2

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.

Nicolas Bourbaki
  • 1,080
  • 9
  • 22
  • *None of the major mathematicians of the last century, or the current one, (at least that I know of), know any statistics.* Unless, you know, you treat statisticians as mathematicians too. – Marc Claesen Feb 09 '15 at 06:30
  • @Marc I do not. How many statisticians have received either the Wolf Prize, Abel Prize, Field's Medal, or any of the other notable mathematical awards? – Nicolas Bourbaki Feb 09 '15 at 06:36
  • Kolmogorov won the Wolf prize off the top of my head. – Marc Claesen Feb 09 '15 at 06:39
  • @Marc Kolmogorov was a mathematician. I have studied him. He was the first person to create a rigorous mathematical theory for probability theory, this is not a subject of statistics. – Nicolas Bourbaki Feb 09 '15 at 06:42
  • 7
    Kolmogorov has several famous statistics results, too. That said, it sounds to me like you have made up your mind before asking the question. – Marc Claesen Feb 09 '15 at 07:03
  • @Marc I have my mind made up a long time ago, but I wanted to hear what statisticians had to say. And the reasons for them saying so. Kolomogorov was a great mathematician. His recognized mathematical achievements have not involved any statistics, however. Gauss has results in both statistics and land surveying. But to say that statistics is mathematics because Gauss was a mathematician, is no different from saying that land surveying is mathematics because Gauss was a mathematician. Is there any recipient of any mathematical award that was entirely a statistician? – Nicolas Bourbaki Feb 09 '15 at 07:09
  • 2
    Related thread: http://stats.stackexchange.com/questions/129264/applied-statistics-vs-mathematical-statistics/129303#129303 – Tim Feb 09 '15 at 07:25
  • 3
    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. – Nick Cox Feb 09 '15 at 10:17
  • 2
    This is perhaps illuminating http://onlinelibrary.wiley.com/doi/10.1111/1467-9884.00130/abstract – Momo Feb 09 '15 at 12:22
  • 1
    Note that *theoretical statistics* is also non-empirical - there's no experiment that would help decide between Bayesian & frequentist ideas - & that extra-mathematical arguments are used in discussion about foundations. – Scortchi - Reinstate Monica Feb 09 '15 at 16:24
  • @NickCox "The bodies that award these medals tend not to include any statisticians, so what is surprising?". Why do they not include any statisticians? Do you really think there is a conspiracy to keep the statistician down, or more simply, that the subject never been recognized as a subject of mathematics. Then you say, "I'd note that several statisticians have been awarded the National Medal of Science in the US". This is not a mathematics award. – Nicolas Bourbaki Feb 09 '15 at 19:03
  • @Scortchi "Note that theoretical statistics is also non-empirical." Neither is theoretical physics, but it is not mathematics. – Nicolas Bourbaki Feb 09 '15 at 19:03
  • 4
    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. – Nick Cox Feb 09 '15 at 19:16
  • @Nick "It's no criticism if mathematicians behave similarly." Actually, this is not true at all. The mathematics community does award all those who do mathematics across the board. Witten, from physics, was awarded the Fields medal. In 2010, I believe, 3/4 of the winners were from probability theory. The award is pretty diverse. You then ask, "You ask for views and then seem to want to shout them all down if you disagree." What is wrong with that? It is called skepticism. I want to hear your reasons for why you say what you say. – Nicolas Bourbaki Feb 09 '15 at 19:24
  • I wish I shared your apparent optimism that this could be a fruitful discussion. – Nick Cox Feb 09 '15 at 19:29
  • @NicolasBourbaki: Well, you brought up "non-empirical" as if to contrast Mathematics with Statistics. I've no opinion on whether some or all formal sciences should be considered part of Mathematics. – Scortchi - Reinstate Monica Feb 09 '15 at 23:38
  • @Scortchi Being non-empirical is not a sufficient conditions for being mathematics, but it is certainly a necessary condition. – Nicolas Bourbaki Feb 10 '15 at 03:52

2 Answers2

5

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.

Nick Cox
  • 48,377
  • 8
  • 110
  • 156
Maarten Buis
  • 19,189
  • 29
  • 59
  • 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? – Nicolas Bourbaki Feb 09 '15 at 08:44
  • 2
    @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. – Marc Claesen Feb 09 '15 at 09:13
  • 2
    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. – Maarten Buis Feb 09 '15 at 12:07
  • Mathematical statistics is clearly mathematics, it fits your description, see texts following definition-theorem-proof style. In applied statistics, as in whatever other branch of applied mathematics, other criteria is also applied. In numerical analysis, you have rigourous theorems about error bounds, but in practice is also used "experience" telling that in some situations methods work much better than the error bounds suggests. In your view, mathematics being created is not mathematics, it becomes mathematics just when research ended and all have become clear. That just doesnt make sense! – kjetil b halvorsen Feb 09 '15 at 12:27
  • @Marc No theoretical physics does not do that, unless it is some very special discipline within it that does, but in general theoretical physics does not fit that description. Theoretical computer is mathematics, yes. So is the mathematical theory of finance. – Nicolas Bourbaki Feb 09 '15 at 19:07
  • 1
    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. – Maarten Buis Feb 10 '15 at 09:01
  • @MaartenBuis The definition I gave above, in paragraph one, is both necessary and sufficient. If some economist formulates economic laws, in a rigorous mathematical way, forgets about the economics, and focuses on the pure axiomatic development of those laws, as his first principles, then I consider him a mathematician. (Cedric Villani, for example). Same with anyone who does quantum mechanics. The problem is that most people who claim to be doing mathematics, upon investigation of the definition I provided, are not. – Nicolas Bourbaki Feb 11 '15 at 03:49
  • 3
    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. – Maarten Buis Feb 11 '15 at 08:41
2

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.

Erik
  • 6,909
  • 20
  • 48