Let me point you to the following recent blog post from Prof Keith Devlin, entitled “What is a proof, really?”
After a lifetime in professional mathematics, during which I have read a lot of proofs, created some of my own, assisted others in creating theirs, and reviewed a fair number for research journals, the one thing I am sure of is that the definition of proof you will find in a book on mathematical logic or see on the board in a college level introductory pure mathematics class doesn’t come close to the reality.
For sure, I have never in my life seen a proof that truly fits the standard definition. Nor has anyone else.
The usual maneuver by which mathematicians leverage that formal notion to capture the arguments they, and all their colleagues, regard as proofs is to say a proof is a finite sequence of assertions that could be filled in to become one of those formal structures.
It’s not a bad approach if the goal is to give someone a general idea of what a proof is. The trouble is, no one has ever carried out that filling-in process. It’s purely hypothetical. How then can anyone know that the purported proof in front of them really is a proof?
Click the link to read the rest of the article. Also read the comments below the article to see what conversation has already been generated.
I won’t be shy in saying that I disagree with Keith Devlin. Maybe I misunderstand the subtle nuance of his argument. Maybe I haven’t done enough advanced mathematics. Please help me understand.
Devlin says that proofs created by the mathematical community (on the blackboard, and in journals) are informal and non-rigorous. I think we all agree with him on this point.
But the main point of his article seems to be that these proofs are non-rigorous and can never be made rigorous. That is, he’s suggesting that there could be holes in the logic of even the most vetted & time-tested proofs. He says that these proofs need to be filled in at a granular level, from first principles. Devlin writes, “no one has ever carried out that filling-in process.”
The trouble is, there is a whole mathematical community devoted to this filling-in process. Many high-level results have been rigorously proven going all the way back to first principles. That’s the entire goal of the metamath project. If you haven’t ever stumbled on this site, it will blow your mind. Click on the previous link, but don’t get too lost. Come back and read the rest of my post!
I’ve reread his blog post multiple times, and the articles he linked to. And I just can’t figure out what he could possibly mean by this. It sounds like Devlin thoroughly understands what the metamath project is all about, and he’s very familiar with proof-checking and mathematical logic. So he definitely isn’t writing his post out of ignorance–he’s a smart guy! Again, I ask, can anyone help me understand?
I know that a statement is only proven true relative to the axioms of the formal system. If you change your axioms, different results arise (like changing Euclid’s Fifth Postulate or removing the Axiom of Choice). And I’ve read enough about Gödel to understand the limits of formal systems. As mathematicians, we choose to make our formal systems consistent at the expense of completeness.
Is Devlin referring to one of these things?
I don’t usually make posts that are so confrontational. My apologies! I didn’t really want to post this to my blog. I would have much rather had this conversation in the comments section of Devlin’s blog. I posted two comments but neither one was approved. I gather that many other comments were censored as well.
Here’s the comment I left on his blog, which still hasn’t shown up. (I also left one small comment saying something similar.)
You said you got a number of comments like Steven’s. Can you approve those comments for public viewing? (one of those comments was mine!)
I think Steven’s comment has less to do with computer *generated* proofs as it does with computer *checked* proofs, like those produced by the http://us.metamath.org/ community.
There’s a big difference between the proof of the Four Color Theorem, which doesn’t really pass our “elegance” test, and the proof of which can be found here: http://us.metamath.org/mpegif/efipi.html
A proof like the one I just linked to is done by humans, but is so rigorous that it can be *checked* by a computer. For me, it satisfies both my hunger for truth AND my hunger to understand *why* the statement is true.
I don’t understand how the metamath project doesn’t meet your criteria for the filling in process. I’ll quote you again, “The trouble is, no one has ever carried out that filling-in process. It’s purely hypothetical. How then can anyone know that the purported proof in front of them really is a proof?”
What is the metamath project, if not the “filling in” process?
If anyone wants to continue this conversation here at my blog, uncensored, please feel free to contribute below :-). Maybe Keith Devlin will even stop by!