Space-Time and 4-Dimensionality

The concept of Space-Time has been that of 4-Dimensionality, i.e. in which Time is affixed to the familiar three dimensions of Space. The problem with this procedure is that a definition of Dimension is lacking--the three of Space can be defined in terms of Perpendicularity, but a mere metaphorical extension of that is inadequate in a rigorous Science.  Ditto for recent further developments, such as in String Theory.  Lacking a definition, any concept of Dimensionality beyond 3 remains a useful mathematical construction, not a description of reality.