“One important issue that we must address is the possibility that there might be numerous quite different, perhaps inequivalent, algorithms that are responsible for the different modes of mathematical understanding that pertain to different individuals. Indeed, one thing is certainly clear from the start, and that is that even amongst practising mathematicians, different individuals often perceive mathematics in quite different ways from one another. To some, visual images are supremely important, whereas to others, it might be precise logical structure, subtle conceptual argument, or perhaps detailed analytic reasoning, or plain algebraic manipulation. In connection with this, it is worth remarking that, for example, geometrical and analytical thinking are believed to take place largely on opposite sides-right and left, respectively-of the brain. Yet the same mathematical truth may often be perceived in either of these ways. On the algorithmic view, it might seem, at first, that there should be a profound inequivalence between the different mathematical algorithms that each individual might possess. But, despite the very differing images that different mathematicians (or other people) may form in order to understand or to communicate mathematical ideas, a very striking fact about mathematicians' perceptions is that when they finally settle upon what they believe to be unassailably true, mathematicians will not disagree, except in such circumstances when a disagreement can be traced to an actual recognizable (correctable) error in on or the other's reasoning-or possibly to their having differences with respect to a very small number of fundamental issues;”