by their discoveries, but which always transcends them and remains open to other systems of relations. Mathematicians’ acts make them, as producers of universal knowledge, universal subjects.
Furthermore, the development of mathematics since the end of the nineteenth century has increasingly freed it from all reference to the empirical intuition that was originally connected with Euclidian geometry and with the physics that had been associated with it. But even in Euclidian geometry, there was, as Descartes pointed out, the distinction between the figure of a triangle and its concept. And non-Euclidian geometries have completely eliminated the intuition of the three-dimensional space in which we move.8 Geometry has thus moved towards increasing abstraction and formalism, without it ever being possible to call into doubt either the reality of its new objects or the truth of their demonstrations. Paradoxically – given its ideal character – mathematics is still the area of knowledge that attests to the capacity of the human mind to produce objective knowledge that transcends time and cultural worlds. Descartes knew this – even if he postulated that mathematical truth was, in the final analysis, grounded in God (the Christian God) when he wrote: ‘For if it happened that an individual, even when asleep, had some very distinct idea, as, for example, if a geometer should discover some new demonstration, the circumstance of his being asleep would not militate against its truth.’9
To conclude this all-too-rapid analysis of the symbolic function, allow me to underscore five important features.
1. Any sign always either stands for something else, or is a function of something else (Hjelmslev).10
2. A sign always obeys a code and is a coded access to a referent. A sign therefore transmits information to those who know the code.
3. A sign as signifier can refer to one or several signifieds at the same time.
4. Signs and the necessary access codes can never contain more information or other types of information than that invested in them by their original inventers and by all those who in turn put them to new uses.
5. The symbolic function exceeds the mind and the body. It pervades everything people do, everything people invest with meaning: churches, temples, statues, mountains, the sun, the moon, et cetera.
All signs, whatever they may be – including mathematical symbols11 – were imagined before being used.
Let us take a few examples of imagining from different contexts.
Ego (man or woman) to him- or herself – inner speech:
– I will always see my mother’s face when she learned of my father’s accident.
– I would never have imagined that of him.
– All these killings; the Mafia is probably behind them.
– I can already see where I’m going to put the table.
Ego (man or woman) to Alter (man or woman) – familiar context, a relative, friend, neighbour:
– Imagine my ex-husband’s face when he saw me with Jacques.
– Who does he think he is? Maybe he imagines we’ve forgotten where he comes from?
– Ten years ago I’d never have imagined that, with all their diplomas, my children would be unemployed.
– You can’t imagine he’s going to get away with it.
Ego (man or woman) in a professional context:
– A philosopher: Imagine a world without God (gods).
– A priest: The tomb was empty; Jesus had risen.
– A general: Let’s imagine the enemy is attacking from this side.
– A mathematician: Let us now suppose that two parallel lines in the same plane meet at infinity; that would contradict one of Euclid’s postulates.
– A politician: When they attacked Iraq and overthrew Saddam Hussein’s regime, the Americans certainly didn’t imagine what would happen next.
– An economist: Great Britain adopting the Euro is unimaginable.
– A poet (Raymond Queneau): ‘If you imagine my girl, my dear, that [love] is going to last, last, last forever.’
To extend this analysis of the forms and effects of the capacity to imagine and to imagine oneself, we need to come back to the analysis of the relationship between the mind and consciousness. We have seen that consciousness of the present is always broader than the present moment, since the immediate past remains attached and present consciousness is already involved in an immediate future. But the capacity to imagine goes beyond these different bits of time clinging to the present. It explores all of the possibilities offered by inner consciousness of time. It allows a human being, at every moment, to be both present and not present.
I am there, in my study; I am looking at an orchid that has come into bloom and, at the same time, I ‘see’ before me once again the greenhouse where my mother used to grow her flowers. When I ‘look at’ the orchid, I am not imagining it; I perceive it, I see it ‘in person’. But at the same time, I see more of it than I actually perceive because even though the flower presents only one of its sides, when I ‘see’ it, knowing that it has more than one side, I represent it to myself, implicitly bearing in mind all the sides I do not perceive. Furthermore, as I look at the orchid, my mind transports me to the greenhouse, now gone, of my mother, who is now gone as well. At the very moment when the orchid manifested itself directly to my consciousness, the image and memory of the greenhouse also filled my conscious mind, and therefore the plant began, for a moment, to exist not as the orchid I was looking at, but as a presence-absence, a presence that made an absence present to me. I thus found myself looking at a flower and imagining a greenhouse that had once existed in my life but no longer exists.
We have just encountered one form of the imagination: that which allows me to remember, to bring back ‘to life’ a past that once existed but no longer does. This indicates a way to continue the inventory of the forms of the imagination. The forms are different and differ according to the nature of the objects, of the imagined realities and according to the time in which the mind places them, situates them, in the past, the present or the future. Applying these criteria, we can distinguish at least eight forms of imaginary activities, which are available at all times to a thinking subject.
The reality imagined either
| 1. really existed, but no longer does; | I can still see my mother’s face. |
| 2. really exists, but somewhere else; | I see exactly where they live. |
| 3. will surely exist in the future; | Next year I will be retired, and I hope to enjoy it. |
| 4. will perhaps exist in the future; | One day I would like to visit South India. |
| 5. did not exist in the past; | Until the twentieth century, no man had flown in the sky. |
| 6. may have existed in the past; | I imagine that the emperor of China may have heard that the Roman Empire existed. |
| 7. cannot now or ever exist; or | No man will ever be able to jump to a height of forty metres. |
| 8. is impossible, but nevertheless possible. | The
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