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In Mathematics, we as an undergraduate are exposed for the first time (at least for me it was the case) to 'rigor'. For example, in Real Analysis classes we often use logical quantifiers in our arguments. However, the curious thing is that even though we are told to use formal reasoning in our arguments we are never told explicitly the reason behind using formal reasoning.

For example, in philosophical arguments the method of reasoning is not always formal and I think that it is more natural to use this kind of reasoning method. Surprisingly enough when we discuss the proofs, we in fact use this kind of informal reasoning but still our professors insist on being able to write formal proofs.

This makes me wonder,

What actually is(are) the importance(s) of formal reasoning in Mathematics? in general in Philosophy?

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    Mathematics is less formal than many people think. The vast majority of mathematical work is informal reasoning, with some kind of tacit background assumption that the arguments could in principle be formalized. – Era Feb 15 '16 at 15:32
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Formal reasoning is powerful for two reasons. The first is the simplest to see: it is easily conveyed in a highly objective manner. When trying to discuss things which must be true for all, the more objective the better.

The second advantage of formal reasoning is that it can be written using formal languages. This permits the fascinating ability to discuss the validity of a claim through symbol manipulation alone. People can agree that a particular set of symbol manipulations correspond to true semantic meaning, and then determine if the semantic meaning is valid merely by looking at the symbols.

My favorite example of where this can be useful is the handling of infinity in mathematics vs. the handling of infinity in Pascal's Wager. The rules of set theory permit a very specific set of behaviors for infinite values, such as the cardinality of the set of all natural numbers. It states that the cardinality of the set of natural numbers is fundamentally smaller than the cardinality of the set of real numbers. Contrast that with Pascal's wager, where Pascal effectively argues that since one infinity appeared in the equation, it clearly had to dominate. It is now trivial to develop mathematical arguments for or against Pascal's wager, using various infinities. It is trivial to discuss maximizing value in the wager by taking the derivative of value and applying l'Hopital's rule (itself a wedging of two infinities against eachother). The only reason these are trivial is because their definitions and proofs are so rigorous that we accept them as valid even if the situation is absurd.

The limit to what formal languages can do was explored by Tarski. You are correct in being bothered that people don't want to discuss why formal reasoning is so important. Tarski demonstrated that the semantics of a formal language must be defined in a metalanguage, suggesting eventually you arrive at a non-formal language to describe how to interpret all the formal languages. However, despite this shortcoming, there is still value in them. They permit arguments like "either you must accept this formal reasoning, or you must come up with your own explanation of why 2+3=5 without using the Peano axioms at the basis of modern math."

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Formal reasoning is an important tool to make subtle differences transparent and explicit. This can be a prerequisite to assess the validity of certain argumentations. E.g., the correct definition of continutiy of a function "f" at an argument "X_0" is

∀epsilon > 0 ∃delta > 0 : |x-x_0| < delta => |f(x) - f(x_0)| < epsilon.

On the opposite, the following statement does not correctly characterise continuity - only the order of epsilon and delta has been interchanged:

∃delta > 0 ∀epsilon > 0 : |x-x_0| < delta => |f(x) - f(x_0)| < epsilon.

By examples like these the beginner learns the differences between and the subtleness of mathematical statements.

In general, it is neither necessary nor desirable to formalize the whole proof of a mathematical theorem. But sometimes it is helpful to formalize a distinguished passage, in order to write down a precise statement.

It is very rare in philosophy to formalize an argument, because already the concepts are defined not precise enough. A famous counterexample is Goedel's formalization of the ontological "proof" for the existence of God. For a nice scetch of the argument see https://en.wikipedia.org/wiki/G%C3%B6del's_ontological_proof

But who wants do read and to check this proof except a student of formal logic?

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The general rule is formal logic for rigor, informal logic for understanding.

If you want to actually decisively establish a result, you do need formal logic, because informal reasoning can be ambiguous and misleading --it's quite possible for something important to be missed. The goal of formal reasoning is to preclude that possibility. However, it's not the way we actually think, so it helps to have an informal apprehension of the problem before tackling the formal proof.

It's not quite the same situation in philosophy, because there is arguably quite a lot of important territory in philosophy that isn't susceptible to formal analysis (I say arguably because it's a controversial topic). On the other hand, everything in mathematics should theoretically be capable of being made formal. In a certain sense, mathematics IS making things formal.

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