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In mathematics, probability means how likely a random event is to occur. But I've also heard mathematicians use this word in other cases too. Here are some examples to make you understand what I mean:

  • For example, They say The Collatz conjecture is "probably" true. There is no random event happening here because the conjecture is either true or false. But when this conjecture is tested for many numbers and there is no counter example, it means that there is a high chance that the conjecture always works.

  • Here is another example: There was a combination question that was about calculating the number of ways that we can sort letters A,B,C and D so that A always comes before B. I solved that and my answer was $\binom 42 \cdot 2$. Then I accidentally realized that this is equal to $\frac {4!}{2}$. So I guessed that "probably" another way of solving the question exists that leads to the second answer, and it did exist.

These statements (e.g: "The Collatz conjecture is probably true") are not mathematical statements. But they are not useless and worthless statements either, and sometimes can help us. So what exactly are they?

Blue
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madfd adfd
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  • Please do not answer in the comments. – Shaun Dec 27 '24 at 09:32
  • Perhaps helpful: https://math.stackexchange.com/questions/5016442/book-on-the-history-of-probability-theory – Ethan Bolker Dec 27 '24 at 20:13
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    You say: "But when this conjecture is tested for many numbers and there is no counter example, it means that there is a high chance that the conjecture always works." <--- Absolutely not (unless you know something about the distribution). – Martin Brandenburg Dec 27 '24 at 23:28
  • Note that @MartinBrandenburg's suspected duplicate is the result of an experiment in using AI to detect duplicates, see this meta post. I don't think it's a duplicate; I can't really see why anyone would think it was. – N. Virgo Dec 28 '24 at 11:20
  • @N.Virgo That's not true. I followed the same procedure as always to find the duplicate. What you describe the context but not how it was done. (Why do you keep misinterpreting?) Also, please read the answers at the linked duplicate. – Martin Brandenburg Dec 28 '24 at 12:41
  • I said this in the meta discussion as well, but I believe this is the closest thing the site has to a policy on whether it's only the answers that matter or the question as well. It's pretty nuanced but it certainly doesn't say that only the answers matter. If anyone is in doubt, please consult it before casting a close vote. – N. Virgo Dec 28 '24 at 14:24

3 Answers3

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I agree with Jair Taylor's (now moderator deleted) comment above. I don't remember the exact text, but it read something to the effect of: "I don't think a professional mathematician is using the word 'probably' in a different way than a lay person would in this context."

This has nothing to do with frequentist vs Bayesian interpretations of probability, but rather with the standard colloquial meaning of the word "probably", which has very little to do with mathematical probability.

Merriam-Webster gives the definition "insofar as seems reasonably true, factual, or to be expected : without much doubt".

When people say "the Collatz conjecture is probably true", they just mean "I think the Collatz conjecture is true, but I'm not sure". They're probably not thinking of any numerical quantification of their level of confidence.

Alex Kruckman
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    I'm not so sure if it's entirely unrelated to the frequentist vs Bayesian interpretations, insofar as the frequentist would probably say that the probability Collatz is true is either 0 or 1 (we can't really "repeat the experiment" as it were) while the Bayesian would give meaningful non-absolute probabilities to Collatz's truth. – Jonathan Beer Dec 26 '24 at 00:14
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    @JonathanBeer Sure, a frequentist and a Bayesian would have different things to say about the question of the probability that Collatz is true. What I'm saying is that this is irrelevant to the question asked, because when someone says "Collatz is probably true", they're probably not thinking about probability. – Alex Kruckman Dec 26 '24 at 00:19
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    If I say “Damn, that’s a huge seagull!” I probably haven’t thought about specific numerical value of its size or weight — but still, the way we think and talk informally about size is strongly informed by the precise ways in which it can be quantified, This is true regardless of how much we’ve explicitly thought about the quantification: pre-modern people had no precise way to quantify sound or temperature, say, but still used informal quantitative scales to describe them. And if you want to analyse such informal usages, it’s helpful to make reference to the precise measures. [cont’d] – Peter LeFanu Lumsdaine Dec 26 '24 at 04:15
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    [cont’d] I think the same holds here: it’s very good to note that mathematicians are usually using the language of probabilistic belief largely informally (just as most people would in many contexts), but that doesn’t render precise models of probabilistic belief irrelevant — they’re exactly aiming towards a more detailed understanding of the intuition underlying the informal use. – Peter LeFanu Lumsdaine Dec 26 '24 at 04:15
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    @JonathanBeer you say "the frequentist would probably say that the probability Collatz is true is either 0 or 1" ... not sure about this. It seems to me there's an alternative framing of this that does fit into a frequentist framework, based on a statement Dan Romik sort-of made in their answer (paraphrasing and simplifying): "when a famous mathematician says some every-day hypothesis is probably true, at least 7 or 8 times out of 10 they're right". That looks to me like a statement a frequentist could get behind (if I understand the label correctly) in a not-just-0-or-1 way. – Don Hatch Dec 26 '24 at 17:55
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Lots has been written about the philosophical meaning of "probability".

The most naive meaning is the first one you state: choosing "at random" from a (finite) set of well defined alternatives. (There's a begged question there about the meaning of "random", which you can't define as "equally probable".)

The (or an) other view is that of subjective probability - your personal degree of belief. Different mathematicians might quantify "the Collatz conjecture is probably true" differently. You could then think about your estimate of that probability by taking into account the extent to which you trusted each of the mathematicians who had that opinion.

Your degree of belief in a conjecture is indeed useful information. If it's high, you spend time and energy looking for a proof. If you're constantly frustrated trying to find one your belief fades. Then you start looking for a counterexample instead. If you don't find one your belief increases ...

