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I am from the mathematics community on StackExchange. I should say that I know absolutely nothing in statistics. I have never taken a statistics course or read any book on it. I am familiar with basic probability theory though, because it is a mathematics subject. After all probability is an application of real analysis to finite measure spaces, and the subject of probability has a rigorous mathematical foundation to it. But what about statistics?

I view statistics as a way to derive quantitative results from observable data. (Please correct me if this is an inaccurate description of statistics.) This is why statistics is very valuable to scientists whose work entirely consists of observable data. But mathematics is entirely non-empirical. None of the major mathematicians of the last century, or the current one, (at least that I know of), know any statistics. This is why I am asking if you, statisticians, agree with the statement that, "statistics is a numerical branch of science, rather than mathematics".

Now I understand that statisticians use mathematics, sometimes even advanced results. But does not necessarily make them mathematics. Physicists use a lot of mathematics too, but they are not mathematicians. For them mathematics is a means to solve some other problem, it is not the math that inherently interests them.

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    $\begingroup$ Kolmogorov has several famous statistics results, too. That said, it sounds to me like you have made up your mind before asking the question. $\endgroup$
    – Marc Claesen
    Feb 9, 2015 at 7:03
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    $\begingroup$ Related thread: stats.stackexchange.com/questions/129264/… $\endgroup$
    – Tim ♦
    Feb 9, 2015 at 7:25
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    $\begingroup$ Corrections: Fields medal, not Field's; Kolmogorov not Kolomogorov. The bodies that award these medals tend not to include any statisticians, so what is surprising? Although this is quiz stuff, I'd note that several statisticians have been awarded the National Medal of Science in the US, for example. $\endgroup$
    – Nick Cox
    Feb 9, 2015 at 10:17
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    $\begingroup$ This is perhaps illuminating onlinelibrary.wiley.com/doi/10.1111/1467-9884.00130/abstract $\endgroup$
    – Momo
    Feb 9, 2015 at 12:22
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    $\begingroup$ That's an absurd suggestion; I imply no conspiracy. It's just how these bodies behave. The leading statistical societies give prizes to statisticians and leading statisticians are active in those, etc., etc. It's no criticism if mathematicians behave similarly. This is a puzzling thread; you ask for views and then seem to want to shout them all down if you disagree. Your countrymen in France (for we know that Bourbaki was/is French, naturally) have an expression: this animal is very bad; if you attack it, it defends itself. $\endgroup$
    – Nick Cox
    Feb 9, 2015 at 19:16

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If you go deeper in any discipline, especially when you get near the boundaries, you will find that those boundaries between disciplines are fairly arbitrary. For example, as a sociologist I would say that economics is just a branch of sociology, they typically don't agree... In practice, the boundaries between disciplines have a lot to do with group identity and university politics. They are an interesting subject for study by sociologist, social psychologist and political scientists, but not much more than that.

So unless you are interested in the sociology of university organization, I would just say that mathematics is whatever mathematicians do, and a mathematician is whoever calls him or herself a mathematician.

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  • $\begingroup$ It is pretty clear what mathematics is. Mathematics is any subject of study that starts with first principles and derives consequences from them in a completely rigorous manner. This is true with probability theory, since the Kolomogrov revolution. Before him probability theory was called considered to be pseudo-mathematics among the math community. The same is true with calculus. Back in the 1600s and 1700s, Calculus was considered to be pseudo-math. It did not have a rigorous theory for it, until Weierstrass. So does statistics fit this criterion to be mathematics? $\endgroup$
    – Nicolas Bourbaki
    Feb 9, 2015 at 8:44
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    $\begingroup$ @NicolasBourbaki if your criterion to be considered maths is to start from first principles (or conjectures, which also happens in math research) and then derive consequences in a completely rigorous manner, then statistics, theoretical physics, theoretical computer science and many more are clear forms of maths. You'll see that theoretical papers in all these domains are essentially maths papers with an application wrapped around it instead of being purely abstract. $\endgroup$
    – Marc Claesen
    Feb 9, 2015 at 9:13
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    $\begingroup$ Theory in general often (not always) fits that definition. So we can add for example theoretical economics and theoretical sociology to the list of disciplines that are actually mathematics. $\endgroup$
    – Maarten Buis
    Feb 9, 2015 at 12:07
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    $\begingroup$ So if you want to move forward you should come up with a clearer definition of what is and is not mathematics. Right now you have a necessary but not a sufficient condition, and Kjetil doubts that it is even a necessary condition. As I stated in my answer, you won't get anywhere with this excercise, as disciplines are just social constructs and thus very messy. Trying to find such a definition that pleases everybody in your own and neighbouring disciplines could be a useful excercise for you to convince yourself that this is indeed a hopeless excercise. Good luck. $\endgroup$
    – Maarten Buis
    Feb 10, 2015 at 9:01
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    $\begingroup$ I suspect others (strongly) disagree with you on that. It is very common to define your own discipline so widely that you consider (parts of) other disciplines as sub-disciplines of ones own discipline (I jokingly claimed economics as part of my discipline in my answer). Such "disciplinary imperialism" is often not welcomed by the other disciplines, to put it mildly. So, if you want to peacefully coexist with the rest of your university I strongly recommend not to be too serious about that. You can quickly end up in a very ugly conflict in which everybody loses. $\endgroup$
    – Maarten Buis
    Feb 11, 2015 at 8:41
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I think it's pretty clear that the theory behind statistics is mostly pure mathematics, while the application is not. For example, showing that the sample mean is the optimal unbiased estimator for the mean of a normal distribution is clearly a rigorous mathematical theorem based on the laws of probability.

In the same way, there are many theorems for statistical algorithms (bootstraping, MCMC) showing that they converge under certain assumptions.

When applying statistics to real world problems there are other more practical elements to discuss. But the theory behind statistics is very profound mathematics.

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    $\begingroup$ Is the mathematical part of statistics an extension of probability theory? Or is it a separate subject in and of itself? $\endgroup$
    – Nicolas Bourbaki
    Feb 9, 2015 at 19:32

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