are not its intended outcome.As all other limits, they are ontologically dependent on their bearer, so that they come into being at the same time as the part.29This means that it is impossible to divide a limit from its bearer: it is impossible to 'peel off' the limit, because the body cannot subsist without a limit and the limit cannot even be conceived without its bearer.Exactly as Epicurus' minima, Aristotle's limits are 'distinguishable, even if not imaginable as existing per se' .
Conceiving of minima under the model of Aristotelian limits is a very powerful tool.Among other things,30 Verde suggests that it might allow Epicurus to block the problem of the arrangement of minima in the atom before it arises: the minima are arranged 'in their own peculiar way' because they do not exist independently of the body to which they belong.This means that there will never be a case in which a plurality of minima need to be put together to create an atom; they are not independent components which need to be arranged.Epicurus might thus think that it is not necessary to look for an explanation of how the minima can come into contact without overlapping, because the situation can never arise.
What interests me most, though, is that thinking of the minima as similar to Aristotelian limits allows Epicurus to explain how the atom can be uncompounded and indivisible, yet internally differentiated: the minima are similar to limits in that they are distinguishable but not separable from the whole.I believe that this insight is crucial, for it shows that Epicurus is working with a notion of divisibility that does not fit precisely into Furley's 28 Ph. iv.8, 215b13. 29 See Metaph. B.5, 1002a34-b5. See Pfeiffer (2018: sect. 6.3) for an account of Aristotle's theory of limits and their relation to the thing they delimit.30 For example, explaining how atoms can move.See Verde (2020) for the details, which go beyond the scope of this paper.distinction between physical and theoretical division.The process by which Aristotle identifies the limits of a body is a special kind of divisibility, which is theoretical (in Furley's sense) in that the parts that result are not physically separated, but that is not geometrical.
It is important to stress that the analogy between Aristotle's limits and Epicurus' minima is not all-encompassing.Because of their lowerdimensionality, Aristotle's limits are indivisible in a very strong sense.Having measure 0 (along at least one dimension), they are absolutely indivisible (along that dimension): it is impossible to divide a point or to slice a breadthless line, both physically and geometrically.On the contrary, Epicurus' minima do have some three-dimensional extension: they are very small, but have the same number of dimensions as the atoms.This is clearly stated in §59, where Epicurus stresses that the minima provide the measure of magnitudes (katametrema).This means that even if Aristotle's limits are in fact both physically and geometrically indivisible, we should not use the analogy to infer that minima are, too.On the contrary, we have good reasons to believe that the minima are geometrically divisible, for they are extended.
What is the point of the analogy, then?I believe that it focuses on a different aspect of Aristotelian limits; not on their geometrical indivisibility, but on the very specific way in which limits are identified as distinct from their bearer.In Metaphysics Δ, Aristotle defines a limit as the 'terminus of each thing, i.e. the primary thing beyond which it is not possible to find anything [of the object], and the primary thing within which everything [of the object] is' .31This definition picks out the limit on the basis of the function that it fulfilsnamely, delimiting its bearer from its surroundings.
This function cannot be fulfilled by the whole, but rather calls for internal differentiations: when we identify the limit, we track back the function to the minimal element that fulfils it.Call this element the 'realiser' of a function.This procedure can be extended to any function of a body or element; given a certain function, there is something that realises it.The realiser of a certain physical function might be the whole body, a part of it, or a sub-element that does not respect the Aristotelian conditions for parthood, such as limits.Identifying such realisers is a mental operation that differs both from physical and from geometrical division.3234 See Furley (1967); Long & Sedley (1987).Notice that this point is different from arguing that theoretical divisibility is implied by the mere possession of shape (cf.section 1).Here, theoretical divisibility is necessary to account for differences in shape, when at least two items are taken into account.This means that the minima can have a shape without being divisible: the only requirement is that they all have the same shape.The same goes for spatial extension and size.
I submit that this mental operation is the one by which Epicurus identifies the minima in the atom.If this is right, it is easier to understand how it might be possible to argue that (i) the atom is theoretically divisible but (ii) not ad infinitum.If the process of theoretical division is not geometrical in character, but consists in tracking and mapping the physical functions of the whole onto the components of the whole that fulfil them, the division must stop as soon as we reach the minimum element that fulfils a given function.This element, the minimal realiser of the function, might happen to be geometrically divisible; if its geometrical parts do not fulfil any independent function, however, it counts as indivisible for Epicurus' purposes.What he is concerned with is the internal 'functional' differentiation of the atom.
My suggestion is that the minima are the smallest elements identifiable in the atom because there is no physical function that needs to be realised by anything smaller than the minima.This is equivalent to claiming that physical processes only need a finite (albeit possibly very large) number of realisers.This is sufficient to justify the imposition of a lower limit to this special kind of divisibility, because it is a process that (i) is theoretical in the sense that the elements are identified by the mind and not physically separated, but (ii) tracks physical functions, so that the modality in question is physical.
Notice that this theory encompasses Vlastos' claim that the minima are the common unit of measure of all atoms, and his physical law L(I).My proposal goes beyond Vlastos' in two respects.The first is that being a unit of measurement is only one of the many functions for which the minima are the minimal realisers -namely, the function of accounting for differences in atomic sizes.Examining all the functions that the minima realise goes beyond the scope of this paper, but a few have been identified in the literature: -The minima delimitate the atoms and allow them to come into contact without overlapping; -The minima allow atoms to move;33 -The minima account for the differences in the shapes and sizes of atoms.34theoretically dividing the whole in physical chunks of matter on the basis of the physical function that they perform.33 See Verde (2013;2020).Each of these functions only needs a limited number of internal differentiations, and thus allows us to identify a finite number of realisers.The minima are the smallest realisers of any atomic function: no further distinctions can be made on the basis of the behaviour of the atoms.Recall that atoms themselves are too small to be visible, and that Epicurean epistemology only allows to posit the existence of imperceptible elements (i) on the basis of an analogy with perceptible objects, or (ii) because they are required to explain some macroscopic phenomenon.If there is no physical function that requires an internal differentiation of the atomic minima, we are not entitled to posit it.
