There's the following very classical paradox in probability theory: Suppose you toss a dart at a dartboard. For each point p on the dartboard, we can ask for the probability that the dart hits that exact point p. Suppose we have no relevant information about the tossing process, so that each point seems equally likely. What is the probability P(p)? Well, it can't be more than zero, because then the total probability, the probability that the dart hits any point, would be, not just greater than 1, but infinite! This is because there are infinitely many points on the dartboard, and any positive number added to itself infinitely many times is infinite. But hold on, it also can't be 0, because that means by a similar argument that the probability that the dart hits any point on the dartboard is also 0 - but it must hit _some_ point.

First I'll note that this problem actually is _not_ entirely resolved by measure theory. Measure theory allows for a uniform probability measure on continuous spaces like the interval [0,1] (by, essentially, not allowing us to add probabilities up in every case. More on this later). But if we just suppose a countable dartboard, we're left with the same problem.

But, hang on, does a dartboard really have an infinite number of points? And what does it mean for the dart to hit a point? Certainly the point of the dart _itself_ has a non-infinitesimal area, so that if we partition the dartboard into "points" of that size, there will be a finite number, each of which we can assign some positive probability without issue. In general, any area on the dartboard, no matter how small, has a positive probability of being struck by the dart, and this remains true no matter how small we make the dart, as well.

In other words, this paradox hinges on an infinitesimal dart, as well as an infinitesimally accurate measuring tape with which to measure the coordinates, which presumably has infinitely long numerals on it (presumably the person assigning these credences has an infinitely large brain as well, to be able to hold in their minds the coordinates pointing out a specific point on the dartboard to infinite precision).

Any mathematician knows that you have to handle a situation like this _very carefully_. You cannot expect to apply your intuition directly and get consistent results. I am actually kind of surprised to find serious philosophers making this kind of argument in 2020. This is on par with saying that Zeno's paradox proves you can't use numbers to measure space (or time).

To be fair to Maudlin here, I do think he has a point: there do seem to be situations where you should be indifferent between a (countably) infinite number of outcomes, and probability theory won't let you do that. Dartboards is just a bad example.

Maudlin's shaky relationship with infinity shows up again in his discussion of the classical Kolmogorov approach to probability. He cites the definition of a _finitely additive_ probability measure, and not the \sigma -additive (or countably additive) version used by every probability theorist and statistician in the world. Then he criticizes it for only being finitely additive!

He also critizises the use of a \sigma -algebra, arguing that it seems irrational to exclude certain sets from consideration. I think Maudlin needs to spend a small amount of time understanding non-measurable sets before arguing they're stupid. For one thing, a nonmeasurable subset of \mathbb {R}^2 (such as we might ask "will the dart hit a point in this subset or not" of), is _extremely weird_. It's consistent with ZF minus Choice that there exist no such sets. Certainly such sets always contain regions of "infinite complexity", in the sense that there are (non-infinitesimal) regions of the dartboard so that, if the dart hits there, no finite amount of measurement will suffice to determine whether it is in the set or out of it. Indeed, it's not too far off the mark to think of the \sigma -algebra as denoting the "determinable" events.

The fact that a probability measure is only _countably_ additive is still a bit weird. You can make philosophical arguments for this choice: To check whether a \in \cup _n A_n, we can check whether a \in A_n for each a - and this is guaranteed to give you an answer in a finite amount of time, but only if you're taking a finite union. There are reasonable counterarguments to this, but Maudlin doesn't engage with this topic at all - it really seems as if he doesn't realize that mathematicians have been taking infinite disjunctions of events all this time[^fn:1].

Maudlin's proposed replacement for numbers is essentially the following idea: it's not required that a person's credences be represented by some definite "thing" - a number or another mathematical object. Instead we should ask about the _structure_ on the system of credences, and what rules this should follow for any "rational" person.

This is actually a very good idea! Structuralism! Defining things extrinsically rather than intrinsically! I like this approach.

Let's go over the basic principles that Maudlin comes up with. The paper is short on rigor, but since it's a philosophy paper and not a math paper, I won't hold that against it.

