Friday, May 10, 2013

On the Claim that Non-Euclidean Geometry Is Needlessly Over-complicated

(This is a paper I was assigned to write on the difference between Euclidean and non-Euclidean Geometry as part of a teacher education program, of all things!)
The mainstream understanding is that the discovery of non-Euclidean geometry by Lobachevsky and Bolyai marked a great step forward for mathematics. However, in the process of researching this paper, I encountered the website of Miles Mathis,, who claims “non-Euclidean geometry, although potentially valid, has been used historically as a cover for bad math.” While this claim at first seems quite inane, reading his work, it is clear he is writing from a place of great passion and dedication to his purpose. After carefully reviewing a number of his papers, I believe Mathis might have identified some fundamental problems in Western society's explanation for how time interacts with space.
For those entrenched within the machinery of the STEM industry, Mathis’s claims will seem outlandish and preposterous, the work of a disaffected eccentric. For me, however, I have long felt there was something amiss inside the trajectory Western culture’s present course of development, and it is partly due to my own pessimism about my culture that opened my mind enough to give Mathis’s work a fair chance.
What exactly does Mathis mean by his claim that non-Euclidean geometry has been a cover for “bad math”?
He means quite simply that it confuses more than it explains. The unnecessary complexity within the theory makes whatever practical usefulness within it impractical to deal routinely with. Looking at the history of the situation will provide some perspective to Mathis’s reasoning.
Non-Euclidean geometry developed as a reaction to the futile effort of medieval European mathematicians to “prove the parallel postulate.” The obvious contrived nature of this dilemma, that it was not addressing any practical problem in the world, did not stop countless Europeans and Arabs from attempting proofs.
In fact, the effort to prove the parallel postulate is analogous to the theological debates of the same time period over how many angels can sit on the head of a pin.
According to Mathis, most of the symbol pushing involved in non-euclidean geometry is analogous to a “proof” that there can’t possibly be more than 1 angel. In other words, hyperbolic and elliptical geometry are not incorrect, but the question they try to answer has limited practical value in the first place.

Why were mathematicians asking such meaningless questions? Likely they were motivated by the same factors that led Thomas Aquinas and Dons Scotus to invest time in the debates over angelology and the head of the pin. To go into the motivation of these men (note that they were all male) would require an extensive reading of Christian theology and its psychological impact on European intellect. Carl Jung writes about this subject extensively in his work on alchemy--the primary passion of Newton’s life--but unfortunately I have not yet read his work, so the topic is outside the scope of this current paper.

Disregarding motivation, we can still ask why no other mathematicians noticed the problems Mathis brings up? To answer this, I will first outline the present-day mainstream understanding of non-Euclidean geometry and its significance.

Non-Euclidean geometry developed as a reaction to the failure of mathematicians to derive the parallel postulate from Euclid’s other four original postulates. The first four postulates can be intuitively grasped as constructions:
"Let the following be postulated":
  • "To draw a straight line from any point to any point."
  • "To extend a finite straight line continuously in a straight line."
  • "To describe a circle with any centre and distance [radius]."
  • "That all right angles are equal to one another."

The 5th postulate, however, cannot be directly observed through construction, because all constructed lines are necessarily finite, whereas the fifth postulate assumes that lines go on infinitely.

For a long time, mathematicians thought that the truth of the 5th postulate could be found (proven) within the logic of the first four postulates. As Wikipedia puts it, “At this time it was widely believed that the universe worked according to the principles of Euclidean geometry.” They thought that the 5th postulate was as inherently a part of the physical universe as the observation that “all right angles are equal”. When trying to describe the physical world, they reasoned that it would be as silly to exclude one as the other.
But Bolyai questioned this assumption, “Bolyai ends his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of the physical universe is Euclidean or non-Euclidean; this is a task for the physical sciences.” -
Bolyai and Lobachevsky actually both independently proved that the 5th postulate could not be deduced from Euclid’s other four postulates. It took 2,000 years for this to happen because mathematicians had to precisely define all the other statements that were logically equivalent to Euclid’s 5th postulate, a task that no one took up for many centuries!
In the 19th century, this work became pressingly relevant because finally the physical sciences--physics and chemistry--were getting to a point where it made sense to ask “What is the shape of the universe?”
In fact, non-Euclidean geometry can be seen as a precursor to Einstein’s discovery that space is interwoven with time so that light travels along curved arcs influenced by gravity, and not in straight Euclidean lines.
But if Mathis is correct, there are actually needless complexities within the present-day mathematical tools of physics, and the glorification of non-Euclidean geometry may be partly to blame.
The implications of Mathis’s claim will be obvious to anyone well-read in mathematics and physics. Non-Euclidean geometry is the mathematics behind Minkowski spaces, which is the most convenient structure for analyzing Einstein’s special relativity. If non-Euclidean geometry is needlessly complex, then special relativity is as well. But special relativity indisputably provides important corrections to Newton’s formulation of motion, so Mathis’s accusations must be directed at Newtonian mechanics as well.

