Descartes and Geometrical Demonstration


Reproduction of a proof from Euclid’s Elements (Public Domain: via Wikipedia Commons)


Those long chains of utterly simple and easy reasonings that geometers commonly use to arrive at their most difficult demonstrations had given me occasion to imagine that all the things that can fall within human knowledge follow from one another in the same way, and that, provided only that one abstain from accepting any of them as true that is not true, and that one always adhered to the order one must follow in deducing the ones from the others, there cannot be any that are so remote that they are not eventually reached nor so hidden that they are not discovered. (Descartes, from the “Discourse on Method”, in Modern Philosophy: An Anthology of Primary Sources (2nd Edition) Ariew & Watkins eds. Hackett 2009, p. 31)

In reply to the suggestion that he put forward the arguments of the Meditations “in geometrical fashion so that the reader could perceive them, as it were, in a single intuition” (A & W, p. 70), Descartes responds that, to some degree, that is precisely what he had already done in the text. Despite this claim, and his subsequent articulation and defense of it, however, Descartes also goes on to present the main line of argumentation followed in the work in an alternative way. That path to his main conclusions involves starting with Definitions, Postulates, and Axioms, and then asserting a Proposition, for which a Demonstration is provided making use of the Definitions, Postulates, and Axioms with which he started. From his reply to the Second Set of Objections, we can infer that he takes this mode of demonstration to be more in line with what his contemporaries generally expect from arguments or proofs given in “geometrical fashion”.

Euclidean Geometry and the True and Immutable Natures of Spatial Things

This makes some sense, given that the paradigm of geometrical argumentation for Descartes’ contemporaries, and for generations of thinkers before them, is provided by Euclid’s Elements. The reproduction provided above is a demonstration of the claim that, for any line segment, there exists an equilateral triangle that includes that line segment as one of its sides. Without being able to read the Greek in which the proof for that claim is provided, we can probably still reconstruct how the proof goes, in a somewhat informal, fashion just by looking at the diagram:

To construct the diagram:

  1. Start with line segment AB
  2. Construct the circle that has AB as a radius and A as its mid-point (call it Δ)
  3. Construct the circle that has AB as a radius and B as its mid-point (call it Ε)
  4. Identify one of the two points of intersection between Δ and Ε (call it Γ)
  5. Construct the line segment AΓ
  6. Construct the line segment BΓ
  7. The result is the triangle ABΓ

Now, using the diagram we can see that:

  1. AΓ is a radius of circle Δ, so its length is equal to that of AB (which is also a radius of Δ)
  2. BΓ is a radius of circle Ε, so its length is equal to that of AB (which is also a radius of Ε)
  3. AΓ = AB = BΓ, so the triangle ABΓ is equilateral

The first 7 steps produce the diagram that allows us to intuit the equality in length of the triangle’s three sides. The next three steps allow us to deduce the very same claim that we also grasp intuitively by a process of reasoning that appeals to first principles. These include i) the principle ‘entities’ involved in the construction (points A and B) and ii) definitions for the terms ‘triangle’ and ‘equilateral’, and iii) a formal axiom of equality (i.e., for any two things, if they are equal to some third thing, they are also equal to one another). The constructive activity based on the principle ‘entities’, however, is really just a matter of making what is ‘already there’ as a formal feature of Euclidean space clear and distinct for the purposes of the argument. We are:

  • Drawing out the line segment these two points already define (since any two points define a line segment)
  • Drawing out each of the two circles that already have the end-points of the line segment as mid-points, and the line segment itself as a radius (any line segment defines two distinct circles that have that line segment as a radius, and one of its end-points as its mid-point)
  • Identifying one of the two points at which these two circles already intersect (two circles defined by the same line segment as a radius, with one taking one end-point of the segment as its radius, and the other taking the other end-point as its radius, will intersect in exactly two points) as a third distinct point of reference
  • Drawing out the triangle that is already defined by this third distinct point, together with the two points we started with (any three distinct points defines a triangle).

