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• CommentRowNumber1.
• CommentAuthorMichael Hardy
• CommentTimeMay 8th 2010
Is probability theory a branch of mathematics?

I've created a page with the title above. Once again, as a newbie, I'm not at all sure whether something like this belongs here or not. This involves mathematics and philosophy, and I'm hoping that if anyone wants to bring physics into it, they'll make that a separate topic unless they surprise me by showing that that's somehow not off topic.
• CommentRowNumber2.
• CommentAuthorEric
• CommentTimeMay 8th 2010

Hi Michael,

For what it’s worth (I’m not on the Steering Committee and not a mathematician and I’m not even working in science anymore - although I used to be an applied physicist), but in my experience the nLab is not the best place to try to start discussions. That is what the n-Forum is for! :)

Nevertheless, the page looks interesting to me and the content seems perfectly suitable for the nLab, but as Andrew pointed out, someone will probably come along and add stuff expressing things from the nPOV :)

If I were to make a suggestion it would probably be to try to turn that page from something that looks like an open question to a page on probability theory. It looks like a really good start to a page on probability theory proper. Once we had at least a stub on probability theory, then we could start talking about open/philosophical questions pertaining to probability theory.

• CommentRowNumber3.
• CommentAuthorMichael Hardy
• CommentTimeMay 8th 2010
Well, I'm curious as to whether the "nPOV" can shed some light on the pages I've created here.
• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeMay 8th 2010

For reasons I’m finding hard to pin down, I’m not overly fond of the page is probability theory a branch of mathematics?. For one thing, it’s the only nLab page that I know of whose title is a question (anyone know another example?). Titles like that are almost always thinly disguised ’no’s, and the unspoken point is that there is an alleged controversy out there and the paper is to be an essay which takes a philosophical position toward it.

But I don’t think I’d like it more if the title were a declarative sentence like “probability theory is not a branch of mathematics”. (By the way, that’s not because I’m opposed to Bayesianism or anything like that – for what it’s worth, and not having pored over this in depth, I find Bayesian intepretations quite acceptable and in conformance with how people use language to express beliefs.)

I like Eric’s suggestion to start a page on probability theory. It would be completely appropriate to have a section on Interpretations, with subsections on frequentist and Bayesian interpretations. I don’t think a long discussion would be necessary to discuss either. The philosophical tension between frequentism and Bayesianism can be acknowledged, as can their respective merits and demerits according to practicing probabilists, but I’d keep that completely factual, concise, and to the point.

Most nLab pages are about mathematical structures, and the nPOV, where applicable, would be a way of considering mathematical structures. I don’t think it has anything to say about Bayesianism particularly (but correct me if I’m wrong!). Interpretations aside, there is a wealth of mathematical structures considered in probability theory (or by probabilists, if you prefer), and it would be absolutely wonderful to have pages on these.

To sum up, I much prefer to keep nLab pages factual, and away from opinions on philosophical questions or other controversies. We do encourage original research, and we aren’t militant about neutrality to the point that all ways of considering mathematical structures must be given equal time, but otherwise we don’t get into taking positions on controversies.

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeMay 8th 2010

Thanks, Todd; I also felt uneasy with this but I was having trouble pinning down what to say. I agree that it would be nice to write about probability mathematically. But I think discussing philosophical arguments is also valuable, and the nLab is officially for philosophy as well as for mathematics and physics. However, I think the discussions would fit better on a page called, say, Bayesian interpretation of probability or something.

Most nLab pages are about mathematical structures, and the nPOV, where applicable, would be a way of considering mathematical structures.

Indeed. We do advance specific points of view, such as the nPOV and those related to it. But it’s likewise unclear to me that the nPOV has anything to say about Bayesianism, and perhaps we don’t want to get into the business of taking sides on controversies that are unrelated to our main focus. However, specific sections and/or specific pages can certainly make arguments in favor of specific sides, as long as the controversy is acknowledged.

What I really don’t understand about the page is why a Bayesian or frequentist interpretation has anything to do with whether probability theory is a branch of mathematics! To me the answer seems to be “yes, obviously” irrespective of one’s philosophical and practical interpretation of that mathematics. (And of course, one’s philosophy and intended applications may suggest different mathematics to do, e.g. study of abstract measure spaces versus selection of uninformative priors, but it’s still mathematics one is doing.) So the Bayesian/frequentist discussion really is about “a way of considering mathematics structures” – isn’t it?

• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeMay 8th 2010

So the Bayesian/frequentist discussion really is about “a way of considering mathematics structures” – isn’t it?

I guess it is, or that it could spawn different sorts of mathematics to do, as you say. I’m really not one to say authoritatively. :-)

What I really don’t understand about the page is why a Bayesian or frequentist interpretation has anything to do with whether probability theory is a branch of mathematics!

At the very least, I’d consider it a heavily mathematicized science, at least as much as physics. My sense is that the whole concept of “randomness” is partly mathematical and partly empirical, maybe part philosophical as well, and that practicing probabilists/statisticians want to think of it that way. But again, I’m not really the guy to ask about that. :-)

Michael H, could your article be read as a reaction against a reductionism which claims that the concept of randomness is “nothing but…” (followed by some purely mathematical concept)? It seems to me there could be a place on the Lab for such a sentiment to be expressed; let’s continue talking about this!

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeMay 8th 2010
• (edited May 8th 2010)

I agree entirely with Eric, Todd and Mike here. I suggest that somebody should take action and reorganize the material as suggested above.

