People report they feel better after they take up exercise / get in shape though. This is not strictly health, but I'd be very surprised if someone tried to argue it's not correlated. I'd also be surprised that everyone is self-deceiving — especially since that would make me one of them ;)
As for cognitive benefits... I'm more skeptical of that. I haven't experienced something profound on that front. But you do better work when you feel better day round. I think that ability to focus for longer periods of time improved slightly.
comment by Self-Embedded Agent
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People report feeling better because of all kinds of reasons. This line of reasoning seems unlikely to convince a skeptic.
Let me throw out an alternate hypothesis, which is a little extreme but ought to be considered: doing sports is a form of health signalling. People signal their health and conscientiousness by sports. Let's call it the Health Causes Sports theory, as opposed to the Sports Causes Health theory. Notice that this neatly explains people feeling good after sports: they succesfully send a difficult-to-fake signal.
A problem with the Sport Causes Health theory is the weak evolutionary story. Why would expending energy and risking bodily harm for no concrete payoff be good for one's health? Many sports carry moderate health risks. Wouldn't it make more sense to not have health tied to activity?
The strongest argument is the Greasing the Gears - theory: by streneous activity people grease their bodily machinery. Too much is bad, but so is too little, and modern humans do too little of it. If this theory is right, it certainly doesn't seem to be any specific muscle [ as the precise sport doesn't seem to matter] but a generalized cardiovascular capacity.
It seems true that modern peoples are much more sedentary than in the past. On the other hand, the amount of activity that people display likely varies wildly over societies& time & place. The 'optimal level of activity' likely also varies with genetic background. If there really was a need to grease physical machinery one would think that the body would try to automatically modulate this, in the same way it has a thermostat for many important physical quantities.
The best argument for the Sport Causes Health theory is probably rats: exercising rats seems to increase their lifespan. There is a plausible story that this has something to do with mitochrondial capacity [I recommend de Grey's book Ending Aging and Nick Lane's 'Mitochrondia' for some possible theories]. Yet lots of things increase lifespan of rats, most famously calorie restriction. Here too, there is a plausible sounding mitochrondial story, as well as other stories. Yet the consensus is that the lifespan increase for humans using calorie restriction is relatively small, if not wholly insignificant [There was a recent post on SSC about this; my takeaway was that even if the effect exists it absolutely wasn't worth it]. Generally, only a tiny fraction of experiments with mice/rats replicate in humans [I've heard 10% being bandied around].
Most supercentenarians don't seem to have engaged in heavy sports, while top sportsmen & women do not exhibit exceptional lifespan increases over the general population. More general, the effect of iq on lifespan is much larger, suggesting underlying biological/genetic reasons dominate for lifespan.
Does this explain all health variance with regard to sports? I would say that's pushing it, yet I certainly would like to see this alternate hypothesis considered. I think there are some good arguments to be made for both theories (Sport Causes Health, Health Causes Sports). All in all, my best guess is that very moderate amount of activity is likely beneficial for health, but sports in excess of that is mostly health-signalling.
See also Hanson about signalling in team sports/competitions: http://www.overcomingbias.com/2019/04/champions-show-war-ability.html
comment by norswap
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Clearing a fully-general counter-argument: Everything is based on some amount of trust - radical doubt just doesn't scale - you couldn't trust most of what your science textbook tells you without running a lot of experiments, which people don't tend to do.
With that out of the way, you can decide who to trust based on other information. So in this case, you can look at the collection of people reporting sports-related improvement, and see how it overlaps with people saying that <dubious thing> made them feel better.
As far as I know, there's a PLETHORA of studies linking exercise to health. You say you assume they are p-hacked etc... hhm why? I know science can be unreliable, but when everything we have points in the same direction, and there's a large volume of it, well, that's certainly some strong evidence.
I also seem to recall that elite athletes do in fact live significantly longer than the general population on average.