19

Is stats maths or not?

Given that it's all numbers, mostly taught by maths departments and you get maths credits for it, I wonder whether people just mean it half-jokingly when they say it, like saying it's a minor part of maths, or just applied maths.

I wonder if something like statistics, where you can't build everything on basic axioms can be considered maths. For example, the $p$-value, which is a concept that arose to make sense of data, but it's not a logical consequence of more basic principles.

Erik
  • 186
  • 3
  • 14
Quora Feans
  • 384
  • 2
  • 3
  • 10
  • 10
    Compulsory XKCD reference: http://xkcd.com/435/ . Anyway, does it really matter? – nico Dec 04 '13 at 21:17
  • 2
    (i) How would we quantify such things? It's not like it's been the subject of a survey! (ii) The calculations almost always involve numbers, but what makes it *statistics*, in my mind, *is usually not in the calculations*. (iii) When I did my undergraduate statistics major, it wasn't in the mathematics department. The place I did my PhD at - under two fairly well known statisticians - wasn't a maths department either. (iv) I don't think it's a joke. It relates to a very important idea - that what makes statistics "statistics" is more about a way of reasoning about particular types of problems. – Glen_b Dec 04 '13 at 21:36
  • I suppose an argument that Statistics isn't just a branch of Maths could go along the following lines: (1) It's a formal science, i.e. not empirically validated. (2) Though Statistics leans heavily on Probability Theory it doesn't follow from it deductively, nor is PT applied ad hoc to each statistical problem; other principles e.g. Sufficiency, Conditionality, have to be introduced to allow the application of PT to inference from data. The existence of alternative inferential paradigms - the several varieties of Bayesianism & Frequentism - supports this point of view. – Scortchi - Reinstate Monica Dec 04 '13 at 21:47
  • 7
    I feel obligated to give a short answer, as I am former pure mathematician (PhD and 3.5 years of postdoc in some kind algebra), and now an applied statistician... well, the kind of stats you learn for applied stats, like "when do I use a $t$-test" or what not, for a mathematician, looks like a recipe book, not like maths. But, for example, van der Vaart’s Asymptotic Statistics is definitely a math book... There are plenty of intermediate levels – some of them not well populated, I think there are not enough books explaining stats with lots of real examples **and** all the mathematical details. – Elvis Dec 04 '13 at 22:19
  • 5
    I don't know what to make of the statement, "the $p$-value, which is a concept that arose to make sense of data, but it's not a logical consequence of more basic principles", I'm not even sure if it can really even be right or wrong. It mostly seems to proceed from confused premises. – gung - Reinstate Monica Dec 04 '13 at 22:25
  • 1
    Similar question [here](http://math.stackexchange.com/questions/286730/math-vs-probability-vs-statistics/286770). – Scortchi - Reinstate Monica Dec 04 '13 at 22:38
  • 1
    @Scortchi Thanks for finding that. In this case, though, I view the two versions of the question--even if they were phrased the same--as being entirely different solely due to the different audiences! On the math site you will get good characterizations of what math is but may also read some ignorant, narrow, or wrong characterizations of statistics (such as "statistics can be seen as an inverse of probability theory"). These reflect uninformed or patronizing points of view that cause so much angst among academic statisticians who have to fit into math departments. We can do better here. – whuber Dec 04 '13 at 23:10
  • @whuber could you elaborate on why you feel that regarding statistics as an "inverse of probability theory in some sense" is uninformed? I feel that it captures the high-level difference in motivations fairly well, in that in statistics we are given data and are tasked with getting at the generating mechanism while probability focuses on a given generating mechanism and asks questions about such as what data to expect from it. Insofar as statistics is a mathematical discipline this seems reasonable. – guy Dec 05 '13 at 00:05
  • 12
    @Guy By analogy, we could characterize chemistry (another "mathematical discipline") as asymptotic distribution theory and C* algebras. Doing so is nominally accurate but so completely misses the essence of what chemistry is and its aims that no chemist would recognize it. Similarly, contrast your characterization with what [leading professional societies say statistics is](http://www.amstat.org/careers/whatisstatistics.cfm): they are worlds apart. "The science of learning from data, and of measuring, controlling, and communicating uncertainty." Not one mention of probability there. – whuber Dec 05 '13 at 07:52
