Yesterday I was talking to one of my mathematics professor regarding the notion of proof in general (whatever the word "general" means to the reader).
In short my claim was,
We can only be certain at the cost of certainty.
Now, let me explain how I came to this seemingly paradoxical idea. Please correct me if my argument is wrong.
When we are talking about the conception of proof, we must be (at least it seems to me) talking about the conception of proof in a language. It seems to me that a proof may be defined (loosely speaking) as a certain set of sentences. However, the problem lies in this linguistic conception of proof. Suppose that in a language L we have proved something. Say, for example, we have proved a mathematical theorem in a formal language. How can we say that the proof is correct? Well, we can say that because we have followed only the logical rules to arrive at the conclusion from premise(s). But the question is, how can we be sure that we have followed "only the logical rules to arrive at the conclusion from premise(s)"? More precisely, how can we be sure that we can indeed use the word "only"? If we can't then we need to prove that we have actually proved what we claim to have proved. The process can be repeated infinitely many times.
When I argued like this the professor told me that he thinks that the question is merely wordplay. When I asked him the reason he wasn't able to give me an answer that satisfies me.
So, basically my questions are two,
Is my question only wordplay? If so, why?
If not then am I correct in claiming what I have claimed at the outset?