• CommentRowNumber8.
• CommentAuthorDavidRoberts
• CommentTimeMay 9th 2010

Well, isn’t probability theory just applied measure theory? :P But if you are asking for actual probability theory, I would lean towards what Todd said: “a highly mathematicized science”. But it’s a spectrum. I had a fellow student at uni doing ’financial maths’ - a lot of stochastic stuff, etc, and it was all operators, SDEs, hard analysis, and I imagine it’s the same for probability. You would likewise get very ’applied’ people in the field, who wouldn’t know a measure if it bit them. In the end the practicing probabilitist has a mental map from the maths to the interpretation, and this influences the sort of maths that goes on, just as in physics.

• CommentRowNumber9.
• CommentAuthorMichael Hardy
• CommentTimeMay 9th 2010
Todd Trimble asked "could your article be read as a reaction against a reductionism which claims that the concept of randomness is "nothing but..." (followed by some purely mathematical concept)?", and a couple of people asked how this implies probability is not simply a branch of mathematics.

The article poses a question belongs to probability theory but is not a mathematical question: how epistemically probably is it that the outcome of the first trial was a "1", given a state of complete ignorance of the relative frequencies of the possible outcomes.

I'm not quite sure yet how to answer Todd Trimble's question, but the word "random" is a red flag in this context. Banishing the word "random" from probability theory would help clear up some confusion.
• CommentRowNumber10.
• CommentAuthorMike Shulman
• CommentTimeMay 9th 2010

The article poses a question belongs to probability theory but is not a mathematical question: how epistemically probable is it that the outcome of the first trial was a “1”, given a state of complete ignorance of the relative frequencies of the possible outcomes.

I thought that to a Bayesian, that was a mathematical question, since “complete ignorance” dictates a maximally uninformative prior from which you can calculate, while to a frequentist, it was a meaningless question, hence not part of probability theory or mathematics.

• CommentRowNumber11.
• CommentAuthorAlexHoffnung
• CommentTimeMay 9th 2010
Right after reading this page I found a page whose title is a question, although it seems to be a personal research page so maybe that does not really count.

• CommentRowNumber12.
• CommentAuthorDavidRoberts
• CommentTimeMay 9th 2010

Yes. That one is my fault - it was meant to be somewhat evocative so as to get Andrew on board (or maybe he was already on board by that time - I forget). I will point out to those who are not in the habit of following random links that it is on my private web. It is also current research, as Alex points out, so falls well outside the realm of an encyclopaedic/descriptive approach.

• CommentRowNumber13.
• CommentAuthorDavid_Corfield
• CommentTimeMay 9th 2010

A fairly standard line amongst philosophers is that Kolmogorov's axioms give a mathematical characterisation of probability, which is then interpretable in several ways. Many of the consequences of the axioms are interpretation independent as pieces of mathematics. On the other hand, the interpretation does tend to lead you to want to carry out different calculations.

A broad brush classification of interpretations has a first dichotomy as to whether probability is out there in the world or instead relates to our state of knowledge. In the first camp we have the frequentists for whom probabilities can only be derived as limit ratios from long series of identical trials. Here there is a clash with an intuition which wants a die destroyed after a single throw to have a probability of showing 6. This may lead to a counterfactual, what the limit frequency ratio would have been had the die been thrown many times.

However, there is another problem. I would like to know my probability of suffering a heart attack in the next 5 years. I need some large class of 'identical' trials to give me an answer. So I look to the number of people who died within 5 years of my age. But then I think there are many factors that might be relevant. I don't smoke, I have a certain diet, I'm married, I have a number of kids, my income is such and such, I don't have to commute far, ... By the time I've finished, I'm the only person in the class of people who have identical health characterisics. And yet I may still feel that there is some number representing the objective chance of my heart attack. This is the view of propensity theorists, such as Popper.

• CommentRowNumber14.
• CommentAuthorDavid_Corfield
• CommentTimeMay 9th 2010

(continued - the software didn’t like long comments)

Now, by the time we’ve specified all of these conditions we might start to think that it’s determined whether or not I will have that heart attack, and that any probability offered merely reflects our ignorance of some of the factors and physiological knowledge. In the case of a coin toss, given initial conditions, it is determined whether head or tail will show. There is no objective probability to show head, it is just a measure of our ignorance.

We have now turned to the other half of the dichotomy, where probabilities reflect states of knowledge. There are broadly two views here:

(1) Subjective Bayesianism: probabilities are degrees of belief, measurable by willingness to take certain bets. Your degrees of belief must be consistent with the axioms of probability theory or else it will be possible to place wagers with you so that you will definitely lose money.

(2) Objective Bayesianism: probabilities are rational degrees of belief relative to a state of knowledge. Given your epistemic state there is a most rational degree of belief in an event.

So if I say an event will take place next door with possible outcomes A or B, the advocate of (1) can have Pr(A) anywhere between 0 and 1, so long as Pr(B) = 1 - Pr(A), the advocate of (2) will have Pr(A) = 0.5 = Pr(B).

Some have quipped that there are $6^6$ varieties of Bayesian, but this dichotomy is a very important one.

• CommentRowNumber15.
• CommentAuthorDavid_Corfield
• CommentTimeMay 9th 2010
• (edited May 9th 2010)

One last thing. Mathematically, the Bayesians will need to do things that the frequentists won’t. I may have a model class with parameters $r, s, t$. I have a prior distribution over these parameters, then data, $d$, arrives and so I’d like to update my distribution of belief over parameter space:

$Pr(r, s, t|d) = Pr(d|r, s, t). Pr(r, s, t)/\int Pr(d|r, s, t). Pr(r, s, t) d r d s d t$.

