Formal calculation

Formal calculation

In mathematical logic, a formal calculation is sometimes defined as a calculation which is "systematic, but without a rigorous justification". This means that we are manipulating the symbols in an expression using a generic substitution, without proving that the necessary conditions hold. Essentially, we are interested in the form of an expression, and not necessarily its underlying meaning. This reasoning can either serve as positive evidence that some statement is true, when it is difficult or unnecessary to provide a proof, or as an inspiration for the creation of new (completely rigorous) definitions.

However, this interpretation of the term formal is not universally accepted, and some consider it to mean quite the opposite: A completely rigorous argument, as in formal mathematical logic.

Examples

A simple example

A somewhat exaggerated example would be to use the equation

:sum_{n=0}^{infty} q^n = frac{1}{1-q}

(which holds under certain conditions) to conclude that

:sum_{n=0}^{infty} 2^n = -1.

This is incorrect according to the usual definition of infinite sums of real numbers, since the related sequence does not converge. However, this result can inspire extending the definition of infinite sums, and the creation of new fields, such as the 2-adic numbers, where the series in question converges and this statement is perfectly valid.

Formal power series

Formal power series is a concept that adopts some properties of convergent power series used in real analysis, and applies them to objects that are similar to power series in form, but have nothing to do with the notion of convergence.

ymbol manipulation

Suppose we want to solve the differential equation

:frac{dy}{dx} = y^2

Treating these symbols as ordinary algebraic ones, and without giving any justification regarding the validity of this step, we take reciprocals of both sides:

:frac{dx}{dy} = frac{1}{y^2}

Now we take a simple antiderivative:

:x = frac{-1}{y} + C

:y = frac{1}{C-x}

Because this is a "formal" calculation, we can also allow ourselves to let C = infty and obtain another solution:

:y = frac{1}{infty - x} = frac{1}{infty} = 0

If we have any doubts about our argument, we can always check the final solutions to confirm that they solve the equation.

ee also

*Formal power series
*Mathematical logic

References

*cite book | author=Stuart S. Antman | title=Nonlinear Problems of Elasticity, Applied Mathematical Sciences vol. 107 | publisher=Springer-Verlag | year=1995 | id=ISBN 0-387-20880-1


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Formal — The term formal has a number of uses, including:General*relating to formality *opposite of informalocial* Formal occasion ** Formal attire worn on such occasions ** Formals are particular meals at some British universities ** In Australian or… …   Wikipedia

  • formal — formal1 formalness, n. /fawr meuhl/, adj. 1. being in accordance with the usual requirements, customs, etc.; conventional: to pay one s formal respects. 2. marked by form or ceremony: a formal occasion. 3. designed for wear or use at occasions or …   Universalium

  • formal logic — the branch of logic concerned exclusively with the principles of deductive reasoning and with the form rather than the content of propositions. [1855 60] * * * Introduction       the abstract study of propositions, statements, or assertively used …   Universalium

  • formal — for•mal [[t]ˈfɔr məl[/t]] adj. 1) being in accordance with the usual requirements, customs, etc.; conventional: to pay one s formal respects[/ex] 2) marked by form or ceremony: a formal occasion[/ex] 3) clo designed for wear or use at elaborate… …   From formal English to slang

  • formal rationality — As defined by Max Weber in his account of the market and economic action, this refers to the extent of impersonal quantitative calculation (that is, risk assessment) which is possible and applied in provisioning for needs. Money is the best means …   Dictionary of sociology

  • Back-Of-The-Envelope Calculation — An informal mathematical computation, often performed on a scrap of paper such as an envelope. A back of the envelope calculation uses estimated and/or rounded numbers to quickly develop a ballpark figure. The result should be more accurate than… …   Investment dictionary

  • mathematics — /math euh mat iks/, n. 1. (used with a sing. v.) the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. 2. (used with a sing. or pl. v.) mathematical procedures,… …   Universalium

  • analysis — /euh nal euh sis/, n., pl. analyses / seez /. 1. the separating of any material or abstract entity into its constituent elements (opposed to synthesis). 2. this process as a method of studying the nature of something or of determining its… …   Universalium

  • Division by zero — This article is about the mathematical concept. For other uses, see Division by zero (disambiguation). The function y = 1/x. As x approaches 0 from the right, y approaches infinity. As x approaches 0 from the left, y approaches negative …   Wikipedia

  • List of mathematics articles (F) — NOTOC F F₄ F algebra F coalgebra F distribution F divergence Fσ set F space F test F theory F. and M. Riesz theorem F1 Score Faà di Bruno s formula Face (geometry) Face configuration Face diagonal Facet (mathematics) Facetting… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”