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One reason why mathematics enjoys special esteem, above all other sciences, is that its laws are absolutely certain and indisputable, while those of other sciences are to some extent debatable and in constant danger of being overthrown by newly discovered facts. —Albert Einstein[1] As opposed to empirical sciences (natural and social), the formal sciences do not involve empirical procedures. They also do not presuppose knowledge of contingent fact, or describe the real world. In this sense, formal sciences are both logically and methodologically a priori, for their content and validity are independent of any empirical procedures. Although formal sciences are conceptual systems, lacking empirical content, this does not mean that they have no relation to the real world. But this relation is such that their formal statements hold in all possible conceivable worlds (see valid formula) – whereas, statements based on empirical theories, such as, say, general relativity or evolutionary biology, do not hold in all possible worlds, and may eventually turn out not to hold in this world as well. That is why formal sciences are applicable in all domains and useful in all empirical sciences. Because of their non-empirical nature, formal sciences are construed by outlining a set of axioms and definitions from which other statements (theorems) are deduced. In other words, theories in formal sciences contain no synthetic statements; all their statements are analytic.[2][3] |
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