Atypical axioms and a further investigation into the concept of a dual identity system

To expound upon the first idea listed in the last blog entry, a notion of a dual identity system, certain things must be covered in greater detail and a set of axioms must be developed. But before those axioms are divulged the very basis of axioms must first be looked into.

To begin an axiom is supposed to be self evident, something so simplistic it cannot be argued against. Of course of Euclids original 5 axioms of geometry the parallels postulate was proven to be false in many circumstances which made it a less than satisfactory axiom. The other axioms speak of two points creating a straight line and circle defined by a centre and a radius, the simplest of stuff. But of course an axioms simplicity is also intrinsically a part of how familiar we are with the subject matter or maybe more appropriately, the subject matter and it's axioms become familiar over time (due to their role in common society and mathematical systems) and so are taken as self evident and described as the foundational blocks of the system (i.e. the 1 + 1 = 2 or 1 + x = successor (x) axiom of standard arithmetic), but in an abstracted realm things are not always what they seem. A straight line, though it is easy enough to imagine or represent geometrically is only a concept. There can be no truly straight line in reality so a truly straight line is only a function of the abstract, and when looked at thusly what exactly is the straight line going through in the abstract sense? Of course we are all very used to the conventions of straight and so on, and we take with it the pratfalls of human error and the imperfection of all corporeal objects and use the geometry to create perfect formulas for these abstract conventions which we have all grown so familiar with. But that is just the point, we are now so familiar with these axioms that there is nothing that can lay a shadow of a doubt into their validity, they have been worked rigorously for thousands of years and have proven their worth time and time again, and so they have locked in their place and are now intrinsic to the whole of civilized humanity (and then some).

So at present I wish to develop a few axioms which will be counterintuitive and seemingly strange for a system which is abstract in the fullest sense of the word and which bears no particular intentional relevance to society and the physical realm.

the axioms are postulated as follows:

let there exist a set of discrete positive numbers such that 0 = 1 

let this dual relationship be described as [0,1] 

there exists no true empty set 

Now a further postulate could be added which incorporates a greater deal of simplicity and proceeds as follows

all numbers greater than 1 have a coefficient of an immutable 1 

However I believe some of the beauty of this system lies in the fact that all numbers can be effectively equated to 0 which equates them to 1 as well, which creates a vast realm of possible solutions, though it may also add complication. If this postulate is abandoned than the following postulate is added as an axiom of the system :

all equations consisting of n elements have a minimum of n + 1  solutions not including [0,1] which is necessarily always a solution as well

This is due to the fact that all numbers have a natural coefficient of 1 attached to them and since 1 is now equal to 0 all numbers are equal to 0 and 1 simultaneously 

so for the simple arithmetical equation of 2x4 we have the following set of answers  = {2,4,8} and [0,1] which can be disregarded as trivial. It will, however, prove a valuable technique for now any element in an equation may be eliminated with ease. 

The obvious downside to this tangential axiom is that it will create a very large number of possible solutions for all equations, creating some deal of confusion. It could also lead to a sort of intrinsic meandering which is my hope for this particular system of mathematics, and which may in and of itself contain intriguing new avenues and beautiful solutions. Further it can be examined that now all equations can be broken down into minimum, median, and maximum solutions. 

Regardless of which of the final two postulates is adopted there is one seemingly obvious pratfall to this whole system, however I believe that said pratfall is actually the systems strongest and most elegant feature. The system quite clearly falls short as a means of precision calculating and it's glaring pratfall is the new vagueness given to all numbers there in. For instance the final postulate which states "equations consisting of n elements have a minimum of n + 1  solutions " gives rise to the question of what do we mean by n + 1 if within this system 1 = 0, therefore n + 1 solutions also equals n + 0 solutions, and furthermore , if said postulate is taken as an axiom of the system n = 0 and 1 as well, so within the language of the system n + 1 elements means essentially nothing. But than of course we are all well aware of what was meant by n + 1 elements because we are already familiar with standard arithmetic, number systems, etc, and the known definition of n + 1 elements is clear to us, despite the paradoxical multitude of solutions given.  The beauty of this systems lies in that paradox, the strength of our knowledge of what the maximum solution is and the simultaneous existence of contrary information which must be accepted as solutions to the problem as well. Of course the minimum and median solutions to the n + 1 postulate can be abandoned because we are speaking of mathematical objects using standard mathematical systems, which becomes the meta-language from which laws and axioms of the new system are derived.

