Prove it!?

Hopefully you already read the brief intro that would motivate this page. If not, please take a quick look here: Home. 

Proof is just a word and we might use it in three related but distinct senses. There might even be more senses of the word, but that's why we have dictionaries. Here are the three of interest to us here in increasing order of strictness or positiveness:

1. Legal proofs

2. Philosophical proofs

3. Logical and mathematical proofs

Notice that there is no category named "scientific proofs." That might be surprising at first. But, consider that even the best scientific hypotheses and even theories are never proved, they are at best not falsified yet. When they make surprising predictions which are later demonstrated (as in Newton's, Darwin's, Einstein's, Dirac's, Bell's and Higgs's cases) to be accurate, we trust the theory even more.

Let's get back to the three categories of proofs and first remark that even in the court of law, there are, my lawyer friends tell me, at least two distinct burdens of proof, one for the civil courts and one for criminal. Regardless, both, I think, depend on evidence. Motive, witnesses, experts, so on and so forth and the more the better. Law and order is a wonderful topic (and a few TV shows), but, let's leave it to them. Here we want to focus on more reliable proofs. 

So, what is a philosophical proof? Actually, they are not even called proofs! They are called philosophical arguments. Remember: 1) Socrates is a man. 2) All men are mortal. Therefore, 3) Socrates is mortal. That's a valid and sound argument. If there is no equivocation then we can call it a proof. This is a fascinating category with many proofs. Here are some of my favorites.

1. It is not good because God loves it. God loves it, because it is good. 

2. I think therefore I am. 

3. Heavens above, moral law within.

4. Justified true belief is not knowledge.

Each deserve at least a page, so we won't get into them here. Let's move on.

Logical and mathematical proofs are the best kind! To start with logic is the foundation of math so why differentiate the two? That question is so good, it actually answers itself. Because logic is the foundation of math, it is well worth the effort to look at some logic proofs before we move on to my favorite examples of mathematical proofs. Before we continue, let me reassure you that you don't need to have a PhD in math to understand fully what we will talk about. Even my 8 year old son understands some of the earlier proofs I have selected for you. If you are interested, if you care enough, all of them will be clear to you. Just try to explain them to your family and friends and in the process you will realize that you do know them, truly! So, go on a read the first here. Or skip to some that are more fun here.