Title: Aristotle's metabasis-prohibition and its reception in late antiquity
Other Titles: Aristotle's metabasis-prohibition and its reception in late antiquity
Authors: Steinkrüger, Philipp Julius
Issue Date: 22-Sep-2015
Abstract: This dissertation deals with an important topic in the history of the theory of scientific knowledge, a theory which became the paradigm for science for the next two millennia. It is well known that Aristotle characterized scientific knowledge with two conditions: first, it must be necessary; and secondly, knowledge is only scientific if the reason or cause of what we know is revealed. To give an example, the theorem that the interior angle-sum of a triangle is 180° is a necessary truth. In order to show why the triangle necessarily has this angle-sum, we have to conduct a demonstration that reveals why the triangle’s angle-sum is necessarily 180°.
Large parts of the first book of the Posterior Analytics are concerned with the foundations and conditions of such demonstrations. One of these conditions has been called the metabasis-prohibition. It demands that a demonstration does not cross (Greek: μεταβαίνειν) from one γένος into another or from one discipline into another. But despite the fact that Aristotle provides a number of examples for such illicit crossings (a demonstration may not cross from arithmetic to geometry; a certain demonstration attempting to determine a square equal in area to a given circle offends the metabasis-prohibition), it is not clear what Aristotle understands under a γένος and why precisely metabasis is problematic for scientific knowledge.
Previous attempts to shed light on this question maintained that Aristotle considered each discipline to be concerned with a specific range or realm of objects (also called ‘domain’) and that the word γένος refers to this realm. This notion of a γένος, its demarcation and internal structure, has been seen as explanatory for the metabasis-prohibition by almost every interpreter in this and the last century. However, making the notion of γένος as the domain of a science explanatory for the metabasis-prohibition (I refer to this interpretation as the ‘domain-interpretation’) leads to severe problems and therefore has to be abandoned.
A textual problem for the domain-interpretation can be found in Posterior Analytics I.7, where Aristotle says that demonstrations prove the per se (καθ’ αὑτό) attributes of a γένος, but that this is not possible in a demonstration that crosses from one γένος to another. But Aristotle’s theory of per se predication (i.e. his theory of essential properties), as exhibited by the foregoing chapters of the Posterior Analytics, does not allow for a realm of objects to have per se attributes. Perhaps this problem could be solved by allowing the term γένος to be used ambiguously in the respective passage (and, in fact, every interpretation, including my own, is to some extend forced to except such an ambiguity). However, there are more pressing, systematic problems.
One of these problems is the following. According to the domain-interpretation, metabasis signifies the crossing from one domain of a science to another and hence the question how a domain is being demarcated has dominated the previously proposed explanations of the metabasis-prohibition. According to the majority of the interpreters, the domain is demarcated by the most general term or terms of the science (in the case of geometry, for instance, this is often said to be ‘magnitude’ or ‘extended quantity’), which subsume the other terms of the science. A different interpretation holds that instead of such a class-inclusion system, Aristotle thinks of the domain of a science as a network, in which subjects and attributes are linked to a term or a class of terms that demarcate the domain (again, this could be ‘magnitude’ for the science of geometry). Both interpretations are faced with a serious problem when the proposed relations between the subjects of the science and their respective highest terms are scrutinized. For if the science of geometry is demarcated by the term magnitude, which subsumes all other subjects (such as triangle, etc.) by way of class-inclusion, then there does not seem to be a non-ad hoc way of preventing the postulation of a more encompassing science whose highest term is quantity. Quantity, however, would subsume both discreet quantity as well as extended quantity (i.e. magnitude) in the same way as magnitude subsumes triangle and hence in this science there would be, according to the domain-interpretation’s assumption, no metabasis between arithmetic and geometry, a result that is plainly in contradiction with Aristotle’s claim. The alternative network-interpretation can be shown to suffer from the same problem. The only way out, embraced but at least one interpreter, is to accept that the domains of sciences are stipulated and hence to accept that whether or not metabasis obtains is a question of stipulation. However, since Aristotle introduces the metabasis-prohibition as a condition for the possibility of scientific knowledge, this is not a viable option.