To clear things up a bit it must be stated that this system will not suffice on it's own, but rather exists as an entity separate, though still related to, our standard mathematical systems. That is, in order to discuss the workings of this new system, we must use our familiar pre-established axiomatic systems. 

Postulated conceptions in regards to non-standard mathematical systems

The prior blog post gives rise to the idea of a whole new scaffolding, new building blocks, for different mathematical systems, and I hope to now elucidate a few possible conceptions as to how this is done. The following systems will be described briefly and synopsized in the following paragraph, for each a full set of axioms should be developed and rigorously analyzed (until a devastating paradox is encountered). Many of the concepts may overlap from one system to another:

The first idea and maybe the simplest concept would be the idea of a dual identity systems. That is where both 1 and 0 are completely interchangeable. Immediately problems arise when it is looked at in the set of all real numbers. So this concept is easiest to accept within the set all positive Integers. This ostensibly creates a mathematical system in which the null set is non existent, because at any time you come up with zero you can also come up with one. Through the lack of the empty set we find that problematic decimals such as 0/0 x/0 and 0/x

In order to make the first idea somewhat more powerful and useful a secondary duality is necessary, certain means of converting number systems. For the interchangeability of 0 and 1 to remain coherent it must be intrinsically connected to another set for which this same duality does not exist. This gives us the ability to move freely between two sets, however since the duality only exists in the set of positive integers if we are to move freely between that system and say the system of all integers (positive and negative) we will have to take the absolute value of said integer and save the sign for until we return back to that system.

This leads to another, broader duality; the notion of moving between a constrained system and an unconstrained system. A constrained system is any non-infinite sum of numbers which can be mapped onto a conjoined system which ranges in the full spectrum of infinity. The constrained system is by nature of it's finite quality can never have a concept of the infinite and the infinite system is aware of the constrained system insofar as it conceives of it as infinitely small part (regardless of how large) of an infinite whole. Though the constrained system will never be able to acknowledge the existence of the infinite system which surrounds it, any equation which exceeds the boundaries of that finite system will allude to the greater whole.

Of course these are all very simplistic prefaces and many of these ideas require further development and a structure of axioms, proof, and logical continuities

On the logical justification for competing mathematical systems

On the logical justification for competing mathematical systems 

First it is important to stop and look at the character of humanity and it's history and time on this planet in relation to the planet itself. The planet we dwell upon, our home, is so old (and of course it is obviously nothing but a blip in time in relation to the universe around it) that the entire span of our existence as a species is but the blink of an eye. And so if it is presumed (hypothetically) that we as a species will live on for a time which will come to pass as a non-infinitesimal ratio to the  that of the earth, than it can be easily stated in that (incredible) time span human beings will have evolved to a whole new level and become something well beyond what we are today. And if to us the pace of evolution is breathtakingly slow it is only because our lives move forward so rapidly and our definition of evolution is skewed  such that we perceive only genetical mutations to corroborate said definition. But in reality all of our technological advances are seeds of the evolutions to come, all of our mechanical reliances, our electronic dependencies, our luxuries, our entertainment, our science, be they for the better or worse, they are rapid expeditors of our natural evolution. And at the root of all these technological advances is mathematics. The digital understructure, the information, the bits. Everything we know is encoded with some form of mathematical scaffolding. And so it must be self evident that the resultant inventions are all exclusively dependent upon that scaffolding. Furthermore it can be stated that though the advancements upon said scaffolding may be approximately infinite, they are still linear, they stem from one source and are thus limited to ever branch off of the central husk, deviating in some ways from the advancements coming before them, but they will necessarily be deviations and not entirely new contrivances. And so in the following paragraphs I shall deduce the necessity to henceforth create different starting points for mathematical systems.

Leopold Kronecker once said that "God created the integers, all else is the work of man" . But we gave definition to those integers. Even if there existence, the existence of any set of numbers, is disassociated with the notion of mathematical creation or the idea of human invention, it can clearly be noted that calculating and quantifying those integers, those numbers in general, is under the umbrella of the work of man. Kronecker poses that the natural numbers were always there, waiting to be found by the probing human mind. A conceptual entity that was as solid as the ground we stand on. And maybe they were. The nature of their existence stems from necessity, the necessity to count objects, the necessity to survive and the wont to prosper, early babylonian tablature, calculations, ways of protecting and organizing resources. So predeceasing pure mathematical thought was pure human necessity, applied mathematics and economics predate conceptual mathematics and abstraction. Of course this neither adds accreditation to applied mathematics nor discredits pure mathematics, but it forms a logical step in the singular invention of abstract mathematical thought. It shows that the root of every piece of modern mathematics stems back to our original need to count , quantify, and organize. And so from those needs a best possible system was developed and that system was moved from culture to culture, sieved through great minds in the great intellectual renaissances, from the babylonians, to pythagorus, to Euclid, to Diophantus,  and up through to Euler and Gauss and so on. And all these great thinkers were engaged in the establishment, refinement, improvement, and abstraction of the basic presiding mathematical system. Guided once again by that same simple scaffolding. 

