A Critique of Euclid's Imposed Definitions
- John Valat de Cordova
- Jul 21
- 4 min read
SANTA FE — Much of the St. John’s College program is dedicated to a pursuit of what could be seen as the truth. A truth in philosophy that we examine in a seminar, a truth in experimentation which we find in our Laboratory tutorials, and a truth of pure and historic mathematics that is pursued in mathematics tutorials. In this essay I plan to examine the pure mathematical concepts and geometry found in Euclid’s Elements by first attempting to pick apart the definitions upon which the book is built upon. And further, by questioning these concepts I want to change your understanding of what mathematics and objective truth is, if its findings can be changed by playing around with defined terms. This analysis will focus on the first set of definitions provided in Euclid's first book.
The first definition in which this essay proposes there is a logical inconsistency is the Euclidean definition of a point. Euclid defines such a thing as “that which has no parts”, which of course, brings up a natural problem: what in this conceivable world has no parts, and can we not break apart every distance conceivable (save, for the Planck distance). As far as Euclid's time we believed there to be things which were the smallest part of a whole, an atom (the natural philosopher Democritus coining the term atomos, or ἄτομος), that being said atoms are microscopic, and purely theoretical at the time of Euclid (and not a particularly popular theory). To add on to even this, physics is well aware of subatomic particles that divide much further down than one atom. Furthermore, even the Planck distance, while not currently being able to be divided further with our modern understanding of mathematics, could still conceptually be shrunk, if not practically at the moment (although this could change at any moment). So the question is, both how precisely measured are our points if they cannot have any parts (and at what point does your precision in a point affect the measurements you receive), but also can anything visible to the human eye, or that can be philosophically conceived, even fit this definition?
To answer the first question, there is nothing that which Euclid had available to make his own definition be adequately filled, to our modern understanding, the Planck length is the smallest possible measure of a distance, but of course, in theory, there could be very little stopping you from being more precise if you were perfectly so. What's to stop me from measuring something in fractions of a Planck length? To answer the second question that arises from the definition, of course not, there is nothing that is visible to the human eye, or that I can be philosophically convinced of existing, which can also be accurately described as having no parts. You can, of course, forgive Euclid for not knowing about subatomic particles, but you cannot use this definition in any serious way without it being discredited in the modern day. There is nothing that cannot be divided further. Which brings us to another question that arises from this definition- is a point nothing?
Of course, a point is something, if we are to use a more practical definition than Euclid's, the only way we could define a point is a representation of something. For instance, a point could represent this author's body and another could represent the classroom in which our mathematics tutorials are held, and if they are accurately put on a map, one could use various equations to figure out the distance of the author from the classroom assuming they have the scale of the map ,accurate locations, and the requisite skills. But the fact that Euclid's definition brings up such a silly question lends much more credibility to the thesis of this paper, that many of his definitions simply are illogical, so much so to the point that our perception of his proofs could be feasibly changed with better ones.
Having belabored the point of my criticism of Euclid’s definition of a point, let us move on to his second definition, that “a line is a breadthless length”. In my opinion, this definition is perfectly fine when discussing an infinite line, as this author first thought Euclid was describing. The breadthless length implies that the line has no end, that it goes on for what is seemingly an unending magnitude. This is a definition close to what this author was first taught as the definition of a line in his honors geometry course his freshman year of High School. That being said, Euclid is defining what we now would call a line segment. Which he attempts to clarify in the definition immediately after this, where he says that “The extremities of lines are points”. The problem with these definitions are simple, if one is to believe that a line segment is what a line is, then Euclid must endeavor to say that his lines are infinite whenever he means them to be, adding a useless struggle and forcing his proposing to be less terse. It is also true that these two definitions are almost contradictory if looked at in a common understanding of English, a thing cannot be breadthless and contained at the same time. One implies a vast infinite and the other defines a narrow segmentation. Not to mention his use of another poorly defined term in this second definition.
There is a natural question that comes from picking apart the logic of definitions in mathematics, which is why do this? If the math makes sense conceptually, why look at the specific wording or word order of a text to pick apart what is otherwise sound logic. The answer is simple; this must be done in the pursuit of that truth which was mentioned at the start of this essay. Without that honest, clear, and logical truth a pursuit of supposedly true mathematics is ridiculous. Euclid cannot claim to be true, or a true pursuit of a geometric existence, without sound logical reasoning and words. With no numbers to go alongside his proofs, all we, the readers millennia later, have are his words. And also, his math doesn't work on earth, as any architect would know.