My thesis provides a new interpretation to help understand Aristotle’s metabasis problem. I argue that Aristotle clearly indicates that the metabasis-prohibition is a consequence of his investigation of per se or essential properties and that we therefore have to explain metabasis with reference to this investigation. A property is essential for an object if the object were not what it is if it did not have this property. For example, a triangle is only a triangle if it is a figure bound by three straight lines with an interior angle-sum of 180°. The properties ‘figure’, ‘bound by three straight lines’ and ‘having an interior angle-sum of 180°’ are all essential properties of the triangle. Aristotle’s way of expressing this is to say that the triangle is a figure in itself or qua itself, i.e. in what it is by itself, or in virtue of itself.
But not all essential properties are equal. Aristotle distinguishes between basic essential properties, on the one hand, and those essential properties that are dependent on them, on the other (i.e. dependent essential properties). In the case of the triangle, ‘figure’ and ‘bound by three straight lines’ are examples of basic properties, while ‘having an interior angle-sum of 180°’ is an example of a dependent essential property. While basic properties enter into the definition of the object, the dependent properties have to be demonstrated to belong to the object by reference to the basic properties. This is the case because the basic properties are the reason or cause why the dependent properties belong to the object, and revealing why a property belongs to an object is the mark of science in Aristotle’s understanding.
I argue that pseudo-demonstrations committing metabasis do so precisely because they do not conduct the demonstration from the essential properties of the object of which they attempt to show that it has a certain property essentially. Such a demonstration, then, will not reveal the reason why a given property belongs to an object. Hence, the kind of knowledge produced by such demonstrations is not scientific knowledge. One immediate consequence of this interpretation is that metabasis can happen within one and the same discipline; previous explanations held that metabasis only happens between disciplines like geometry and arithmetic. But that is not the case, because a discipline comprises many different objects (like triangle, circle, square) and the basic essential properties of one such object will not be the reason or cause why certain properties belong to another object. Because of its emphasis on the relations between essential properties of a kind, I call my interpretation the essence-interpretation.
In this interpretation, then, the term γένος primarily signifies the respective kinds (e.g. triangle, square) the demonstration is about and which the demonstration shows to possess certain properties in virtue of the kinds’ basic properties. Apart from the above-mentioned passage in Posterior Analytics I.7, in which Aristotle explicitly says that a demonstration shows that a γένος possesses certain properties – and which turned out to be problematic for the domain-interpretation – there are further passages in I.4-6, which corroborate this use of the word γένος. However, the essence-interpretation does not deny that Aristotle had a notion of a domain of a discipline and that he intended the metabasis-prohibition to also restrict crossings from one such domain to another. However, such domains are not explanatory for the metabasis-prohibition; rather, a crossing between two such domains is seen to be derivative, since, if we suppose, e.g., a genus-species ordering the various kinds of reality, a crossing between domain will be a crossing between kinds.
This interpretation can be shown to deal much better with the above-mentioned problems. A possible change in the meaning of the term γένος can be explained with reference to the primary-derivative relation that I just pointed out. The problem of the expansion of domains and the related problem of stipulation can also be avoided: since metabasis can also happen within a discipline, an expansion of the domain of a science does not, in the essence-interpretation, validate a demonstration previously seen to be invalid because of metabasis. And since this is not the case, stipulating what kinds a given discipline deals with does no longer affect the conditions of the possibility of knowledge.
Moreover, as I try to show in the remainder of the first part of the thesis, none of the passages of the Posterior Analytics that have a direct relation to the metabasis-prohibition is incompatible with my interpretation, and often we gain an exegetical advantage and a better understanding of the text. The examples that Aristotle mentions in I.7 and I.9 are now open to better, often straightforward explanations. Furthermore, two claims that stand in a direct relation with the metabasis-prohibition, and which take a central place in the Posterior Analytics, can be better explained from the viewpoint of the essence-interpretation, namely the claim about so-called subordinate sciences and the claim that the principles of a science cannot be demonstrated. Especially the former posed great problems to the domain-interpretation; the genus-species and network-interpretation have a hard time showing why subordination proofs are exempt from the metabasis-prohibition, as Aristotle says. The essence-interpretation, on the other hand, can explain the exception with reference to the fundamental requirements of scientific knowledge, as they are stated by Aristotle in the beginning of the Posterior Analytics and drawn out in his theory of essential belonging.
