A quick reflection and Reza’s post: Euclid’s Elements, a philosophical thriller (Toy Philosophy repost)
Insightful as ever and just plain interesting, Reza from Toy Philosophy blog analyzes Euclid's Elements in Part 1 of a philosophy of mathematics thriller. Thus, I repost the below but - as always, please visit the original as well, even if only to survey the layout. (As an aside, the student taking my Philosophy of Mathematics independent study has been accepted to a paid internship at MIT this summer. She is brilliant and while procrastinates, produces beautifully elegant work. She and I are both learning much about the deep and at times murky philosophical theory which underpins all of mathematical thinking.)
In reposting the below I leave the post in its original form. Typically I italicize posts to indicate that they are not mine, when I repost others' blog posts. In this rare case I leave as-is, only because the post has so many important titles and translations that italicizing it would take away from its coherence.
Among the greatest mathematical treatises in the antiquity and beyond, no title matches Euclid's Elements in simplicity, elegance, popularity and sheer hair-raising brilliance of analytic imagination. It is a book that is accessible to any person who wishes to initiate into that fathomless realm we call mathematics. But the same thing can be said about Elements's philosophical depth. Elements is in fact a book in which the boundaries between mathematics and philosophy completely fade. In this marriage between philosophy and mathematics via the geometric method, we see a form of intuitive mathematics whose results are both sophisticated and non-trivial even in terms of modern mathematics, and a philosophy which points towards possibilities of formal and systematic thinking.
The aim of this series is to dissect the tissue between geometrical and philosophical problems and tropes in Elements, using concepts and ideas situated at the nebulous interstices between philosophy, logic and mathematics. The first few installments will focus on the overall characteristics of the Euclidean universe. But as we proceed, we will shift the attention toward analysis of particular examples.
If you remember the school days, you recall your teacher treated Elements in terms of analytic geometry alone. But Elements is equally a work of philosophy. While it is quite controversial to claim that Euclid was a Platonist, we can imagine that the philosophical climate during the life of Euclid was saturated with Platonic ideas, above all the doctrine of forms or ideas. Even though Euclid might not be at the end of day, a platonist, he nevertheless is preoccupied by the same philosophical concerns which preoccupied the Plato of the late period, particularly the Plato who revised his early doctrine of forms beginning with Theaetetus and brought it to maturity in Philebus.
Suffice to say that the very core of Elements deals with the dialectics between universal forms and fleeting particularities. But this dialectic as I will elaborate is not Aristotelian insofar as it involves something more than the existential interpretation of mathematics or to be more specific analytical geometry—i.e. the correlation between the genus and species—where all the proof of a general concept demands is finding or constructing a particular instance that can be subsumed under the general concept, like this rectangle and rectangularity as such. The framework in which the Euclidean oscillations between particulars and universals is expressed is what Plato calls craftsmanship or world-building which is an enterprise undertaken by the mind. The mindcraft draws as its raw material physical becoming which is only endowed with forms of elemental powers and no higher forms. The process of the craft itself proceeds by way of patterns. A pattern is, however, not a thing, but that by which a thing is structured, made or designed. Moreover, patterns are not discrete. For at each level in the hierarchy of particularities and universalities (e.g., these straight segements-cum-acute angles, this triangle, this particular kind-of-traingle and triangularity as such), there are such uniformities as patterns mainly by virtue of how—rather than merely what—things hang together—that is, the question of structure as the designation of Being.
Yet patterns are not exhausted by how and what things hang together, for they can be patterns of how different patterns hang together. For example, think of Book 1, propositions 2 (henceforth, I.2) in Elements which we will have the occasion to examine in the next installments. In this demonstration, you require not only the patterns by which lines hang together, but also how circles and a straight segment hang together (for the purpose of constructing an equilateral triangle). Furthermore, you should know how different circles, and the vertices of an equilateral triangle—i.e. composite patterns and simple patterns—can hang together in the right way so as to build a diagram that demonstrates the proposition.