  • @gung, yes, I expressed myself poorly. I wanted to say that in maths, you start with axioms or postulates and build your theory upon it. No matter how useless or useful they are, everything has to be tied together. In statistics, you might have a concept that summarized your data set, that's useful, but you don't exactly know why. – Quora Feans Dec 05 '13 at 17:59
  • @QuoraFea I think you are incorrect to say that concepts in statistics are not based on fundamental axioms and proved results. – bdeonovic Dec 05 '13 at 21:28
  • @QuoraFea: What I was trying to say is that statistical reasoning is based on *additional* 'axioms' to those of Mathematics. – Scortchi - Reinstate Monica Dec 05 '13 at 21:41
  • @whuber I don't think that analogy is totally fair, because saying that statistics is "an inverse of probability" actually captures what statisticians are doing - given random data generated from a process, we attempt to learn things about that process. I don't see how this is that far away from what a practicing statistician is doing (e.g. I'm given data from a clinical trial and I wish to generalize the effect of a drug to the whole population). – guy Dec 05 '13 at 22:24
  • @Benjamin: they might be proved results, but not non-empirical conclusions from logical axioms. An average mean of a set might be more meaningful at times than a median, and both might be correctly calculated (from a mathematical perspective), but there's nothing in math to say that one is better than the other. This part is all non-mathematics, although we cannot dismiss it when it comes down to practical results. – Quora Feans Dec 05 '13 at 23:24
  • @whuber isn't "communicating uncertainty" the job of probability? If I have a 1/6 chance of rolling 6 with a standard dice the "uncertainty" that I will roll a 6 is ~83%, or am I missing something? – Erik Sep 17 '15 at 02:56
  • @Erik Uncertainty goes far beyond estimating or quoting probabilities. It includes assessing the quality of data, evaluating model uncertainty, developing confidence and prediction intervals, eliciting information about loss functions and prior distributions, and much more. Communicating these things includes identifying, quantifying, presenting, interpreting, and explaining statistical models, procedures, and their results. Although much of this involves probability at some level, it goes well beyond it. – whuber Sep 17 '15 at 04:47
  • Statistics is a subfield of applied mathematics (for me the most representative one). Some pure mathematicians don't regard applied math as mathematics per se, but more as "applications of mathematics" to other disciplines (e.g. Physics). I guess for those people, stats is not maths. Usually, though, those are people who are not well familiar with statistics and don't know that the theory of statistical inference relies solely on pure math. – Digio Nov 08 '15 at 12:48
  • 1
    "For example, the p-value, which is a concept that arose to make sense of data, but it's not a logical consequence of more basic principles." – Digio Apr 04 '16 at 10:21
  • 1
    @Digio I'd be interested in seeing an axiomatic treatment of statistics where the P-value pops out as the right thing to look at. Their motivation is philosophical - "either the null is wrong or I got really unlucky." But that line of reasoning is not something that the math itself suggests is reasonable. Contrast this with Bayesian logic - there you have an axiomatic treatment where the mathematics tells you precisely what to do (use posterior probability). Essentially, math tells you the properties of P-values, but it does not advocate for their usage; justifying their use is philosophical. – guy Aug 20 '16 at 15:08
  • 1
    @guy Generally speaking, there is an axiomatic treatment to suggest that a probability may lead to a right or false decision using possibility theory and fuzzy logic but this is admittedly outside the scope of classical statistics (except maybe the scenario where a p-value is almost equal to zero). I never argued on Bayesian inference being more or less scientific, I don't consider myself an expert on that topic. – Digio Aug 22 '16 at 14:14

8 Answers8

16

Mathematics deals with idealized abstractions that (almost always) have absolute solutions, or the fact that no such solution exists can generally be described fully. It is the science of discovering complex but necessary consequences from simple axioms.

