Speaking as one with a PhD in Pure Math, who goes out and gives talks about how exciting and interesting math can be, who knows and has regular contact with working mathematicians such as Tim Gowers, I can unequivocally say yes, theorem proving is an art.
But there is a stage where the art finishes, and you have to convert the proof into a communicable description of the proof. This is where you have to create a complete narrative, starting from known and agreed statements and deriving a sequence of statements using agreed deductive principles.
The artistry is in finding the why of your theorem, in selecting your path through the infinitely large set of exponentially branching possibilities, and being guided some sense of "rightness," or beauty.
I've always (for some definition of "always") felt that finding conjectures tends to be a science. You gather data, you find patterns, you form hypotheses, you make predictions, you test your predictions, you gain confidence, and eventually you state your conjecture. So far you've been doing science.
Then you have to proof your result. Without an indefinable sense of a guiding "truth" you are then lost. Mechanical manipulations are of little value, you need a sense of where you are going.
There is the art.
Finally, when you have your proof, you need to give directions, to take others through the steps and thus convince them. That is exposition (of a sort), which is a combination of art and science, but a different art and a different science.
So yes, proving theorems is an art, and as with all arts there is much to be gained from training, but there must be the spark underneath to achieve real distinction.
That's what I lacked, although I have enough to see and appreciate it in others.
But there is a stage where the art finishes, and you have to convert the proof into a communicable description of the proof. This is where you have to create a complete narrative, starting from known and agreed statements and deriving a sequence of statements using agreed deductive principles.
The artistry is in finding the why of your theorem, in selecting your path through the infinitely large set of exponentially branching possibilities, and being guided some sense of "rightness," or beauty.
I've always (for some definition of "always") felt that finding conjectures tends to be a science. You gather data, you find patterns, you form hypotheses, you make predictions, you test your predictions, you gain confidence, and eventually you state your conjecture. So far you've been doing science.
Then you have to proof your result. Without an indefinable sense of a guiding "truth" you are then lost. Mechanical manipulations are of little value, you need a sense of where you are going.
There is the art.
Finally, when you have your proof, you need to give directions, to take others through the steps and thus convince them. That is exposition (of a sort), which is a combination of art and science, but a different art and a different science.
So yes, proving theorems is an art, and as with all arts there is much to be gained from training, but there must be the spark underneath to achieve real distinction.
That's what I lacked, although I have enough to see and appreciate it in others.