Source: Wilfrid Sellars, The Soul as Craftsman
From this brief discussion on the world of Plato-Euclid mindcraft, we can conclude that the process of the craft consists of sensory stuff, patterns after which things are made, patterning patterns (patterns for organizing and structuring other patterns which are greater wholes) and recipes which are instructions concerning how and what patterns pertaining to what material ingredients and/or lower patterns should be mixed together. But the objective of a recipe is to make a product that can in turn be incorporated as an ingredient into another recipe. Therefore, in addition to the above components, there should also be something like a craft test or demonstration whereby
Thus recipes are, broadly speaking, objective principles or practical intelligibilities which have as their ingredients even theoretical intelligibilities as well as more mundane ingredients (e.g., sensations, material things which might be in fact the products of other materials-cum-forms-cum-recipes such as a tanned leather with tumble finishing or in the case of Elements, an equilateral and equiangular pentagon).
Source: Oliver Byrne's edition of Euclid's Elements (II.7)
In short, recipes whose equivalents in Elements are procedural diagrammatic constructions represent the engines of the craft by which not only we can make things but also, demonstrate how materials, products, and even single recipes hang together such that the ensuing craft is a universe—a world-soul—in which all (spatial) relations between things (particular instances) and forms (universals), or forms and forms are articulated and rendered intelligible. But this resulting craft or universe can also be imagined as a universe in which ever more complex forms or higher mixtures (to mikton) can be made. An apposite metaphor for this universe, is a river whose source is a mountain. The limits of the mountain is the earthly ground and a given sky which is demarcated by the snowy peaks. Even though the river's origin is limited by material sediments and heavenly forms—the melting snow—the river soon finds its path along the geodetic path to the sea where strange fauna, forms and adventures await us. But the course of the river is always tortuous as it passes through forests of intermediary forms before it shapes estuaries where the tidal waves of complex forms and discrete instances and patterns meet together. This is nothing other than Plato's vision of the revised doctrine of ideas—the craftsmanship of the soul—where the craft of the mind coincides with a new bottomless expanse of forms. The possibility of constructing a new world or a nested hierarchy of forms from the limited resources of the existing world is the sure conclusion of this vision.
In this respect, the recipe or the ongoing instruction regarding how to navigate between the particular and the universal, local and global has something more than just material ingredients and forms. The recipe consists of elementary ratios and proportions which in Euclid's universe can be compared with principles in Elements which are common notions (principles1) and postulates (principles2) which respectively signify undergirding assertions and elementary construction recipes. Common notions are quantitive assertions or intuitive axioms such as 'Things which are equal to the same thing are also equal to one another.' Postulates, on the other hand, instruct certain kinds of elementary constructions like Postulate 2 that states, 'To produce a finite straight line continuously in a straight line.' Whereas, the relations between principles1 and theorems are deductive to the extent that the truth of the conclusion is contained in the truth of the premise, the relations between principles2 and problems are not deductive for the construction cannot be considered as a deductive inference from the postulate.
Moreover, the focus of a recipe is not restricted to pure construction. The idea of craft as Plato has suggested also entails a function called 'limiting' (to peras). In Theaetetus, Plato speaks of a function that 'freezes or fixes the flux of things' (183a7), or 'make things stand still' (157b7) and limits that which is unlimited, or more precisely, indeterminate (apeiron), thus bringing it into determination and intelligibility. This limiting or determining function is attributed to that of language and logos and is closely associated with measure (metron) which depending on the context can be epistemological, ontological or axiological. In the epistemological context, metron signifies the quantification of the apeironic flux or the continuum of greater-and-smaller into intelligible degrees or grades (e.g., being hot, warm, lukewarm and cold, or being extended this-such and being extended that-such). It is precisely the study of this limiting function that later on via the influence of neo-Platonists on scholastic philosophy culminates in Nicole Oresme's work (Tractatus de configurationibus qualitatum et motuum) on diagrammatic configurations known as latitudes of forms—intensive and extensive elaborations of qualities— which in turn paved the road for articulation of differential equations of motion that scaffolded the revolutions of Copernicus and Kepler.
It is, however, important to realize that quantification for Plato so as for Euclid is not exclusive to the domain of numbers but can also include geometrical-spatial extensions. Once the limiting or determining measure in the latter sense (e.g., line as the limit1 of surface, or the definition of angle as the limit2 of its construction) is established, we can derive determining spatial relations between determined or limited geometrical figures. Only when such determinate spatial relations are available, a diagram can be constructed on previous diagrams so that we can move from one proposition or problem to another.
