Arithmetic, enumerative induction and size bias

Number theory abounds with conjectures asserting that every natural number has some arithmetic property. An example is Goldbach’s Conjecture, which states that every even number greater than 2 is the sum of two primes. Enumerative inductive evidence for such conjectures usually consists of small cases. In the absence of supporting reasons, mathematicians mistrust such evidence for arithmetical generalisations, more so than most other forms of non-deductive evidence. Some philosophers have also expressed scepticism about the value of enumerative inductive evidence in arithmetic. But why? Perhaps the best argument is that known instances of an arithmetical conjecture are almost always small: they appear at the start of the natural number sequence. Evidence of this kind consequently suffers from size bias. My essay shows that this sort of scepticism comes in many different flavours, raises some challenges for them all, and explores their respective responses.

Body size, sampling completeness, and extinction risk in the marine fossil record

Larger body size has long been assumed to correlate with greater risk of extinction, helping to shape body-size distributions across the tree of life, but a lack of comprehensive size data for fossil taxa has left this hypothesis untested for most higher taxa across the vast majority of evolutionary time. Here we assess the relationship between body size and extinction using a data set comprising the body sizes, stratigraphic ranges, and occurrence patterns of 9408 genera of fossil marine animals spanning eight Linnaean classes across the past 485 Myr. We find that preferential extinction of smaller-bodied genera within classes is substantially more common than expected due to chance and that there is little evidence for preferential extinction of larger-bodied genera. Using a capture–mark–recapture analysis, we find that this size bias of extinction persists even after accounting for a pervasive bias against the sampling of smaller-bodied genera within classes. The size bias in extinction also persists after including geographic range as an additional predictor of extinction, indicating that correlation between body size and geographic range does not provide a simple explanation for the association between size and extinction. Regardless of the underlying causes, the preferential extinction of smaller-bodied genera across many higher taxa and most of geologic time indicates that the selective loss of large-bodied animals is the exception, rather than the rule, in the evolution of marine animals.

Reducing module size bias of participation coefficient

Both natural and engineered networks are often modular. Whether a network node interacts with only nodes from its own module or nodes from multiple modules provides insight into its functional role. The participation coeffcient (PC) is typically used to measure this attribute although its value also depends on the size of the module it belongs to, often leading to non-intuitive identification of highly connected nodes. Here, we develop a normalized PC that overcomes the module size bias associated with the conventional PC. Using brain, C.elegans, airport and simulated networks, we show that our measure of participation alleviates the module size bias, while preserving conceptual and mathematical properties, of the classic formulation of PC. Unlike the conventional PC, we identify London and New York as high participators in the air traffic network and demonstrate stronger associations with working memory in human brain networks, yielding new insights into nodal participation across network modules.