Mathis is well aware of all these consequences and has written essays addressing them all, links to which can be found in the bibliography.

Assuming Mathis is correct, why then is non-Euclidean geometry so well-established and nearly universally supported by professional mathematicians today? Why did no one noticed the unnecessary complications and obfuscations in non-Euclidean geometry once it had been established? This paper will not go into the psychology of the situation, but it will outline how mathematical errors became established and entrenched, and thereby perpetuated the problem once it had occurred.

A major cause is due to a fundamental misunderstanding of what a “proof” is in the first place. Mathis explains that the problem arose from the Greeks definition of “proof” as being distinct from intuition or immediate empirical evidence:
Very often we choose definitions and axioms from a pool of statements that are empirically or intuitively very strong. That is, we can point to something that is innate or experienced as proof. But this kind of proof has never been accepted by philosophers, for various (not so good) reasons. -
Instead the Greeks insisted that proofs had to be established through logical deduction from the axioms and definitions. This attitude still lingers in mainstream mathematics to this day, and leads to the blind acceptance of unnecessarily complicated theories, simply because the axioms are considered firm and untouchable. As Mathis explains:
[The ancient Greeks reasoned that] you proved things by showing that your results were logically contained in your axioms. You did not prove anything by modern standards, since you found no “new information.” You only clarified old information. You re-expanded your definitions, reminding yourself of their full content. And yet the Greeks saw this as true learning, as true intelligence, as real knowledge, as non-trivial. -
Mathis observes that proving a conjecture is “not provable” is never very practically helpful, because you have only shown that the conjecture is not provable within the limited set of axioms you start with. The conjecture may still be consistent with another set of axioms. The relevant question is “how is the conjecture relevant?”

So, we can now see that the idea that the parallel postulate is something to be “proven” is quite ridiculous to begin with. The issue of whether the parallel postulate is an axiom or is deducible as a theorem has no practical real-world significance. The parallel postulate can be accepted, or it can be rejected. But, much like many medieval theological debates, rejecting the parallel postulate serves very little purpose, other than to create unnecessarily complex mathematical theories to occupy people’s time. All the applications of non-Euclidean geometry can be accomplished through much simpler means. (Note that Euclidean geometry does not rule out the results of spherical geometry. Movement along a sphere is quite consistent with a three-dimensional Euclidean space.)

As Mathis argues, all the effort to “establish firm axioms” that Hilbert, Russell, Whitehead, and countless other talented mathematicians exerted was largely a waste. Mathis agrees with Karl Popper that “all creative math and science is based on hypothesis and is not provable.” All advances in both mathematics and science do not need to be formally proven from axioms--all that matters is that real people in the field see that the results are true and/or useful! If other people in the field are convinced that a conjecture is true, then whether or not you can write other statements which logically imply the result is meaningless.
Mathis explains:
Popper believed that tautologies escaped his falsifiability principle. ... “That thing I am pointing at is a white dog.” Which is of course to call a white dog a white dog. A = A. How could you be wrong?
A tautology is a statement that cannot be falsified, or disproven by counterexample. “I name this dog ‘Spot’.” This statement, and all tautologies are true-by-definition. The truth of a tautology comes by virtue of its own claim. Tautologies are statements for which it would be meaningless to compare it with other accepted truths or observations in order to determine its consistency or inconsistency.
In this way we get beyond both Gödel and Popper. We can “prove” a great number of things by showing them to be tautologies. The birds migrating is another example. “I define those things that you see flying as birds. I define migrating as doing what they are doing. Therefore they are migrating.” ... It turns out that a large percentage of language and even science can be shown to be tautological. ...
You will say that this just means that it is trivial: it is empty of content. But it is not empty of content. It is full of definitions and axioms, which are quite rich in content. For instance, in the white dog example, you learned what a dog is and what white is, at least according to me. ...
What else is learning, in most cases, but discovering what things are according to other people—parents, teachers, historians, writers, scientists, etc. As a person, you learn a huge list of definitions. A large percentage of what anyone knows is definitional. ... “That is a dog; a dog is that.” A = A. So we already have a huge body of statements that are true because they are tautologies, and yet they are rich in content.

So here, Mathis is vehemently agreeing with Bolyai’s quote that “it is not possible to decide through mathematical reasoning alone if the geometry of the physical universe is Euclidean or non-Euclidean.” But Mathis and Popper take it even further and say that no statement can be positively verified at all. There will always be an unbridgeable gap between the physical universe and the language we use to describe it. This fact can be seen time and time again in the history of science. Ptolemy’s classification of planetary motion was replaced by Kepler’s and Newton’s, whose classification was replaced by Einstein’s. We can define aspects of the physical universe and construct all sorts of elaborate classifications, but ultimately, we can only compare the effectiveness of one classification to another. There is no way to judge the absolute truth of any of our statements. All of the classifications in our theories boil down to tautological definitions.

