# Formality and mathematics

Why is it important to be formal in mathematics? Is formality beneficial for students? Or is it just to scare students away from mathematics?

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I prefer to use intuition to solve math problems. But formality is useful to express your answers. I do think some people get lost in symbol manipulation without seeing the bigger picture, though. I really liked the book "The Loss of Certainty" by Morris Kline, which talks about the difficulty of taking formality to its limit. –  Stephen Montgomery-Smith Nov 28 '13 at 18:39

I will not be so ambitious as to write a full answer to this question. That would require a table of contents listing lots of long chapters.

If you have a class of calculus students in a state university, it would be a very serious mistake to think "These kids understand well the intuitive idea that a continuous function on a closed bounded interval assumes all intermediate values; I must show them that intuition is unreliable and there's a rigorous view of the question."

Rather, developing their intuition for the matter, which is severely deficient when they start the course, should be a priority.

Rigor is used for checking the correctness of the finished product, not for creating it in the first place.

PS: But I think it's also true that when one computes something "formally" (not in the sense of "logically rigorously", although that may be right in some instances, but in the sense of following the form) one might have no intuition telling one where the whole thing is going, and that might come later.

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"Rigor is used for checking the correctness of the finished product, not for creating it in the first place." This is good. –  Hammerite Nov 28 '13 at 22:26

It is not important, or even desirable, to always be formal in mathematics. It is important to sometimes be formal in mathematics, in order to check that our intuition has not led us astray. The hypothetical possibility of formalizing all mathematics that has been done so far is what keeps mathematics together and "on track" as a discipline. We must take care not to stray so far into informality that this becomes impossible.

Speaking sociologically, a mathematical proof of a proposition is something that can convince all other mathematicians (given sufficient attention on their part) that the proposition is correct. Certainly a proof does not have to formal to achieve this goal. However, the possibility of formalizing the proof must exist.

If a proof were essentially intuitive, it might appeal to my intuition but not yours, or vice versa. If mathematicians accepted the conclusions of reasoning that could not be formalized, then mathematics would run the risk of fragmenting into mutually incomprehensible sub-disciplines, none of them with an indisputable claim to mathematical knowledge.

In teaching mathematics, one should be especially conscious of the downsides of excessive formality, such as "scaring students away." However, because the role of formality in mathematics is ultimately an important one, it would be a mistake to avoid formality entirely when teaching mathematics. What is taught in a mathematics classroom should contain something of the essential nature of mathematical practice, even if it would attract more students to explain vector calculus through, say, interpretive dance.

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http://www.refsmmat.com/articles/unreasonable-math.html

This article does an excellent job of explaining why mathematics is so technical: it's a precise set of instructions that, when combined in different ways, can be used to describe diverse real-life phenomena, like energy transfer or the accumulation of interest. It is exactly the precision that allows them to be combined reliably, a property which informal languages lack.

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Why ? Because, given the fact that $\displaystyle\sum_{n=0}^\infty\frac{x^n}{n!}=\lim_{n\to\infty}\left(1+\frac xn\right)^n=e^x$ , one might be tempted to say that $\displaystyle\lim_{n\to\infty}e^{-n}\cdot\sum_{k=0}^n\frac{n^k}{k!}=\lim_{n\to\infty}\left(1+\frac1n\right)^{n^2}\cdot e^{-n}=1$. Both of which are false ! Same here.

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