What are some strategies to challenge an appeal to authority fallacy in a respectful way?

What happens when the "authorities" cited are mathematical proofs? Scientists and engineers study math as a tool to prove correctness of results.

I think part of this also has to do with the human need to be apart of the ‘herd’ when we see someone with perceived authority or power we instantly believe that it’s more true. Individually we will think logically but when in a group more often than not we will stay in the group.

Technologies and institutions viewed as authority figures also fall under this fallacy (and others). Not knowing anything about the people and systems composing them adds several degrees of risk. Establishing and maintaining a constant and present sense of trustworthiness for every source and piece of information we (each) interact with isn't realistic. This fallacy is a natural part of the imperfect human experience counterbalanced by seeking assurance our trust isn't misplaced where it matters to us.

It is sad and pitiful because I have seen victims of authority fallacy all over the place. Some are beaten by it so they never question anything, they even praise the authorities. Is it fear? Is it flattering? I don't know. I think it may relate to self-confidence in some circumstances, but it seems a lack of individual thinking may be the case. All these lead to insensitivity and ignorance. Such vices will gnaw at the consciousness piece by piece until it becomes numb. I work with one living proof right now, folks. So, take heart!

I find that building authority is important when you are showing potential customers why your product works. Influencer marketing is super useful, but as a business owner and as a consumer you must be aware of the effectiveness of this as well as using it in an ethical way

I rather use the word, "make sense." In my experience, there are various situations worthy of being skeptical, and this skill would prove the ability of risk assessment. If the authority cannot endorse extraordinary skills, that means this management is incompetent or vainglorious. Logically and respectfully express facts and potential results in a suggestive manner in combination with a sincere attitude, laying reasonable solutions but leaving the decision to the authority. Then, you can say, "I told you so."

In my experience as a young (still new at the time) employee working in the research field, it is hard to leave room for the authority fallacy, but it definitely still happens. I noticed this article didn't cover is the response of "Because I said so," or "I have been doing this longer than you have." Maybe this a different topic, but just because someone has years of experience handling a problem or challenge, does not mean they are correct or have identified the best way to succeed. You should always confidently explain your perspective. If it's not met with the same level of respect or politeness, it's also important to know when to stop questioning or challenging because you're not going to make a meaningful breakthrough.

The missing piece in this well written, informative piece is what do you do when the powers that be are in love with the "authority" and their thoughts? Do you buck the system and cry BS, or do you cite the authority figure's position and bite your tongue? I would go with my best judgement as to what the best answer is and have a pocketful of reasons as to why I did.