Following my last post, I will next cover the idea's of modern formal axiomatic systems, giving a basic explanation of them. This will allow me to cover the Godel Incompleteness Theorem and why it is not so bad as one would think and finally end with the consequences of Chaitins number which allows one to state that there is No GUT and if such an idea is at all reasonable.
To answer the last however, I have to develop some ideas in realtime about the concept of numbers and stuctures. This will lead to conclusions such as there is no such thing as "1", which is every bit an abstraction as "a+ib". The prior only seems like it is "real" because it is most easily relatable to sensational reality - reality as experienced by the senses - and experiences. I will have to go into a bit of philosophy and history to give this proper coverage. I will start with a post on quaternions I made in the math forums, backtrack to a post on the development of numbers I made some months ago and then move forward into naturals, rationals, irrationals, complex, quaternions, octonions and sedenoids; noting how these all relate to reality.
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Introduction: The Build up to the Quaternions
Quaternions are very interesting objects and not just because of their mathematical properties. Quaternions have a rich history and I urge anyone with an interest in them to dig back some 130 years or more and read the works of the people then. The modern treatments are all dry and incredibly lacking in comparison. In fact, I argue that it is not possible to truly understand them without their historical backdrop. I will give here a brief summary.
Quaternionic methods were introduced a few decades before the vector analysis we are now used to and by someone of no small fame: Sir William Rowan Hamilton. Compared to other ideas, such as the vector analysis, the concept gained acclaim fairly quickly due to the status of its creator. This is impressive especially considering its originality. Nonetheless there was still resistance due to its abandonment of the commutative property and its excessive and seemingly flippant use of the "imaginary" numbers. Quaternions as well, had strange rotational properties not seen in anywhere in nature (although now the spins or intrinsic angular momentum of point particles- can be detailed such). You have to turn a quaternion around two full times in order for it to return to its original orientation. All this led to the widespread discomfort with the system despite the many new avenues of exploration it allowed.
But it seems I have gotten ahead of myself. However, before I continue on I will emphasize that the imaginary numbers are not imaginary, they are every bit as real (or not real) as the so called real numbers. One would better call them the Lateral or Adjunct Numbers. One must realize that the real number line was created to solve certain problems encountered in mathematics: addition, subtraction etc. When one arrives at the case of solving certain quadratics or polynomials, the square root of negative numbers - numbers which are neither less than, equal to or greater than zero are encountered. They have no place on the real line.
However, one can show that these numbers can be treated as couples in a Real, R2, Vector space. Hence the complex numbers are 2 dimensional in scope. They can be treated as 2D vectors in a real vector space. They are an advancement and extension of the concept of number to meet the advancing needs of man.
The Search for Quaternions
Quaternions were a result of the success of a geometric treatment (as couples e.g (a,b) ) of the complex numbers as done by Argand, Gauss and the like. The natural idea of course, was an attempt to create triplets of numbers which contained the same arithmetic properties as complex and real numbers (allowing one to study 3 Space). Many tried and it was Hamilton who at last got it right by realizing that multiplication of complex numbers could be treated as if they were based on the products of the magnitudes and included angles. You could define rotations with multiplication of complex numbers! To rotate a point in the plane one but has to muliply by the unit complex number (cos(?)+isin(?), on a unit circle). In order to carry such a treatment to 3 dimensions he noticed that it is not sufficient to use just the angles and lengths, but also the plane on which these angles were on! In 2D, a magnitude and an angle is sufficient but in 3D, you need an axis as well, which tells you how to turn.
You need more information than a system of triplets would allow, you need a quadruplet system! With this Hamilton developed his quaternionic system and vector analysis was born. He added the two imaginary/adjunct axes j and k in addition to the already existing i and R (the reals) such that a 4 dimensional space was created. Since i2 = -1, and k2 = j2 = i2 then it follows that ijk = -1 (since due to the non-commutative nature of quaternion multiplication, ij=-ji=k, a point that is not often emphasized), the infamous Hamilton relation. Now the ability to analyze 3D space was born.
