Reification as the birth of metaphor

reflective acts of understanding (which may involve grasping of finitary propositions) are simply an extension of our understanding in this more basic sense of 'having a world'. (p. 102) The intimate understanding we are talking about is best explained through a comparison to the way people comprehend basic aspects of the physical world. The 'experiential' comprehension gives people an ability to anticipate behaviors of material objects without reflection. Indeed, when in a blink of an eye we jump to save a leaning glass of water from falling, it is not because we have recalled the law of gravity, confronted it with empirical data at hand and made an appropriate inference. Our understanding expresses itself in the ability to know what is going to happen without even being aware of the way in which the prediction was made. Having this kind of understanding renders our method of handling abstract ideas all the characteristics which, according to Fischbein (1987) are typical of intuitive thinking: our knowledge is self-evident, coercive, global, and extrapolative. At this point, the central question is what are the sources of this overpowering feeling of obviousness and inevitability of properties and relations which have not been deductively derived from known facts. How can a mathematician anticipate 'behaviors' of abstract structures which have never been seen before? It seems quite obvious that this special mode of reasoning, let us call it a direct grasp, becomes only possible when a metaphor has been constructed to give concepts their meaning. It is an embodied schema projected from another area which brings the anticipatory insight. After all, this schema is built on mathematicians' previous experience and thus its inner logic and other properties are inherited from this earlier experience (it would be in point here to make a reference to Johnson-Laird's (1983) work on human reasoning; metaphors seem to play a role similar to that which the British psychologist ascribes to the constructs he calls mental models). This 'hereditary' mechanism which underlies the construction of metaphors has, obviously, some disadvantages. First, because of the experiential origins of the hierarchical sequence of metaphors, the different constraints on our imagination, these basic side effects of embodiment, are carried like genetic traits from one generation of abstract concepts to another. Some confinements may have to be alleviated to make the movement toward the more abstract ideas possible; nevertheless much of them will be preserved along the way and will continue to delimit mathematical thought. Second disadvantage has to do with the principle on which the direct grasp is based. Once the abstract objects emerge and their embodied schemata are constructed, our abstract reasoning becomes much like the reasoning induced