What makes for a rigorous proof?

Question

As an undergraduate student, who wants to solidify his mathematical skills, I want to understand what exactly determines if a proof is rigorous.

Answer 1

To me, a proof is rigorous if I understand every step and the conclusion follows without any gaps from the premises.

This means, of course, that some proofs that other consider rigorous are not considered such by me because of my lack of understanding or knowledge.

Answer 2

A rigorous proof is understood to be a proof that you typically find in mathematical papers and mathematical analysis books. In other words, it's a proof that you can't poke holes in, so the conjecture must necessarily be true because you have covered all possible objections one might come up with. My analysis prof made the analogy of being a lawyer, and arguing before a jury, whom you are trying to convince that what you are saying is true, and that the opposing counsel is lying.

In practical terms for the undergrad student and/or newbie, there are two broad types of formal/rigorous proofs

(i) constructive proofs

(ii) proofs by contradiction

With proof by contradiction, you suppose that the conjecture is false and consequently arrive at a contradiction.

With constructive proofs, you build your arguments and show why necessarily a conjecture must be true.

These types of proofs are common in mathematics because conjectures are written to lend themselves to such style of proof - in the sense that conjectures make an assertion about the conditions under which a given statement is true.

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In a more concrete sense, lets take the example of continuity, specifically a continuous function.

Informally, a function is continuous on an interval if we can draw the graph from start to finish without ever once picking up our pencil. In other words, a function is continuous if its graph has no holes or breaks in it.

Formally, we have to go into more detail, meaning:

• define a set $A$ as a subset of the set of real numbers $\mathbb{R}$, i.e. $S$ $\subseteq$ $\mathbb{R}$
• define the function $f$ as being a map from the set $A$ to $\mathbb{R}$, i.e. $f:A \rightarrow \mathbb{R}$
• define a constant number $c$ as being an element of $A$, i.e. $ c \in$ $A$
• $f(x)$ is continuous at point $c$ if and only if for every number $\epsilon>0$, there exists a number $\delta>0$ such that $|x-c|<\delta$ $\Rightarrow$ $|f(x)-f(c)|<\epsilon$ (with $\epsilon \in \mathbb{R}$ and $\delta\in \mathbb{R}$)
• if the above statement about $f$ holds for all points in the set $A$, then the function $f$ is continuous on the set $A$.

So you see in the formal sense, we have expanded our "intuitive" definition and provided a constructive algorithm for how you can determine if a function is continuous or not. The things that you should especially notice is that first we picked a single point in the set $A$ and then expanded to the entire set $A$ (which has infinitely many points). Quantifiers (statements like for all, for one, etc.) are key to getting the grip on how to prove things.

A more detailed write-up of how to write rigorous proofs is written by Prof. Hutchings at UC Berkley here: https://math.berkeley.edu/~hutching/teach/proofs.pdf. The book "How to Read and Do Proofs: An Introduction to Mathematical Thought Processes" by Daniel Solow is also a great read to get you used to thinking about rigorous proofs.