- The set of "credences" should have a partial order[^fn:2]. - The map \operatorname {Cr} which assigns an event its credence should be order-preserving - in other words, if event P entails the event Q, then \operatorname {Cr}(P) \leq \operatorname {Cr}(Q). - There should be a least credence \bot and a greatest credence \top , corresponding respectively to the credence in an event which is certain not to happen or certain to happen (say, a contradiction and a tautology). - The map \mathrm {Cr} should have the following property: if P \Rightarrow Q, \operatorname {Cr}(P) = \operatorname {Cr}(Q),

then \operatorname {Cr}(Q \wedge \neg P) = \bot . This is the so-called "Euclidean property" that Maudlin cannot stop going on about: "the whole is greater than the part", or in this case, if P entails Q, and it's possible that Q but not P (i.e. Q does not entail P), then our credence in Q must be _strictly greater_ than our credence in P.

We can define a partial addition operation on credences, denoted \oplus in the paper, by setting \operatorname {Cr}(P) \oplus \operatorname {Cr}(Q) = \operatorname {Cr}(P \vee Q) whenever \Cr (P \wedge Q) = \bot . It's not exactly clear in the paper whether we extend this using the equality on credences - for example, whether we can assign a value to \operatorname {Cr}(P) \oplus \operatorname {Cr}(P) by finding some other event Q so that \operatorname {Cr}(P) = \operatorname {Cr}(Q), \operatorname {Cr}(P \wedge Q) = \bot , and setting \operatorname {Cr}(P) \oplus \operatorname {Cr}(P) = \operatorname {Cr}(P \vee Q). This seems a harmless enough extension (if we really regard credences as _equal_, and not just equivalent in some sense, it seems inevitable), but Maudlin doesn't really spell it out. Nevertheless the addition is still _partial_, since, for example, when rolling a fair dice, we can't add our credence in the event "The result will be one of 1,2,3,4" to itself, since there is no mutually exclusive event which is equally likely. This is not really a bug - indeed, while you can add probabilities and get a number greater than 1, you can't really treat that number _as a probability_.

So far, Maudlin's approach differs from the Kolmogorovian approach chiefly in that he lets the poset of credences be any ordered set, not just numbers, and that he doesn't require the order on credences to be _total_ - we can have no opinion about the relative plausibility of two events without declaring them _equally_ plausible. This is reasonable enough. Probably the weirdest thing here is that the "Euclidean property" is _also satisfied by Kolmogorov probability!_ The reason that adding a point to a set doesn't change the probability that the dart will land there is simply that it's impossible that the dart will land at that precise point. This is obviously counterintuitive (and there are ways of defending it, and counterarguments to those, and so on - but Maudlin does not engage with this at all), but that is the classical approach. It does not seem to violate the Euclidean property at all.

Having the addition in hand, we can actually reformulate the Euclidean property simply as "\oplus is cancellative", i.e if C \oplus D= C \oplus D' then D = D'. This means that \operatorname {Cr}(Q) = \operatorname {Cr}(P) \oplus \operatorname {Cr}(Q \wedge \neg P), and if this equals \operatorname {Cr}(Q) then \operatorname {Cr}(Q \wedge \neg P) = \bot , since \bot is the unit of addition.

Now Maudlin wants to talk about _relations_ between credences. First of all I have to go on a rant about his whole discussion about ratios. Mathematics has come a long way since Euclid, and citing him like you actually believe his axioms provide a sound formal basis of geometry is not gonna convince anyone that you know what you're talking about. Maudlin goes oddly pedantic and says that the ratio of a circle's circumference to its diameter is not \pi , but rather, the ratio of the number \pi to the number 1, and this is in general what people mean when they identify a ratio with a number. I'm glad to hear that Maudlin has found a solution to the long-standing problem of the identity of mathematical objects (presumably resolving it in favor of a _very_ strong form of Platonism?), but I couldn't find this in the bibliography of the paper or any of his other work. Until I do, I have to insist that "loose talk" is when you insist that two mathematical objects identified under isomorphism cannot possibly be called equal, not the other way around.

