Simonton (1988) Age and Outstanding Achievement: What Do We Know After a Century of Research? Psychological Bulletin, Vol. 104, No. 2, 251-267.

Short version: the productivity for mathematicians seems to peak around late 20s or early 30s, with the productivity after the peak falling to less than one-quarter the maximum. However, the average quality of a contribution does not seem to vary with age, and exceptional researchers (in any field) tend to remain unusually profilic, as compared to an average researcher of the same age, even after passing their peaks.

Long version:

In the first place, the location of the peak, as well as the magnitude of the postpeak decline, tends to vary depending on the domain of creative achievement. At one extreme, some fields are characterized by relatively early peaks, usually around the early 30s or even late 20s in chronological units, with somewhat steep descents thereafter, so that the output rate becomes less than one-quarter the maximum. This agewise pattern apparently holds for such endeavors as lyric poetry, pure mathematics, and theoretical physics, for example (Adams, 1946; Dennis, 1966; Lehman, 1953a; Moulin, 1955; Roe, 1972b; Simonton, 1975a; Van Heeringen & Dijkwel, 1987). At the contrary extreme, the typical trends in other endeavors may display a leisurely rise to a comparatively late peak, in the late 40s or even 50s chronologically, with a minimal if not largely absent drop-off afterward. This more elongated curve holds for such domains as novel writing, history, philosophy, medicine, and general scholarship, for instance (Adams, 1946; Richard A. Davis, 1987; Dennis, 1966; Lehman, 1953a; Simonton, 1975a). Of course, many disciplines exhibit age curves somewhat between these two outer limits, with a maximum output rate around chronological age 40 and a notable yet moderate decline thereafter (see, e.g., Fulton & Trow, 1974; Hermann, 1988; Mc- Dowell, 1982; Zhao & Jiang, 1986). Output in the last years appears at about half the rate observed in the peak years. Productive contributions in psychology, as an example, tend to adopt this temporal pattern (Homer et al., 1986; Lehman, 1953b; Over, 1982a, 1982b; Zusne, 1976).

It must be stressed that these interdisciplinary contrasts do not appear to be arbitrary but instead have been shown to be invariant across different cultures and distinct historical periods (Lehman, 1962). As a case in point, the gap between the expected peaks for poets and prose authors has been found in every major literary tradition throughout the world and for both living and dead languages (Simonton, 1975a). Indeed, because an earlier productive optimum means that a writer can die younger without loss to his or her ultimate reputation, poets exhibit a life expectancy, across the globe and through history, about a half dozen years less than prose writers do (Simonton, 1975a). This cross-cultural and transhistorical invariance strongly suggests that the age curves reflect underlying psychological universals rather than arbitrary sociocultural determinants. In other words, the age functions for productivity may result from intrinsic information-processing requirements rather than extrinsic pressures due to age stereotypes about older contributors, a point that we shall return to in the theoretical section (see also Bayer & Dutton, 1977).

[...]

Generally, the top 10% of the most prolific elite can be credited with around 50% of all contributions, whereas the bottom 50% of the least productive workers can claim only 15% of the total work, and the most productive contributor is usually about 100 times more prolific than the least (Dennis, 1954b, 1955; also see Lotka, 1926; Price, 1963, chap. 2). Now from a purely logical perspective, there are three distinct ways of achieving an impressive lifetime output that enables a creator to dominate an artistic or scientific enterprise. First, the individual may exhibit exceptional precocity, beginning contributions at an uncommonly early age. Second, the individual may attain a notable lifetime total by producing until quite late in life, and thereby display productive longevity. Third, the individual may boast phenomenal output rates throughout a career, without regard to the career's onset and termination. These three components are mathematically distinct and so may have almost any arbitrary correlation whatsoever with each other, whether positive, negative, or zero, without altering their respective contributions to total productivity. In precise terms, it is clear that O = R(L - P), where O is lifetime output, R is the mean rate of output throughout the career, L is the age at which the career ended (longevity), and P is the age at which the career began (precocity). The correlations among these three variables may adopt a wide range of arbitrary values without violating this identity. For example, the difference L - P, which defines the length of a career, may be more or less constant, mandating that lifetime output results largely from the average output rate R, given that those who begin earlier, end earlier, and those who begin later, end later. Or output rates may be more or less constant, forcing the final score to be a function solely of precocity and longevity, either singly or in conjunction. In short, R, L, and P, or output rate, longevity, and precocity, comprise largely orthogonal components of O, the gauge of total contributions.

When we turn to actual empirical data, we can observe two points. First, as might be expected, precocity, longevity, and output rate are each strongly associated with final lifetime output, that is, those who generate the most contributions at the end of a career also tend to have begun their careers at earlier ages, ended their careers at later ages, and produced at extraordinary rates throughout their careers (e.g., Albert, 1975; Blackburn et al., 1978; Bloom, 1963; Clemente, 1973; S. Cole, 1979; Richard A. Davis, 1987; Dennis, 1954a, 1954b; Helson & Crutchfield, 1970; Lehman, 1953a; Over, 1982a, 1982b; Raskin, 1936; Roe, 1965, 1972a, 1972b; Segal, Busse, & Mansfield, 1980; R. J. Simon, 1974; Simonton, 1977c; Zhao & Jiang, 1986). Second, these three components are conspicuously linked with each other: Those who are precocious also tend to display longevity, and both precocity and longevity are positively associated with high output rates per age unit (Blackburn et al., 1978; Dennis, 1954a, 1954b, 1956b; Horner et al., 1986; Lehman, 1953a, 1958; Lyons, 1968; Roe, 1952; Simonton, 1977c; Zuckerman, 1977). [...]

