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| In [[mathematical logic]], a '''formal calculation''' is 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.
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| 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 [[mathematical logic|formal mathematical logic]].
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| ==Examples==
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| ===A simple example===
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| A somewhat exaggerated example would be to use the equation
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| :<math>\sum_{n=0}^{\infty} q^n = \frac{1}{1-q}</math> | |
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| (which holds under certain conditions) to conclude that
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| :<math>\sum_{n=0}^{\infty} 2^n = -1.</math>
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| 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 [[p-adic number|2-adic numbers]], where the series in question converges and this statement is perfectly valid.
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| ===Formal power series===
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| [[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.
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| ===Symbol manipulation===
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| Suppose we want to solve the [[differential equation]]
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| :<math>\frac{dy}{dx} = y^2</math>
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| Treating these symbols as ordinary algebraic ones, and without giving any justification regarding the validity of this step, we take reciprocals of both sides:
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| :<math>\frac{dx}{dy} = \frac{1}{y^2}</math>
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| Now we take a simple [[antiderivative]]:
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| :<math>x = \frac{-1}{y} + C</math> | |
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| :<math>y = \frac{1}{C-x}</math>
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| Because this is a ''formal'' calculation, we can also allow ourselves to let <math>C = \infty</math> and obtain another solution:
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| :<math>y = \frac{1}{\infty - x} = \frac{1}{\infty} = 0</math>
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| If we have any doubts about our argument, we can always check the final solutions to confirm that they solve the equation.
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| ==See also==
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| *[[Formal power series]]
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| *[[Mathematical logic]]
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| ==References==
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| *{{cite book | author=Stuart S. Antman | title=Nonlinear Problems of Elasticity, Applied Mathematical Sciences vol. 107 | publisher=Springer-Verlag | year=1995 | isbn=0-387-20880-1}}
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| [[Category:Mathematical logic]]
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