Finally, in addition to the recipe, there should be such things as craft tests or in Euclid's world, demonstrations. If the Euclidean construction is understood as effecting what we aim to effect via diagrams, demonstration can be thought as a stepwise procedure for confirming that the construction has indeed effected what it says it has. Throughout the course of demonstration that covers every step of the construction rather than only the final result, tests can be executed either as objections (enstasis) or cases made against the current construction or diagram. If the former i.e. objection wins, the entire construction is null and void. But if the case—which can be understood as a diagram model that serves or effects the same purpose in a different context—wins, the construction is not necessarily erroneous, since it might prove or demonstrate the same thing in another diagram or geometrical context (allos).
Moreover, demonstrations are applied to two different aspects of the diagrams or products of the craft:
(1) those attributes of diagrams which pertain to participation (methexis) of elements or part-whole relationships. Such methexis-related aspects can include mereological relationships between regions, and segments or lines which demarcate boundaries as in the case of the notorious diagram in I.1 where two circles whose centers are the two endpoints of the same straight segment should intersect at exactly one point. But there is no explicitly stated rule in Elements guaranteeing that such a configuration would invariably result in an intersection point. Imagine circles made with lines with different breath or thickness, or made of squiggly lines. The result won't be guaranteed to yield an exactly one intersection point. Yet if we see the implicit desideratum of intersecting circles in terms of how the components should hang together from a mereological perspective we can say that given such and such regions and boundaries appear to participate mereologically, the two circles should in fact intersect.
(2) The second aspects are what can be dubbed as analogical (analogon) attributes in the sense that Plato defines them ('ana ton auton logon'), namely, ratios, proportionalities and the equality of non-identicals. Whereas methexis-related aspects are based on the appearance of diagrams, analogical aspects are not concerned with how diagrams look like.
Attributes (1) and (2), therefore, roughly correspond to what Kenneth Manders in Diagram-Based Geometric Practice calls exact (analogical aspects) and co-exact (methexis-related aspects of diagrams) attributes.
The final products of the craft—i.e. constructions which have withstood demonstration or validation—are mixtures (mikton) or determinate complexes which are demonstrated diagrams. Only once such mixtures are available, it is possible to use them as ingredients of another craft or construction.
At this point, it is perhaps necessary to make a brief point about the nature of Euclidean demonstrations as Platonic craft tests. So far I have used the words demonstrations and proofs interchangeably. But demonstrations are not exactly proofs in a technical modern sense—only in the very loose sense of proof (we will return to this point in next installments). Furthermore, even the word demonstration is not accurate for describing the system of Euclid. The phrase quod erat demonstrandum not only should not be translated as that which was require to prove, but also itself is an inaccurate Latin translation of the Greek verb deiknumi whose precise translation is the Latin monstrare i.e. to show. In Second Analytics, Aristotle fully distinguishes deiknumi as an informal and epistemological investigation from apodeiknumi or apodeixis (proof) which has an exact connotation within the lexicon of syllogistic logic as an inference that draws certain conclusions from certain premises. While this comment might appear as a petty etymological indulgence, as Andrei Rodin has detailed in The Axiomatic Method and Category Theory, it indeed has a significant implication given Euclid's own remarks and Proclus's commentary on Elements. The difference between monstration (Euclid's focus) and demonstration vis-à-vis proof suggests that we can arrive at sound and non-trivial results in mathematics without relying on an axiomatic method in the sense we understand it today. Even the Euclidean givens (data) are not exactly formal axioms since not only they are underdefined / undefined but also all the rules for building on the data are not explicitly stated.
Within the framework, we can now see that the genius of Euclid's Elements is not as much in devising new feats of proof and demonstration as it is in setting up a generative space—a unified process of craft—that accommodates all previous works done in analytic geometry. 1. Plato-Euclid's World of Mindcraft
Having gone through this brief introduction, we should now ask: what is exactly Platonic about the universe of Elements? Absent a a more detailed response, the above introduction—particularly, the comparison between the role of construction in Elements and the process of craft in the late dialogues—would be hardly anything other than an impressionistic account. Yet to answer this question, it is also imperative to suspend some of the most dogmatic clichés about the work of Plato inherited from the misinterpretations of Aristotle and neo-Platonists (e.g., the Third Man, the equivocations of ideas with numbers a la Pythagorean arithmosophy and the misrepresentation of the Good as the divine demuirge) as well as their almost exclusive attention to the dialogues of the early and the middle periods. Plato is notorious for being the most watchful and unforgiving critic of himself. So the answer simultaneously calls for a direct engagement with the dialogues, particularly, the later ones and a critical correction of Aristotelian-neoPlatonic commentaries which make almost the entire body of Platonic studies until the late nineteenth century—a trend that comes to an end with the rise of Marburg, Tübingen and analytical schools of Platonic studies as represented by figures such as Natorp, Reale and Vlastos.
