Ostensibly, at the end of the day, math should be just about what can be proven right from wrong. Thus, we may reason that we shouldn’t care about someone’s credentials in order to cite them — we can just read their proof and check the validity of their words ourselves.
But despite this, I have often found myself evaluating the authoritativeness of authors through things like credentials when trying to pick resources to learn from. Why do I end up doing this? Are there in fact aspects of doing math for which credentials become important? For these aspects, can we devise new methods to reduce dependence on credentials, or even eliminate it, in favor of judging just based on content?
Off the bat, one of the most often-cited reasons for considering credentials, at least in research, seems to be simply practicality: there are just too many papers to read thoroughly, so people make initial judgment calls, based on credentials, that are meant to filter out some of the papers most likely to be flawed. In general, this is the purpose of something like arXiv’s endorsement system: it is a quick-and-dirty approach to filtering out some “probably invalid” content, in a platform that prioritizes quick publication over delayed processes with peer-review guarantees.
This is something that will hopefully be addressed by a system that can automatically peer-review math, which I believe is achievable technologically (although we are not there yet.) I discuss this a little more in my post on automated verification of math.
This point doesn’t make sense for learning, since it is in the student’s best interest to read everything and study the material as thoroughly as possible, verifying things themselves. Thus, credentials wouldn’t be as important when it comes to practicalities of time for learners.
Credentials can matter though for mathematical knowledge besides the logic of the content (definitions, theorems, and proofs) — for what we can call “non-logical” knowledge. For example:
- Notation and terminology — specifically, knowledge of what the standard notation and terminology are for a concept.
- The research frontier — specifically, knowledge of what is already known to the math community and what is not yet known.
The research frontier is clearly helpful to know for researchers as well as learners who are planning to do research. In turn, this is better enabled by knowing the existing terminology, since that allows better discovery of information about the concept, including the research frontier. (However, terminology may not be sufficient for the research frontier, since I’ve heard that there can sometimes be a delay in publishing research findings, and thus knowledge may be known to various members of the math community in a colloquial, undocumented way before it’s written up and officially published.) Terminology is also useful for learners even if they are not planning to do research, since it allows for better discovery regardless.
I have seen the impacts of these two aspects of non-logical knowledge in my explorations of my own mathematical ideas, where I often wonder if they are suitable to be published as research. For example, some time ago, I was curious about whether the characteristic of a ring could be extended to more general algebraic structures. I was in fact able to do this, and I wrote a definition that generalized the concept from rings to monoids. I called this the “characteristic of a monoid,” which I explored in this post. However, I soon learned that there was already a different existing term for this: it had already been generalized from rings to groups under the name of “exponent.” Granted, monoids are even more general than groups, but nevertheless much of my work was already known to the math community in the context of groups, and many of the existing results and proofs translated directly to monoids.
So, can we devise methods to reduce dependence on credentials for these two aspects of non-logical knowledge?
Well, what I’ve realized recently is that the ideal of a single, consistent bank of terminology used by everyone is not actually what always happens in practice. For example, people studying abstract algebraic logic have studied the exact same mathematical object that people studying closure operators have, under the different name of “consequence operator.” While there are standards bodies that attempt to standardize certain things, there is so much activity going on across all of math that not everything is fully standardized. Furthermore, it’s not like every mathematician knows all of math. Every person will know a subset, and if they do work outside what they typically study, then it is very likely that they may not be aware that their definitions are equivalent to concepts already being studied by others. In general, mathematical activity is more like a distributed system than a centralized one, with different people doing work at the same time and then syncing and coordinating without the use of one governing body.
In fact, historically, it is likely that the terminology that became standardly used was just that put forth by those with the “loudest voices” or the “most connections.” (Not to undermine the work of the “loudest” or “most connected” people, but it is worth critically examining any relatively large emphasis on their work too.) In modern times, with the advent of the Internet and search engines, a lot of this has transformed since mathematicians are able to much more easily discover what others have been doing and how that compares to their ideas. But certain things have been still relatively hard to discover, even with these technologies: for example, searching “is there a standard term for [definition X]” is much less fruitful than “what is known about [standard term].” Recently, with the introduction of technologies like ChatGPT and LLMs, even these “reverse search” kinds of queries are now often possible. For example, I’m now able to have an interaction with ChatGPT like what I showcased in my post on non-standard axioms of various math structures. But I don’t think we’re fully there yet, in terms of technology that fully enables the standardization of mathematical terminology. LLMs still are known to hallucinate, so they are not currently at the level of reliability that would be important going forward for math. I think instead that in the next 10-20 years, machine learning techniques like LLMs combined with deep mathematical knowledge like automated theorem proving algorithms will allow us to build a system that can truly answer questions like “is there a term for [definition X]” with the fidelity that is expected for math.
Then, with better ability to standardize terminology independent of authors’ credentials, combined with the vast availability of information already enabled by existing technologies like the Internet and large databases, knowledge of the research frontier will become less dependent on credentials as well. (Knowledge that is known to a few people and not published anywhere will always exist for a brief period of time, and that will remain very hard to surface with technology. Instead, people should publish faster and more often, even if it’s just partial progress, so that the whole math community can benefit.) With this kind of continual progress in technology for math, including automated verification, systemized standardization of terminology, and possibly other things too, we can move closer and closer to reducing dependence on credentials in math. In turn, this will enhance the practice of math, making it overall more accurate and more inclusive.