The construction does not bring these entities into existence. They already exist according to the definitions of ‘points’, ‘lines’, ‘line segments’, ‘radius’, ‘circle’, etc.–we are simply ‘tracing over’ the features of space that are most relevant for our purposes–fixing them clearly and distinguishing them from all the other countless formal features that space has–and assigning conventional symbols to them for the purposes of reference as we proceed from one step to another.

Now, think about this: A and B are not points within any determinate coordinate system. There is no particular value assigned to the length of line segment AB. If we assign A the value of 0 and B the value of 1 (or -1), we thereby determine the length of AB to be 1, but there is nothing compelling us to do this in this particular case; i.e., nothing in the proof requires us to do that and we don’t get anything extra out of doing it. That means A can be any point anywhere in any space, B can be any other point anywhere else in that same space, and AB can be a line segment of any length whatsoever. All we are really dealing with are the formal relations between: any one point in space and any other point in space; any two circles that each have one endpoint of the same line segment as midpoint and the line segment itself as radius, and between the three sides of any triangle that is defined in this manner. Because of this, whatever we can conclude about AB through this process holds equally for any line segment whatsoever. In other words, we have demonstrated that, for any given line segment, there exists an equilateral triangle that includes that line segment as one its sides.

A Generally Platonic (Neo-Platonic/Augustinian/Cartesian) Perspective Applied to the Euclidean Example

So, what we ‘see with our eyes’ when we look at the diagram is simply a single drawing with two circular figures, a triangular figure, and some conventional symbols taken from a couple of different alphabets… what we intuit with the mind if we are thinking/looking at it in the right way, however, is a universal truth that can be expressed in a countless variety of ways in an equally countless variety of ‘concrete’ things that are produced over a period of time, maintained for some time, and destroyed over a period of time. These ‘concrete’ spatial things come to be and pass away. Their geometrical forms, however, are eternal and unchanging. These forms are not grasped by the senses–they are not transmitted from the concrete bodies that have them into our sensible bodies, where they then inform our minds in ways that give us ideas. These forms are grasped by the intellect through its own act of reflection on the geometrical features of the space in which it presents things to itself as bodies.

These formal features are involved just as much when the mind imagines some possible body somewhere in space, or even when it imagines some space supposedly devoid of any bodies, as when it ‘sees a body’; i.e., takes itself to be focusing on some actual thing that is currently interacting with the body with which the mind is somehow unified in such a way that the overall process is causing sensations in that body of which the mind is directly aware, or sensory ideas in the mind. According to Plato, studying geometry is a necessary prerequisite to doing philosophy, in part, because it trains the mind to move away from the continuously changing qualities of ‘actual’ material objects, first, to the unchanging spatial forms that these objects instantiate and, then, to a grasp of the basic principles of these spatial forms. This provides a step in our intellectual grasp of what is eternal and unchanging, though geometry itself doesn’t get us all the way to a grasp of metaphysically first principles–for that we need training in dialectic (we’ll get to that later…).

So How Does All of this Relate to a Geometrical Mode for Meditations on First Philosophy?

We get something resembling the proof structure outlined here in Descartes’ reformulation of his main line of argument in the Meditations. However, we do not get anything like the corresponding diagram, when it comes to proofs for the existence of God or for the real distinction between the mind and the body. Of course, we don’t get these diagrams in the Meditations either. Why is that and how do the Meditations provide something analogous that is designed to let the reader “perceive [its arguments], as it were, in a single intuition”?