Statistics uses math, but it is not math. It's educated guesswork. It's gambling.

Statistics does not deal with idealized abstractions (although it does use some as tools), it deals with real world phenomena. Statistical tools often make simplifying assumptions to reduce the messy real world data to something that fits into the problem domain of a solved mathematical abstraction. This allows us to make educated guesses, but that's really all that statistics is: the art of making very well informed guesses.

Consider hypothesis testing with p-values. Let's say we are testing some hypothesis with significance $\alpha = 0.01$, and after gathering data we find a p-value of $0.001$. So we reject the null hypothesis in favor of an alternative hypothesis.

But what is this p-value really? What is the significance? Our test statistic was developed such that it conformed to a particular distribution, probably student's t. Under the null hypothesis, the percentile of our observed test statistic is the p-value. In other words, the p-value gives the probability that we would get a value as far from the expectation of the distribution (or farther) as the observed test statistic. The signficance level is a fairly arbitrary rule-of-thumb cutoff: setting it to $0.01$ is equivalent to saying, "it's acceptable if 1 in 100 repetitions of this experiment suggest that we reject the null, even if the null is in fact true."

The p-value gives us the probability that we observe the data at hand given that the null is true (or rather, getting a bit more technical, that we observe data under the null hypothesis that gives us at least as extreme a value of the tested statistic as that which we found). If we're going to reject the null, then we want this probability to be small, to approach zero. In our specific example, we found that the probability of observing the data we gathered if the null hypothesis were true was just $0.1\%$, so we rejected the null. This was an educated guess. We never really know for sure that the null hypothesis is false using these methods, we just develop a measurement for how strongly our evidence supports the alternative.

Did we use math to calculate the p-value? Sure. But math did not give us our conclusion. Based on the evidence, we formed an educated opinion, but it's still a gamble. We've found these tools to be extremely effective over the last 100 years, but the people of the future may wonder in horror at the fragility of our methods.

David Marx
  • 6,647
  • 1
  • 25
  • 43
  • 8
    The p-value is not the probability that we are wrong when we reject the null hypothesis, as that also depends on H1 which does not enter into the calculation of the p-value (well illustrated by http://i.stack.imgur.com/tStr4.png - the probability that H0 is wrong and that the sun has exploded is rather less than p = 1/36). – Dikran Marsupial Dec 06 '13 at 09:05
  • Could you suggest a better simple language interpretation of the p-value? "The probability that we observe the data at hand given the null is true" perhaps? I've already delved much deeper in the p-value example than I was intending to. My intention was to make a point about statistics, not provide a tutorial on interpreting p-values. I don't want to get too derailed. Thanks for pointing that out, in any event. – David Marx Dec 06 '13 at 17:29
  • 2
    The p-value is the probability of a result at least as extreme as that observed **if** the null hypothesis is true. The point that the link between the plausibility of the null hypothesis and the p-value being largely subjective, rather than a logical necessity, is a good point though (+1). I have been wondering lately whether frequentist hypothesis testing is any less subjective than the Bayesian approach, where at least the subjectivity is made more explicit. – Dikran Marsupial Dec 06 '13 at 18:21
  • It's not clear to me how your p-value interpretation/definition differs from the alternative I offered in my last comment. There's certainly a degree of subjectivity in frequentist hypothesis testing, but it's the same kind of subjectivity that gets invoked when interpreting a Bayes Factor. And it's not like significance level isn't communicated (i.e. the subjectivity is made explicit here too), it's just often chosen based on convention, whereas there's usually more thought put into choosing (informative) Bayesian priors. – David Marx Dec 07 '13 at 03:43
  • I was just intending to re-emphasise the "if", which should be emphasised strongly when discussing what a p-value is, as the importance of that condition is key. The problem with the significance level is that it is often based **only** on convention, rather than including what the observations say about H1 (which *is* included in the Bayes factor). Frequentist statistics was intended to do away with the supposed subjectivity of the Bayesian approach, but in this case it seems to gloss over the subjectivity, but it is still there. However, it is still a case of horses for courses. – Dikran Marsupial Dec 07 '13 at 12:18
  • The subjectivity is not in the p-value, but in the interpretation of the p-value in setting the "significance" level. The assumptions that go into deciding the significance level are very rarely explicitly stated in studies where hypothesis tests are used in science. Just stating the significance level does not communicate the subjectivity involved in justifying the significance level. – Dikran Marsupial Dec 07 '13 at 12:22
  • 1
    @David: The "at least as extreme" makes a great difference - the probability of the observed value under the null is not in general the p-value, even for discrete test statistics where it makes sense. I know it's tangential to the point you were making, but if [Wikipedia](http://en.wikipedia.org/wiki/P-value) can get it right, we should be able to on Cross Validated. – Scortchi - Reinstate Monica Dec 07 '13 at 18:53
  • I agree that the decision wasn't math, but surely the math was. All of the math performed was the *statistics* part (in my opinion). The translating/interpreting of the statistics is the non-math part. I feel like you're commingling the two parts. – Erik Sep 17 '15 at 03:29
  • For an explanation on the p-value see http://stats.stackexchange.com/questions/166323/misunderstanding-a-p-value/166327#166327 –  Sep 17 '15 at 16:20
  • This answer is nice to read but I think it's defining statistics way too broadly. People (and other organisms) have been perfecting the art of making educated guesses for billions of years before the field of statistics existed. Were they all doing statistics? Is it doing physics if you make a "law" stating that things fall to the ground when dropped? – Paul Sep 17 '15 at 16:57
10

