Recently I had a conversation where I defended the rationality behind my being skeptical of the validity of the proofs and conclusions constructed in very abstracted, and not experimentally or formally verified math fields.

To my surprise, this provoked a very heated debate, where I was criticized for being overly confident in my assessments of fields I have very little contact with (I was expecting begrudging agreement). But there was very little rebuttal of my points! The rest of my conversation group had three arguments:

1. Results which much of a given field in math rests get significant attention.
2. Mathematicians would be highly renowned for falsifying an assumption many others were basing their work on.
3. You know very little about math as a field, so you should trust mathematicians when they tell you something is likely true.

These each seem flawed. My responses:

1. It is not always apparent on which results a given field in math rests, and many fields rely on many other fields for their conclusions, so I'm skeptical of the implicit claim that there are often very few results imported from other fields in a particular field, and that it is easy to enumerate those assumptions. Surely easier than in other fields, but still not absolutely easy. Its not like math papers come with an easy to read requirements.txt file. And even if they did, from programming we know its still not so easy.

2. Maybe this is true, maybe not. What is certain is attempting to falsify results won't get you renown, so mathematicians need to choose between taking the boring & un-glamorous gamble of attempting to falsify someone's results, or choose the far more engaging & glamorous option of proving a new theorem. I expect the super-majority choose to prove new theorems. It is easy to prove a theorem, it is far harder to find the one rotten proof in a pile of reasonable proofs (and even then you still need to go through the work of trying to correct it!).

3. I agree I know very little about math as an academic discipline, but this does not mean I should blindly trust mathematicians. Instead, I should invoke the lessons from Inadequate Equilibria, and ask whether math seems the type of field whose incentives sufficiently reward shorts in the market on math results compared to the difficulty & cost of such shorts. And I don't think it does, as argued in 2^[1].

The argument was unfortunately cut short, as we began arguing in circles. I don't think my interlocutors ever understood the points I was making, though I also don't think they tried very hard. They may have just been too shocked at my criticism of two sacred objects of nerd culture (academia, and math) to be able to hear my arguments. But I could have also been bad at explaining myself.