Getting it Straight
For nearly two millennia the sense of Euclid’s own definition of a straight line seems to have escaped the commentators on Euclid’s Elements. Since Euclid’s definition of a straight line as ‘a line that lies evenly with the points of itself’ turns out to be the only genuine definition of a straight line and the only one that has a place or a role in geometry, this failure calls for a serious examination because it suggests that behind it may be an important misconception about the nature of geometry as a piece of human knowledge. The trouble comes with the conception of geometry as a body of knowledge that stands outside and above the concrete practical world. This prevented the commentators from seeing that Euclid’s definition was founded on and related to a practice – that of draftsmen who check the straightness of their straightedge by drawing a line and then rotating the straightedge and drawing a second line on top of the first. If, and only if, the points of the two ‘lie evenly with themselves is the straightedge straight.
Euclid’s definition allows us to move directly to the proposition that ‘Two straight lines cannot have a common segment’ - something which those who didn’t see the sense of his definition often thought had to be added as an axiom if geometry were to have validity. From Euclid’s definition it follows that if lines have two points is common then they have them all in common and are not two lines at all and only one.
The standard would-be definition ‘a straight line is the shortest distance between two points’ may give us a truth about straight lines but hardly tells us what straightness consists in. And it has no place in a geometry that does not involve measurement or comparison of lengths. It would also have no role there in establishing theorems. It is scandalous that such a false definition still appears in school text-books that are pretending to introduce students to rigorous reasoning through geometry.
The failure to understand Euclid’s definition was bad enough but even more damaging to geometry was the would-be definition that was offered in its place. ‘The shortest distance between two points’ is neither a definition nor has it any place in a geometry that does not involve measurement or comparison of length. That ‘definition’ can enter into no proofs – unlike Euclid’s, which immediately establishes the theorem that ‘Two straight lines cannot have a common segment.’ From Euclid’s definition it follows that if they have two points in common they have them all in common and are identical and the same line and not two.
Because he did not understand that definition of Euclid’s and see its function in his proofs, the early commentator, Zeno of Sidon wrongly thought that certain of Euclid’s proofs were faulty and needed the supplement of further axioms such as that ‘two straight lines cannot have a common segment’ to ensure their validity. If he had understood Euclid’s definition he would have seen that it ensured that if two lines had more than one point in common they would have all their points in common and would not be two lines at all but only one.
Reflecting on the reasons why the commentators have been unable to understand and accept Euclid’s definition can tell us a lot about the framework of thought that has dominated the modern era and which needs thorough criticism that can put it behind us and free us up for a better understanding of things. I think that one reason that it was difficult for the commentators to see the sense of Euclid’s definition was that from very early on geometry was seen as a body of ‘higher truth’ lifted well above the practice of builders or carpenters, and coming to us from the heavens and antecedent to any practice. This is why they could not allow a fundamental concept of geometry to derive from and reflect that practice of draftsmen and others to check straightness by that rotational method.
This conception of geometry as a ‘higher truth’ standing above the material world and shaping it is pretty obviously why Plato had ‘Let No One Ignorant of Geometry Enter Here’ engraved in the stones above the entrance to the Academy. It is for that reason completely ironic that Plato himself offered a definition of straightness that not only was derived from a practice of carpenters but was in addition defective and could not have any function in geometry. It left many theorems without a foundation and therefore undermined geometry’s rigor. Plato defined the straight as ‘that of which the middle obscures the ends’. The first thing to say about this is that it comes from the practice of carpenters of sighting along a plank to see if it is straight. To try to turn this practice into a definition requires us to assume that light travels in straight lines – but even to assume this we have already to have a concept of straightness that is prior to the would-be definition. Also that operation of sighting could play no role in geometric proofs and would leave them defective. So at the same time as he was offering geometry as a model of truth and rigor, he was subverting that very rigor with his defective and useless definition of straightness.
The other standard definition: ‘The shortest distance between two points’ also comes from a practice – the carpenter’s chalk line method of laying out a straight line, and it is equally useless in establishing theorems and postulates. It is in fact scandalous that the ‘shortest distance’ definition still appears in textbooks of a subject that is held up as a model of rigor. That definition equally subverts that very rigor in its attempt to detach geometry from human practice and derive it from the heavens. There is no way in which the ‘shortest distance’ definition could enter into a geometrical proof. Geometry does not involve measurement of that kind. It may show lines to be equal or unequal but the process of proof of that would already involve and depend on the definition of straight line.
Euclid’s definition not only enters into the proofs of theorems – for example, by obviating the need for the supplement that Zeno thought was necessary for their validity, it also reflects the role that straightness plays in human life. We make things straight so that there will be no gaps in the floorboards or between the door and the frame. We want the points on the door and the points on the frame to ‘lie evenly’ with one another. Geometry was, to begin with, a practical subject helping builders and carpenters to do their work well and accurately. But it was very early raised up and given an almost religious significance. In the Middle East there was even a sect that built cubes all over the countryside in ancient times, regarding them as objects of reverence. As soon as geometry was given that sort of role in human life, it had to be lifted above the practical role it previously had and the practical definitions that went with that role had to be surpassed. But if we leave those practical definitions behind and rule them out we leave ourselves in a mess. Thomas Heath in his book Euclid published in the early part of the last century which gives a thorough and comprehensive account of Euclid and his commentators was even moved to say that Euclid ‘was attempting the impossible’ in setting out to define something so fundamental as straightness in lines. But it turned out to be not impossible and Euclid did it perfectly well. His definition is the only genuine one that both tells us what a straight line is and allows us to move forward to generate theorems about them. What the commentators could not take in is what was expressed so clearly by Newton in the preface to the Principia; ‘The foundation of geometry lies in mechanical practice.’ This precisely characterizes Euclid’s definition – it is founding geometry in ‘mechanical practice’. It was because later theoreticians saw geometry as necessarily standing before practice and determining it that they could not accept a definition that founded itself on practice and thereby undermined that picture.