Tongue firmly in cheek:

Einstein apparently wrote

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.

so statistics is the branch of maths that describes reality. ;o)

I'd say statistics is a branch of mathematics in the same way that logic is a branch of mathematics. It certainly includes an element of philosophy, but I don't think it is the only branch of mathematics where that is the case (see e.g. Morris Kline, "Mathematics - The Loss of Certainty", Oxford University Press, 1980).

Dikran Marsupial
  • 46,962
  • 5
  • 121
  • 178
  • 2
    *Is* Logic a branch of Mathematics? Including three-valued logics & modal logics, or just first-order predicate calculus? Are all formal sciences somehow Mathematics? – Scortchi - Reinstate Monica Dec 06 '13 at 17:33
  • I would view the study of any system for manipulating symbols according to a set of rules (e.g. formal languages) to be a variety of mathematics, so yes, I suppose I probably would. The trouble with labels is that they are not always fully descriptive of everything to which they are applied (I wouldn't say I was exactly a mathematician, a statistician or a computer scientist, but I have some aspects of all three). Similarly the same thing can often be placed in more than one hierarchy, so perhaps there isn't a unique solution to the question! – Dikran Marsupial Dec 06 '13 at 19:10
  • By your argument statistics, as a description of reality, comprises geometry and quantum field theory, too, but it does not include hypothesis testing (because most hypotheses are contra-factual--they are intended to be falsified--and therefore plainly do *not* "describe reality"). – whuber Dec 06 '13 at 20:51
  • The Einstein quote was the tongue in cheek bit, and wasn't meant to be taken seriously; I'm pretty sure it isn't quite what Einstein actually had in mind! – Dikran Marsupial Dec 06 '13 at 21:31
6

Well if you say that "something like statistics, where you can't build everything on basic axioms" then you should probably read about Kolmogorov's axiomatic theory of probability. Kolmogorov defines probability in an abstract and axiomatic way as you can see in this pdf on page 42 or here at the bottom of page 1 and next pages.

