While reading books in Mathematics and Theoretical Computer Science, usually the term Formal Statement props up. What is meant by that?
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1This is a second repetition of a statement that was previously made informally, i.e. imprecisely. – vadim123 Jan 27 '14 at 02:00
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See http://en.wikipedia.org/wiki/First-order_logic#Formation_rules.. – Berci Jan 27 '14 at 02:06
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better if you gave a longer excerpt from a mathematics book you have in mind. – Will Jagy Jan 27 '14 at 02:28
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It's a tricky question because "formal" can mean two different exactly opposite things depending on the context. Sometimes an author will carry out "formal manipulations", meaning manipulations that look reasonable or intuitive but are not rigorous. On the other hand, formal can also mean extra rigorous in some contexts, such as "formal systems" in logic.
Can you post an example of where formal is used that is confusing, along with the context?
Nick Alger
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What is formal depends on who you ask. Something that is formal to someone who studies "applied" mathematics may seem quite informal to someone who studies mathematics for mathematics sake. Formal is usually used to mean "meeting some standard of rigor".
An informal proof that $x^3 \in \mathbb O (x^4)$ might simply say "because $x^3$ is smaller than $x^4$". Someone might call $(\forall x > 1)\, x^4 > x^3 \rightarrow x^3 \in \mathbb O (x^4)$ a more formal statement.
Some (myself included) only use "formal" to mean a mathematical statement that is part of a (usually electronic computer) proof engine, and use the word "rigorous" for anything less.
DanielV
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