Rigour in the context of "Consistency"

A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning that establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning that establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.

Proofs employ logic expressed in mathematical symbols, along with natural language that usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics, oral traditions in the mainstream mathematical community or in other cultures. The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.

Logical reasoning is a mental activity that aims to arrive at a conclusion in a rigorous way. It happens in the form of inferences or arguments by starting from a set of premises and reasoning to a conclusion supported by these premises. The premises and the conclusion are propositions, i.e. true or false claims about what is the case. Together, they form an argument. Logical reasoning is norm-governed in the sense that it aims to formulate correct arguments that any rational person would find convincing. The main discipline studying logical reasoning is logic.

Distinct types of logical reasoning differ from each other concerning the norms they employ and the certainty of the conclusion they arrive at. Deductive reasoning offers the strongest support: the premises ensure the conclusion, meaning that it is impossible for the conclusion to be false if all the premises are true. Such an argument is called a valid argument, for example: all men are mortal; Socrates is a man; therefore, Socrates is mortal. For valid arguments, it is not important whether the premises are actually true but only that, if they were true, the conclusion could not be false. Valid arguments follow a rule of inference, such as modus ponens or modus tollens. Deductive reasoning plays a central role in formal logic and mathematics.

The scholarly method or scholarship is the body of principles and practices used by scholars and academics to make their claims about their subjects of expertise as valid and trustworthy as possible, and to make them known to the scholarly public. It comprises the methods that systemically advance the teaching, research, and practice of a scholarly or academic field of study through rigorous inquiry. Scholarship is creative, can be documented, can be replicated or elaborated, and can be and is peer reviewed through various methods. The scholarly method includes the subcategories of the scientific method, with which scientists bolster their claims, and the historical method, with which historians verify their claims.

In mathematics and computer science, an algorithm (/ˈælɡərɪðəm/ ) is a finite sequence of mathematically rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning).

In contrast, a heuristic is an approach to solving problems without well-defined correct or optimal results. For example, although social media recommender systems are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation.

The language of mathematics has a wide vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in lectures, and sometimes in print, as informal shorthand for rigorous arguments or precise ideas. Much of this uses common English words, but with a specific non-obvious meaning when used in a mathematical sense.

Some phrases, like "in general", appear below in more than one section.

Hard science and soft science are colloquial terms used to compare scientific fields on the basis of perceived methodological rigor, exactitude, and objectivity. In general, the formal sciences and natural sciences are considered hard science by their practitioners, whereas the social sciences and other sciences are described by them as soft science.

Precise definitions vary, but features often cited as characteristic of hard science include producing testable predictions, performing controlled experiments, relying on quantifiable data and mathematical models, a high degree of accuracy and objectivity, higher levels of consensus, faster progression of the field, greater explanatory success, cumulativeness, replicability, and generally applying a purer form of the scientific method. A closely related idea (originating in the nineteenth century with Auguste Comte) is that scientific disciplines can be arranged into a hierarchy of hard to soft on the basis of factors such as rigor, "development", and whether they are basic or applied.