For example, failure to demonstrate positive transfer between mathematical training and general reasoning ability could be attributed to ineffective teaching of mathematics; in such case, the results need not be interpreted as refuting the theory of formal discipline.
I was a graduate student in applied mathematics, and he seemed to worry that I wasn't getting any rigorous mathematical training there; this was true enough, but it was unimportant, since my interest at PHRI was merely in learning about Japanese port operations and policy.
Every kid naturally figures this out also without mathematical training.
It was from his father that Battista received his mathematical training.
It also has a relatively low learning burden for users without formal mathematical training.
From the first he proved himself an agile theorist at a time when it was still possible to be an economic theorist without extensive mathematical training.
This doesn't require any form of mathematical training; thanks to the internet, anyone with a curious mind can check around statistics to see what the real story is.
When the reasoning abilities of groups of mathematics students in secondary schools were compared with those of other equally talented students who had not had the same mathematical training, no differences in general logical effectiveness were observed between the groups.
If the then-traditional manner of teaching mathematics could be changed to emphasize logical thinking (rather than routinized application of formulas), it was argued that perhaps mathematical training could improve reasoning ability in general.
Yet any mathematician will immediately recognize the argument as mathematical, while people without mathematical training will probably find difficulty in following the argument, though not because of unfamiliarity with the subject matter.
Often you have to do some digging to find out they are actually... sociologists.The years of graduate-school seminars and rigorous mathematical training empowers PhD economists to converse with each other in a language all our own.
