Philosophy of Proof Analysis

Desire to know is the fundamental drive of human ingenuity and progress. In the realm of epistemology (theory of knowledge), there is a definite demarcation between facts that are self evident, a priori, and those that are proved to be true from employing systematic processes of verification. This, in more technical term, is referred to as the scientific methodology. The scientific method is the systematic process of certifying that an assumption concerning a set of phenomena made based on a single or few observations can be used to determine the general and universal behavior of like phenomena when exposed to variables similar to the earlier observation. The process passes through stages. First an observation is made about a set of phenomena. Then a hypothesis is established. Subsequently more investigation is carried out concerning the said phenomena leading to the gathering of data. The gathered data are then tested in a variety of ways. When this happens, the earlier hypothesis is either proved to be true based on these tests or discarded as not true. In the event that the earlier hypothesis is proved true, then it becomes a theory. This scientific technique of truth verification starts first from a single observation to a generalization.

It is not to say that science has been totally oblivious of the so called self-evident truths. These are facts which are accepted by all as needing no verification since they are in their nature purely self evident or a priori. Such truths are used to make formal judgments about particular things. The statement, “all men are mortal”, assumes a priori that every living man must someday die and so based on this one can say that since there is a man named Pete, that he is mortal and therefore must one day die. This is accepted as true based on the universal statement “all men are mortal”. However, some such formerly established universal statements have been overturned by science. This was possible through the breakthroughs of science. It was formerly established that “whatever goes up, must come down”. This at one point was self-evident and incontrovertible. But right now, based on the better knowledge of the planetary arrangements and space exploration, this hitherto universal statement has been nullified. In a sense, the notion of “up” or “down” are no more seen from the same perspective as was hitherto the case. Those terms are rather contingent upon relative points of reference in the cosmos.
Any study therefore on techniques of proof hinges on these two dimensions of scientific inquiry. The one progresses from the part to the whole; the other assumes the whole as self-evident and now makes judgments about the part from this a priori knowledge of the whole. Proof is simply the process of how propositions and conclusions are derived from employing specific procedures. Those procedures are the techniques of proof. The technique employed in proving a given proposition is largely determined by the nature of field of study within which the said proof is sought. Proof in logic and mathematics is different, (procedurally and in the nature of interacting variables), from proof in the physical sciences. Both are different from proof in Law. But in all, the process tries to establish the truth of propositions based on specific core assumptions and peculiar behaviour of interacting variables.

Proof in mathematics and logic follows the patterns by which premises are made and based on the nature of the interaction among these premises, conclusions are drawn.

Let us examine the following argument.

All S is P, X is S, therefore X is P.

The process assumes first that everything in the class of S is contained in the class of P. Now if X is part of the class of S which is wholly part of P, it therefore follows that everything about X, (insofar as X is part of S) is also part of P.
This form of proof is employed in logic and mathematics, though in the latter, the premises can be referred to as the ‘demonstrates’. In mathematics, the interacting variables by which proofs are made are mostly made up of numbers and equations, as in algebra. Algebra, allows symbols (usually letters) to represent unknown numbers in mathematical equations. Algebra enables the basic operations of arithmetic, such as addition, subtraction, and multiplication, to be performed without using specific numbers. People use algebra constantly in everyday life, for everything from calculating how much money would be needed to fund a child’s tuition in a given year, to figuring out how long it will take to travel by car at a certain speed to a destination that is a specific distance away. Here the value of the numbers and symbols employed in proving a given mathematical proposition is mostly virtual.

Let us take the Pythagorean theorem for example, which states that “in a right angled triangle, the square on the hypotenuse, is equal to the sum of the squares in the other two sides”. This theorem is represented as a2 + b2 = c2 . Now to prove this theorem, the variables a, b, and c would be furnished with values which are merely virtual and do not have any real representation in any citable concrete experience and from the interaction among the variables, proof is made.
Proof in science goes beyond the pure philosophical question of ‘why’ of things to the ‘how’. Scientific methodology is rigorous and seeks to formulate theories by which universal proposition and generalizations can be made without danger of error. Science goes beyond the elementary assumptions of common sense, to seek universal grounds on which a particular phenomenon or set of phenomena can be predicated.
In law, proof is the process by which evidences are used to establish that a particular issue (or issues) as debated on the floor of the court is either true or false. Proof is therefore essentially concerned with clarifications on the ‘how’ of a given statement, but the very letters of the procedure of this clarification differ from field to field.

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