Where does convergence collide?

A sequence converges when its terms come and stay arbitrarily close to a fixed value, its limit, as the index grows. In Rudin's Principles of Mathematical Analysis the definition is exact: a sequence converges to a point when, for every distance chosen in advance, all terms past some index fall within that distance of the point. The referent is a sequence of numbers or points in a metric space, and the unit is whether that formal condition holds, not how fast, not how alike, and not who caught up to whom. Analysis further splits the word into pointwise, uniform, and absolute convergence, each a distinct condition on the same term. This is convergence as a proved property of an infinite list, and it shares only the image of things drawing together with the other meanings.[1]

Unit

A sequence or series in a metric space

Recovery means

Not applicable

Measured by

Whether the epsilon-N limit condition is satisfied, proved rather than estimated

Convergent evolution is the independent arrival of similar traits in lineages that did not inherit them from a shared ancestor, as when wings appear separately in birds, bats, and insects. Losos, reviewing the idea in 2011, treats convergence as evidence that selection can push different starting points toward the same solution, while cautioning that similarity can also come from shared constraint rather than shared adaptation. The referent is two or more separate lineages and the trait they came to share, and the unit is the repeated trait, not a limit and not an economy. Nothing here approaches a fixed value; the resemblance is between organisms, and it is read backward from the tree of life rather than proved forward. This meaning and the mathematical one share the word and almost nothing else.[2]

Unit

Independent lineages and a shared trait

Recovery means

Not applicable

Measured by

Phylogenetic and morphological comparison showing a trait arose more than once

In growth economics, convergence is the tendency for poorer economies to grow faster than richer ones and so close the income gap over time. Barro and Sala-i-Martin, in their 1992 study of US states and countries, distinguished beta convergence, where low starting income predicts faster growth, from sigma convergence, where the spread of incomes across economies narrows. The referent is a set of economies and their income per person, and the unit is the rate of catch-up, conditional on saving and technology. Here things do draw together, like a sequence toward a limit, but the drawing together is an empirical claim about regions that may or may not hold, not a proved property, and it says nothing about traits or media. Whether convergence occurs is contested and often only conditional.[3]

Unit

Economies and income per person over time

Recovery means

Not applicable

Measured by

Regressions of growth on initial income (beta) and the cross-economy dispersion of income (sigma)

Media convergence is the flow of content across once-separate media, industries, and devices, together with the audience behaviour that follows it from one platform to the next. Jenkins, in Convergence Culture (2006), argues it is less a technical endpoint than a cultural shift, a change in how content is produced and consumed as old and new media collide. The referent is media forms, companies, and audiences, and the unit is the merging of previously distinct channels, not a numerical limit, not a trait, and not an income gap. The word again names things coming together, but the things are platforms and practices, and the claim is interpretive rather than proved or estimated. A poor economy catching up and a phone that is also a camera and a newspaper are different constructs under one word.[4]

Unit

Media forms, industries, and audiences

Recovery means

Not applicable

Measured by

Case study and cultural analysis of content and audiences moving across platforms