This is meaningless to the frequentist, for whom there can be no distribution of the parameters. It leads the Bayesian to lots of intractable integrals, often resolved by Monte Carlo methods.

If we have pages on probability theory here from the nPOV, then we’ll have to include the stuff about the Giry monad. Looking back, I see that I claimed it went nicely with a Bayesian point of view. I still think there’s something right about that.

• CommentRowNumber16.
• CommentAuthorHarry Gindi
• CommentTimeMay 9th 2010
• (edited May 9th 2010)

Probability theory is really just measure theory in disguise where they use funny words like "almost always" instead of "almost everywhere" and "expectation" instead of the integral over X, "random variable" for "measurable function", etc. Measure theory is a (sub-)branch of mathematics, and so it follows easily that probability theory is a (further sub-)branch of mathematics.

• CommentRowNumber17.
• CommentAuthorTodd_Trimble
• CommentTimeMay 9th 2010

@Alex #9: no, that page doesn’t count, because while it’s part of ncatlab.org, it’s not part of nLab. :-)

• CommentRowNumber18.
• CommentAuthorMichael Hardy
• CommentTimeMay 9th 2010
Harry Gindi: you're merely repeating a cliché and giving no argument. I proposed a counterexample to what you're asserting.

It seems tempting to say that one reason why probabilists pay so little attention to some things that statisticians deal with is that the subject matter dealt with by statisticians differs. But sometimes I wonder if some of it is due to their habit of treating probability theory as if it's _only_ measure theory.
• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeMay 9th 2010
• (edited May 9th 2010)

1. concerning the organization of the pages: one point of the wiki is that not only does it provide information, but it provides also the context of the information by giving links back and forth between subjects that touch on each other. Currently it seems the page in question is not connected much. That should be changed. I think it is good to split off pages from their mother pages (such as probability theory) if they become long. The mother page should in that case however

1. exist in the first place or be created if they do not yet

2. link to the split-off pages.

Here we would need a remark somewhere on a page titled probability theory: For a discussion of the status of probability theory in math see the dedicated page “Is probability theory a branch of mathematics”.

2. Concerning the content: it has been said before (at the entry in question, notably), but I repeat what I think is right: the relation of probability to math is precisely that of physics to math. In fact, it seems to me it can be seen as a special case of the relation of physics to math: on the one hand we have the mathematical model (measure theory here) which can be studied all by itself, even disregarding its phyiscal/probability theoretic interpretation. On the other hand we have some at least vague idea how from looking at that model we can deduce statements about the observable world.

So similarly I could ask: is Hamiltonian mechanics a part of mathematics? As a model certainly it is: it is just the study of symplectic geometry. But when I want to use that model to predict where and when the next solar ecplise will be observable on earth, then in some way or other I am going beyond just math.

• CommentRowNumber20.
• CommentAuthorIan_Durham
• CommentTimeMay 9th 2010
I have a colleague in our math department who is a probability theorist and who did his PhD under Michael Keane. Keane is supposedly fairly well-known in that area (something I didn't know until we hired my colleague Steve). Anyway, they consider probability theory to be what they do but they are also pure mathematicians, so who knows. I suppose it is a matter of opinion.

Bayesianism is becoming increasingly important in quantum mechanics and a month or so ago I added a short blurb on quantum Bayesianism (QBism) under quantum state. Personally, I think that a mixture of QBism and relationalism (similar to Rovelli's stuff) offers a more empirically satisfying explanation of physics in general and I think that categories seem to be natural for describing relational theories.
• CommentRowNumber21.
• CommentAuthorHarry Gindi
• CommentTimeMay 9th 2010
• (edited May 9th 2010)

Probability theory is precisely the study of measure spaces with total measure 1 (measurable functions, changing measures, etc.). It is a formal branch of formal mathematics. "Probability theory" as practiced by "Probabilists" or "Operations Research Dudes" or "Statisticians" is not a formal branch of formal mathematics. Your argument is akin to saying that differential geometry is not an area of mathematics because engineers, physicists, etc., use it.

Anyway, if your objection is to the frequentist approach formalized by Kolmogorov, then we can say that frequentism is mathematics and Bayesianism is not...

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeMay 9th 2010
• (edited May 9th 2010)
• CommentRowNumber23.
• CommentAuthorIan_Durham
• CommentTimeMay 9th 2010
@Harry: just curious who you were responding to. If it was to someone other than me, then just ignore this. If it was to me, then I think you misinterpreted what I was saying. I think it absolutely is a branch of pure mathematics and obviously you are correct in that it is really just measure theory (or seems to be from my limited knowledge of measure theory).

Personally I think both the frequentist as well as Bayesian approaches have validity (clearly the frequentist approach has validity since I would guess that, without knowing it, that's what most Casinos use to calculate odds). Quantum Bayesianism is a bit more than just straight Bayesianism in that it is a bit more relational (which is why I like it) and denies the ontological status of quantum states. It has been accused of being solipsistic, but I think Chris Fuchs has done a fairly good job of refuting that accusation. Just my opinion.
• CommentRowNumber24.
• CommentAuthorHarry Gindi
• CommentTimeMay 9th 2010

I was responding to Michael Hardy.

• CommentRowNumber25.
• CommentAuthorMichael Hardy
• CommentTimeMay 9th 2010
Harry Gindi: Why would frequentism be mathematics while Bayesianism is not?? Both are philosophical positions on how to apply mathematics to statistical inference. Kolmogorov's system fits with Bayesianism better than it does with frequentism, notwithstanding the fact that Kolmogorov was a frequentist.