While specifying the associations among the three components of lifetime output, we have seemingly neglected the expected peak productive age. Those creators who make the most contributions tend to start early, end late, and produce at above average rates, but are the anticipated career peaks unchanged, earlier, or later in comparison to what is seen for their less prolific colleagues? [...]

If old and young mathematicians have different strengths and weaknesses maybe it's best to have a few of both.

(part 2)

if one calculates the age curves separately for major and minor works within careers, the resulting functions are basically identical. Both follow the same second-order polynomial (as seen in Equation 1), with roughly equal parameters. Second, if the overall age trend is removed from the within-career tabulations of both quantity and quality, minor and major contributions still fluctuate together. Those periods in a creator's life that see the most masterpieces also witness the greatest number of easily forgotten productions, on the average. Another way of saying the same thing is to note that the "quality ratio," or the proportion of major products to total output per age unit, tends to fluctuate randomly over the course of any career. The quality ratio neither increases nor decreases with age nor does it assume some curvilinear form. These outcomes are valid for both artistic (e.g., Simonton, 1977a) and scientific (e.g., Simonton, 1985b) modes of creative contribution (see also Alpaugh, Renner,& Birren, 1976, p. 28). What these two results signify is that if we select the contribution rather than the age period as the unit of analysis, then age becomes irrelevant to determining the success of a particular contribution. For instance, the number of citations received by a single scientific article is not contingent upon the age of the researcher (Oromaner, 1977).

The longitudinal linkage between quantity and quality can be subsumed under the more general "constant-probability-ofsuccess model" of creative output (Simonton, 1977a, 1984b, 1985b, 1988b, chap. 4). According to this hypothesis, creativity is a probabilistic consequence of productivity, a relationship that holds both within and across careers. Within single careers, the count of major works per age period will be a positive function of total works generated each period, yielding a quality ratio that exhibits no systematic developmental trends. And across careers, those individual creators who are the most productive will also tend, on the average, to be the most creative: Individual variation in quantity is positively associated with variation in quality. There is abundant evidence for the application of the constant-probability-of-success model to cross-sectional contrasts in quantity and quality of output (Richard A. Davis, 1987; Simonton, 1984b, chap. 6; 1985b, 1988b, chap. 4). In the sciences, for example, the reputation of a nineteenthcentury scientist in the twentieth century, as judged by entries in standard reference works, is positively correlated with the total number of publications that can be claimed (Dennis, 1954a; Simonton, 1981 a; see also Dennis, 1954c). Similarly, the number of citations a scientist receives, which is a key indicator of achievement, is a positive function of total publications (Crandall, 1978; Richard A. Davis, 1987; Myers, 1970; Rushton, 1984), and total productivity even correlates positively with the citations earned by a scientist's three best publications (J. R. Cole & S. Cole, 1973, chap. 4). [...]

The constant-probability-of-success model has an important implication for helping us understand the relation between total lifetime output and the location of the peak age for creative achievement within a single career (Simonton, 1987a, 1988b, chap. 4). Because total lifetime output is positively related to total creative contributions and hence to ultimate eminence, and given that a creator's most distinguished work will appear in those career periods when productivity is highest, the peak age for creative impact should not vary as a function of either the success of the particular contribution or the final fame of the creator. Considerable empirical evidence indeed demonstrates the stability of the career peak (Simonton, 1987a). In the sciences, for instance, the correlation between the eminence of psychologists and the age at which they contribute their most influential work is almost exactly zero (Zusne, 1976; see also Lehman, 1966b; of. Homer et al., 1986). And in the arts, such as literary and musical creativity, the age at which a masterpiece is generated is largely independent of the magnitude of the achievement (Simonton, 1975a, 1977a, 1977c). Thus, even though an impressive lifetime output of works, and subsequent distinction, is tied to precocity, longevity, and production rate, the expected age optimum for quantity and quality of contribution is dependent solely on the particular form of creative expression (also see Raskin, 1936).

Galenson's book on artists fascinated me: he identified two clusters, experimental artists who liked to sketch and rework things and whose quality increased with age, and conceptual artists, who liked doing preparatory work and outsourcing the actual production, who made massive contributions when young but whose productivity rapidly tapered off.

With art, there's room for both types, but I imagine that math and related fields are heavily biased towards the conceptual style, especially the theoretical components of those fields.

Depends on the surface area of unbroken ground. I understand there are quite a few marginal areas in mathematics where you can come up with novel approaches that will be quite impressive to the other five people working on that specific sub-sub-sub-area, but not necessarily that much to mathematics at large. Also, contemporary mathematicians whose names are actually recognizable by popular science literate non-mathematicians are a very small group even compared to the sort of top researchers who are working with the sort stuff the apocryphal wisdom about needing to be in your 20s seems to apply to.

Though I'd also like to see more arguments about how above 30 mathematicians can do all sorts of useful stuff when you don't get fixated on paradigm-upending world-class results, and what sort of stuff this is.