For those of you truly interested in better understanding "the core" and it's very specific involvement in movement and exercise, try this very simple demonstration.
In a standing position, lift one foot from the floor and take note of how readily you are able to balance on the supporting leg. Now do the same on the opposite leg. Having tested your ability to balance on each leg, repeat the above process, but this time consider what you understand by the core, and activate the relevant muscles.
What is your reaction to this shift in focus? If you have adequately engaged the correct muscles, you will notice a sense of improved stability, prolonging your capacity to balance, and minimising any urgent need to lower the raised leg back to the floor. A deeper insight into these complementary core engagement and balance functions can be gained by also performing this demonstration with your eyes closed.
Observation: The above demonstration allows us to consider the global concept of exercise. Not from the traditional possibility that general strength training may incidentally affect the core, but in terms of how the deliberate and sustained inclusion of the core will expressly benefit fundamental strength training outcomes. In other words, optimal expansions in desired strength & fitness outcomes, stimulated by training that originates and extends from and through the core.