Is there an implicit age limit for mathematical productivity?

[nightdove]

Some mathematicians note that their intellectual powers (at least where mathematics is concerned) seem to diminish with age, for instance Hardy. Was this griping a mere excuse for their lack of talent to begin with? Other prodigies appeared to have retained their mathematical fecundity into old age - Euler, Gauss and Newton, while others only bloomed in their fourties, for example, Weierstrass.

Does anyone here think that age-related declines in fluid intelligence, or 'g', affect mathematicians more than experts in other disciplines? More importantly, can a high level of 'g' (above the level needed for scientific productivity) be maintained into old age? Can experience really compensate for deficits in computational power and clarity of mind?

 

[andytoh]

That's an interesting question. Knowledge should improve with age, but forgetfulness may increase with age as well. In terms of problem solving, I think that peaks at a certain age and then declines after that. The question is what age is this peak?

 

[Werg22]

The human brain capacity is infinite. I believe as long as it stays active it won't lose it's capacities.

 

[arildno]

Weierstrass did NOT bloom in his forties; most of his seminal work on function theory was developed in the years when he was a lowly teacher much earlier.

He even had the lack of foresight to present this material to the parents of prospective students in order to attract pupils to the school (none of them could understand any of it).

 

[mathman]

Erdos was productive until he died (age 83). He is considered one of the best mathematicians of the 20th century.

 

[Gib Z]

Probably different for each person, and perhaps even each field of mathematics.

Though I think its easily decided Erdos was the better mathematician between him and Hardy.

 

[J77]

Mathematical productivity?

Surely, that's simply inversely related to the amount of teaching one has to do!

It's completely independent of age :tongue:

 

[andytoh]

Mathematical productivity?

Surely, that's simply inversely related to the amount of teaching one has to do!

It's completely independent of age :tongue:

OP is referring not to amount of publications overall, but how productive the mind is during the research periods.

 

[complexPHILOSOPHY]

While I regard psychometrics as utterly useless in some of it’s applications, there is some empirical support for general intelligence, (g factor). While fluid intelligence is assumed to be responsible for ‘general intelligence,’ such as learning, critical thinking, analysis and pattern-recognition, crystalized intelligence is assumed to be responsible for the allocation, recollection and application of acquired knowledge, associations, perceptions and experiences. I know that some researchers posit a peak in fluid intelligence during early-to-middle adulthood and a gradual decline shortly after. Crystalized intelligence is assumed to remain relatively stable over one’s adult life and only begins to gradually decrease around the age of 60.

While there is some strong correlation between aspects of the physical structure of the brain and general intelligence, the problem is still very open. It is postulated (through the extrapolation of data from research), that fluid intelligence involves the dorsolateral prefrontal cortex, the anterior cingulate cortex and other systems related to short-term memory and attention. Crystallized intelligence is assumed to have a strong correlation with brain regions that involve the storage and usage of long-term memories, such as the hippocampus. However, there is no general consensus amont cognitivsts or neuroscientists, regarding intelligence. In fact, there is no consensus on what intelligence is (for instance, a tribe in the amazon would not value 'mathematical intelligence' and an ability to do riemannian geometry in your head but would value quite highly, an intelligence associated with survival or harvesting of food). Without a general consensus regarding intelligence, it is hard to discover the causal pathways between electrobiochemical reactions/physical structure and intelligence. I do believe that the g factor has strong empirical support, it is only through specific presupposed axioms or postulates about intelligence, that it can be derived. Without strong empirical support for physical causal relationships with intelligence or even some concenus on what intelligence is, I remain skeptical.

Tests can derive heritability between specific aspects of general intelligence and physical structures of the brain, (of which there is documented research for) as well as a correlation between g factor and rats. While there is a positive correlation between g factor and academic/career performance, the causal pathways are largely unknown. Research supporting strong correlates includes, overall brain mass, glucose metabolization rate within the brain, and the mass of the prefrontal lobe. There is conflicting evidence regarding the correlation between g and peripheral nerve conduction velocity (which is calculated by measuring the distance between electrodes and the time it takes for electrical impulses to travel between electrodes), with some reports of significant positive correlations, and others of no or even negative correlations. The fact that there is conflicting data and research is not a surprise (it is a necessary aspect of research), however, it does suggest a lack of unification between the g factor and intelligence (or perhaps the consensus among researchers).

