Declaring a method when creating an object

Java is statically typed. When you say object.a() it is looking for the method a in the Object class which is not present. Hence it does not compile.

What you can do is get the method of object using reflection as shown below :

Object object = new Object() {
  public void a() {
     System.out.println("In a");
  }
}

Method method = object.getClass().getDeclaredMethod("a");
method.invoke(object, null);

This would print

In a

java.lang.Object has no a methods declared (2), while the anonymous class returned by the class instance creation expression new Object() { public void a() {} } does (1).

Use Java 10's local variable type inference (var) to make the second option as valid as the first one.

var object = new Object() {
    public void a() {}
};
object.a();

In the second option, you assign your new object to a reference of type Object. Because of this, only methods defined in java.lang.Object could be called on that reference.

And in the first option, you basically create new object of anonymous class that extends java.lang.Object. That anonymous class has the additional method a(), which is why you can call it.