But is the tautological learning that occurs in non-Euclidean Geometry rich in content? Or is it mostly meaningless symbol pushing? To judge, we must examine the amount of usefulness against the amount of unnecessary complication--ultimately these are the only variables that history will judge it by.

Mathis claims that non-Euclidean geometry’s “needless complexity, its definitional opacity and incompleteness, and its inherent lack of rigor have opened it up to broad and one might say universal misuse.” So how is it needlessly complex? How is it opaque, incomplete, and lacking in rigor? Mathis explains,
Any problem that can be solved with curved geometry can be solved with straight geometry, and it can be solved much more quickly and transparently with straight geometry. ...
Curved geometry, If it is performed scrupulously, it may be just as valid, but that is very far from saying it is just as good. It is less clear, more unwieldy, less efficient, and far easier to fake.

To check whether Euclidean methods can be used to solve any problem as well as non-Euclidean methods, we need to examine which problems are being solved with non-Euclidean methods. It turns out just to be one type of problem--Minkowski space, which is a theory designed to conveniently accommodate Einstein’s equation. Mathis’s claim, however, is that such a construct is impractical and needlessly complex. According to Mathis, equations that are valid in static situations, need to be significantly altered in kinematic situations. In other words, movement and time fundamentally alter the way that the dimensions of space interact. Einstein showed that time and space are inseparable, but Mathis argues that Einstein did not go far enough. Einstein was still using the mathematics of Newton, which is a space-only based analytical tool. He should have rejected the calculus and other mathematics that are based on the assumption that two-dimensional curves can be treated as the limit of a series of infinitely small one-dimensional line segments. Mathis argues that in kinematic situations, curves cannot be equated to lines at all. Movement in one spatial direction fundamentally alters how the object in question should be analyzed in a dimension perpendicular to the movement.

Einstein did not fully realize the implications of his own results. If time and space are really not separable, then our analytical cannot pretend that they are. We have to be very careful when dealing with time and movement. As Mathis points out, the modern conception of time, based on clocks rather than on stars, is a fairly recent invention in human history. Time is not absolute--it is always the comparison of one movement to another. Mathis explains:
Whenever we measure time, we measure movement. ... Every clock measures movement: the vibration of a cesium atom, the swing of pendulum, the movement of a second hand. ...
So time is not a measurement of "time." Time is a measurement of the movement in or on a given clock. And this given clock is uniform only by definition. It is uniform relative to a standard clock. One that has been defined as uniform. This standard clock cannot be proven to be uniform. It is only believed to be more uniform, based on previous definitions and previous clocks.
Mathis goes on:  "When we measure distance, we measure movement. ... When we measure time, we measure the same thing, but give it another name. Why would we do this? Why give two names and two concepts to the same thing?"  Why? Mathis answers: "in order to compare one to the other. Time is just a second, comparative, measurement of distance."

Time, we see, is another metric for talking about our movement through space. Our ability to experience time is actually a sixth sense, a mental mechanism that verifies data of the external physical reality in a way that our other senses cannot.
Mathis continues,
Even the second is operationally a length. For instance, look at Minkowski’s four-vector field. All his basic variables or functions in that field are lengths. As I have shown, x, y, and z are differentials, and differentials are lengths. The time variable is also an interval, which is operationally a length; so when it is transformed by i to make the field symmetrical, it must take its character with it. Time is operationally a length both before and after Minkowski makes it imaginary. My point with all this is that lengths, like numbers, should not be stretchy. Once we are given a certain object to measure, the length is no longer a variable, it is an unknown. A variable only varies in a general equation, but once we apply that equation to a certain object or event, the variable no longer varies. It stands for an unknown number, and unknown numbers are just as stable and invariable as known numbers. But in many curved manipulations, you will find numbers, both known and unknown, varying. This is a sure sign that you are in the presence of hocus-pocus.
In this paper, Mathis goes into further detail on how Minkowski’s constructions are reckless and dishonest - . Other than Minkowski spaces, no applications of non-Euclidean geometry are claimed to exist.

Mathis’s work is the first systematic attempt to correct the problematic formulations regarding curves that have been central to mathematics since Newton’s and Leibniz’s formulation of a calculus derived from curves. At one level, I still find his analysis hard to believe, although I believe I have quite an analytical mind and his writing strikes me as completely authentic. The mainstream understanding of calculus and non-Euclidean geometry is so entrenched within the Western education system that Mathis’s task is truly Herculean in scope. However, not more Herculean than many other worthy causes, such as women’s rights, marriage equality, and establishing truly democratic decision-making policies within communities.

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