What we have to be clear about is that though the founding definitions may derive from practice and in this way give geometry a relation to the practical world, once they are adopted and given that status as founding definitions, they are no longer shaped by practice but are given the role of shaping practice. They now determine our practice and are not determined by it. It is our own decisions and commitments that create this seemingly ‘higher’ world that dictates to us.
Newton turns out to be a Marxist before his time in putting mechanical practice at the foundation of knowledge, while at the same time getting Euclid out of the shadows and allowing the sense of his definition to be seen and to be seen as the only possible foundational definition. It may well be that Newton anticipated me in seeing the sense of Euclid’s definition, and I am willing to believe it though I have no direct evidence.
Heath thought that Euclid was ‘attempting the impossible’ in trying to give a definition of an absolutely fundamental concept such as straight line, but that was because he did not consider the possibility that he could refer to a practice in his definition. Heath was of that school that saw geometry as something above the material world and human practice. And how was it to be raised to those celestial heights? On this view, the definitions of the fundamental terms would have to relate them to things beyond this world, fixed and eternal, things that generally get the title, ‘transcendentals’. The trouble with tramscendentals is that their proper home is in a religious context and not a secular one. They cannot be reached by the senses or checked out by experimental means. They belong in the realm of faith not the world of secular facts. When the founders of modern philosophy thought they could secularize the previous framework of thought by turning nature into a transcendental standing outside the material world yet governing it according to immutable laws, they created a monster beyond human understanding. Even religious faith could not help here because that transcendental monster was meant by them to impact on and control the material world and it was impossible to say how that was meant to take place. The best we can say is that though the definitions that we put at the foundation of geometry may arise out of practices once they are adopted and made foundational, we are bound by those decisions and they generate a system that stands above and guides further practice and does not merely reflect it.
The point we have to seize on here is that it would actually be impossible to give a definition of the most fundamental of the concepts of geometry except by reference to practice. To say that they are fundamental is to say that there are no deeper concepts in terms of which they can be defined. Practice is the only thing that can get those fundamental concepts off the ground and give them substance and content. Newton saw this when he said ‘To draw a straight line is a problem, but it is not a geometrical problem. The solution to this problem is required from mechanics….Therefore geometry is founded in mechanical practice.’ But it is not just the drawing of straight lines that requires ‘mechanical practice’ we need to refer to that practice in order to say what a straight line consists in as well as how to produce them.
This whole issue of Euclid’s definition and what it shows us about how the foundation of geometry lies in practice is of the utmost importance in the challenge it issues to the whole framework of thought that has dominated and defined the modern era. This framework sought the starting points for knowledge and truth in things detached and prior to human practices, things set in the heavens as transcendentals. It was this framework of thought that prevented the commentators from being able to see the sense of, and to accept Euclid’s definition. As I have pointed out elsewhere, this modern framework arose from the attempt of the founding fathers of modern philosophy to save themselves work in creating a new secular framework suitable for the modern era that followed on the feudal. They thought they could simply take the theological framework of the previous era and simply substitute Nature for God.
This attempt to save themselves work landed them with far more work - with insolvable problems and impossible projects. The adapting of the old theological framework of the previous era left them looking for ultimate foundations and absolute starting points to put in the place of God – starting points that stood apart from, and confronted humanity from some unspecifiable place beyond. These were the transcendentals that were thought to be necessary foundation any genuine knowledge. The proper home of transcendentals is in a religious context and the attempt to make a home for them in a secular context is richly productive of mystery and confusion. The attempt to treat Nature as a transcendental governing the material world from somewhere outside created a monster beyond understanding. That is, Nature was held to impose its eternal laws on the material world by means that are never specified and never investigated – for very good reasons – they are beyond understanding.
Something that is claimed to be outside the material world and not part of it, has for that reason got to be something immaterial. Yet that immaterial would-be entity is pictured as controlling the material world and making it adhere to those eternal laws. How something immaterial can be taken to impose a pattern of behavior on something concrete and material is beyond secular understanding – and for that very good reason, the question was never raised by the philosophers who were trying to make Nature a secular substitute for God. In the context of religious faith, God is taken to be something beyond the world yet capable of acting on it. God’s powers were allowed to be mysterious because religious faith is at bottom an acknowledgement of the limitations of human understanding in the face of the universe. But the secular framework that was being constructed could not allow such mysterious powers to operate as part of its system of secular explanation whose whole object was to eliminate mystery and to seek timeless truths that were not the product of faith but had objective roots in the material world.
But that whole framework of thought and its projects and standards are still with us and are the precise reason why it was impossible for the commentators to understand and see the force of Euclid’s definition of a straight line and that human practices were the only place one could start in getting the founding conceptions off the ground. We need to abandon that search for transcendentals as the necessary foundations for truth and human knowledge and accept it that human knowledge grows out of human practices in interacting with the material world and is not founded on mysterious non-entities located in some nowhereland beyond us.