While I am certain and aware that documented cases exist regarding a correlation between a decrease in academic/career performance and the g factor, I still feel as though it is largely inconclusive.

I would approach psychometrics with a weary perspective and be certain that you have looked through some of the research pertaining to it. If you read through the research and determine that you still embrace psychometric testing, than all is well. I, however, I yet to be convinced.

So my answer is that, no, I do not think I can conclusively support your hypothesis.

 

[nightdove]

It wasn't MY hypothesis, but rather an intriguing hypothesis worth testing, given the amount of anecdotal evidence, sometimes personally offered by scientists and mathematicians, in it's favour. I would in fact prefer if the hypothesis weren't true. After all, the psychologists who support it have also accumulated a mass of evidence stating that fluid IQ peaks in the neighbourhood of 24 years, which is precisely my age, which could only mean that my intelligence will only go downhill from henceforth. Moreover, I find it difficult to refute that substantial physical deterioration occur in response to aging independent of any environmental causes. Unfortunately, the human intelligence and all its attributes are housed in a lump of flesh (the brain) whose structural integrity varies with time, just like the rest of the human body.

While you often notice some senior athletes who are arguably fitter (in their chosen sport) than many ordinary younger blokes, you cannot argue the fact that compared against their younger peers, these senior athletes tend to be less physically capable. As with the body, with the brain.

It is precisely the interaction of the dorsolateral PFC and anterior cingulate that are so important for mathematicians. My own hypothesis is that the anterior cingulate is responsible for the maintenance of large objects in working memory, while the dorsolateral PFC plays a particularly important role in the manipulation of the aforementioned information. It is precisely these structures that decline the quickest and are the most vulnerable to damage, whereas the temporal lobes (the knowledge store) is relatively unaffected. This makes sense, the more advanced the age, the greater the number of opportunities to acquire knowledge (and build wisdom), but the less the capacity to perform the de-novo computations mediated by the PFC and anterior cingulate needed to be an effective researcher.

I am wondering if some of the age-related declines in the structural integrity of the frontal lobes are reversible. I know it seems premature for a 24 year old to think about this.

 

[mathwonk]

I was most productive when I worked about 14-18 hours a day, which coincidentally was also when i was young enough to manage this level of exertion. Basically all i did was work, exercise, eat a little and sleep a little.

Mathematical work takes a lot of effort and time. So it can diminish if you were stealing time from activities that you become less inclined to steal from, like family.

also as you get older you may get health problems. But there are older mathematicians who are just smart, and well organized, and work in a controlled intelligent way which they can sustain.

There is also a psychological side. Some people maintain a youthful level of enthusiasm and curiosity about math into very old age. But there are certainly factors making it harder to produce as much in old age in any field.

I am unqualified to compare different fields, but i know there are definitely mathematicians who find some other fields significantly easier than mathematics. Some of my acquaintances have gone into mathematical physics e.g., as a way of easing their workload, or computer science.

others have changed fields simply because their interests changed. maybe it is not a matter of those fields being easier, but merely being different, hence freshly interesting.

there are many fields in which a knowledge of mathematics is a huge advantage, and one can work there, especially with collaborators, and trade for a long time on ones residual knowledge of mathematics.

This would not work for me as those fields are to me harder than math and i am totally ignorant of them. I on the other hand am enjoying pontificating here about math to young people. Obviously while here I am not proving any theorems.

 

[mathwonk]

Practically however, theories about age related math work should not be used to discourage people from doing math because they are "too old". This is nonsense.

If one enjoys math one should pursue it. There is nothing we can do to make ourselves younger, but much we can do to make ourselves more active and maybe more curious. certainly more knowledgeable.

even if i might have achieved more had i begun research activity at 22 instead of 35, still i am having fun at it.

 

[mathwonk]

also people like mumford and grothendieck, bequeathed projects just before leaving algebraic geometry, that have kept people busy afterwards for decades.