Just to give you a flavour of his abstract definitions, he defines a random variable as a 'measurable' function as explained in a more 'intuitive' way here : If a random variable is a function, then how do we define a function of a random variable

With a very limited number of axioms and using results from (again maths) measure theory he can define concepts are random variables, distributions, conditional probability, ... in an abstract way and derive all well known results like the law of large numbers, ... from this set of axioms. I advise you to give it a try and you will be surprised about the mathematical beauty of it.

For an explanation on p-values I refer to: Misunderstanding a P-value?

Community
  • 1
  • 2
    Isn't there still an important distinction, though, between Probability Theory (Maths) & its application to problems of inference (Statistics)? The Bayesian & frequentist approaches show the same mathematical apparatus ([typically, or almost](http://stats.stackexchange.com/q/126056/17230)) used with quite different concepts of probability. – Scortchi - Reinstate Monica Sep 18 '15 at 11:05
  • @Scortchi: I am not sure whether the concepts of probability are different for frequentists and Bayesians; see http://stats.stackexchange.com/questions/230415/is-there-any-mathematical-basis-for-the-bayesian-vs-frequentist-debate/230943#230943 –  Aug 23 '16 at 06:13
  • I don't see any disagreement between my comment & your answer to [Is there any *mathematical* basis for the Bayesian vs frequentist debate?](http://stats.stackexchange.com/q/230415/17230). By "mathematical apparatus" I mean what follows from Kolmogorov's axioms; by "concepts" I mean the interpretations as limiting frequency, degree of belief, &c. – Scortchi - Reinstate Monica Aug 24 '16 at 08:58
4

I have no rigorous or philosophical basis for answering this, but I've heard the "stats is not math" complaint often from people, usually physics types. I think people want guarantees certainty from their math, and statistics (usually) offers only probabilistic conclusions with associated p values. Actually, this is exactly what I love about stats. We live in a fundamentally uncertain world, and we do the best we can to understand it. And we do a great job, all things considered.

Jordan
  • 95
  • 1
  • 6
2

Maybe its because I'm a plebe and haven't taken any advanced mathematical courses, but I don't see why statistics isn't mathematics. The arguments here and on a duplicate question seem to argue two primary points as to why statistics isn't mathematics*.

  1. It isn't exact/certain, and as such relies on assumptions.
  2. It applies math to problems and anytime you apply math it is no longer math.

Isn't exact and uses assumptions

Assumptions/approximations are useful for lots of math.

The properties of a triangle that I learned about in grade school I believe are considered true math, even though they don't hold true in non-Elucidean geometry. So clearly an admission of the limits, or stated another way "assuming XYZ the following is valid", to a branch of math doesn't disqualify the branch from being "true" math.

Calculus I'm certain would be considered a pure form of math, but limits are the core tool we built it on. We can keep calculating up to the limit, just as we can keep making a sample size larger, but neither give increased insight past a certain threshold.

Once you apply math it isn't math

The obvious contradiction here is we use math to prove mathematical theorems, and no one argues that proving mathematical theorems isn't math.

The next statement might be that thing x isn't math if you use math to get a result. That doesn't make any sense either.

The statement I would agree with is that when you use the results of a calculation to make a decision then the decision isn't math. That doesn't mean that the analysis leading up to the decision isn't math.

I think when we use statistical analysis all the math performed is real math. It is only once we hand the results to someone for interpretation does statistics exit mathematics. As such statistics and statisticians are doing real mathematics and are real mathematicians. It is the interpretation done by the business and/or the translation of the results to the business by the statistician that isn't math.

From the comments:

whuber said:

If you were to replace "statistics" by "chemistry," "economics," "engineering," or any other field that employs mathematics (such as home economics), it appears none of your argument would change.

I think the key difference between "chemistry", "engineering", and "balancing my checkbook" is that those fields just use existing mathematical concepts. It is my understanding that statisticians like Guass expanded the body of mathematical concepts. I believe (this might be blatantly wrong) that in order to earn a PhD in statistics you have to contribute, in some way, to expanding the body of mathematical concepts. Chemistry/Engineering PhD candidates don't have that requirement to my knowledge.

