In this essay I wish to discuss what could seem to be a rather esoteric question, but is in fact, as I intend to show, a really important one: Do straight lines really exist?

It was the Swiss philosopher, poet and mystic Raymond Lulus Parzilasus who said that the question of straight lines should stand in the centre of philosophy and human culture as it stands at a crossroads where many other questions meet, and, in a proper way, can be solved.

Unfortunately, Lulus’s house and his rich library ,including most of his writings ,were burned in a fire, that some researchers (F.G Kunway for example) say was set by order of the church. Anyway, few of his writings were saved, and are spread mainly as fragments in his pupils’ books. F.G Kunway made a huge contibution collecting every fragment quoting lulus’s books, and confirming its’ authenticity. Kunway argues that lulus is the greatest philosopher of his time and shows it very clearly in the first part of his important book – “On truth, cloaks and cuckoo clocks”.

In the following pages I wish to trace Kunway’s path in order to understand what makes the “straight lines issue” so important, and maybe to figure out whether straight lines really exist.

The first to address this issue, though not directly, was Aurides, the less-known pupil of Parmenides. The best known pupil of Parmenides was Xenon who is responsible for one of the famous paradoxes of philosophy – the paradox of Achilles and the turtle. Xenon showed that Achilles can never beat a turtle in a race, if the turtle begins the race only a little before him, thus proving that motion is impossible. Aurides argued that Achilles can reach the turtle if he runs around (a-round).

This answer is not well known and did not get the right attention it deserved even in its’ time, perhaps because Aurides was the son of a maid. What emerges, however, from Aurides’ claim, that motion is possible but not in straight lines, is that straight lines which are man-made are falsely treated as such, when in fact they are only approximately so. However, no serious thought was given to this implication of Aurides’ sayings, supposedly because of the great importance of geometry in that time.

The next time that this issue was raised again was only about 1,000 years later. It is difficult, if not impossible, to point out the person who brought the question to discussion, but we surely know that in the 11’Th century there was a great debate on the status that straight lines should be given. The church claimed that only angels can move along straight lines, and man can only imitate purely their movement. The debate was whether one should try to do so, or refrain from it. I should mention that the reflections of this debate presented themselves even hundreds of years later and can be seen, for example, in the different types of Gothic cathedrals.

The debate reached its’ height at the beginning of the modern age. The church still enjoyed a great influence and power and according to it’s doctrine, only God and his entourage deserve the honor of using perfectly straight lines. As Clemens the sixth (1340-1421) phrased it in his announcement from 1411 “If man could make and use a perfectly straight line it would be an insult toward mighty god”. Some esoteric cults even argued that the divinity of Jesus can be seen by the straight lines of his earthly body nailed to the cross – two straight lines perpendicular one to another.

Two main streams of thought emerged from and against the church’s doctrine. The one, represented by Pico de la Mirandola – in his “Theses” – is that man has the potential ability to reach divinity, and this can be seen in his handiwork. There are no straight lines in nature, says Mirandola, but man can draw them, and this is what differentiates him from all that surrounds him, and gives him the ability to be god-like. Some scholars in Picos’ time claimed that Pico had not noticed the contradiction found in his writings. If there are no straight lines in nature, as Mirandola claims, then god did not create them, and if man can make straight lines, then this action does not bring him closer to god, but on the contrary – distances him from God actions. Other scholars tried to solve this problem by arguing that god unites in himself both elements, or by saying that what makes man god-like, is not  mimicking god, but the ability to draw straight lines as a new creation.

The second stream of thought that emerged at the beginning of modern times was the scientific one. Its’ representatives argued that the issue should be dealt with, only in scientific ways. The discovery of America was, for some of them a proof that what we think to be a straight line is actually not so. The famous scholar, Joachim Maximillian (1511-1602), claimed that we imagine straight lines to exist and then pretend to see them wherever we go, but this only illusion, and, as Columbus has shown, if we stop imagining straight lines we will begin to see the true nature of the world, and most of the scientific problems could be solved.

Heinrich Getz (1780-1843) followed the footsteps of Maximillian but claimed that Maximillian did not understand that straight lines are a primary condition to our recognition and thought. We can’t see the world in any other way than by straight lines, says Getz. Even the lines that are not straight can be seen as such only in comparison with straight lines. “Purifying our recognition from straight lines, as Maximillian demanded us to do, means erasing our perception totally”, claimed Getz.

Lulus claims that no genuine argument concerning “the straight lines issue” was raised after Getz’s, and what proceeded Getz’s death was merely recycling and combining old ideas. For example, In the Romantic era people tended to see straight lines as artificially man-made creation that separates man from nature. Nature does not contain straight lines, claimed the romantics and therefore one should not use them. We can easily see in this concept traces of the arguments that were given earlier.

As I mentioned before, F.G. Kunway’s book contains two parts. The first is Kunways’ great work on R.L’s writings (though some scholars think that R.L is a fictional creation made by Kunway, in order to give his research an historical flavour). In the second part of the book, Kunway tries to complete the work Lulus started and takes the discussion to a new level.

“When we start to deal with this question”, claims Kunway, “we should first clarify to ourselves what we are looking for”. Based on the definition of straight line as the shortest line connecting two dots, Kunway suggests that there must be at least one straight line between any given two dots, and the important word here is “at least”. We are used to think that two dots can be connected only by one straight line, but according to the definition of straight line, there can be two, three or fifty straight lines between any two points.

This short argument is what stands at the foundation of the new geometries that were developed at the 19’Th century, and has paved the way for Einstein’s theories of the relationship between space and time, but the important thing that Kunway realized, was that the notion of the straight line is an empty one. Indeed, some philosophers argue that the question of straight lines is merely a linguistic question: We can’t decide whether straight lines exist in reality. All that we can do is to investigate the use of this notion and its meanings in different human languages and cultures.

F. Metoux (1929-) argues that a straight line is a masculine notion that men all over history used in order to strengthen their power over women. Women, according to her, should use soft lines. Others highlight the meaning of straight as heterosexuality, and call to abandon the too common use of this word as signifies something good for itself.

In this essay I gave a few examples of what can be said on this issue. I hope I succeeded in showing the importance of this question, even to our daily lives, or as Borges phrased it, “The imaginative notion called a straight line is one of the most truely influential notions of human culture”.


המאמר שלעיל נכתב אי אז ב-1998 כתרגיל בכתיבה פסאודו-אקדמית. למרות חסרונותיו הניכרים, והאנגלית העילגת של כותב המאמר, מצאתי עניין מסוים בפרסומו. למען הסר כל ספק. אם יש אמת באחת העובדות המובאות במאמר, אין זה אלא מקרה לא מוצלח.
נגישות