 

[complexPHILOSOPHY]

Is 15-18 hours necessary for undergraduate studies or was this for graduate school (or was it just for your own independent gain)? That number seems ridiculous. You get my respect if you have the concentration and endurance to sustain a study session for that many hours in a day (regardless of whether it is simultaneous).

 

[neurocomp2003]

i would assume productivity of the brain is correlated to how well you maintain your lifestyle to keep your brain sharp.

 

[complexPHILOSOPHY]

i would assume productivity of the brain is correlated to how well you maintain your lifestyle to keep your brain sharp.

What lifestyle would you say is condusive to maintaining a 'sharp brain'? I won't divulge my lifestlye on here because it might violate PF rules but I have a pretty crazy lifestyle and still maintain a solid GPA.

 

[mathwonk]

certainly not undergraduate school, where i seldom even went to class. but after decades of lazy and wild goofing off, i had to work extremely hard to try to catch up, and i really never did.

you can't believe those old stories about walking through the snow to school anyway.

i will tell one true story about my work life in the early 1980's. I commuted 130 miles round trip to work but wanted to be home every night to look in on my family. one night i arrived home after a very long work day only at 5am, slept 45 minutes, got up and drove back to work another entire day.

of course this is insane, but how else does one come from a temporary instructor at a state college with no phd, and a meat lugger, to a PhD postdoc at harvard in only 5 years?

this gives you an idea of how hard it is to excel some of these people out there, and many of them are also smart. or it may also suggest how much one can accomplish with will power. any good grad student or profesional can maintain a 4.0 gpa whenever they want to, its all about dedication. they are on another level entirely.

when i went back to gradschool at 32, i not only had a 4.0 every year for 3 years, but only missed solving one homework problem in those 3 years. One of my proofs in algebraic topology was used the next year by the professor who had solved it incorrectly himself. when i spoke in the faculty seminar in several complex variables, i resented kodaira's original proof of the vanishing theorem, a result that none of the faculty knew, and apparently none were willing or able to read.

It took me 5 days to read, learn, and prepare over 60 pages of sheaf theory, harmonic analysis, topology and metric geometry. But this did not mean I was anywhere near having a thesis.

gpa measures very little about a profesional. it means you can learn, ok but what can you DO?

 

[mathwonk]

heres a more famous example. hyman bass is an outstanding mathematician, a professor at columbia who collaborated with j.p.serre, and especially once on solving a famous problem. Hyman said they worked all day, he explained what he could do and Serre said he could do the rest, and they explained their work to each other, and afterwards bass, a powerful man who once enjoyed working on boats, said he was exhausted and went home to sleep.

the next day he saw serre, a relatively slight man, who had gone home and written the whole thing up, before going to bed!

 

[complexPHILOSOPHY]

Right now, all I can do is learn because I am very behind everyone else and trying my best to catch up. I have the dedication and passion to work as hard and as much as I need to, to be regarded as a resource. I just want to teach and do research! Those are my dreams.

I love your stories mathwonk, always gives me inspiration.

YOU ARE THE HOMIE!

 

[mathwonk]

as for a sharp brain, i would say trying to do research is best, i.e. grapple with and try to solve really hard problems, ones that take months or years to solve. when i return to research after a layoff, it feels like returning to exercise after sitting on the couch for a season, all the mental muscles are stiff and slow and it seems almost hopeless to get it moving again.

there seems to be a part of my brain which is never used except for research in algebraic geometry. so when i don't do any, it essentially atrophies.

 

[mathwonk]

sometimes i think the age limit for math is more like 8 or 10 or 12. I.e. if we lose them before then, they never seem to come back to math in college or later.

 

[Diffeomorphic]

sometimes i think the age limit for math is more like 8 or 10 or 12. I.e. if we lose them before then, they never seem to come back to math in college or later.

I'm not sure about that. I was not a successful student in Math ever until Grade 12 when I began to try and got 100% in all my Math and Physics classes and now I currently have a 4.0GPA in my undergraduate. In my Grade 10 and 11 Math classes I did poorly (60%, 70%) and even had to get tutored for a class. Now, I've done research, published a paper, and I think I will have a future in research.

 

[smyth]

so ... is there a chance to understand how physics (acctuallly) work beigining with 31? not to begin a new career, but out of curiosity, as a hobby