Andrew Dynneson September 2015 This is a brief history of ...utminers.utep.edu/.../dynneson-SpaceAndForm.pdfThis is a brief history of the philosophy of space. Philosophers have been

Andrew Dynneson September 2015

This is a brief history of the philosophy of space. Philosophers have been discussing these issues for centuries. Today, I will be touching on a few of the most notable thinkers,and neglecting many others.

Zeno's Paradox.

Zeno was a Greek Philosopher from Elea, believed to be active around 490-460 BC. Most of what is known about Zeno is from second-hand accounts, noted by Plato in his work Permenides, about half a century after Zeno, and he was also mentioned in Aristotle's Physics, about century after Zeno, and a few others.

Zeno is most often quoted as proposing the following paradox, as it appears in Aristotle's Physics (VI:9, 239b15), and translated into English:

“In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.”

[email protected] University of Texas at El Paso

Illustration 1: https://www.pinterest.com/marjaoilinki/the-paradox-archive/

mailto:[email protected]


This was an early indiction that common-sense notions such as distance and space begin to break down when one begins to devise a system, that is at once both formally-logical and consistent with real-life, in order to build mathematics that includes the Physics of motion as part of its construction.

If the tortoise receives a head-start, the hair much reach to the point at whichthe tortoise began, this is the point “x0.” At this time, the tortoise has moved further along his track, and so the hair must reach this new point, “x1.” But by this time, the tortoise has reached “x2,” and so-on, ad-infinitum.

This can be used as a critique of the modern-familiar Euclidian (c.300BC) definition of a point: “A point is that which has no part.” This can also be usedas a critique of the Cartesian (17th Century AD) system, which uses some assumptions which are identical to the Euclidean system. And a point has zero dimensions. If two points are dimensionless, then how can point A ever reach point B if they are both in-motion? Therefore there will always be some distance between the two moving points.

This is even a critique of Modern Calculus, towitt we have very little answer, other than crossing our fingers and hoping that additional problems do not arise (they do).

This argument gives credence to the Atomists, who believe that there is a smallest possible unit for which matter is indivisible. Did Early Greek Philosophers distinguish between matter and space? If so, how did they draw this distinction?

Zeno's Paradox is an example of a philosophical argument which is ad-absurdum. One supposes the given assumptions, then presents a situation for which those assumptions are shown to be absurd.




Plato.

Plato was an influential Athenian Philosopher which has had a profound effect on Western Philosophy. It is difficult to distinguish between the philosophies of Plato or his mentor Socrates, since it is possible that Socrates did not write anything of his own down on paper. Much of Plato's writings are devoted to preserving the dialogues of Socrates, and it is unclear how much bias from Plato entered those lines.

Although it may be impossible to construct an absolutely perfect circle in the real-world, Plato would have argued that there is an abstract form, which is a perfect circle, which “exists,” and from which all circles draw upon this perfect form to exist as a circle-like object. Although trees have missing branches or carvings upon their bark, there is an abstract-form of “Tree,” from which all trees derive their myriad-forms.

The mathematics of a circle hence is drawn upon this perfect-circle-form, and for which circles in our world are approximate constructions. This goes against mathematical Constructivism, which holds that for something to be proven to exist, it must be constructible. Although even Constructivism can be argued to have its roots in Platonism.


Illustration 2: Raphael's Depiction of the School of Athens



Rene Descartes.

The familiar x-y axes, and the corresponding mathematical system surrounding these are given the name “Cartesian,” after this influential 17th Century philosopher. Especially when concepts of Geometry are prevalent, they are also given the name “Euclidean,”since the two systems are analogous.

Descartes is often quoted as having coined the phrase, “I think, therefore I am.”Although I find some of his reasoning to be inadequate to hold that Solopsism is true, that is, that the self is the only thing that can be known to exist. However, if I begin doubt even my own existence, this goes beyond thescope of this lecture.

Decartes image https://traveltoeat.com/wp-content/uploads/2013/06/image.jpg




Calculus.

Newton and Leibniz were late 17th and early 18th century philosophers. Newton was also an alchemist. It is not clear if Newton stole some of his Calculus from Leibniz without attribution, or if Leibniz stole some of his Calculus from Newton, and passed it off on his own. It is clear that the two knew about eachother, possibly even read eachothers' work. However, it is not clear who should be credited with the invention of Calculus. There was much controversy at the time. Some people believe it is possible that they both invented Calculus simultaneously, and no offense was intended. It is clear that both mathematicians provided valuable insights, and added to the theory in their own ways. As students, we learn from this situation, and give proper attribution to ideas!

