The human-scale physical world is relatively easy to understand, and we may have evolved or learned to perform the trickier computations using specialized modules, such as perhaps recognizing parabolas to predict where a thrown object will land. You get far with linear models, for instance, assuming that the distance something will move is proportional to the force that you hit it with, or that the damage done is proportional to the size of the rock you hit something with. You rarely come across any trajectory where the second derivative changes sign.
The social world, the economic world, ecology, game theory, predicting the future, and politics are harder to understand. There are a lot of nonlinear and even non-differentiable interactions. To understanding a phenomenon qualitatively, it’s helpful to perform a stability analysis, and recognize likely stable areas, and also unstable regions where you have phase transitions, period doublings, and other catastrophes, You usually can’t do the math and solve one of these systems; but if you’ve worked with a lot of toy systems mathematically, you’ll understand the kind of behaviors you might see, and know something about how the number of variables and the correlations between them affect the limits of linear extrapolation. So you won’t assume that a global warming rate of .017C/year will lead to a temperature increase of 1.7C in 100 years.
I’m making this up as I go; I don’t have any good evidence at hand. I have the impression that I use math a lot to understand the world (but not the “physical” world of kinematics). I haven’t observed myself and counted how often it happens.
The human-scale physical world is relatively easy to understand, and we may have evolved or learned to perform the trickier computations using specialized modules, such as perhaps recognizing parabolas to predict where a thrown object will land. You get far with linear models, for instance, assuming that the distance something will move is proportional to the force that you hit it with, or that the damage done is proportional to the size of the rock you hit something with. You rarely come across any trajectory where the second derivative changes sign.
The social world, the economic world, ecology, game theory, predicting the future, and politics are harder to understand. There are a lot of nonlinear and even non-differentiable interactions. To understanding a phenomenon qualitatively, it’s helpful to perform a stability analysis, and recognize likely stable areas, and also unstable regions where you have phase transitions, period doublings, and other catastrophes, You usually can’t do the math and solve one of these systems; but if you’ve worked with a lot of toy systems mathematically, you’ll understand the kind of behaviors you might see, and know something about how the number of variables and the correlations between them affect the limits of linear extrapolation. So you won’t assume that a global warming rate of .017C/year will lead to a temperature increase of 1.7C in 100 years.
I’m making this up as I go; I don’t have any good evidence at hand. I have the impression that I use math a lot to understand the world (but not the “physical” world of kinematics). I haven’t observed myself and counted how often it happens.