One of Kolmogorov's axioms is countable additivity. Some probabilists (e.g. William Sudderth, and possibly Leonard Jimmy Savage) have held that only finite additivity should have been positited axiomatically. Would you say that what they're doing when they argue for that position fails to be probability theory on the grounds that it's not within Kolmogorov's axiomatic system?
• CommentRowNumber26.
• CommentAuthorIan_Durham
• CommentTimeMay 10th 2010
@Harry: Okie dokie. Wasn't sure.
• CommentRowNumber27.
• CommentAuthorHarry Gindi
• CommentTimeMay 10th 2010

@Michael Hardy: I am open to new axiomatizations of probability theory, but the development of such axiomatizations does not fall within the realm of probability theory proper. Probability theory as a pure mathematical pursuit only includes those parts of the theory that are developed rigorously.

• CommentRowNumber28.
• CommentAuthorMike Shulman
• CommentTimeMay 10th 2010

I think Urs #19 is spot on about how statistics, frequentist/Bayesian arguments, and inference relate to mathematics, and at the same time Harry #21 is right that the purely mathematical parts of the theory are of course mathematics.

One can of course argue about whether the phrase “probability theory” refers to the purely mathematical theory in question, or to the non-mathematical parts of the theory involving its application to real problems. Here there is a distinction with “physics,” I think, because people do at least sometimes use “probability theory” to refer largely to the mathematics, whereas when studying Lorentzian manifolds purely mathematically I think they would call it “differential geometry” or at best perhaps “mathematical physics.” But evidently “probability theory” is also used to encompass the nonmathematical parts of the theory, so the eventual entry we end up with should make that clear too.

And I still think that all of that applies just as well to any type of probability, be it frequentist, Kolmogorov, Bayesian, etc. The nonmathematical parts of the two theories are different, leading them to also care about different parts of mathematics, but the relationship between the theory and the mathematics is analogous.

• CommentRowNumber29.
• CommentAuthorHarry Gindi
• CommentTimeMay 10th 2010

I feel that it's condescending to call any subject where one rigorously proves theorems from axioms anything other than mathematics. I am willing to include any such subject under the umbrella of mathematics, as long as the acceptance of a result in said subject is conditional on the presentation of rigorous proof.

Then the parts of probability theory that are formal and rigorous are without a doubt mathematics. In fact, this is what I consider "probability theory proper". There are areas of "mathematics" (principally applied mathematics) where formal proof is not required, but I do not consider them to be a part of "mathematics proper".

• CommentRowNumber30.
• CommentAuthorDavidRoberts
• CommentTimeMay 10th 2010

There are areas of “mathematics” (principally applied mathematics) where formal proof is not required, but I do not consider them to be a part of “mathematics proper”.

ouch. What about experimental mathematics? (this is so off topic, but perhaps we need a page on that ;-)

• CommentRowNumber31.
• CommentAuthorHarry Gindi
• CommentTimeMay 10th 2010
• (edited May 10th 2010)

Experimental mathematics is a heuristic to discover formal proofs, but it is not in and of itself "proper mathematics".

• CommentRowNumber32.
• CommentAuthorIan_Durham
• CommentTimeMay 10th 2010
What's experimental mathematics?
• CommentRowNumber33.
• CommentAuthorDavidRoberts
• CommentTimeMay 10th 2010

Things like where people run massive numerical computations of integrals or combinatorial problems and try to figure out what the exact answer might be. Or wading through structures that are too numerous to actually compute all of them, but seeing what results could be gleaned from the small corner that can be examined. This it not really an informed view, but roughly the right idea. In short: mathematical reasoning done on massive problems with the aid of computers in areas that are at present out of our reach as far as pen and paper proofs go.

• CommentRowNumber34.
• CommentAuthorTim_van_Beek
• CommentTimeMay 10th 2010
• (edited May 10th 2010)

What’s experimental mathematics?

Example:

If you try to “solve stochastic differential equations numerically” you need random numbers, but if you use a deterministic algorithm to generate numbers, these will not be “truly random” (therefore the numbers are called pseudorandom numbers and the algorithms pseudorandom generators). Some mathematicians that employ this technique like to stress this fact by using the term “(computer) experiment”. See e.g.

• CommentRowNumber35.
• CommentAuthorMichael Hardy
• CommentTimeMay 10th 2010
"I am open to new axiomatizations of probability theory, but the development of such axiomatizations does not fall within the realm of probability theory proper. Probability theory as a pure mathematical pursuit only includes those parts of the theory that are developed rigorously."

That is circular reasoning. Probability theory as a mathematical pursuit includes only those things that are in fact mathematics; therefore everything within probability theory is mathematics.

And you still don't appear to have noticed my particular example, proposed in the page this thread was originally about, of a question that is not a mathematical problem at all but clearly belongs within probability theory.
• CommentRowNumber36.
• CommentAuthorIan_Durham
• CommentTimeMay 10th 2010

In short: mathematical reasoning done on massive problems with the aid of computers in areas that are at present out of our reach as far as pen and paper proofs go.

Ah, ok. I've never heard it called "experimental mathematics" before. I've always thought of that as computational mathematics, like the flyspeck project for instance (the one trying to prove Kepler's Conjecture).
• CommentRowNumber37.
• CommentAuthorHarry Gindi
• CommentTimeMay 10th 2010

And you still don't appear to have noticed my particular example, proposed in the page this thread was originally about, of a question that is not a mathematical problem at all but clearly belongs within probability theory.

I don't see how this is circular. I'm defining "probability theory proper" to be the part of probability theory that is rigorously axiomatized and proven. When mathematicians talk about probability theory, this is usually the part of the theory they are talking about.

