Chapter 4: Cartesian Geometry

Rene Descartes (1596 – 1650) was an interesting character. Educated as a lawyer, he never practiced law. Instead, he determined to become a mathematician, and later, a philosopher. Back during the time that Descartes lived, unlike today, people of great learning and intellect were treated like celebrities. Many were afforded envious lifestyles by the royal families just due to their notoriety and perceived wisdom, much as our society today rewards athletes and entertainers. How many reading this know more than they’d like to admit about Miley Cyrus yet cannot name even one active, current astronaut or scientist?

Descartes participated in and made contributions to the Scientific Revolution along with his peers, Galileo, Copernicus and Kepler. Among other innovations, the Scientific Revolution put an end to accepting certain precepts regarding the fundamental nature of matter, such as Aristotle’s claim that the world was made of earth, fire, air and water; and that the sun travelled around the Earth. Until this time, pretty much everybody still practiced what we like to call today ‘Magical Thinking’. Magical Thinking is the thought process that, because of which, many ’facts’ were accepted as being true, without proof, just because someone had said so, or because some ancient text stated something as fact, even though to a rational mind that statement of ‘fact’ might be perfectly unreasonable. Witness the countless hours of thought and experimentation during the Middle Ages that Alchemists spent in the search for the method that would convert lead (or any other element for that matter) into gold. It made little difference that the searchers were unalterably convinced in the validity of their task at hand, or that they had great faith in their methods and/or the skills that they might have had. All of their research and confidence amounted to naught because as we know today, the only way to do convert lead into gold is to alter its nucleus and atomic structure with a particle accelerator. That quixotic quest was respected ‘science’ during the Middle Ages.

Descartes is most famous for his philosophical postulate: “I think, therefore I am”, which is, by any measure, a brilliant summation of an historic philosophical principle using very few words. Of course, it’s probably famous because it’s short and catchy, and fits well into modern entertainment media like the phrase “To be or not to be…”.

And while philosophy has moved on, the mental framework that Descartes imposed on civilization with his other most notable creation remains at the core of our current understanding of space and time. That creation is the geometry that retains his name and it is called “Cartesian Geometry”.

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Descartes realized, better than anyone else at the time, that (as pointed out previously) Euclidean Geometry, although capable of locating any point in space, could not describe all space simultaneously. By referencing only positive numbers, even if zero was included, one did not have the tools to reference points that were more than 900 away from an original referenced point. Had Descartes chosen to, he could have added a directional reference to his coordinate system. He did not.

Instead, he gave us negative numbers.

In order to describe the whole of the space surrounding any location, he took Euclidean Geometry to one of its logical conclusions, and extended the coordinate axes to infinity in both directions. This left one half of each line on the opposite side of zero, that is, in the opposite direction of the positive values. Therefore, to enumerate and describe that portion of the axis, he created the set of negative numbers, that all have values that are less than zero. The true implications of this choice will be discussed in the following and sporadically in later chapters, but in this section, we’ll limit the discussion to the physical aspects of the construction.

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As intended in this construction, the negative sign in front of a number was utilized to infer direction. Little thought was given to the fact that a truly alternate meaning was given to number value. Even today, negative numbers are considered to be a component of the ‘Real Number’ set, as if one could actually have 3 less than zero oranges considering only the present.

This geometrical advance then, carved space into octants, each of which was conceived as being identical, with the exception of its ordinate numbers. The three ordinate axes, generally labeled x, y and z, also created planes (x-y, x-z and z-y) that served as the boundaries between one quadrant and the next. This construct fits in perfectly with the ‘flat earth’ concept.  The fact that Descartes created this almost 100 years after Columbus made his historic voyage proving that the world was spherical is somewhat surprising.

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Descartes chose to describe locations by listing the distance of each point from the origin in sequence, starting with the x axis, placing the number values within a single equation and notating the axis associated with each number with an indicator letter, i, j and k, respectively. This method created a vector (r) that located the desired point and took the form: r = xi + yj + zk, where x, y and z are variables used to represent the numerical axial values to be individually determined and can best be described as creating a path from the known location (zero, the origin) to the point to be identified The designators were used to show that the numerical values could not actually be added, unless, of course, it had the same designator. 5i + 6j = 5i + 6j (and also = 6j + 5i) and cannot be reduced, but 3i + 2i = 5i.

This system proved to be extremely flexible, succinct and singularly descriptive. However, there was at least one fly in the ointment. Although the addition and subtraction of negative numbers, by themselves or in combination with positive values is straight forward and logical, multiplication and division needed new rules. The most prominent and far reaching rules he created were these:

1. A positive value multiplied by a positive value yields a positive value.

2. Both a positive value multiplied by a negative value and a negative value multiplied by a positive value yields a negative value.

3. A negative value multiplied by a negative value yields a positive value.

These rules were applied without regard to true, inherent value (greater or lesser) or order, so that the operations would remain commutative. Most of us have been taught this so long ago that we never think about or examine the logic of this rule set, we just blindly apply them to achieve the commonly assumed and codified result, but do these rules actually make sense in a physical world?

Of course, the first statement does. This is a founding principle of mathematics and can be demonstrated geometrically as shown in the diagram below.

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The proof for rule 2 gets a little more dicey, however. Since the system is commutative, (-4 * 5) = (-5 * 4) = -20. All well and good, however, when applied geometrically, these equations give areas located on opposite sides of the origin, as shown below. This also introduces the concept of negative area, one that will be discussed at greater length later, and is a conundrum in its own right.

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Rule 3 however, requires a complete leap (or maybe even a pole vault) of faith.

3. A negative value multiplied by a negative value yields a positive value.

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Here we have two values that are each clearly less than zero. Geometrically speaking, this puts the area in the lower left quadrant of the two dimensional plane, clearly in the negative realm. Yet, we are told to believe that this operation, somehow, magically, creates a positive value (area). How does this happen? What was Descartes thinking? Why do we accept this? Two values, each of which is less than nothing in value, when multiplied together, create a positive, definite, describable quantity. This is truly amazing! Something large and positive can be created from less than nothing! Alchemy revived! The transformation is complete.

Of course, there were good reasons to make these rules the way Descartes did. Once accepted, these rules create a geometrical/mathematical/analytical system that is very useful, and almost internally consistent. There is one element of this system, however, that makes everything blow up. A single, useful, irrational number that epitomizes the inherent logical flaws of Cartesian geometry. A number that is the logical descendant of the previous discussion, and that, while purely fanciful in nature, has been accepted and utilized in complex equations to explain complicated things.

That number would be: the square root of negative one.

This ends the first published selection from Dear Dr. Einstein. The author realizes that much, if not most of the material presented here is review for most of his readers, but it is important to understand where we are, intellectually, before we can move on to a better understanding. There are many subtle mathematical rules that we as a civilization take for granted that, when viewed from a more detached perspective, don’t make a whole lot of sense, so to understand the need for change, they must be examined using the cold light of logic. This first installment will be followed by others, on a schedule that has yet to be determined, until the final chapters have been released. 

So bear with me. This book has actually been completed, so we won’t get part of the way through and abandon the project, and once we’ve managed to dispense with creating the need for change, we’ll get into the concepts for a better, more realistic system. In the next chapters we’ll be dealing with imaginary numbers, our concept of time, and our ability (or lack thereof) to perceive motion.

Take care until then

– O. Penurmind (@O_Penurmind)

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