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.
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Examples
A simple example
A somewhat exaggerated example would be to use the equation
- <math>\sum_{n=0}^{\infty} q^n = \frac{1}{1-q}</math>
(which holds under certain conditions) to conclude that
- <math>\sum_{n=0}^{\infty} 2^n = -1.</math>
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.
Symbol manipulation
Suppose we want to solve the differential equation
- <math>\frac{dy}{dx} = y^2</math>
Treating these symbols as ordinary algebraic ones, and without giving any justification regarding the validity of this step, we take reciprocals of both sides:
- <math>\frac{dx}{dy} = \frac{1}{y^2}</math>
Now we take a simple antiderivative:
- <math>x = \frac{-1}{y} + C</math>
- <math>y = \frac{1}{C-x}</math>
Because this is a formal calculation, we can also allow ourselves to let <math>C = \infty</math> and obtain another solution:
- <math>y = \frac{1}{\infty - x} = \frac{1}{\infty} = 0</math>
If we have any doubts about our argument, we can always check the final solutions to confirm that they solve the equation.
See also
References
- Stuart S. Antman (1995). Nonlinear Problems of Elasticity, Applied Mathematical Sciences vol. 107. Springer-Verlag. ISBN 0-387-20880-1.