• CommentRowNumber38.
• CommentAuthorIan_Durham
• CommentTimeMay 10th 2010
I thought about the example given and I think I see both Michael's point as well as Harry's point. Ultimately I think it's really related to the broader question of "what is mathematics?" One view holds that it is as close to the Platonic ideal as one can get and should be free from "interpretation," i.e. being rigorously axiomatized, there should be no need for interpretation. On the other hand, in certain branches of mathematics including probability theory, as Michael has shown, there appear to be places where different interpretations might potentially lead to different mathematical results. The question for the nLab (I would think) is whether categories help shed any light on this issue.
• CommentRowNumber39.
• CommentAuthorHarry Gindi
• CommentTimeMay 10th 2010
• (edited May 10th 2010)

No, the view I'm espousing is formalism, not platonism. However, the way I view formalism is something like "relative platonism".

• CommentRowNumber40.
• CommentAuthorIan_Durham
• CommentTimeMay 10th 2010
Hmmm, OK, I see what you're saying. Well, I didn't mean to imply you were necessarily a platonist. I suppose there's probably a continuous spectrum of views on the nature of mathematics.
• CommentRowNumber41.
• CommentAuthorMichael Hardy
• CommentTimeMay 10th 2010
Harry Gindi: If you really only mean you want to stipulate what a certain verbal expression refers to because of convention, then that would seem to make your comments vacuous and irrelevant to the discussion.

Ian Durham: I don't think my example shows that different mathematical results follow from different interpretations; rather it shows that "probability theory" should be construed as encompassing more than just mathematics.

As a tentative definition I'd say mathematics deals with abstract structures. E.g. it doesn't matter whether the chess pieces are made of ivory or electronic images on a screen; the "abstract structure" is the same either way since the concrete structure are isomorphic to each other. Thus mathematics is the study of those truths that are preserved by isomorphism. Whether 2 + 3 = 5 doesn't depend on whether you're counting pennies or occurrences of a particular classical Greek word in the writings of Plato; in this case the isomorphism is merely a one-to-one correspondence. When Russell said (but correct me if he didn't say it) that mathematics is the subject in which we don't know what we're talking about, I take that to mean it doesn't matter which concrete instance of the abstract structure we're talking about; 2 + 3 = 5 either way.
• CommentRowNumber42.
• CommentAuthorHarry Gindi
• CommentTimeMay 10th 2010
• (edited May 10th 2010)

Harry Gindi: If you really only mean you want to stipulate what a certain verbal expression refers to because of convention, then that would seem to make your comments vacuous and irrelevant to the discussion.

It appears that you are implying that this discussion had any relevance to the nPOV in the first place. I feel like this entire thread is "subjective and argumentative" (and would vote to close it as such if it had been on MO). I'm using the definition of Probability theory that is accepted by mathematicians. Perhaps we could introduce the following convention:

"Probability theory" refers to the formal mathematical study of probability. It is a full-fledged branch of mathematics and has the full strength of mathematics with rigorous proofs etc.

"Probability" refers to the informal study that you're harping on about. It is not actually important to mathematicians and consists of a superset of Probability theory including all of the heuristics that statisticians and "Probability dudes" use.

To be honest, I don't care what the statisticians and "Probability dudes" think. I'd rather piss them off and tell them that they're not doing probability theory than tell the mathematicians who work in probability theory that they aren't doing mathematics (a stupid, insulting, and untrue statement). The point here is that the difference between "probability theorists" (in the sense above) and "probability dudes" is far greater than the difference between "probability theorists" and people working in analysis.

As a tentative definition I'd say mathematics deals with abstract structures. E.g. it doesn't matter whether the chess pieces are made of ivory or electronic images on a screen; the "abstract structure" is the same either way since the concrete structure are isomorphic to each other. Thus mathematics is the study of those truths that are preserved by isomorphism. Whether 2 + 3 = 5 doesn't depend on whether you're counting pennies or occurrences of a particular classical Greek word in the writings of Plato; in this case the isomorphism is merely a one-to-one correspondence. When Russell said (but correct me if he didn't say it) that mathematics is the subject in which we don't know what we're talking about, I take that to mean it doesn't matter which concrete instance of the abstract structure we're talking about; 2 + 3 = 5 either way.

Isn't this obvious to anyone who has done mathematics since Bourbaki? This is what axiomatization of structures and properties does. I'm pretty sure that everyone here understands this point (the nForum is not the talk page on wikipedia).

• CommentRowNumber43.
• CommentAuthorIan_Durham
• CommentTimeMay 11th 2010
Well, then, I guess I would fall in Harry's camp then in that I would think of it this way: mathematics is entirely self-consistent. Interpretation should have no affect on the outcome. That is true in science only in so much as good scientific theories must match existing experimental results, but can - and often do - differ on their predictions of future experimental results. In other words, mathematics ought not include any ambiguity in its purest form (though whether it attains this form at all times is debatable). Science, on the other hand, always has ambiguous "edges" to it (particularly when we push its boundaries).
• CommentRowNumber44.
• CommentAuthorMike Shulman
• CommentTimeMay 11th 2010

Harry, why so insistent that something with “theory” in its name should refer only to mathematics? Surely you don’t want to insist that physicists studying “relativity theory” or “quantum theory” find a new name for their subject?

And I would be very much surprised if the applications of mathematical probability are completely unimportant to the mathematicians who study probability. If so, then those mathematicians are probably the poorer for it.

