Proofs and sufficient arguments.
Proving something without any doubt is an elusive task. What does it mean to actually prove something? When is the argument good enough? All these questions may have popped up in our heads during our tenure as high school or university students. Moreover, these are questions that may appear throughout our lives in different contexts and situations. While the current post concerns mostly mathematical proofs and formal proofs, I believe that knowing from where the idea of proving something comes, may help the reader in different situations. This is particularly important in times in which truths are being distorted in all kind of ways.
The simplest definition of proof I could find is from Wikipedia (what a shocker): “A proof is sufficient evidence or a sufficient argument for the truth of a proposition”. This definition is rather interesting, but it introduces a term that by itself does not say much: sufficient evidence. I have always interpreted this notion as a state of being where the claim you are trying to prove is self-evident; not only for yourself, but for every person who sees it. I believe then, that the cornerstone of proving is indeed knowing when exactly our argument is enough for validating our claim. What I find really exciting about this idea is that it applies to many aspects of life and many different settings. While it is not the same to prove a mathematical theorem than proving someone is guilty of a crime, the question we are answering in the end is the same: when is our evidence good enough?
Luckily for us, a long time ago a bunch of people, who will probably be part of a different post, came up with a great tool for knowing exactly when an argument is enough: logic.
Note: You can skip the rambling and go directly to the tidbits and recommendations for formal proving by going to the end of the page.
Logic: the study of arguments.
There are many definitions of logic, but I find the one given above, to the simplest of all. Logic is purely and simply the study of arguments. Logic allow us to take a person’s argument and break it down in such a way that it is possible to verify whether the argument makes sense or it is complete nonsense. In this text, we will consider arguments to be the track of thought a person follows when trying to claim something is true (prove!). Now, let us begin with an simple thought experiment:
Try to imagine the physical shape an argument would have and ask yourself what shape do you think of, is it a tree? a graph? a circle? a square? After you have done this, try to picture the components of an argument and think whether there is a certain pattern that allow the make the connection between them.
In my case I often see arguments as tree shaped structures where each node correspond to a proposition and each edge indicates that the next proposition is a consequence of the father node. If reasoning has an structure, is it not reasonable to want to analyze it? Indeed, the object of study of formal logic is the study of the structure of arguments. In particular, in logic we do not care about the meanings of each argument, we only look at its shape, and using determine whether the argument is valid or not.
When doing mathematical proofs, it is logic what dictates whether the shape of our arguments is correct or not. This, obviously is enhanced by adding a meaning to each proposition that we use in the proof.
Example: The sum of two integers equals an integer.
Consider two even integers $x$ and $y$. Proof that is also an even number.
We depart from the premise $x,y$ are even. This implies then that both $x,y$ can be written as multiples of $2$:
therefore, we can rewrite $x+y$ by replacing $(1),(2)$ as follows:
Since any number multiplied by $2$ is an even number (I am not proving this, but it can be done), then we can safely assume that $x+y$ is an even number an therefore, we have finished the proof. QED
Graphically, we could see this reasoning as a tree composed of 4 nodes and 3 edges. The first node is the assumption “$x,y$ are even integers” and each
edge represents a step obtained by using the “semantic” of the predicate. In other words we are using the concrete definition of each predicate to move to the next step. First, we go with the definition of an even number: a number that is a multiple of 2. From this definition we can infer the shape of the new predicate, since this shape is given by equality,
we use it in the predicate:
which means that both $x,y$ are multiples of $2$.
After the rambling that has been this post, I can then enumerate some of my own recommendations when being confronted with a formal proof:
- Write down your assumptions. Key words in the statement you want to prove: if, assume, supppose, etc.
- When you have accumulated all the assumptions from the statement, start by tracing a path to the conclusion you want to reach.
- Once you have an idea of that path, try to see if the assumptions you have can generate new nodes in the mental tree of the proof. If they can, then keep repeating the same with the newly generated statements until you reach the desired conclusion.
- Do not get stressed if the proof is not working out, there are many techniques for proving (I may do a tutorial on this in another blog).
- Sometimes statements are just not true (duh…), in this case you may want to revisit your hypotheses and see if maybe something is missing.
- This is important: HAVE FUN! Proving is like playing to be a detective, you are trying to find solutions and the way you device the solution is usually surprising and can fill you with satisfaction.
After these small recommendations I close this post. It was a bit difficult to write, nonetheless I had tons of fun writing it. Have fun! and happy proving!