I’d like to see more counterarguments to the thing about mathematicians being much less useful for ground-breaking work after their 20s that don’t rely on extreme outliers like Witten, Andrew Wiles or Paul Erdös.
That would be difficult, since “groundbreaking work” automatically implies “extreme outlier”.
In fact, I would expect that typical mathematicians are much more useful above 30 than below—to a greater extent than is the case for the extreme outliers.
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? [...]
...and after posting that comment, I remembered that I had made an earlier post citing studies that said that it’s the middle-aged and not young scientists who are the most productive, which is in conflict with the results I just quoted. I feel silly now. I guess I should re-read the studies that I referenced three years ago to figure out what version is correct.
I guess I should re-read the studies that I referenced three years ago to figure out what version is correct.
Just to make the obvious point, your earlier post seems to draw on citations using mostly post-60s and later data, while that 1988 paper uses many citations from the 60s or earlier.
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).
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.
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.
Actually, one of the first things that new researchers have to learn is that just thinking about a problem and coming up with ideas will get you nowhere—you have to actually get your hands dirty and try things out to make progress.
Oh, definitely. I don’t mean to imply that, say, Warhol never got his hands dirty- but that Rembrandt’s skill was in the realm of dirty hands and that Warhol’s skill was in the realm of insight.
(I know in my research the act of sitting down and writing out an idea or sitting down and coding an algorithm or sitting down and going through the math has been indispensable, and strongly recommend it to anyone else.)
I’d like to see more counterarguments to the thing about mathematicians being much less useful for ground-breaking work after their 20s that don’t rely on extreme outliers like Witten, Andrew Wiles or Paul Erdös.
That would be difficult, since “groundbreaking work” automatically implies “extreme outlier”.
In fact, I would expect that typical mathematicians are much more useful above 30 than below—to a greater extent than is the case for the extreme outliers.
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:
...and after posting that comment, I remembered that I had made an earlier post citing studies that said that it’s the middle-aged and not young scientists who are the most productive, which is in conflict with the results I just quoted. I feel silly now. I guess I should re-read the studies that I referenced three years ago to figure out what version is correct.
Just to make the obvious point, your earlier post seems to draw on citations using mostly post-60s and later data, while that 1988 paper uses many citations from the 60s or earlier.
If old and young mathematicians have different strengths and weaknesses maybe it’s best to have a few of both.
(part 2)
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.
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.
Actually, one of the first things that new researchers have to learn is that just thinking about a problem and coming up with ideas will get you nowhere—you have to actually get your hands dirty and try things out to make progress.
Oh, definitely. I don’t mean to imply that, say, Warhol never got his hands dirty- but that Rembrandt’s skill was in the realm of dirty hands and that Warhol’s skill was in the realm of insight.
(I know in my research the act of sitting down and writing out an idea or sitting down and coding an algorithm or sitting down and going through the math has been indispensable, and strongly recommend it to anyone else.)