See the Wikipedia pages on probability interpretations and Bayesian probability.

J. W. Tanner
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Ethan Bolker
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Research mathematicians are in the habit of constantly forming opinions about a "level of belief" that the mathematician attaches to various hypotheses (that is, mathematical statements whose truth value is unknown) that are related to the problem one is trying to make progress on. These opinions are based on intuition, factoring in what the mathematician currently knows about the problem, and their accumulated body of experience with research and problem-solving.

Developing such opinions, keeping track of them, and updating them as one learns more about the problem one is trying to solve, are all critical skills that have a large effect on the ability of the mathematician to make progress and come up with good research.

These "levels of belief" are not "probabilities" in any real sense, except in the very loose sense that the most successful mathematicians have developed efficient methods for accurately predicting whether hypotheses are true or not. These successful mathematicians don't always get it right, but they typically get it right more frequently than less successful mathematicians.

When mathematicians talk about research with other mathematicians, they often want to communicate how strong their level of belief in some mathematical statement being discussed is. Being human, they sometimes employ imprecise, metaphorical language. In such a setting it is not uncommon for a mathematician to use the language of probability and say that a mathematical statement is "probably" true. Everyone understands that this is not a literal statement about probabilities; rather, it is simply shorthand for the fact that the mathematician has a high level of (intuitive, non-probabilistic) belief in the statement.

Two final thoughts related to your specific example of the Collatz conjecture:

  1. The idea I mentioned that the most successful mathematicians are reliable predictors of what is "probably" true applies to "everyday" hypotheses - the sort of small, non-famous questions that come up every time one is working on a research problem. If you have a discussion with a famous mathematician about some research problem and 10 "everyday hypotheses" come up that the famous mathematician says are "probably" true, I'd wager they will be right about at least 7 or 8 of the 10.

    By contrast, in the context of a famous problem like the Collatz conjecture, I don't think anyone in the world -- even the smartest mathematicians alive -- can reliably predict whether the conjecture is true or not. The problem is that hard, and that different from anything that any mathematician who has ever lived has ever successfully understood, that personally, I wouldn't assume that any mathematician's opinion about it, however famous or successful that mathematician is, is a reliable indicator of anything. (I mean the opinion by itself - of course, the reasons that led that famous mathematician to arrive at their opinion may be good ones that make the opinion more likely than not to be true.)

  2. Another issue with your example is that the Collatz conjecture is a conjecture. When a mathematical statement becomes a widely known conjecture, the fact that it is referred to as a conjecture is already an indication that there is a widely held belief that it is "probably" true, in the same sense discussed above that most serious mathematicians who have thought about it have a reasonably high level of belief that it's true. In the case of the Collatz conjecture, the reason for this is that we have some nice heuristic arguments supporting it, and strong numerical evidence, whereas no one has proposed any convincing arguments for why it shouldn't be true.

    For this reason, the statement "the Collatz conjecture is probably true" is effectively a tautology. Of course, there are contexts where it would make sense for a mathematician to say something like that, but it still comes across as a bit redundant and funny to my ears.

Dan Romik
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    I'm skeptical of your (and other answers') assertion "These "levels of belief" are not "probabilities" in any real sense". The fact that you would "wager they will be right about at least 7 or 8 of the 10", which I assume you mean literally, seems to indicate, to me, that you are indeed talking about a real probability. We could say you're betting on a conditional probability- that is, it's conditional on the information you have, which does not include the truth or falsity of collatz, but which does include the mathematician's estimate and your estimate of the mathematician's reliability. – Don Hatch Dec 26 '24 at 14:11
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    @DonHatch well, I said they are not probabilities “_except in the very loose sense that …_”. So yes, depending on one’s philosophical view of what “probability” means exactly, one can reasonably ascribe some probabilistic meaning to these beliefs. Personally I don’t find that a particularly helpful point of view however. I have long been of the opinion that people overuse probabilistic ideas to describe aspects of the world that simply don’t lend themselves to such descriptions. This is particularly true IMO when it comes to estimating the “probability” that famous conjectures are true. – Dan Romik Dec 26 '24 at 15:50
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    Ok...and my perception seems to be the exact opposite of yours: personally I don't find the dismissal of such probabilistic statements, calling them "not meaningful" or "meaningful only a very loose sense" to be a particularly helpful point of view. I'm rather mystified, actually, since the fact that you're willing to bet on odds of something tells me, I think, that you do believe there's a perfectly meaningful / real-life probability at play here, as I understand and use the word.I wonder if perhaps the phrase "not meaningful" could be replaced by a term we both understand and agree on. – Don Hatch Dec 26 '24 at 17:17
  • @DonHatch I guess I just enjoy betting. It doesn't mean that my betting decisions mean anything. :-) – Dan Romik Dec 26 '24 at 18:29
  • You write "The problem is ... that different from anything that any mathematician who has ever lived has ever successfully understood." Actually, John Conway successfully understood a very interesting generalization of the Collatz conjecture – Lee Mosher Dec 27 '24 at 15:12
  • @LeeMosher I am aware of Conway’s result, and stand by what I said. Conway’s result doesn’t come anywhere close to the sort of understanding that would be needed to crack Collatz. In fact, his undesirability result arguably reinforces what I said about the conjecture being “that hard”. – Dan Romik Dec 27 '24 at 16:06
  • As I understand things, mathematical probability is a completely abstract concept that cannot be said to be the same as this use of the word "probably," but there is a valid interpretation of mathematical probability as a level of belief, which is the same kind of thing as this use of "probably." The remaining distinction is that the levels of belief in unproved theorems usually (I suspect) have not been quantified with the kind of precision necessary to use them in mathematical calculations. – David K Dec 30 '24 at 18:22