I will take a step back and highlight the problems of a triplet system as seen at the time. The idea was to develop a system of numbers such that: associativity for multiplication and division, commutativity of addition and multiplication, non ambiguous division, distribution, a kind of absolute value, usefuleness in 3D analysis was possible. Using a system of triplets did not allow on ambiguous division, a method for finding absolute value/modulus, nor was there commutivity, properties which were all important to Hamilton.
'Every morning in the early part of the above-cited month, on my coming down to breakfast, your (then) little brother William Edwin, and yourself, used to ask me, "Well, Papa, can you multiply triplets?" Whereto I was always obliged to reply, with a sad shake of the head: "No, I can only add and subtract them."'
--Sir William Rowan Hamilton
Quaternions as Mother of Modern Vector Analysis
However, Hamilton realized that based on the properties of compositions of rotations in 3D, it was not necessary that they commute, all the other requirements however, were met (unlike modern vectors which do not allow sensible division, and non-associativity of one of the types of multiplication, no moduli concept). Indeed, Hamilton realized that he could define how multiplication worked, there was no natural way in which it must occur. Hamilton developed his quaternions over 20 years and coined the terms "vector", the imaginary/complex part of the quaternion and "scalar", the real part, as he divided his quaternions into a scalar and vector part. He defined the multiplication for quaternions and noted when both had a scalar (the real) portion equal zero, a scalar (negative of modern version) and "vector product" is had. That is, when a pure quaternion is multiplied by another the rules of quartenion multiplication dictate that the quaternion be formed in such a way that a rule for complex portion (vector) only and real (scalar) only multiplication could be conceived.
For example (0,v) * (0,w) = (-v.w, v x w) where v or w is constructed such: (x = ai, y = bj , z = ck) - hence i, j and k define imaginary/adjunct axes and x,y,z are thus adjunct components/numbers. v o w= -(v.x * w.x + v.y * w.y+ v.z * w.z) and v x w = i(v.y * w.z- v.z * w.y) + j(v.z * w.x- v.x * w.z) + k(v.x * w.y - v.y * w.x).
Again, notice how the quaternion dot product is the negative of the modern concept. Hamilton found many ways to perform analysis with quaternions. Indeed, Hamilton wrote an 800 page highly grandiloquent and bombastic work on the quaternions where phrases such as "quadrantal versor, which is of course a semi-inversor" were quite common.
The Strange Quaternions
Quaternions though, are limited to only being most suited to a system of 3 dimensions. It should be emphasized that quaternions unlike vectors, are 4 dimensional numbers, you can exponentiate, divide, take logs etc of them; the scalar part denotes the magnitude of an arrow in 4 space. Indeed the concept of number we are used to (properties mentioned above) can only be defined to hold in the real, complex and Quaternion field, that is only in 1,2 and 4 dimensions. 5 or 45 dimensional numbers are not allowed for fairly involved reasons (whose great implication i will explore some future time), although in 16 dimensions all properties but associative are retained. Quaternions have the strange property of being a kind of scalar vector hybrid. Indeed the quaternion and its strange concept are not very intuitive for the mind although they were built for the express purpose of ease and simple encoding of rotations in 3 dimensions in mind. Hence their overwhelming application in the area in modern times.
The vector part of a quaternion is of course not the modern notion of a vector, it has strange rotational properties being due to the noncommutative nature of the quaternion basis and it being a 3 dimensional surface defined on on a 4 dimensional hypersurface that is isomorphic to the group SU(2). In fact, the pure quaternion (zero for the scalar portion) are a representation of a rotation by 180 degrees. Consider the following to aid in visualization of the concept that I quite like.
Your Hand behaves as a Quaternion!
