A formal science is a branch of knowledge that is concerned with formal systems, for instance, logic, mathematics, systems theory and the theoretical aspects of computer science, information theory, decision theory, statistics, and linguistics.
Contents 
The formal sciences are made up of symbols and theoretical rules  which are based on actions and reactions. They can often be transferred into reality through the use of the applied sciences and thus, are proved to be very useful.
It is fair to say is that there is no system (that we know of) which contains no inputs or outputs. ^{[1]}
Problems occur in formal science mostly when dealing with subjectiveness. For example, a computer program will find it hard handling emotion, unless you are able to define the effects in relation to an envelope's threshold. ^{[2]}
Logistics, scales, variables and addition, are all product of human thought but frequently, they can help to create a theoretical relationship between different phenomena.
People sometimes make the mistake of confusing theoretical systems with reality, applying theoretical models as if they represent reality perfectly. Of course, no model represents reality perfectly. Much like a word, letter, number or symbol, a map is good example of a theoretical representation of reality.^{[3]}
Applied mathematics is the transfer of mathematical 'rules' from 'fairly accurate' theoretical models into new 'fairly accurate' theoretical models. Of course, they can then be tested to see if relationships actually exist.
In the field of computer science, theoretical models can be tested almost instantaneously. Ever more increasingly, computers are able measure electrical signals and perform logical tests, processing the data in next to realtime. ^{[4]}
The difference between formal science and natural science is that the formal scientists concern themselves mostly with storing, encoding, decoding and manipulating input & output of data whereas the natural scientists concern themselves solely with relevance of the observable data alone.
A good example of this how natural and social scientists seem to think a hypothesis and evaluation is a fixed requirement for use within a scientific method.
A hypothesis usually defines falsifiable terms in order to 'prove' or 'disprove' a prediction which can be based on subjective conjecture, observations or even just an "educated guess" and an evaluation usually 'tests' the falsifiable terms, to see if the prediction was 'correct' or 'incorrect'.
Formal science does not attempt to "guess" what "will happen"^{[5]} during a test. It does not believe in 'proving' or 'disproving' a biased prediction 'right' or 'wrong' and thus only relies on the most objective factors of scientific method such as methodology and result(s).
Formal scientists believe that people can make their own mind up and draw their own conclusions from the actions and consequences themselves. In short, they tend to look at on what happens rather than attempting to justify why it happens.
Mathematics, logic, words and symbols are frequently used by natural scientists, thus formal science often provides a natural scientist's toolkit.
Formal science's began before the formulation of scientific method, with the most ancient mathematical texts dating back to 1800 BC (Babylonian mathematics), 1600 BC (Egyptian mathematics) and 1000 BC (Indian mathematics). From then on different cultures such as the Indian, Greek and Islamic mathematicians made major contributions to mathematics, while the Chinese and Japanese independently developed their own mathematical tradition.
Besides mathematics, logic is another example of one of oldest subjects in the field of the formal science's. As an explicit analysis of the methods of reasoning, logic received sustained development originally in three places: India from the 6th century BC, China in the 5th century BC, and Greece between the 4th century BC and the 1st century BC. The formally sophisticated treatment of modern logic descends from the Greek tradition, being informed from the transmission of Aristotelian logic, which was then further developed by Islamic logicians. The Indian tradition also continued into the early modern period. The native Chinese tradition did not survive beyond antiquity, though Indian logic was later adopted in medieval China.
As a number of other disciplines of formal science rely heavily on mathematics, they did not exist until mathematics had developed into a relatively advanced level. Pierre de Fermat and Blaise Pascal (1654), and Christiaan Huygens (1657) started the earliest study of probability theory (statistics) in the 17th century.
In the midtwentieth century, mathematicallybased studies such as operations research and systems engineering were developed. The rise of the computer gave a great impetus to these sciences and to theoretical computer science and information theory, allowing the study of complex systems beyond the range of traditional mathematical techniques. The rise of these disciplines made it clear that mathematics was only one of a range of formal or mathematical sciences.
