2) Rationale for choosing principles
3) Consequences of such principles
I. There exists a physical world independent of us.
II. Systems can observe and interact with this physical world.
III. Through observation, cognitive and interaction processes in some systems it can be shown that phenomena in the physical world obey mathematical constructions.
IV. The relation between a phenomenon and its mathematical construction cannot be proven true with absolute certainty.
Rationale for Principle I.
We cannot inductively or logically prove the existence of an external world. It is merely assumed, as we can only know of sense impressions which appear internally in our brains. However, this assumption seems valid to hold as the rejection of such an assumption leads one to try to explain the origin of our sense impressions and the problem of how one even exists. Such grounds are left for the ancient philosopher or the insane.
Axiom I. has enabled explanation of the origin of sense impressions, of the beginning of ones existence (birth) and many other simple phenomena. (Theories on optics, anatomy and general biology requires the force of an external environment to explain biological phenomena)
Rationale for Principle II.
Systems can be defined to be physical systems which observe a finite region of space with sensory equipment. These systems can manipulate some finite region of space with other physical equipment.
These systems include biological systems, computational machines and robotics. Obviously there is much variety in the mechanisms for observation and interaction but the principle logically covers these.
Rationale for Principle III.
This axiom is much more complicated. First it can be said that in our definition of a system it is possible for a system to develop a component which has a structure such that information can be communicated and processed into more different components. For example a human brain is such a component of a system. It is also possible for this system to devise communicative methods to interact with other systems of the same or similar type. This sort of logic continues until we arrive at complex adaptive systems such as ourselves which have developed science and other means of investigating the world. Through science up until now natural phenomena seem to elegantly obey mathematical constructions such as the nature of space-time (relativity), Quantum mechanics, Financial systems, biological systems and plantation etc.
I say mathematical constructions because in the future we may not just use algebra, geometry, matrix algebra etc. there may be more sophisticated mathematical techniques to describe nature.
If we come across a phenomenon in nature which does not seem to obey our current mathematical techniques at the time. It may be that more innovation or discovery must first come in mathematics. I hold almost no doubt (some skepticism should remain as principle IV will highlight) that mathematics can describe nature in impeccable detail and form. By obeying mathematical constructions I mean that the phenomenon in question can be described or modeled in some way by a mathematical construction. The mathematics does not necessarily dictate the phenomenon (although it could in principle if it were shown that the physical world is constructed of an information basis).
Rationale for Principle IV.
The physical world which we, systems, observe and interact with is not identical to the abstract mathematical world we witness in mathematics. They seem to originate from fundamentally different origins and we seem to interact with them in different manners. One with sensation the other with logic, internal cognition and reason. The apparent certainties and completeness within mathematics is due to it being merely a construction based upon axioms and logical deductions. It is trivial in this sense. Nature may not be necessarily be like this and until we know every specification of nature we must hesitate to be absolutely certain about any mathematical theory about nature or a part of nature.
Until we have witnessed every observed phenomenon in our universe and in all universes we cannot accept any logical certainty applied to the laws of nature. It is also impossible for us to not observe every phenomenon as we are finite systems with restricted mechanisms e.g. we may not view the world at the speed of light or we may not observe the event horizon in a black hole and communicate it back.
Because of these limitations we have thus provided rationale for principle IV. However, these principles themselves are not certain as they refer to nature. This implies that they may turn out false under more investigation or they may remain true but they may not eternally be true.
Consequences of such principles:
These four principles can be chosen as axioms to the empirical method, that is, as fundamental assumptions made by science at this current time and most probably for a long time.
These principles imply that we can attempt to understand everything we observe through mathematical modeling and theories. We can through these models try and predict to varying degrees of certainty certain events in nature. Through these predictions and their verification or rejection by nature theories develop and science develops. For example Paul Dirac's Quantum theory of the electron predicted anti particles and hence anti matter. These could be tested and were verified. More work could be produced based or around that theory. If the theory would have been false then physics would develop in a different way, nature would look different.
Principle I and II imply that the meta-physical objects do not exist. Meta-physical means something which is independent of the physical world. As such objects cannot be observed or interacted with there existence cannot be explained mathematically or even verbally. They cease to exist. This then implies the non-existence of God if God is meta-physical. If he is physical then we can in principle verify God however there is nothing specific to predict about God, nothing quantifiable or detectable therefore it seems God cannot be verified even if he is physical. To make this clearer: If I told you that there was DUFFY and I think that DUFFY exists, you would say where is he? If I saw everywhere or anywhere then you would be confused. Vagueness such as this cannot be useful in verifying such objects. It seems DUFFY does not exist or its existence is negligible. So we do not hold Go'd's existence to be true on this fact. This applies to other such objects or things which are either 1) metaphysical 2) or non-verifiable.
We could in principle use these mathematical constructions and interact with the universe in such a way as to benefit us, the observing system. This is what we humans have done. We have formulated medicine and technology which have benefited us (in the sense of prolonging our life expectancy, intelligence and general health and well-being)*.
It must also be said of the importance of the study of mathematics, science and the application of them. As they are the source of our ability to overcome environmental struggles with the environment and with the inevitable event of individual death.
It seems other subjects are not on the same level. This may sound narrow-minded but it is true, without science and mathematics our race would be extremely primitive. Other subjects are great for intellectual pleasures and past-times etc. but if they vanished we would still be able to conquer our environment and other such problems. However this does not mean they should just vanish. Some subjects such as art, philosophy, theology and literature represent our past in the sense that they were used to overcome such problems but improved methods have evolved and stemmed from them. The evolution of our intelligence as a species is a very interesting and gradual process.
I hereby present to you the basic principles of the scientific method. For which you may adopt, adapt them or discard them.
*I am talking on an aggregate level as we are in better health to those in the Tudor times and more of us can read and write etc..