How do you know if you are a mathematician?

Also, what’s the best way to anger your readers? For the latter, the best way to anger your readers is to get into discussions about personal identity, which is what this article is about. So … eep.

I want to lay down some thoughts on an important matter about identity and label in the world of mathematics. This is a difficult discussion to be had because when defining a label we are bound to do at least one of the following:

- exclude people who have identified themselves with that label
- include people who might not be wanted to be included by those who already have that label
- conflate an existing label with another label and in so doing strip some people of their identity and bestow on them a different identity with which they do not agree

I’m sure there are other effects, but the above three are what I often see when having a conversation about what it means to be a mathematician.

In a somewhat circular manner, let’s approach this *like* a mathematician in a loose form of axioms, definitions, theorems. I posit that there are at least four systems by which one begets the title of “mathematician”. Experientially, I have found that most people adopt at least one of these four systems if not some intersection of them. Some systems are subsets of other systems.

In no particular order let’s go through these and see what ends up happening in terms of inclusion, exclusion, and conflation, per the above.

### System 1 — The PhD or equivalent

*You are a mathematician if and only if you have earned a PhD or equivalent from an accredited university program.*

This is probably the most annoying and elitist definition of a mathematician. However, it is probably also the cleanest definition in that there is a third-party objectivity to it with its rites and rituals which ultimately produced an authoritative piece of paper [the doctorate].

This system includes and excludes people in a black-and-white manner — either you got the paper or you didn’t and “your feels” don’t matter. It can feel cold and bureaucratic. Some will argue that’s how society works. A physicist is one who has earned a PhD in Physics. Same goes for chemist, biologist, botanist, metallurgist, geologist, archaeologist, etc. Additionally, in this system we have clean definitions for subspecialties — topologist, algebraist, probabilist, number theorist, etc. And from this we have further squabbles about how much of a mathematician one really is if they are an “applied” mathematician versus a “pure” mathematician. Delving in the world of computational mathematics can get the up-turned nose of the pure mathematician. This in turn leads to calls of elitism by the applied mathematician. Humans have a penchant for infinitely divisible hierarchies.

But of course, there can be a deep injustice in this system. Personally, I know more than a few people who have gone through the rigors of a doctoral program only having to abandon it for some reason unrelated to their mathematical abilities — budget constraints, advisor politics, university politics, health, to name a few. These people have, by almost all other measures of talent, have the skills of a mathematician, just not the paper.

This leads us into the matter of practice.

### System 2 — Practice of mathematics

* If you practice mathematics, in any capacity, then you are a mathematician.*

Born out of a rejection of the bureaucratic definition of mathematician, this is a generally liberal definition of mathematician in that almost anyone can profess unto themselves the title of mathematician. An offshoot of this system modifies the statement as “When you practice mathematics, in any capacity, you are a mathematician for that time period.” In both versions, “practicing mathematics” can mean something as “basic” as performing arithmetic or as complex as working on the Riemann hypothesis. In this system, second graders are mathematicians as are university professors.

As a highly populist system, it has a tendency to draw the ire of those who have the title via System 1. This ire then is viewed as elitist. There is no third-party standard and regardless of talent, merit, or commitment, everyone is a mathematician.

This system, though, has its critics not just from the “System 1 elites” — those not involved in the field of mathematics tend to scoff at System 2. To what proportion these individuals exist, I am unsure, as I have not done a formal study, but anecdotally, the proportion feels non-negligible. There is a difference in being a student of mathematics and being a System 1 mathematician. Second graders are students of mathematics or mathematicians in the making, but mathematicians they are not.

With all this said, a purpose of the System 2 definition is to create a larger community through which mathematics may flourish. Mathematics has an uphill PR battle as it is in attracting and retaining talent [this is the “Math Misery” blog for a reason!]. By endowing those who engage with mathematics the title of mathematician perhaps they are more apt to keep with it and by extension will encourage those around them to pursue it. Furthermore, it doesn’t actually affect “real” mathematicians [an interesting term I’ve heard in conversation with System 2 mathematicians] since everyone knows where in the hierarchy they reside. And again, hierarchies present themselves even in this system of “everyone is a mathematician”.

