Mathematics and Logic

Logic plays a double role in mathematics. What most people think of as its role, but it is really rather unimportant for most mathematicians, that it provides the foundations on which the subject is built. No mathematician ever writes out a long complicated argument by going back to the notation and formalism of logic, but every mathematician must have the confidence that he/she could do if it were demanded. Reasoning and logic are to each other as health is to medicine, or better as conduct is to morality. Reasoning refers to a range of natural thought processes in the everyday world. Logic is how we ought to think if objective truth is our goal and the everyday world is very little concerned with the objective truth. Logic is the science of the argument of conclusions we have reached by natural reasoning. My point is that, for such natural reasoning to occur, consciousness is not required. The very reason we need logic at all is because most reasoning is not conscious. Thus it might be said that mathematics naturally proceeds by truth-feelings, but with logic always there as a last resort. The second and much more important role of logic is as a branch of mathematics, on a par with number theory or algebraic geometry, it develops by using the common culture of mathematics, and makes its own rather important contributions to this culture.

One may ask whether logic is part of philosophy or independent of it. According to Bochenski, this issue is nowhere explicitly raised in the writings of Aristotle. However, Aristotle did go to great pains to formulate the basic concepts of logic (terms, premises, syllogisms, etc.) in a neutral way, independent of any particular philosophical orientation. Thus Aristotle seems to have viewed logic not as part of philosophy but rather as a tool or instrument to be used by philosophers and scientists alike. This attitude about logic is in agreement with the modern view, according to which the predicate calculus is a general method or framework not only for philosophical reasoning but also for reasoning about any subject matter whatsoever.

Logic is the science of correct reasoning. What then is reasoning? According to Aristotle, Topics, reasoning is any argument in which certain assumptions or premises are laid down and then something other than these necessarily follows. Thus logic is the science of necessary inference. However, when logic is applied to specific subject matter, it is important to note that not all logical inference constitutes a scientifically valid demonstration. This is because a piece of formally correct reasoning is not scientifically valid unless it is based on a true and primary starting point. Furthermore, any decisions about what is true and primary do not pertain to logic but rather to the specific subject matter under consideration. In this way we limit the scope of logic, maintaining a sharp distinction between logic and the other sciences. All reasoning, both scientific and non-scientific, must take place within the logical framework, but it is only a framework, nothing more. This is what is meant by saying that logic is a formal science.

A logical system of any kind has the following parts. First here is an alphabet of symbols from which formulas will be built. These may include symbols for variables, logical symbols like connectives and quantifiers, and mathematical symbols standing for relations, constants and functions. A formula is a finite string of symbols, but not every finite string of symbols is a formula. There should be a rule, mechanical in nature, for deciding whether a string of symbols is a valid formula. Such a rule is often given by specifying “atomic” formulae and giving rules for building more complicated formulae from simpler ones, together with a proof that any valid formula can be built up in a unique and recognisable way by the rules. Logic is the science of formal principles of reasoning or correct inference. Historically, logic originated with the ancient Greek philosopher Aristotle. Logic was further developed and systematized by the Stoics and by the medieval scholastic philosophers. In the late 19th and 20th centuries, logic saw explosive growth, which has continued up to the present.

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