2. Natural Right:
This mathematical logical foundation invented by Aristotle had been a dominant ethical principle of the Western philosophy as well as the law principle of Catholic church. Nevertheless, in 17th century, a historically remarkable philosophical rebellion against Aristotlean ethical principle which had dominated for approximately 2 millenniums (2,000 years)! John Locke, a physicist as well as the father of Liberalism, challenged against Aristotle's foundation of ethics.
The name of this most remarkable rebellious philosopher is John Locke who was mathematician and physician as well as an ethical philosopher. He pointed out that our living world is composed of not only all what are already existing (A=ℝ\0) and nothing (A=0) but also something not discovered yet.
He criticised that the perspective of Antiquity only regards of the physical substances already discovered in this natural world and then ignores both those which have not yet discovered and those which can be invented. For instance, Schliemann was motivated to discover the ruins of Troy by assuming that Troy was not purely a mythological city and so existed in real.
Furthermore, from his physician's perspective, he argued that engineers and scientists often try to invent something they want it to exist but does not really exist yet. At this process, they always have an imagination of what they wish to invent. Therefore, without any imagination of something which have not existed, any new invention will not be created. For instance, when we wished if we could have an instant verbal communication with people living in the place which is several miles away from where we live, the motivation to create an international telecommunication tool was born, and then this wish became true.
Of course, the new innovation will be created as long as it is possible to exist in the reality of this natural world. So, the resources and the tools, these inventors can use to use the new substance which has never existed, must be the physical substances which are existing. But, without their imagination of something which has never existed, their motivation to discover and invent cannot exist. All in all, Locke insisted that it is essential to have an imagination of anything unreal and mythological which we wish if it can exist. But, he also claimed that this desired discovery or invention will become to really exist in this natural world as long as the already existing resources and tools are used and the law of nature allows it to exist.
John Locke was also famous as a progressive Christian. He challenged against the scholar philosophy of Catholicism as well as tried to conserve teachings of the holy bible. He insisted that stories written on the bible are unreal and superstitious, but they are important to remember them as the source of our imagination. He also argued that some of the moral teachings in the bible might not be helpful and should be amended or even ignored, but these teachings have provided us with the chance to think and discuss about these issues. For example, sex outside marriage is considered as sin by means of the original bible teaching meanwhile it is largely tolerated and even encouraged in the current world. Without this bible's teaching, human-beings would even not have think and discuss about the morality concerning sexual acts. In various savage cultures where the holy bible was not introduced to the inhabitants there, these inhabitants passively commit their sexual acts without considering about the meaning of their sexual acts.
This ethical principle can be explained as shown in the graph below:
When A exists, there is A. In the logical analysis, as there are A and A, the interaction of A and A which is denoted as
. Furthermore, the existence of neither A nor A which is denoted as 
may exist. When the knowledge of the world is limited to only the surface of the earth, the longest distance possible to know is limited within the earth. But, when the existence of both the outer space and the cosmos is discovered, the longest distance possible to know becomes way much longer. Furthermore, when people realise that the solar system is only the part of one galaxy and there are several galaxym, the longest distance they know become further longer. Then, it is imaginable to think that there can be a further longer distance which has yet not discovered, and it may become possible to discover by inventing a new technology to measure it. Therefore, the possibility of discovery is infinite, and the infinite length is imaginable as well as maybe possible to discover.
When the knowledge of the world is limited what can be seen in an unaided eye, it is very difficult to imagine or even unimaginable to know what cannot be seen in an unaided vision. But, when the tool which enables to see very small things which can be measured by micrometre is invented, the shortest length possible to know and imagine become further shorter.
As we know that we know A=ℝ\0 and A=0 exist as true, we can infer the existence of the infinity. By means of mathematical axiom, any real number ( ℝ ) divided by zero ( 0 ) is the infinity ( ∞ ). This is inferred by the theorem that the number becomes bigger when it is dividing by smaller number. So, as we know that we know there is nothing ( 0 ) and a very long length may be possible to exist, the existence of the infinity is imaginable, and then it is plausible to denote that the infinity might exist in real. All in all, the infinity should be denoted as
, the interaction between A and A. As we know that A x A = A2, the square root of A2 is A. So, the square root of + (positive) A ( √A , √+A ) exists. In mathematics, there is the number called "Complex Number" which is the root of a negative real number i.e. √-A. The complex number is neither A nor A, but still composed of A and A. So, it should be denoted as
. By means of mathematical axiom, the complex number is definable, and imaginable to create the formula. However, it is very difficult to imagine and draw a picture of its existence, and it is unimaginable to figure out how it exists in our real world. So, the complex number only existed in mathematics, and there was no physical substance really existing in this natural world.
The complex number has become to be used to draw computer graphics shown on the screen, and the mathematical model indicating the wave of earth quake. The computer graphics, the technicians frequently use the complex number as a part of their mathematical formulas creating their complicated computer graphics. In order to show the graph of the complicated movement such as the waves of earth quake, it requires a unique formula composed of the complex number to model this trend. All in all, the complex number enables to draw the computer graphics showing a virtual world more real like this natural world than it used to be able to show, and then draw the mathematical models showing the trend of the scientific observations which are complicated to describe.
John Locke applied this principle to his ethics in order to claim that we should aspire to establish a fairer, more equal, and peaceful world which is imaginable as well as practically possible establish. The utopia may be impossible to become true in our natural world due to the limitation of natural resources and knowledge available for us in this natural world, and the barrier by the law of nature. However, he suggested that we should keep an utopia as our ideal world we wish if we could create. He admitted that we constantly suffer from the dilemma between what is available and what is wanted. But, he strongly insisted that we must always attempt to invent the new methodology of establishing a more ideal world than how the current real world is by using the resources available which the law of nature permits to use.
The idea of Natural Right was born of this theory. Natural Right regards that all human-beings have the naturally endowed rights which should prevail for all humans under any circumstances. It claims that the very basic right which can be protected even within a very limited availability of resources and technology must be provided for all civic individuals who can cohabit together under a consensus of social contract among them. During the time when the fathers of Natural Right, most notably John Locke and Thomas Paine, these rights for civic individuals were the equality of opportunity, freedom of choice within the limit of availability, and property right, must be provided to all human-beings. As the time passes and available technologies become more advanced, the categories of which were claimed to be considered as the natural rights increased.
There are two mainstreams of Natural Law ethical theory. The one regards of the freedom of choice and protecting the private property right as the fundamental Natural Right, and claim for a very small size or even none of government, state's authority, and any form of public sector intervention to individuals' choice and life style. Another supports for an optimum size of government, state's authority, and any form of public sector, and their intervention to individuals' private life in order to support them living well at the best possibility, and preserving the public property as well as, or even more than. the private property right. The former became the principle of Libertarianism, and the latter became the principle of Modern-Liberalism.