• CommentRowNumber45.
• CommentAuthorHarry Gindi
• CommentTimeMay 11th 2010
• (edited May 11th 2010)

Probability theory is not intrinsically an empirical science. I would call the application of probability theory as an empirical science the study of statistics.

And with respect to your second paragraph, I would argue that they are not interested in the applications while they are wearing their mathematician hats.

• CommentRowNumber46.
• CommentAuthorIan_Durham
• CommentTimeMay 11th 2010

And with respect to your second paragraph, I would argue that they are not interested in the applications while they are wearing their mathematician hats.

Some of the greatest discoveries in mathematics have been developed directly from applications.
• CommentRowNumber47.
• CommentAuthorMichael Hardy
• CommentTimeMay 11th 2010
Harry Gindi wrote: "Probability theory is not intrinsically an empirical science. I would call the application of probability theory as an empirical science the study of statistics.

"And with respect to your second paragraph, I would argue that they are not interested in the applications while they are wearing their mathematician hats."

I don't think I ever suggested that probability is an empirical science.

Supposing your second paragraph to be true, what is its relevance?
• CommentRowNumber48.
• CommentAuthorMike Shulman
• CommentTimeMay 11th 2010

Harry, I feel as though you have a habit of saying something, then when someone objects to it, in response you say something different, but act as though the different thing were what you had said in the first place. Perhaps it is true that when mathematical probabilists are studying probability purely as mathematics, they are not interested in the applications, but I don’t think that’s what you said in #42; you said that the non-mathematical parts of probability theory “are not important to mathematicians,” and that’s what I objected to.

And I don’t think that anyone has claimed that probability is empirical, so your first sentence in #45 is also attacking a straw man. All we’ve said is that probability theory includes things which are not purely mathematics, but also involve the interpretation of mathematics. The world is not partitioned between mathematics and empirical science. And to me, “statistics” has a specific meaning that is less general than “probability.”

• CommentRowNumber49.
• CommentAuthorMichael Hardy
• CommentTimeMay 11th 2010
Mike Shulman: Interesting that you say statistics is "less general than" probability. It's hard to imagine anything in mathematical probability theory that would not potentially be of interest to statisticians, but there are many things in statistics that probabilists generally ignore (design of experiments, exponential families, estimation, imputation, prediction (....um.... OK, maybe probabilists do pay some attention to prediction), sufficiency (in Fisher's sense), confidence intervals, ...... OK, I haven't prepared on this, so for now I'll stop with this very short list.)

The way I think of it is that a probabilist starts with a model (e.g. coin tosses are independent, "heads" appears 40% of the time) and can tell you the probability of some event ("HHHH" occurs before the second time "HTHT" occurs) whereas a statistician observes the outcome and then tries to figure out what the model was.

Statisticians more frequently than probabilists read about things like epistemology, the problem of universals, the problem of induction, etc.
• CommentRowNumber50.
• CommentAuthorHarry Gindi
• CommentTimeMay 11th 2010

@Mike Shulman: The reason I was talking about empirical science was a response to things like "relativity theory" and "quantum theory", which are empirically supported (parts of) science. I was drawing the distinction between the use of "theory" for rigorous mathematics and empirical science. Since probability is not an empirical science and arguments are given a priori, the "theory" part should be reserved only for those rigorously proven results.

And with respect to your second paragraph, I was clarifying my unclear statement. I will make sure to make it clear when something is a clarification.

• CommentRowNumber51.
• CommentAuthorMike Shulman
• CommentTimeMay 11th 2010

@Harry: It sounds to me like you’re saying that it’s okay to call something empirical a “theory,” but if it’s not empirical, then “theory” should mean only rigorous mathematics. But for the same reason that the world is not partitioned between mathematics and empirical science, I see no reason why the space of things dignified by the name “theory” should be so partitioned. (-:

@Michael (do you prefer “Michael”? Or should we distinguish between ourselves in some way other than by use/nonuse of a nickname?): Well I could be wrong in my feeling of what words mean! Especially an area where I know very little. You make a good point. However I thought that you were arguing for the inclusion of such “non-mathematical” aspects of statistics/probability under the umbrella of “probability theory.” Was I mistaken?

• CommentRowNumber52.
• CommentAuthorHarry Gindi
• CommentTimeMay 11th 2010
• (edited May 11th 2010)

@Mike S: I would argue that rational discourse is partitioned precisely into "a priori arguments" and "a posteriori arguments". In fields where justifications are given a priori, the word "theory" should be reserved for "a field of study attempting to exhaustively describe a particular class of constructs", but in fields where justifications are given a posteriori, the distinction of a "theory" should be given to "a coherent statement or set of statements that attempts to explain observed phenomena".

Source: Wiktionary

My assertions are at least consistent with the dictionary definition of a theory. If we accept that mathematics is the study of systems of axioms and their formal consequences, then the distinction follows from the definition. Then the only propositions one must accept for the distinction to hold from the above argument are:

a.) The definition of the word "theory"

b.) The definition of the field of mathematics

I am reasonably certain that we could significantly strengthen the definition of the field of mathematics to be as restrictive as the nPOV requires without actually disturbing this distinction, and I am also confident that you accept these definitions of the word "theory". This is my justification for making the distinction. Anyway, I definitely disagree that it is totally artificial and have hopefully supported my argument with enough evidence to bring you over to the dark side..

• CommentRowNumber53.
• CommentAuthorUrs
• CommentTimeMay 11th 2010

you are all aware that despite this lengthy discussion, an nLab page probability theory has still not come into existence?

• CommentRowNumber54.
• CommentAuthorHarry Gindi
• CommentTimeMay 11th 2010

Well, we're currently debating what is meant precisely by the term "probability theory".

