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I have been wondering for several years why I just can't trust my own proofs in math or when giving some logical arguments. I keep thinking I might be making a mistake at some step of the proof and then I doubt every step.
After reading a bit of Descartes I suspect that the reason I am not confident in my proofs is that I doubt my logical reasoning. May I ask what is the problem here? How can I stop doubting my reasoning?
Sherrilyn Roush describes epistemic self-doubt as the special case of doubt "where what we doubt is our ability to achieve an epistemically favorable state, for example, to achieve true beliefs". The problem with self-doubt is "one is using one’s judgment to make a negative assessment of one’s judgment".
Not only Descartes but also Socrates is known for self-doubt. For Socrates, Roush claims
his acknowledgment that he did not know had the salutary effect of making it possible for him to find out. If he were sure that he already knew, then he would not have motivation to look for the answer.
Not all self-doubt is "constructive":
...Descartes’ epistemic self-doubt was extreme in undermining trust in a belief-forming faculty, and in the wide scope of beliefs that were thereby called into question...
Rousch presents five questions underlying the "fit between one’s beliefs (first-order beliefs) and one’s beliefs about one’s beliefs (second-order beliefs)":
Questions about epistemic self-doubt can be organized into five over-arching questions: 1) Can the doubting itself, a state of having a belief state and doubting that it is the right one to have, be rational? 2) What is the source of the authority of second-order beliefs? 3) Are there general rules for deciding which level should win the tug of war? If so, what is their justification? 4) What does the matching relation this adjudication is aiming at consist in? 5) If mismatch between the levels can be rational when one first acquires reason to doubt, is it also rationally permitted to remain in a level-splitting state—also known as epistemic akrasia (Owens 2002)—in which the self-doubting conflict is maintained?
Given this summary, let's consider the OP's concern:
After reading a bit of Descartes I suspect that the reason I am not confident in my proofs is that I doubt my logical reasoning. May I ask what is the problem here? How can I stop doubting my reasoning?
The doubting of logical reasoning suggests the OP's concern fits Rouch's first question about whether such doubt is rational. Given the examples of both Socrates and Descartes and evident human fallibility there is nothing wrong with such doubts. The final concern about how to stop doubting suggests that the OP is also concerned with the rationality of remaining in a "level-splitting state".
Rousch offers four models for approaching those five questions. The final one described as "calibration" may be helpful. Our beliefs (proofs) are like reading a thermometer. When we doubt the reliability of the thermometer itself (logical reasoning) the thermometer needs calibration.
Regarding math proofs or logical arguments one way to calibrate the "thermometer" is to use a proof checker.
Roush, Sherrilyn, "Epistemic Self-Doubt", The Stanford Encyclopedia of Philosophy (Winter 2017 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/win2017/entries/epistemic-self-doubt/.
Excellent reasoning. The answer is: you can't. Doubt about the foundations of knowledge is the natural consequence of knowledge itself.
Kant proposed the idea that our perception defines our reality, but it could be inherently flawed. That is because we live upon a tautology. There's no ultimate rule that would validate any other rules. Truth is defined by our perceptions, and our perception tells us what is truth.
Descartes already noticed the problem, so he tried to find a fundamental truth. And that truth is existence. We cannot doubt existence. My interpretation of cogito ergo sum is this: something exists if we can interact with it in some way. So, if I interact with myself, I can be sure that I exist, so I think (of myself), then I exist.
From my personal point of view, sciences are viced with presuppositions based on the kantian tautology. Mainly, the effect of perception upon knowledge. You can read my personal criticism to science exemplified with thermodynamics here: https://philosophy.stackexchange.com/a/57718/23407