First off, we don’t get diagrams because such things are useful only when what we are trying to intuit is some universal or particular feature of the outward forms of sensible things. Diagrams depend on a kind of visual mapping relationship between the parts of an abstract representation (drawing, blueprint, model) and the parts of some multiplicity of (real or possible) things that can express the same visible form in concreto. I can draw a diagram the parts of which correspond to the physical parts of an object I want to build, or I can build a plastic model the parts of which correspond to the parts of a human heart. In these cases, there may be differences in size, scale, dimensions, and raw materials between the digram and the object, and between the model and the real thing, but there are crucial similarities that allow the one to stand in for the other in certain respects. These similarities have to do with the fact that each of the objects involved is represented as extended–a drawing or a blueprint provides a 2-D representation of a (real or imaginary) 3-D object; a model provides a 3-D representation of a (real or imaginary) 3-D object.

If we are committed to the view that to be is to exist as some particular thing extended in space (or as some actual property or modification of some thing that exists in this way) this feature of diagrams or models won’t be likely to bother us that much. However, this view of being seems like an optional (if not entirely arbitrary) assumption, rather than the result of some convincing demonstration or explanation. There are all sorts of reasons why one might wonder if that is true, or wonder how we might be able to know that it was true (if it were), and many people take themselves to have pretty convincing reasons to believe that it is actually false. We don’t need to go into all of these considerations in any detail here, but we can think about the burden of proof as lying with someone who claims to rule out the very possibility that anything could exist without being a particular body, or a property of a body, or a relation between bodies. Even if it may sound plausible on the basis of our experience, given our natural preoccupation with bodies, it is hard to grasp exactly what the argument for such a strong ontological conclusion could look like. Materialism (as far as I can tell) always has been and, presumably, always will be an attractive position for a whole lot of reasons. It is also a position that (again, as far as I can tell) always has been and, presumably, always will be open to serious objections on a number of important points.

Anyway, the point here is that it is not obvious to everybody who appears to have their wits about them and who appears to understand things like math and physics (in the 5th Century BCE or in the 17th or 21st Century of the Common Era) that everything that is real is extended in three-dimensions and divisible into parts. So, can we even think about the possibility and or the existence of such things? Can we know that such things are really real (or that they exist) and are not just imaginary things that aren’t even really possible, or really possible things that don’t happen to exist? Descartes clearly thinks we can: I am, I exist, I am a thinking thing, or a mind; I have an idea of infinite perfection, which leads me to accept that there is another mind incomparably greater than my own that exists; I have all these other ideas of the true and immutable natures of things and of their outward forms–these ideas would still be real, and they would still allow me to determine the objective features of possible things, even if no existing things ever have been or ever will be designated by these ideas. The mind, its thoughts, and the ideas it focuses on when it is thinking can certainly represent space and bodies to itself, but does the mind have to be in space as a body, or as a modification of some body, in order to do this?

My mind would seem to be aware of itself less as an extended thing among other extended things, or as a property or modification of some such extended thing, and more as a point from which the extension of bodies (including my own) can be perceived from the outside, and the inner workings of my own body can be perceived from a privileged viewpoint that is not available to other minds (at least as far as I know). I intuit my body as having parts that can be distinguished from one another and that can be divided into further parts (that can be divided into further parts, etc.) but I do not intuit my mind in the same way. I have states of mind, but they give way to one another with no clear divisions or breaks, and these states don’t present themselves as divisible into other states (that are divisible into other states, etc.). I have ideas that represent complexities of all kinds, but these ideas don’t present themselves as being complex–i.e., they are simple, indivisible, unities that are present to the mind all at once. You cannot divide my mental representation of a body into a bunch of smaller representations of its parts in the way, for example, that you can divide a piece of paper with a diagram of a building into a bunch of smaller pieces of paper with diagrams of the building’s parts. The division of the thing I am perceiving does not amount to the division of the perception–my perceptive awareness of the thing being divided involves, first, a unified perception of the thing as unified, then, a unified perception of the thing that was one as now split in two, then, a unified perception of the thing that was one until it was split into two as now split into four, etc.