We know that after the second trip to Syracuse, Plato became critical of his early doctrine of forms (e.g., Parminedes) as represented in the works of the middle period such as The Republic. He began to see forms as classificatory universals, namely, categories or ta koina (see Theatetus and Sophist). As ta koina, forms or ideas no longer have the earlier characteristics of the Socratic and Pythagorean theories of forms, or at least such characteristics are not prominent anymore. The inception of this new doctrine of forms or ideas begins with the transitional dialogue Theatetus, but it is only in Philebus that Plato gives a complete account of his new doctrine.
According to this new thesis, the aim of the doctrine of forms is Demiurgen, world-construction or craftsmanship of the mind. In Timaeus, we are dealing with god as the Demiurge but in Philebus, this abstract divinity is suddenly replaced with a neutral word, to demiurgen. It is now the human mind that is akin to the good which is beyond all gods and beings and even truth and beauty, and not the god as the ideal of the nous. This manifestation of the good is like a recipe or an objective principle for building worlds. It is a recipe precisely in the sense that Wilfrid Sellars talks about a recipe for making a cake, a recipe consisting of theoretical and practical intelligibilities.
If you have made a cake from scratch, you know very well that it is not an easy task. For a a recipe for making a cake—unlike a recipe for making a soup—involves precise ratios, proportions and stages of how and what elements should be added together. The formulas of this recipe are what called objective principles or rules as in contrast to the social nomos or conventions. To build a house as a shelter (the external purpose of the construction), we ought to abide by such and such principles like taking care of the foundation, beams, etc. The specific formula of how we lay the ground or what beams—made of out of what materials—we use might change over time, but the objective principles endure. A house needs a foundation and a ceiling even if the foundation is bottomless and the ceiling extends to the sky. These principles pertain to the domain of forms or ideas. While the nomos is always prone to corruption (as in the case of the codes of building issued by a corrupt builders guild which dictates that all houses should be built out of the material ingredients over which it has sole monopoly), objective principles are genuine objects of rational examination and revision.
Parallel to this Platonic account, the fleeting shadows on the wall of the human cave could not even be recognized if some dim light was not present in the cavern. This light is not a literal analogy for purity, it is rather a metaphor for intermediating forms or universals, the mathematicals or analytic idealities. These are construction principles which intermediate between pure ideas and eikones or sensory shadows. In this sense, Plato is the enemy number one against the myth of the given, for he thinks that the structuring factor is not within the domain of sensory fluxes—the fleeting shadows or eikones—but in the dim light i.e., intermediating forms which imitate the light of the sun qua pure forms or generalized structures: that is to say, mind as the dimension of structure.
In Philebus, Plato makes that claim of impiety for which Socrates was executed. He says the human mind is akin to the Good. We know that what Plato means by the Good in Philebus is the principle of structure (the kernel of intelligibility and intelligence which is even more fundamental than truth, beauty or justice). A few pages later Plato tops up his thesis with a new claim, 'and the good is beyond all being'. In other words, Plato suggests the structure—or the mind as a configuring or constituative element—is the very factor by which Being comes to the fore and can be talked about coherently. Plato's articulation of Being in terms of intelligence or mind is quite similar to the view of the mature Parmenides who has relinquished the early Eleatic confusion of Being and thinking, and instead interprets the thesis of 'Being and thinking are one' as thinking or structure being the very designation of Being. To speak of Being without the dimension of structure or mind is the apotheosis of sophistry and the aporia of the unintelligible (cf. Lorenz Puntel's Structure and Being).
However, the dimension of structure or in Plato's terms the limiting (to peras) is not an index of solipsistic idealism, for it requires a fourfold view of the universe qua structure where episteme not only gains traction upon an external world but also thoughts or more generally, intelligence (nous) is no longer passive. Intelligence is now defined in terms of what it does—the unfolding of the intelligible even that of itself or the enrichment of reality—and not in terms of passive receptivity of an external reality. Accordingly, the Platonic fourfold view is defined in terms of an activity called craftsmanship whereby through various ingredients, structuring factors (logoi) and principles (dialectica) intelligence makes itself and reveal the intelligible dimension which is that of Being. But insofar as there is no a priori limit to the intelligible, there is no limit to the self-cultivation of intelligence or the poesies of mindcraft either. The twist in this scenario is that the mindcraft or intelligence posits qua an an active rather than a passive factor of intelligibility, it also has the capacity—as Rosemary Desjardins elaborated—to posit (tithemi) a new kind of reality pertaining to both Being and itself (see Plato and the Good, p.61).