The distinction that statistics contributes to the body of mathematical concepts is what sets it apart from the other fields that merely use mathematical concepts.

*: The notable exception is this answer that effectively states the boundaries are artificial due to various social reasons. I think that is the only true answer, but where is the fun in that? ;)

Community
  • 1
Erik
  • 186
  • 3
  • 14
  • 1
    If you were to replace "statistics" by "chemistry," "economics," "engineering," or any other field that employs mathematics (such as home economics), it appears none of your argument would change. As such it seems to be without any substance. – whuber Sep 17 '15 at 04:52
  • Statistics PhDs do not have to "contribute to the body of mathematical concepts." Most stats PhDs are awarded for contributions to statistical *methodology* and *statistical* theory. (Few mathematicians, if any, pay attention to the statistical literature. It just isn't a good source of new or fruitful mathematical ideas in general. I am *not* referring to literature in probability theory here.) Moreover, chemists, engineers, physicists, etc. often do create (or, usually, re-create) mathematical ideas in their work; that does not automatically turn their fields into branches of mathematics. – whuber Sep 17 '15 at 17:56
  • @whuber That is very interesting. It appears as if I don't have a leg to stand on. – Erik Sep 17 '15 at 18:20
  • 1
    For the record, I have not downvoted your contribution. This is a sensitive topic for many--for example, many college math departments are still trying to treat statisticians as mathematicians, to the detriment of both--and so it's likely to elicit some strong reactions. – whuber Sep 17 '15 at 20:16
  • 2
    @whuber I'm tough enough to stand a few down-votes regardless. :) I think you were respectful at all times, so don't worry about that. Besides voting is anonymous for a reason. No need to go on the record. – Erik Sep 17 '15 at 20:26
2

Statistical tests, models, and inference tools are formulated in the language of mathematics, and statisticians have mathematically proven thick books of very important and interesting results about them. In many cases, the proofs provide compelling evidence that the statistical tools in question are reliable and/or powerful.

Statistics and its community may not be "pure" enough for mathematicians of a certain taste, but it is definitely invested in math extremely deeply, and theoretical statistics is just as much a branch of mathematics as theoretical physics or theoretical computer science.

Paul
  • 9,773
  • 1
  • 25
  • 51
  • 3
    Hi Paul, as you say, stats is full of nice theorems and proofs (+1), there is even an axiomatic theory of probability, developed by Kolmogorov, as I explain in my answer. –  Sep 17 '15 at 16:43
-3

The "difference" relies on: Inductive reasoning vs. Deductive reasoning vs. Inference. For instance, no mathematical theorem can tell what distribution or prior you can use for your data/model.

By the way, Bayesian statistics is an axiomatised area.

Compay Segundo
  • 11
  • 2
-3

This may be a very unpopular opinion, but given the history and formulation of concepts of statistics (and probability theory), I consider statistics to be a subbranch of physics.

Indeed, Gauss initially formalized the least squares regression model in astronomical predictions. The majority of contributions to statistics before Fisher were from Physicists (or highly applied mathematicians whose work would be called Physics by today's standards): Lyapunov, De Moivre, Gauss, and one or more of the Bernoullis.

The overarching principle is the characterization of errors and seeming randomness propagated from an infinite number of unmeasured sources of variation. As experiments became harder to control, experimental errors needed to be formally described and accounted for to calibrate the preponderance of experimental evidence against the proposed mathematical model. Later, as particle physics delved into quantum physics, formalizing particles as random distributions gave a much more concise language to describe the seemingly uncontrollable randomness with photons and electrons.

The properties of estimators such as their mean (center of mass) and standard deviation (second moment of deviations) are very intuitive to physicists. The majority of limit theorems can be loosely connected to Murphy's law, i.e. that the limiting normal distribution is maximum entropy.

So statistics is a subbranch of physics.