I finally consider two of Aristotle’s treatises on natural science and show that he appeals to the metabasis-prohibition in his criticisms of certain demonstrations of his fellow natural scientists. In the Generation of Animals, we find Aristotle criticising a certain proof about the sterility of mules and he unequivocally points out that the problem of this proof is metabasis. According to the domain-interpretation, this proof should somehow violate the borders of the domain of the science of biology or zoology by crossing over from another domain of a science. However, in Aristotle’s own description of the offending proof, this does not seem to be the case: all terms are zoological and there is no mention of a different science. Rather, the problem of the proof is that it did not show why the property ‘sterile’ belongs to mules from the basic essential properties of the kind to which sterility belongs. A similar result can be drawn from the analysis of a number of passages of the De Caelo. Hence, the essence-interpretation proves to be superior to the previous explanations of metabasis here, too.
In the second part of the dissertation, I turn to the ancient legacy of the metabasis-prohibition. There are two reasons for this. First, by considering the two texts that deal directly with the Posterior Analytics, namely a paraphrase written by Themistius and a commentary written by Philoponus, I show that there are reasons to believe that both authors understood the metabasis-prohibition along lines similar to the essence-interpretation (Philoponus more clearly so than Themistius). Secondly, it appears that the metabasis-prohibition poses a problem for the late ancient philosophers. This comes especially to the fore in the works of Proclus, who refers to the metabasis-prohibition in his Commentary on Plato’s Timaeus (a work on cosmology and natural philosophy) and in his Commentary on Euclid’s Elements. While Proclus explicitly agrees with the metabasis-prohibition in some passages, he appears to violate it in others. This is evident when he, for instance, claims that demonstrations in natural philosophy employ theological premises and when he says that the automorphism of the numbers 5 and 6 has to be demonstrated with reference to the circle. One particular area of importance in this respect is Proclus’ doctrine of ‘geometrical atomism’, which is part of his natural philosophy. This doctrine states that the elements out of which all material reality is constructed are certain geometrical bodies (fire, for instance, is a pyramid) and that the properties of the elements (e.g. being hot) depend on the geometrical properties of these geometrical bodies. Here again it seems that we have an obvious case of metabasis (from geometry to natural philosophy). Moreover, in some sense one should even expect Proclus to violate and indeed to reject the metabasis-prohibition, for the Neoplatonists emphasize that reality and knowledge about reality possess a great pervasive unity.
In seeking to solve the tension in Proclus’ view of metabasis – accepting the prohibition but also apparently violating it – my analysis came to the following conclusion. Since the ultimate justification of the metabasis-prohibition lies in the theory of essential properties, one should expect that the demonstrations that appear to be guilty of metabasis somehow violate the strictures of this metaphysical background theory. However, it turns out that this is not the case. For Proclus’ theory of essential properties, and in particular his theory of dependence of properties, differs from that of Aristotle. The way certain objects come to possess properties can be described from two directions. From the point of view of the object, it can be said that an object possess a property because it participates in a reality that is, in the hierarchy of Neoplatonic reality, higher than itself. This is the bottom-up perspective. The same relation can also be described from a top-down perspective: every part of reality derives its being from a procession from higher realities with the One, the highest Neoplatonic principle, which is located at the top of this hierarchy. One distinctive feature of this metaphysical system is that the higher element from which a lower element receives one of its properties does not have to possess this same property itself. Rather, what this higher element possesses is the power to bring about the respective property of a lower element, or even this very element itself.