The second way in which my proposal goes beyond Vlastos' is that I believe that Epicurus takes this physical theory to be able to block the paradoxes of divisibility, which (as we saw) are of some concern to him.One might wonder how a physical theory is supposed to achieve this result, especially given that the minimal realisers identified by the procedure are (by explicit admission of Epicurus) spatially extended and thus geometrically divisible.Under my interpretation, Epicurus' strategy does not require him to deny that the minima are geometrically divisible.To block the paradoxes, he only needs to establish that (i) tomē reaches an end, and that (ii) metabasis only involves a finite number of steps.He can do that by positing some constraints on what count as a legitimate outcome of tomē, and what counts as a step of a metabasis.If I am right, Epicurus wants to interpret tomē as the progressive analysis of a whole and identification of the (smaller and smaller) realisers of its functions, and metabasis as the operation of passing over all these realisers, as identified by the tomē.The physical thesis that all physical processes involving atoms only require a finite number of internal differentiations does provide a legitimate justification for positing a lower limit to both these operations.
Is this a legitimate response to the paradoxes?Does this actually solve the problems?Answering these questions goes beyond the goal of this article.However, I would like to point out two things, that might lend some prima facie credibility to my proposal.First, it should be noted that the formulation of the paradoxes was far from fixed, and scholars are still divided as to which kind of divisibility is invoked by the paradoxes.It is thus not so surprising that Epicurus would propose a new understanding of 'division into parts' rather than immediately understanding the paradoxes as arising from geometrical divisibility and looking for a solution to the geometrical version of the paradoxes.Second, it should be noted that Aristotle's own acclaimed solution can be presented in similar terms.35Aristotle agrees that it is impossible to traverse infinitely many parts, but he points out that although a continuous magnitude is potentially infinitely divisible, it is never, in fact, actually divided into an infinity of parts.This intuition allows him to block the paradoxes because it puts some constraints on what counts as a part: something that is merely geometrically divisible does not count as a part until it has been in fact divided, either by a physical or a mental act.The metabasis of a continuous magnitude requires a finite number of steps, even if the magnitude itself is in fact infinitely divisible, because the metabasis only requires one to go through the parts that have actually been identified.Obviously Aristotle's solution differs from Epicurus' , for he allows for the possibility of operating artificial and arbitrary divisions -thus allowing for tomē to go on ad infinitum.The theory that I am attributing to Epicurus, on the contrary, restricts the possible divisions of a whole to those that single out the realisers of physical functions: thus, both tomē and metabasis are finitary processes.

Conclusion
The goal of this paper was to find a way to salvage two apparently incompatible claims: Furley's claim that the minima are meant to block the paradoxes of divisibility, and Vlastos' claim that the minima should be justified in physical (and not geometrical) terms.To do this, I tried to refine Furley's distinction between physical and theoretical divisibility: while Furley understands theoretical divisibility in geometrical terms, I argued that other, nongeometrical, ways of identifying parts in a physically intact whole are possible.I believe that when Epicurus states that atoms are divisible but atomic minima are not, he is appealing to one of these theoretical, but non-geometrical, kinds of divisibility.
To understand how this special kind of divisibility works, I started from Verde's suggestion that Epicurus' minima should be understood in the light of Aristotelian limits.Limits are distinguishable, but not separable from the whole that they delimit.This is the special kind of divisibility that we are looking for.However, Verde's analysis does not try to provide a justification for the indivisibility of the minima.I explained how pinning down the specific grounds that make the Aristotelian limits distinguishable from the body that they delimit also allows us to understand why the Epicurean minima are not further analysable.My suggestion is that their distinguishability is based on the fact that they fulfil some specific function, that is not fulfilled by the whole: the limits of a body are 'realisers' of a specific physical function, namely delimiting the body and allowing for contact.I argued that this is what grounds the analogy between Epicurean minima and Aristotelian limits: minima should be considered as the 'minimal realisers' of the physical functions of the atom.They are distinguishable from the atom because they fulfil some physical function that the atom as a whole does not, but no further parts can be distinguished within them because there is no physical function that requires their internal diversification.The main advantage of this interpretation is that it becomes possible to claim that there is a finite number of minima without having to adopt an alternative geometry, nor to posit shapeless bits of matter.
Moreover, this suggestion is compatible with the texts.First, it makes sense of Epicurus' claim that the minima are limits, as limits are indeed the minimal realisers of essential functions of each body: they delimitate it from the outside and allow for contact.Second, the introduction of minima can still be seen as the Epicurean response to the paradoxes of divisibility, which, as I showed, are clearly in the background of Ep.Hdt.56-57.
Finally, this reading fits nicely with other aspects of Epicurean physics, and in particular with the later introduction of quantized motion.Positing that there is a finite number of minima in the atom is equivalent to claiming that all physical processes only involve a finite number of realisers.As a consequence, physical processes can only have a finite number of steps.Motion cannot be continuous, for otherwise the minima themselves would need to be part in one state and part in another: for the minima to be functionally undifferentiated, motion must be quantized.A theory of quantized motion is not present in the Letter to Herodotus, but it is famously attributed to the Epicurean school by Simplicius and Themistius.36Interpreting the minima as minimal realisers of the physical functions that involve the atom allows us to track the seeds of this theory back to Epicurus' own work.