How could we prove things like the non-spatiality or immateriality of the mind and the existence of an infinitely perfect immaterial being by way of a method that is so closely tied to proving things about the spatial extension these things are expressly thought to lack? Well, first things first. That is, the order is one of the central features of geometrical proofs. Start with what is known on its own and then proceed to the knowledge of what follows from that. Apart from this general rule of order, however, there is the question of the mode of the proof, which can be of two different kinds:

Analysis = shows the path to discovery in a way that others can follow and understand as if they had discovered it themselves (a priori). Here what is first is where we happen to start; what is known on its own is only whatever we are immediately aware of.

This kind of proof won’t persuade a hostile reader (who digs in and refuses to budge), or an inattentive reader (who doesn’t carefully gather and retain what is going on at each step)

But this is how first principles are demonstrated to us—i.e., this is how we come to grasp the necessity of the truths expressed in the definitions, postulates, and axioms. 

Synthesis = demonstrates conclusions in the opposite way. Once we are in possession of the definitions and axioms (in the sense of grasping their truth for ourselves) what is first is the definitions, postulates, and axioms. Our prior understanding of these first principles allows us to deduce theorems on which the truth of the other claims to be demonstrated depend, and to do so in such a way that we carry the intuitive grasp of the first principles through every step and are able to grasp the truth of these dependent claims with equal clarity.

This kind of proof is capable of wresting assent from a hostile reader

But it is incapable of satisfying the minds of those who want to learn–we don’t discover anything genuinely new in this way, all we do is take the things we already know to be true on other grounds and provide logical derivations of their truth.

This is how first principles are used to demonstrate other truths to us—i.e., we come to grasp the necessity of what we may have previously thought to be merely contingently the case, by understanding how it follows from necessary first principles.

These modes aren’t generally held apart and appreciated for the significant differences involved when the method is used in geometry, since what is first with respect to synthetic demonstrations in geometry are things that “accord with our use of the senses”. It isn’t really that we don’t have to discover them via analysis, if we want to have real insight into what we are doing with them in synthetic demonstrations, but it is easier to get by in the synthetic part without this insight and to wrest assent from those who lack it through requiring their adherence to principles and rules.  

In metaphysics, it takes a lot of effort to grasp first principles clearly and distinctly. Instead of disciplining ourselves to think carefully about what defines something as being a substance, or as being a cause (i.e., what is the true and immutable nature of substantiality, or of causality) we lapse back into thinking about the things we commonly accept to be substances and causes (i.e., bodies of particular kinds). It takes a lot more work to get to the point where we can start from metaphysical first principles and proceed synthetically from there.

The analytical path of the Meditations takes off from (((my obscure and confused) awareness of myself) as being sensibly aware of) the qualitative features of bodies and leads me to discover that the first thing that is known on its from this starting point is my existence as a thinking thing. What follows from that, again in this analytical mode, is my discovery of a criterion for truth–whatever I clearly and distinctly perceive to belong to a thing really does belong to that thing. I clearly and distinctly perceive that an idea of infinite perfection belongs to my mind, so an idea of infinite perfection really does belong to my mind. I clearly and distinctly perceive that the cause of some idea must be at least as real as what is represented in the idea, so the cause of my idea of God must be at least as real as God. I clearly and distinctly perceive that I am not infinite and perfect, so I am not infinite and perfect (i.e., I am not God and am not the cause of the idea of God). This analytical mode of proof is designed to lead us to the simultaneous grasp of:

  • The existence of God
  • The nature of a substance (to be in itself and to be conceivable through itself)
  • The nature of causality (formal reality through which something else receives formal reality)
  • The nature of the human soul (finite thinking substance), and
  • The nature of bodies (extended substance)  

and to do it “in geometrical fashion so that the reader could perceive them, as it were, in a single intuition”. 

If you would have preferred to skip the analytical part and just start with Definitions, Axioms, and Postulates–whether because you already had the insight Descartes is trying to lead you to, or because you want to have your assent wrested from you, instead of freely giving it–Spinoza is the way to go from here.





















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