Building on this metaphysical background theory, Proclus aims to show how certain objects a given science deals with receive their identities and properties from higher elements of reality. In the Euclid Commentary, he outlines how demonstrations of these relations should be conceived. For instance, the automorphism of the numbers 5 and 6 has its cause in the circle and hence a demonstration why these numbers are automorphic has to start from the essence of the circle. This appears, from an Aristotelian perspective, to be a metabasis-demonstration, and indeed one of the crossings that Aristotle explicitly forbids in the Posterior Analytics (namely from geometry to arithmetic). Proclus, however, argues that automorphism is a form of circularity and that the geometrical circle is likewise a form of circularity. Both the numbers 5 and 6 as well as the geometrical circle receive their respective properties of circularity from a higher principle that neither belongs to arithmetic nor to geometry, but possess circularity in the purest way. Since Proclus holds that the circularity of the geometrical circle and the circularity of the numbers 5 and 6 have their cause in this higher principle, a demonstration revealing this relation takes the form of a subordination demonstration and is not guilty of metabasis. On the contrary, the relations that these properties have to their higher principle demand that a demonstration, if it aims at obtaining the highest form of scientific knowledge, does make the respective reference.
This result suggests that the metabasis-prohibition should be understood as a purely formal theorem. The prohibition is, in accordance with the analysis of Aristotle’s argument for it, a consequence of the theory of essential properties. The concrete effects of the prohibition, i.e. the question which demonstrations are seen to be violating it, depend on the given theory of essential properties. Since Proclus’ theory differs from Aristotle’s, the two philosophers will see different demonstrations as violating the metabasis-prohibition.
However, even taking into account the competing theories of essential properties, there still remains room for differing views on the question which demonstrations violate the metabasis-prohibition. For two philosophers who agree on the theory of essential properties might still give different answers to the question what the basic essential properties of a given object are. For instance, it is not at all self-evident that light propagates along straight lines, as Aristotle held. For this reason, according to Aristotle, demonstrations about the propagation of light (the science of optics) need to refer to the geometrical properties of straight lines. But a different philosopher could, while agreeing with Aristotle’s theory of essential properties, hold that light propagates along curved lines and hence he would see demonstrations about the propagation of light that make references to straight lines as committing metabasis.
The metabasis-prohibition, then, is formal. The effects of its application are determined by two factors: one, the theory of essential properties; and the other, the particular views a given scientist takes towards the essence or identity of the objects he wants to conduct demonstrations about.
Table of Contents: Part One: Aristotle on Metabasis

1. The Theory of the Posterior Analytics
1.1. An Initial Look at Metabasis
1.2 Literature Review
1.3. Analysis of the metabasis-prohibition in Posterior Analytics I.7: Aristotle's claim that it is a consequence of the doctrine of per se belonging
1.4. Per se belonging
1.4.1 Background of the investigation of per se belonging: the notion of science
1.4.2. Necessity and the three conditions in APo I.4
1.4.3. Per se belonging and metabasis
1.4.4. The argument continued in APo 1.5 and 1.6
1.5. The essence-interpretation: summary and evaluation
1.6. Applying the essence-interpretation to the Posterior Analytics
1.6.1. Metabasis and the examples
1.6.2. Subordination
1.6.3. The principles cannot be demonstrated
1.7. Counterevidence
2. Metabasis in other works of Aristotle
2.1. De Caelo
2.2. On the Generation of Animals
3. Summary of the first part

Part two: Late Ancient Responses

4. Introduction
5. Works dedicated to the Posterior Analytics
5.1. Themistius' paraphrase of the Posterior Analytics
5.2. Philoponus' Commentary on the Posterior Analytics
6. Proclus on metabasis
6.1. Proclus' acknowledgement of the metabasis-prohibition in the in Eucl. and in Tim.
6.2. Violations of metabasis?
6.2.1. Proclus' claim that the principles are received from above
6.2.2. Geometrical atomism Plato's geometrical atomism Aristotle's criticism Simplicius' and Proclus' responses to Aristotle's criticism
6.3. Modification of the concept of metabasis vs. modification of (the concept of) essences
6.3.1. First option: modification of the concept of metabasis
6.3.2. Second option: modification of the (concept of) essences Metaphysical background Demonstration of principles from higher principles Aristotelian subordination Proclean subordination
7. Concluding remarks on the nature of the metabasis-prohibition
8. Appendix: A translation of Themistius' Paraphrasis on APo. I.6-9
Publication status: published
KU Leuven publication type: TH
Appears in Collections:De Wulf-Mansion Centre for Ancient, Medieval and Renaissance Philosophy

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