The Platonic fourfold as presented in Philebus is nothing but a new interpretation of the analogy of the divided line in The Republic. The divided line is a diagram of how global conditions of thinking, action and value can be related to the local conditions. It consists of four segments which give us four domains with their corresponding modes of cognition/sensation, episteme (knowing) and their objects. From segment one to the segment four we have eikasia(eikones), aisthesi or pistis (aisthêta), logos dianoia(mathêmatika) and epistêmê(ideai).
The genius of this diagramatic analogy is in identifying the extreme segments (segments 1 and 4) under two modes of relations to time. The true forms or ideai are timeless or time-general whereas the sensory eikones are time-specific or temporal. In a sense, the divided line is about how what is timeless connected with what is temporal, how the oneness is mixed with the multiple, or how pure forms gain traction upon and are connected with the sensory shadows. The answer lies in the intermiadting domains or segements which are represented in the divided line as mathêmatika and aisthêta / pistis.
So what is the significance of these intermediating levels? Recall that sensory fluxes of eikones or imagistic impressions are too transitory to be arrested as anything you might call a sensible object. Pure ideas in a similar vein are too detached from particularities to gain traction, by themselves, on the worldly or the cavernous affairs. Another problem is the question of how oneness (of pure forms) as an organizing principle comes into contact with the multiplicity of things. The ideas are multiples but individually each idea is always a unique kind of form (i.e. it is one). On the other hand, eikones or what you might call registers of the apeiron—that is, the indeterminate and transitory flux of smallers and greaters. At the level of the first segment which is that of eikasia whose objects are the fleeting imagistic impressions eikones, there is no such a thing as multiplicity of things. Why? Because even multiplicity of things require a principle of unification. It is only when we organize the fleeting sensations as the affects impinged upon us by one and the same object (here, the object is the higher principle closer to ideas or formal constitution) that the fleeting sensible shadows become multiple things, this shadow-puppet, that shadow-puppet, etc. So the question of multiplicity does not even arise at the level of pure sensation. It only arises at the level of opinions or dogmas regarding the appearance of objects. In otherwords, it is only when the mind posits a thingly whole (object or in Kant's sense gegenstand) which binds together different properties that we can talk about multiplicity of either properties or sensible things. The following quote by Desjardins should shed some further light on the matter:
For, on the one hand, a physical object seems to be distinctly different from any or all of its properties: they are quite separate kinds of things; on the other hand, what is exactly a physical object over and above its physical properties? While there is no difficulty in thinking of a physical object that has no actually perceived properties, our notion of an object seems nevertheless to be such that it does not make sense to talk of a physical object that has no perceivable properties: such a notion of a bare particular seems incomprehensible. This of course, only exacerbates the question, however, for what then is the relation between an object and its properties? We seem to be hoist on a dilemma in which, on the one hand, we want to say both that, in some elusive sense, the object and its properties are different, and that, in no less an elusive sense, they are somehow the same; and on the other we want to say that the object is neither simply the same as, nor simply different from, its properties. But, as the Parmenides suggests, if the relation between two things is neither sameness nor difference, then perhaps it is that of whole and part (l46b3-5). Plato's model for such a relation does seem in fact to be what he conceives of as a whole of parts, where on the one hand, the whole is nothing other than the parts (there is nothing added to the parts), on the other, the whole is indeed other than (i.e., more than the sum of) its parts. In short, while a whole is analyzable into its parts, it is not reducible to those parts. Thus as I understand Plato, while a physical object is analyzable into its physical properties, it is nevertheless not reducible to those properties.
Thus, the whole of the sensible object—like the moving shadow on the wall—we can conclude, is not given by sensory fleetings, but is in fact the product of what can be called transcendental constitution—a semblance of what Plato calls intermediary forms qua mathematicals. Therefore, the multiplicity of the physical furniture of the world is not given to us through sensory eikones, it is engineered—a la positing a new kind of reality—by the semblance of the higher principles which are mathematicals qua objects of logos dianoia.
But now a new question raises its head: What are mathematicals and what is their role?
I will answer this question in the next installment, until then, ciao.