This leads to System 3

### System 3 — Research of mathematics

*You are a mathematician if you conduct and publish peer-reviewed research in Mathematics.*

In a need for refinement from System 2, but not as much bureaucracy as System 1, we have the scientific definition of mathematician. It is not enough to simply fiddle with mathematics to earn the mathematician title, but rather one must be engaged with the subject matter at an expert level.

How is this not System 1? For starters, those in graduate programs in Mathematics typically publish research papers before they are awarded their doctorate. So, they can be System 3 mathematicians before they are System 1 mathematicians. Additionally, a PhD is not a requirement for one to conduct expert level mathematics research. It is possible to have had no formal [System 1] background in mathematics and to make a contribution to the body of mathematics. In this manner, one has become a mathematician.

The reality though is that while in technicality this is not System 1, in practice it is overwhelmingly System 1. However, System 3 retains the populist feel of System 2 without the bureaucracy of System 1 all the while not sacrificing ethos. In some sense, System 3 is more restrictive than System 1 since not all PhD holders are active researchers. For example, I haven’t published a paper in 7 or 8 years. A relaxed version of System 3 is *You are a mathematician if you have ever published peer-reviewed research in Mathematics.*

Still some find System 3 to be hokey in that any joker can produce some result in mathematics in some peer-reviewed journal [of possibly dubious quality]. Enter economics.

### System 4 — Show me the money

*You are a mathematician if you earn, as primary income, through the practice of mathematics.*

So now this opens up a few can of worms. Immediate criticism is that while certain non-academic professions require the use of mathematics, that mathematics is nothing more than arithmetic gymnastics handled by spreadsheets and other software. These people are not mathematicians, but rather users of mathematics. On the other hand, if your job title is something like “Quantitative Analyst”, “Quant Trader”, “Mathematician”, “Math Analyst”, “Data Scientist”, “Statistician”, etc. *and* the job requirement was that you hold some degree in mathematics [typically Bachelor degree or above] or “quantitative” field, then perhaps, possibly, you could claim the title of mathematician.

Some of these non-academic positions do require System 1 mathematicians, some don’t. But the general argument here is that there is an objective third-party that values your services. That third-party is “the market”. If you are able to earn an income by selling mathematics, then certainly you have employable talent and that is enough to literally earn you the title.

But, let’s not forget hierarchy. What of math teachers then? Not math *professors* (hierarchy, remember). Math professors are mathematicians; there’s almost no argument there in any system. But are math teachers — those who teach mathematics without the PhD — mathematicians? What about those who don’t teach at the college / university level but still hold a PhD? Anyone who teaches mathematics and earns, as primary income, from this act is certainly *selling* mathematics. Are they not mathematicians?

Plenty of math teachers [again my experience with math teachers before they know I am a mathematician a la System 1] do not believe they are mathematicians because they believe in the System 1 definition of mathematician. Yet, System 2, 3, or 4 believers will thrust onto them the title of mathematician. On the other hand, there are plenty of math teachers who are not System 1 mathematicians, but believe they are mathematicians by some version of System 2, 3, or 4.

What if one is a math education researcher? Are they not a mathematician? There is some turf war here in that somehow the business of education is not enough for the mathematician title. Via System 1, the doctorate is in Mathematics Education, not Mathematics. They are two separate disciplines with different levels of rigor and focus. The former isn’t really about the study of mathematics, it’s about the study of the study of mathematics. At least, that’s the thought process of some and how we have delineated these fields in our universities.

### Conclusion

So there you have it. Are you a mathematician? It depends if you are willing to allow for a third-party definition or if you are going to self-define. Regardless, it seems that in order to be a mathematician you must first pick your system and then fight with others.

There is a fifth system — the dictionary definition — Merriam-Webster gives “a specialist or expert in mathematics”. And this is unhelpful as that is at the heart of the debate.

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