AdamO
  • 52,330
  • 5
  • 104
  • 209
  • 5
    This thesis is as implausible as it is illogical. As Stephen Stigler points out in his books, psychologists, economists, and most other social scientists did *not* adopt the physicists' methods for up to another century due to real doubts about their applicability and their interpretation. That is *prima facie* evidence that statistics is far more than a branch of physics. Other disciplines, ranging from engineering through biology, also employ physical methods and physical theories, but that does not make them branches of physics either--at least not in any meaningful or insightful way. – whuber Dec 06 '13 at 20:47
  • Didn't the Bernoulli's interest in probability stem from gambling rather than physics? – Dikran Marsupial Dec 06 '13 at 21:36
  • @whuber As with my field, biostatistics, I'm keenly aware that these applied sciences existed in various forms prior to their distinct identification as a field of science. I believe these fields, though, were formally preceded by the field of statistics itself. This of course is not the case for physics. The one central theme in these applied sciences the formulation of a process as a model relating some predictor to a response. Perhaps the language of statistics was in part born out of the need to generalize such concepts as to apply to these fields. – AdamO Dec 06 '13 at 21:59
  • @DikranMarsupial I sure the (many) Bernoullis' interests were varied indeed. I'd be curious to know which one in particular dealt with gambling problems, probably Daniel or Nicholas who tackled the St. Petersburg paradox, though others still derived limit theorems and formulations of some random processes in statistics and probability theory. – AdamO Dec 06 '13 at 22:04
  • That last statement is a more plausible and interesting hypothesis, but it does not hold up well in light of simple examples. For instance, the terminology of "regression" comes from biology and anthropometry, not physics. Stigler's thesis is that astronomers were early adopters of statistical techniques because it made sense to them that observations could be combined in a meaningful way, whereas it was not at all apparent that averaging data about any number of human beings had any meaning whatsoever. That reveals statistics as a *tool* of the sciences, but not a branch of any one of them. – whuber Dec 06 '13 at 22:04
  • 1
    You're thinking of *Jacobus Bernoulli,* posthumous author of *ars conjectandi* (ed. Nicholaus Bernoulli, 1713). Probably the last people who *seemed* to be motivated by gambling problems were Pascal and Fermat in 1654, but even then it appears they were using certain gambling problems (the "problem of the points") only as a motivational example and not as the focus of their investigation. (Modern scholarship actually traces the problem of the points to Islamic contract law c. 1200.) The last mathematician of note who truly was motivated by gambling probably was Cardano (1501-1576). – whuber Dec 06 '13 at 22:08
  • @whuber "The last mathematician of note who truly was motivated by gambling probably was Cardano" not Persi Diaconis? – AdamO Dec 06 '13 at 22:13
  • @whuber your last reply to me makes a great deal of sense. I think you bring up a point that shows a shortcoming in my explanation of statistics. While physicists were interested in characterizing *error*, they were less concerned with estimation and limiting distributions of sample estimates. This is exactly the kind of language that bothers me when a statistician claims that they "modeled the *error*" rather than "modeled the *trend*". Without a doubt, the lineage of Galton, Pearson, and (lastly) Fisher was the major rise of modern statistics as we know it. – AdamO Dec 06 '13 at 22:18
  • 1
    Diaconis the *magician*? I wouldn't conflate gambling with showmanship! You have a point, but you could push back a little better by suggesting that many "investors" are actually gamblers, whence many theoreticians in mathematical finance might truly be motivated by that form of gambling. Just a thought... Anyway, it is clear that by the time Huygens published his little treatise in 1657 that people were creating a theory of probability (and statistics) for reasons much more profound and far-reaching than doing better at the gambling tables. – whuber Dec 06 '13 at 22:18
  • (And I fully agree about the terminology problems. My pet peeve concerns references to non-existent "populations" in many applications. Ah... there's some linguistic evidence that some of the roots of statistics lie in sampling rather than observations of the cosmos.) – whuber Dec 06 '13 at 22:22