Okay, rant done. We can define the ratios between credences by addition. For example, we can say that credence C is twice as big as credence D if C = D \oplus D. Maudlin, quoting Euclid: "Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another". There is however a serious problem with this: the partiality of addition means that you can't _compare_ these ratios, and many pairs of credences have no ratio to one another, even though Maudlin insists that they do.

Let's pull back here a bit. Suppose I have an ordered (commutative) monoid (M,\leq ,+). Then I can say that m,m' have a ratio to each other if there exist n,n' \in \mathbb {N} so that n.m \geq m', and n'.n' \geq m. If this is true, we can actually obtain a unique real-numbered ratio "m/m'" as the supremum of all n/n' \in \mathbb {Q} so that n.m' \leq n'.m.[^fn:3] Here I'm using n.m to denote the n-fold sum m + m + \cdots + m.

Now, this _doesn't_ work for credences - even in situations where it feels like it should[^fn:4]. The increasingly accurate rational approximations to the "true" real ratio requires longer and longer chains of addition, but this is generally impossible to define for credences. We can define what it means for C/C' to be an integer - namely that n.C' = C. We can also _sometimes_ define what it means for C/C' to be a rational number - for example if 2.C = 3.C', we can say C/C' = 2/3. But if e.g. C = \top , there's no way to make this work. And, crucially, we can't usually compare ratios with each other either - otherwise, we might have argued that expecting number representations for each ratio is too much.

As it stands, by Maudlin's (or rather, Euclid's) definition of ratio, there is no ratio between \top and any event with "probability greater than 1/2" (speaking informally - of course credences aren't probabilities and so on), in contrast with how Maudlin uses it.

This is an area where the paper could really have benefited from some rigor. It's not clear whether Maudlin means to tack on "ratios" as a primitive notion - so that, in addition to having a "more likely than" relation on credences, we have ratios between certain pairs of credences, which can be compared.

All these issues could be solved _very easily_, by imposing some extra structure on the model. For instance we could add "formal sums" of credences which don't have a sum, and say that the statement 2.P \geq Q makes sense even when 2.P is not defined as a credence.

As I mentioned above, I actually, genuinely think it's a good idea to think about replacements for classical probability theory. In some sense my own work on Markov categories is a version of this, a sort of "synthetic probability theory". Much like Maudlin, Markov categories start by asking "what structure do we actually need to do the job of probabilities?". They go in a very different direction, not least because we actually care about the job that probabilities do for people in the real world, but the spirit is actually similar.

But stress-testing your theory by applying it to situations involving infinity - without working rigorously with infinity _at all_ - is bound to get you into trouble. Frankly, if the only problem your theory solves is that it makes sense of infinite conjunctions, it's basically useless. Infinite conjunctions only come up in abstrac mathematical models, and trust me, those guys aren't really looking for a new theory. In the real world, it's very hard to write down infinitely many statements and think about your credence in the statement "one of these is true".

The mathematical idea in this paper - which, again, I actually don't think is a stupid idea - could have fit in a couple of pagers, including a fair amount of exposition and the easy fixes to the problem mentioned above. The philosophical work here seems mainly to be a takedown of traditional probability, which falls woefully flat.

[^fn:1]: A less elegant reason for asking for countable composition but not general infinitary composition is that it leads to a useful, theoretically convenient theory, but still allows for measures like the uniform measure on the interval. This is just an argument from practicality - we take these axioms instead of some others because they make the calculations work out - so I won't hold it against Maudlin if he doesn't take this argument very seriously [^fn:2]: Maudlin seems to not know this term, which probably could have cut several pages from his exposition [^fn:3]: Ironically, in the usual construction of real numbers, they are limits of ratios (between integers), so one could make a convincing argument that a real number really is literally a ratio. [^fn:4]: Credences can be "infintesimal, in the sense that we might have n.C \leq C' for ALL n. That's not what I'm talking about."