Some schools teach that taijiquan lacks pushes!
Others teach very little but pushes.
What is the place of the push in taijiquan?
In the case of Yang style taijiquan a push is the safest way to issue energy into a training partner. Pushes are also very common in actual fights, but no actual fight relies upon pushes in general.
Some schools teach pushing as a way to lead into other moves, the force of the push can also be directed at the ground. This is much like judo and aikijutsu, the throws that are common are rarely fatal and yet if the direction and angle of the throw are changed a little then the person thrown literally breaks on the ground or is severely disabled. In training the emphasis is on safety, this is true also for taiji. However the same moves that are safe bounces in the training can be used as exceedingly dangerous attacks. Moreover there are multiple methods for striking a downed opponent in taiji.
So what is the place of pushes in taiji? Certainly not as a primary technique, but there is definitely a place for them.
To even get to the point where a push is possible however many things need to happen. If the idea is that the duifang attacks and then gets pushed then disaster will occur when the duifang is of such skill that his attack cannot be neutralized. My teachers teacher gave a demonstration of applications once where he stated that if you always wait for an opponents attack then you will die, that sometimes in order to defend the self one must attack the attacker before he can complete his or her attack. Otherwise it will be to late to act.
So when it is said that there are no pushes in taijiquan, this is both true and not true. Taijiquan is not pushing, though in training it occurs a great deal, this is for a reason and not because pushing is the goal. The goal is to listen/stick/adhere/follow and to employ those skills, which can only be gained by practice, to create and utilize openings in your duifangs attacks and movements. However the difference between training situations and actual violent encounters should be considered. To seek to push is to have an intention all ones own, a person with intent cannot employ stick/adhere/listen/follow because they are preoccupied with thier intention of pushing.
So in a way there are no pushes in taijiquan, just like there are no applications. However this means that there is no set mental posture or by the numbers application, rather the possibilities are nearly limitless. The idea of no applications does not mean that taiji lacks applications, just that to have an application in mind means that there is no taiji, no relationship of yang and yin in regard to the moment. For taiji to be taiji it must encompass yang and yin, it must be undifferentiated and in the now, otherwise it is a pale imitation of the art regardless of how effective it is.