>It can be that the PhD's "view of how the system under examination works" indicates that your mathematics doesn't actually apply to the system. At that point, it doesn't matter how error-free your mathematics is.

If they pointed out that it doesn't apply, sure.

However, this is how it usually happened (meant to put in the original post but forgot):

He looks at his colleague and asks "Have you ever heard of this?" The other person shakes his head. Hence, rejected.

That's why the chapter is called social proof.

No commentary about my mathematics, or how applicable it is.

I'm not talking really complex stuff. We had a model of a physical process (equations) that they had put into their software. The equations were in a reference document we all had access to. Occasionally they would say something like "This cannot be modeled because the equation in our model is not monotonic". At which point I take the equation, compute the derivative (sorry, this example did have calculus), and show that it is monotonic.

Response: "Look, everyone knows such a system is not monotonic!" (note again the socialness of their proof)

I'll give you a reverse example.

(Details varied to simplify the example).

We had a circuit (netlist) whose output (e.g. current) we were interested in. I was tasked with tweaking some of the components such that we hit a target current. I did it, but did not modify any components' capacitance. However, some of the frequency output was impacted, which we normally control by varying the capacitances in the circuit.

Their response: You screwed up - we told you not to change the capacitance!

Me: The capacitances are all the same. They are unchanged. You can verify for yourself.

Them: Impossible. I've been doing this for 15 years, and have never heard of the frequency changing for reasons other than capacitance.

Me: Here are the actual equations for the frequency measurement that you're worried has changed (I know them because I coded them into the system!). Capacitance is not an explicit input, but can creep in indirectly. It's not obvious to me from the equations what role capacitance even plays here (linear, quadratic, exponential, lognormal, etc). Can you point out to me why you're so certain?

Them: Look. The frequency never changes unless you change the capacitance. Everyone who has done this for years knows that (and he was right - everyone did say that). Go redo all this work.

So I redid it with the exact same result (wasn't really hard - I version controlled my work).

Them: Unacceptable. I will not accept this work unless you can explain to me why the frequency is changing when it shouldn't.

Me: It's a complex circuit. I didn't design it. I'm not a circuits guy. I don't know the intricacies.

Them: You're going to have to figure it out.

Me: I'll go to the circuit designer (in another team).

(Walk to his cube - he's out for a week on vacation).

Them: Sorry, we cannot continue this work unless this is resolved.

(Twiddle my thumbs for a week till the designer returns. Then ask him).

Circuit Designer: Of course it can change even if the capacitances don't change. Why are they saying it only changes with capacitance? Based on what?

Me: Based on (making the quotes symbol with hands) "everyone knows". (Yes, I really did respond that way - the absurdity was getting to me by then).

Designer: Let me talk to them.

Overall, 2 weeks wasted because "everyone knows". I think in the whole team, I was the only one who questioned the tribal wisdom. Whenever someone joined the team, they were taught this incorrect tribal wisdom. I was in the (un)fortunate position to have done some work that just happened to go against the tribal wisdom. I had to defend myself, and that forced me to question the wisdom (once I determined I had done all the steps correctly).

But they did not have to explain why they believed what they did. I had to explain why I did not believe what they did. They were the ones making an assertion about the relationship between capacitance and frequency. However, it was my job to disprove it - not their job to prove it. Essentially, I was put in a position to prove a negative, because they already had their proof (social proof). My proof was very clear: I had a clear counterexample to their theory, but it didn't hold up to their proof. They were not willing to examine my counterexample.

The guy grilling me who wasted 2 weeks wasn't just anyone. He was one of the most senior engineers in the company. Very sharp guy who deserved his post. Not an idiot.

But even they fall prey to social proof.