rigor

Mathematical rigour is often cited as a kind of gold standard for mathematical proof. It has a history traced back to Greek mathematics, in the work of Euclid. This refers to the axiomatic method. Mathematical rigour in the modern sense originates no later than the generation of Bolzano, Cauchy, and Abel. Cauchy's starting point was the rejection of the principle of the generality of algebra as practiced by Euler, Lagrange, and other luminaries of earlier generations. Starting in the 1870s, the term gradually came to be associated with an implementation of the Weierstrass (ε, δ)-definition of limit and Cantorian set theory.

Mathematical rigour can be defined as amenability to algorithmic proof checking. Indeed, with the aid of computers, it is possible to check proofs mechanically by noting that possible flaws arise from either an incorrect proof or machine errors (which are extremely rare).^[3] Formal rigour is the introduction of high degrees of completeness by means of a formal language where such proofs can be codified using set theories such as ZFC (see automated theorem proving).

Most mathematical arguments are presented as prototypes of formally rigorous proofs. The reason often cited for this is that completely rigorous proofs, which tend to be longer and more unwieldy, may obscure what is being demonstrated. Steps which are obvious to a human mind may have fairly long formal derivations from the axioms. Under this argument, there is a trade-off between rigour and comprehension. Some argue that the use of formal languages to institute complete mathematical rigour might make theories which are commonly disputed or misinterpreted completely unambiguous by revealing flaws in reasoning.

In relation to physics

The role of mathematical rigour in relation to physics is twofold.

First, there is the general question, sometimes called Wigner's Puzzle,^[4] "how it is that mathematics, quite generally, is applicable to nature?" However, scientists assume its successful application to nature justifies the study of mathematical physics.

Second, there is the question regarding the role and status of mathematically rigorous results and relations^{[clarification needed]}. This question is particularly vexing in relation to quantum field theory.

Both aspects of mathematical rigour in physics have attracted considerable attention in philosophy of science. (See, for example, ref.^[5] and works quoted therein.)

In relation to the classroom

Rigour in the classroom is a hotly debated topic amongst educators. Generally speaking, however, classroom rigour consists of multi-faceted, challenging instruction and correct placement of the student. Students excelling in formal operational thought tend to excel in classes for gifted students^{[citation needed]}. Students who have not reached that final stage of cognitive development, according to developmental psychologist Jean Piaget, can build upon those skills with the help of a properly trained teacher.^[6]

Rigor in the classroom is commonly referred to as rigorous instruction. It is instruction that requires students to construct meaning for themselves, impose structure on information, integrate individual skills into processes, operate within but at the outer edge of their abilities, and apply what they learn in more than one context and to unpredictable situations ^[7]

See also

• Honesty
• Intellectual
• Intellectual dishonesty
• Pedant
• Scientific method
• Self-deception
• Sophistry

References

1. ^ Bartlett, John, comp. Familiar Quotations, 10th ed, rev. and enl. by Nathan Haskell Dole. Boston: Little, Brown, 1919; Bartleby.com, 2000. http://www.bartleby.com/100/343.html. Retrieved Oct. 25, 2006.
2. ^ Wiener, N. (1985). Intellectual honesty and the contemporary scientist. In P. Masani (Ed.), Norbert Wiener: Collected works and commentary (pp. 725- 729).
3. ^ Hardware memory errors are caused by high-energy radiation from outer space, and can generally be expected to affect one bit of data per month, per gigabyte of DRAM.[1].
4. ^ This refers to the 1960 paper The Unreasonable Effectiveness of Mathematics in the Natural Sciences by Eugene Wigner.
5. ^ Gelfert, Axel, 'Mathematical Rigor in Physics: Putting Exact Results in Their Place', Philosophy of Science, 72 (2005) 723-738.
6. ^ Forum: Academic Rigor, in: UNIversitas: The University of Northern Iowa Journal of Research, Scholarship, and Creative Activity 1.1 (Fall 2005).
7. ^ Jackson, R. (2011). How to Plan Rigorous Instruction. Alexandria, VA.: ASCD.