Part of Calculus is an attempt to deal with indeterminant forms, for example 0/0. Calculus also attempts to deal with infinities ( ∞ ). The way Modern Calculus does this is by replacing the act of dividing by zero with the act of “nearly” dividing by zero, and sees what is happening with the curves. It also replaces the act of reaching the infinite, with the act of achieving the “arbitrarily large.” Since I am not familiar with the original literature, I cannot comment much on the Original Calculus of Newton and Leibniz, compared with the form that it takes today.




Also, regarding the writings of Leibniz, there is this quote:

"...[T]he folk reading simply ignores Leibniz's metaphysics of motion, it commits Leibniz to a mathematical howler regarding his laws, and it is arguable whether it is the best rendering of his pronouncements concerning relativity; it certainly cannot be accepted unquestioningly."

http://plato.stanford.edu/entries/spacetime-theories/#1

In the future, I wish to read the work of both philosophers thoroughly, instead of relying merely on folklore.

Absolutism vs. Relativism.

Let us flash-forward to modern times, and take a look at the 20th century, as well as recent developments. Since there were a number of contradictions apparent in the Newtonian theories, for example stellar abberation, which produces a counter-intuitive movement of cosmic objects (stars), and this is something which is measurable-observable, and goes against the notions of “absolute space,” for which Newton, and I would have to guess, most philosophers of earier centuries.

For a good long while, it was believed that Euclid's 5th Postulate: “If a straight


http://plato.stanford.edu/entries/spacetime-theories/#1



line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles.”

Was not only true, but actually Provable from the previous four postulates! Another way of thinking about this postulate is that lines that are not parallel must cross. Another way of thinking about this is that lines which are parallel may not ever meet.

A mathematician supposed that this 5th postulate was actually not true, and attempted to prove that this was absurd by showing contradictions in the mathematics (ad absurdum), along with Euclid's four postulates. Not only did he fail to prove it, but this helped to form Non-Euclidean Geometry by helping mathematicians to consider that the 5th may actually be false (Exploring Chaos: A Guide to the New Science of Disorder).

Einstein was certainly not the first to propose that space and time were not separate entities: see Lorentz, Minkowski, Poincare, Riemann, and many others proposed a geomety that was space/time. Einstein receives the lion's share of the credit because he devised a most complete Theory of Relativity.




According to this theory, time does not move at the same rate for all observers, nor does space have the same length for all observers, relative to eachother. Furthermore, space itself has curvature, thereby utilizing a geometry for space/time that is not Euclidean. For parallel lines, not only do they cross, but those lines will be curved differently, depending on where they are located in the universe.

Einstein's theory has also been confirmed by experiment. During an eclipse ofthe Sun, the Sun's gravity acted sort of like a magnifying glass, bending light.




This measurable observation was consistent with the theory. Atomic clocks have also been placed on airplanes, and their times did not agree with the clocks at the airport when the plane landed. The amount of time-difference was precisely the amount of time predicted by Einstein. [Take a look at these studies if you are interested]

Then Quantum Physics happened, which frustrated Einstein, even though he helped to found it. If I measure the temperature of my hot cup of coffee, sincethe thermometer is at room-temperature, in order for the mercury to rise, it must absorb some of the heat from my coffee. In essense, the act of measuring my coffee causes that same coffee to cool a little-bit. Therefore, the act of measuring the temperature affected the temperature itself. Ordinarily, this small but “predictable” difference does not affect it very much, and so it has been neglected in the past.

However, to observe a single particle, one must observe that particle. To observe a particle requires light or some means of “looking at it.” If a single wave of light hits the particle, the particle moves. Therefore, neither where a particle is located nor where a particle is going can be measured exactly. And this effect of the light is unpredictable, it comes in a wave, and so all we can do really is to talk about probabilities and statistics.

There are open philosophical questions at this juncture. Other experiments seem to indicate that matter exists as waves when not measured/observed. When matter is not observed, does it exist as a wave, or as a particle, some superposition of both, or are the Solopsists correct that we cannot know?

Light also comes in minimal packages, called quanta, in essence this is the absolute minimum unit that we can use to measure space. It is possible that the Atomists were correct that the universe has a smallest induividual unit which is bigger not-zero. It is also possible that even though we cannot measure anything smaller than this, that it exists. It is an open question. Do you find yourself leaning more towards Platonic thinking, or do you think that the only things that can exist are things that can be measured or constructed?

The Theories of Relativity and Quantum have so far been inconsistent with eachother because mathematicians have been unable to derive Relativity from Quantum predictions, however Physicists believe that both theories “should” be true.




Poincare helped to found Topology in response to the Chaos that was emerging in the Sciences (Exploring Chaos). It is a truly beautiful artwork of Theoretical Mathematics.


Illustration 3: rippletunes.com


Documents

Andrew Dynneson September 2015 This is a brief history of ...utminers.utep.edu/.../dynneson-SpaceAndForm.pdfThis is a brief history of the philosophy of space. Philosophers have been