• CommentRowNumber55.
• CommentAuthorDavid_Corfield
• CommentTimeMay 11th 2010

I would keep any page on probability theory very brief. It has a very different scope depending on your stance. For Jaynes, probability theory is very much like a form of logic, but reasoning on the basis of uncertain knowledge. His best known book is called ’Probability Theory: The Logic of Science’ (well worth reading). For him, statistics and decision theory fall within its remit (statistics being done the Bayesian way, avoiding hypothesis testing, etc.). For the frequentist, statistics and probability theory don’t overlap.

Unless the nPOV has something particular to say on the subject, I don’t see why nLab should wade into this controversy.

• CommentRowNumber56.
• CommentAuthorHarry Gindi
• CommentTimeMay 11th 2010

I feel like it's consistent with the nPOV to talk about a "mathematical theory of probability", whatever one wishes to call it. I have given a (hopefully) decent argument that this "mathematical theory of probability" should be called "Probability theory" (which should follow from the nPOV as long as the nPOV has something to say about the definition of mathematics itself).

Even if one does not accept my argument, I offer an argument that takes a different approach: The nLab is a site for pure mathematicians and theoretical physicists. The "mathematical theory of probability", if the distinction must be made, is the part of the "greater probability theory" that is important to the intended audience. Since the other parts of the "greater theory of probability" are not really relevant, this lends support to the idea that a page on "probability theory" should be focused principally if not exclusively on the "mathematical theory of probability".

• CommentRowNumber57.
• CommentAuthorUrs
• CommentTimeMay 11th 2010

Unless the nPOV has something particular to say on the subject, I don’t see why nLab should wade into this controversy.

I don’t think there is need to wade into any controversy. I don’t see a deep controversy anyway.

But, David, I am hoping one day we’ll have a decent page probability theory that, among other things, describes the category-theoretic approaches to it, which you have been disucssing so often on the nCafe.

• CommentRowNumber58.
• CommentAuthorDavid_Corfield
• CommentTimeMay 11th 2010

OK. So I’ve made a small start there, pasting in some cafe material. We need an introduction, which could be category theory free.

• CommentRowNumber59.
• CommentAuthorTim_van_Beek
• CommentTimeMay 11th 2010

I had a go at it :-) Is the book “stochastic relations” by Doberkat of relevance? I saw that David mentioned it on the nCafé and skimmed the table of contents, but all I have learned yet is that it obviously is about stochastics and uses the language of category theory.

• CommentRowNumber60.
• CommentAuthorAndrew Stacey
• CommentTimeMay 11th 2010

(Although the last few comments have gotten back to discussing actual nLab pages, this discussion has veered sufficiently away from actual nLab pages that I’ve moved it into the Atrium.)

• CommentRowNumber61.
• CommentAuthorUrs
• CommentTimeMay 11th 2010

Thanks David!

Very nice. Finally some constructive progress here.

I have further expanded the Idea section at probability theory a bit, and then I added plenty of hyperlinks to the nPOV material that you had added.

• CommentRowNumber62.
• CommentAuthorIan_Durham
• CommentTimeMay 11th 2010
Please don't shoot me for doing so, but I added a query at the probability theory page.
• CommentRowNumber63.
• CommentAuthorHarry Gindi
• CommentTimeMay 11th 2010
• (edited May 11th 2010)

From the idea section (could somebody put the code for a query box on the sidebar? I always have to go to another page to check it, and I can never remember it.):

Notice that in this respect probability theory has a similar status as (other(?!)) theories of physics: there is a mathematical model (measure theory here as the model for probability theory, or for instance symplectic geometry as a model for classical mechanics) which can be studied all in itself, and then there is in addition a more or less concrete idea of how from that model one may deduce statements about the observable world (the average outcome of a dice role using probability theory, or the observability of the next solar eclipse using Hamiltonian mechanics)...

This is not true. Unlike the "(other?!) theories of physics", probability theory is not attempting to model the real world (it is an a priori, not an a posteriori field of discourse, which I thought we all agreed on (at least earlier in this thread)). The analogy is much closer with higher category theory, where there are different models of (oo,1)-categories that all describe fundamentally the same objects.

Probability differs from physics in the following way: results derived in probability theory are independent from the universe in which we live. They are mathematical truths, rather than sketches of physical laws that can vary from time to time and place to place and can never be judged to be completely accurate (this is why mathematics is the "queen of the sciences". It is not subject to the same philosophical issues that one encounters in the empirical sciences).

Anyway, in my opinion, the "Idea" section currently detracts from the value of the rest of the page.

• CommentRowNumber64.
• CommentAuthorTim_van_Beek
• CommentTimeMay 11th 2010

@Ian: No problem, but that is a question that I would like to “outsource” to a different page, we can do that if we take “random” as given on the probability theory page and discuss its different interpretations somewhere else (like is it “insufficient knowledge” or “god playing dice” :-)

I know some mathematicians specializing in mathematical statistics who firmly believe that our physical world is deterministic (I think they still do, even after I told them about quantum mechanics). To them it’s really only about mathematical models that help you decide what to do in certain situations, without any epistemologic relevance.

• CommentRowNumber65.
• CommentAuthorHarry Gindi
• CommentTimeMay 11th 2010
• (edited May 11th 2010)

@TvB: Gerard 't Hooft (not a crackpot) put a paper up on the ArXiv a few years ago talking about a mathematical basis for a deterministic theory of QM http://arxiv.org/abs/quant-ph/0604008

I don't know what became of it, but since he's a nobel prize winner, it can't be that bad, right?