And so Discrete mathematics gives way to continuous mathematics and mathematical physics gives way to quantum physics and sometimes we stumble on some data that we just don't understand and just doesn't compute with our maths and so we make a little note of it and keep going. And if we had to put a shape to everything, well it might just look like a pyramid (or better yet pascal's triangle) standing on it's nose. And though it exists in realm beyond balance and measure, it is still less than an optimal shape, it's limiting and confining and somewhat rigid in it's possible growth. It also has the disadvantage of becoming too diffuse and too disparate (and anyone can easily see that the realm of higher mathematics today is so vat that it has nearly as many specialized fields as it has mathematicians.) Of course there have been so many great breakthroughs in history, and we all reap their bounty today in the digital age. Great leaps of mind which have changed the very course of humanity, Geralomo's complex numbers and Newtons Calculus and Shannon's bit and so many others. Without these amendments to the greater trunk of mathematics as a whole, well our culture would certainly not have progressed as rapidly and we would not be living in this technological wonderland that we are so lucky to be a part of.  

So it is quite apparent that historically speaking the silent engine of evolution (we will term it a "technological evolution" to avoid complications in definition) has always been the mathematics. Leaps of mind in the field of numbers and symbols which have most often been in the form of solving one thought problem or another, something entirely unrelated to the corporeal realm of objects and physical entities. An abstraction of the concept of numbers. An abstraction of an abstraction. And yet there is always a safe footing rooted in the historical foundation of it all. And so I will henceforth pose the idea of new foundations and it's logical benefit. 

Though at first the notion of system which defies almost all of the familiar rules seems, if nothing else, counter intuitive. And though I firmly believe it is logical, I will not deny the fact that it is necessarily counter intuitive. As human beings our mathematical system has been founding on the principles of counting and quantities. The manipulation of abstract symbols has had it's heart and soul planted in the firm soil of a couple basic axiomatic principles such as 1 + 1 = 2 or 1 + x = successor (x). Of course there are ways of developing systems which can defy one rule or another. But never is the whole world of mathematics re-imagined never are the first principles ignored and the axioms changed in favor of a whole different realm. Never is a bit strong of a word, as non euclidian geometry changed the truth value of the parallels postulate and George Boole's algebra subscribes to it's own list of axiomatic statements. Boole being the prime example here as his investigation into human thought led him to create an entirely different and equally logical system  (designed for a specific purpose) which eschewed any relation to the world of numbers and standard arithmetic. But now I am proposing that it is necessarily of great value to the evolution of human thought that further abstractions be promoted. That the greatest discoveries come not in an over saturated science or a field which is so complex that even those people making discoveries are wont to admit they don't understand their own results. That the greatest possible value for change comes from new ways of looking at things. This prior statement can be viewed as an axiom of the evolution of thought, or even human evolution. It is easily shown that historically the works which are often viewed as crazy, heretical, or worthless because they do not fit into a standard mold often pop up, years later and prove they are worth their weight in gold many times over. That the concepts generated when people think so far out of the box they are frowned upon by the current reigning councils of thought, are the very ideas which change the landscape entirely. And so to develop some new basis upon which new concepts can stand, be they entirely unrelated to the notion of quantity and fall purely in the realm of manipulation of numbers, be they systems derived solely upon links in prime numbers with new meanings to all symbols and new axioms which hold up the very structure of all this new work. It can be stated by the following simple equivalence: if all the math that we have established today is founded upon simple laws handed down from basic needs thousands of years ago, than it is natural to assume that a new math based purely on the creative nature of the human mind will flourish ten fold. This will be where humanity will again see a great burst of growth, a great tide of thought which can again evolve the very shape of humanity once more. 

Mathematical truth is one of the only things which all humanity can agree and likely it is one of the most resounding things that places our species above and beyond