Straighten your left arm and make your hand into a fist, the axis i,j and k are represented by pointing to the left, up and forward. Rotations are described by moving the arm about, save that a rotation in 3 Space - that is, moving your arm about some imagined a point - corresponds to half a rotation of the quaternion representation. Although you are rotating about in 3 dimensions, your hand is actually behaving as a quaternion with its strange properties. For example, put any object in your hand to serve as reference, now twist your arm so that you have rotated it by 360 degrees. Your arm should be in an uncomfortable position with your elbow pointed upwards. To get the hand back to its original orientation you must rotate a further 360 degrees. To "do this" bring your arm towards you until you have a sort of hinge shape, now bring your arm up (to a sort of L shape, it should kind of look like the robot dance) and then extend it out. You have just performed a 720 degree rotation as described by quaternions. It can be said that the first rotation is described by some quaternion v and the second by w.
A rotation of 180 degrees by a quaternion in fact corrsponds simply to a flip or a multiplication by -1. Objects that have this property in actuality and not in the form of a trick are the point particles, specifically, their spin. See here on the Dirac belt trick and here as well. The image below is from the latter of the two sites.
Using a unit quaternion Z, a rotation in 3D can be performed by rotating some pure quaternion v into another, w. This is done by w = ZvZ*, where Z* is the conjugate of Z and iff Z = a+bi+cj+dk then Z* = a-bi-cj-dk.
I have the highest admiration for the notion of a quaternion; but, as I consider the full moon far more beautiful than any moonlit view, so I regard the notion of a quaternion as far more beautiful than any of its applications. As another illustration, I compare a quaternion formula to a pocket-map---a capital thing to put in one's pocket, but which for use must be unfolded: the formula, to be understood, must be translated into coordinates.
--Arthur Cayley
Modern vectors were developed when people like Heaviside and Gibbs sought to make the quaternions more vectorial and less scalar while also extending the applications and use of quaternions outside "quaternions". The view of them was that they had little natural application though Hamilton had used them to study heat,optics and other mechanical system, it was not believed to be natural.
"Hamilton's extraordinary Preface to his first great book shows how from Double Algebras, through Triplets, Triads, and Sets, he finally reached Quaternions. This was the genesis of the Quaternions of the forties, and the creature thus produced is still essentially the Quaternion of Prof. Cayley. It is a magnificent analytical conception; but it is nothing more than the full development of the system of imaginaries i, j, k; defined by the equations,i2= j2=k2=ijk=-1. with the associative, but not the commutative, law for the factors. The novel and splendid points in it were the treatment of all directions in space as essentially alike in character, and the recognition of the unit vector's claim to rank also as a quadrantal versor. These were indeed inventions of the first magnitude, and of vast importance. And here I thoroughly agree with Prof. Cayley in his admiration. Considered as an analytical system, based throughout on pure imaginaries, the Quaternion method is elegant in the extreme. But, unless it had been also something more, something very different and much higher in the scale of development, I should have been content to admire it;---and to pass it by."
--Tait
The vector approach was created, based on the idea of dropping the scalar and seperating components. It borrowed heavily from the quaternion analysis (for example cross and dot product, etc. Maxwell had defined such thing as curl, divergence etc., although already his was a more vector based approach as was necessary if one wanted to study physical systems). The problem with the acceptance of vectors was that not only were they not commutative they were also not non-ambigiously divisible, they had 2 ways to multiply - one of which was not associative, they just were not very number like. And people did not like this. Nonetheless their great applicability (many problems in physics such as electricity, fluids etc.) , extensibiltty and neceessity to study many new physical systems easily led to their gradual wide acceptance versus the quaternions. More on modern vectors later.
"Quaternions, on the other hand, are like the elephant's trunk, ready at any moment for anything, be it to pick up a crumb or a field-gun, to strangle a tiger, or uproot a tree; portable in the extreme, applicable anywhere---alike in the trackless jungle and in the barrack square---directed by a little native who requires no special skill or training, and who can be transferred from one elephant to another without much hesitation. Surely this, which adapts itself to its work, is the grander instrument. But then, it is the natural, the other, the artificial one."
--Tait
More on quaternions: Curious quaternions