Edit: Here's a more recent paper of his with similar ideas: http://xxx.lanl.gov/abs/0908.3408

Edit 2: Oh wait, Urs probably knows him!

• CommentRowNumber66.
• CommentAuthorUrs
• CommentTimeMay 11th 2010

Ian, since your query box is not at all about what probability theory describes but whether or not quantum mechanics is a probabilistic theory, I have removed the query box and instead reproduce it here, for whatever discussion may here come out of it:

Ian Durham: At the risk of starting another argument, I would just like to point out that there are theoretical physicists for whom the phenomena described by probability theory are not random. Specifically, both the epistemic and consistent histories approaches to quantum theory do not view the quantum processes that are described by probability theory as being random. Their view is that we use probability theory simply due to a lack of knowledge on our part, i.e. some things, which are perfectly non-random, are nevertheless unknowable. Just my two cents.

• CommentRowNumber67.
• CommentAuthorUrs
• CommentTimeMay 11th 2010

probability theory is not attempting to model the real world

Sure it is. What do the think the word “frequency” in “frequentist interpretation” comes from?

But I agree with your other point: in as far as probability theory is about the observable world, it is different from most other theory of physics, much more universal, yes. That’s why I tried to be careful, wrote “similar status” and put that funny “other(?!)”.

• CommentRowNumber68.
• CommentAuthorIan_Durham
• CommentTimeMay 11th 2010
OK. Quantum theory was just my personal example, though. As Tim pointed out, some mathematicians see the world as deterministic (independent of any physical theory). But, I see your point nevertheless.
• CommentRowNumber69.
• CommentAuthorUrs
• CommentTimeMay 11th 2010

Okay, I see. I reworked the first sentence at probability theory to

Probability theory is concerned with mathematical models of phenomena that exhibit randomness , or more generally phenomena about which one has incomplete information.

• CommentRowNumber70.
• CommentAuthorIan_Durham
• CommentTimeMay 11th 2010
• (edited May 11th 2010)
For anyone interested in such things, some colleagues of mine in quantum information put out a paper last year that examines determinism in general probabilistic theories:

http://arxiv.org/abs/0909.5075

Rob Spekkens, one of the co-authors, has developed an epistemic theory of QM that has gained some traction and the underlying processes are deterministic in his theory to some extent.
• CommentRowNumber71.
• CommentAuthorIan_Durham
• CommentTimeMay 11th 2010

Probability theory is concerned with mathematical models of phenomena that exhibit randomness , or more generally phenomena about which one has incomplete information.

Perfect, thanks!
• CommentRowNumber72.
• CommentAuthorTim_van_Beek
• CommentTimeMay 11th 2010

Harry wrote:

Gerard ’t Hooft (not a crackpot) put a paper up on the ArXiv a few years ago talking about a mathematical basis for a deterministic theory of QM

Oh, interesting, I did not know that.

My last comment may seem to imply that I think that quantum mechanics enforces an interpretation including some sort of randomness, but the situation I had in mind was a little bit different: My friends did not know that any physical theory existed with any non-deterministic interpretations around, that was completly new to them.

• CommentRowNumber73.
• CommentAuthorUrs
• CommentTimeMay 11th 2010

Concerning the t’Hooft-article:

there are of course many attempts in the literature to realize standard quantum mechanics as the coarse-grained version of a non-probabilistic theory. Great minds have tried themselves on this, not the least t’Hooft. But it’s one of those dangerous questions to follow…

• CommentRowNumber74.
• CommentAuthorEric
• CommentTimeMay 11th 2010

@Harry #65: Thanks for the references. Very interesting.

• CommentRowNumber75.
• CommentAuthorHarry Gindi
• CommentTimeMay 11th 2010
• (edited May 11th 2010)

@Urs: Could you at some point write up an article for the cafe with an account of some attempts at making QM deterministic and what went wrong in those attempts (written for mathematicians who aren't trained in physics)? I'm really interested in the subject, but I don't ever plan to be a "physics guy", and I assume that it would be interesting to a lot of the readers at the cafe who are pure mathematicians and would get pretty lost actually trying to read through the literature. Just an idea for a post at some point in the future.

• CommentRowNumber76.
• CommentAuthorUrs
• CommentTimeMay 11th 2010
• (edited May 11th 2010)

Harry,

it would be desireable to have such an entry, but I am afraid I won’t be able to do it justice without going back and reminding myself of lots of literature.

One apparently well thought-through approach is that by Adler. There is this book review which gives a useful impression.

• CommentRowNumber77.
• CommentAuthorMichael Hardy
• CommentTimeMay 12th 2010
Here is a paper I wrote on a topic related to those in this thread. It could bear some examination from a category-theoretic point of view:

http://arxiv.org/pdf/math.PR/0203249
• CommentRowNumber78.
• CommentAuthorIan_Durham
• CommentTimeMay 12th 2010
Thanks for that reference Michael. It looks quite illuminating. (I had no idea Jaynes worked for Socony Oil!)
• CommentRowNumber79.
• CommentAuthorMichael Hardy
• CommentTimeMay 12th 2010
I think he only briefly visited Socony Oil, but I'm not sure.
• CommentRowNumber80.
• CommentAuthorIan_Durham
• CommentTimeMay 13th 2010
• (edited May 13th 2010)
Even so, I find that very interesting. It's a bit sad to think that the glory days of corporate R and D seem to have faded. With a few exceptions (Google, Microsoft, IBM), companies are gutting their R and D departments. What's happened to Bell Labs is almost a crime.