I would like to add a remark which relates only indirectly to the foregoing; we have seen above the importance of the Analysis Situs and I explained that this is the real domain of geometric intuition. Does this intuition exist? I will recall that we tried to do without it and that Mr. Hilbert sought to found a geometry that has been called rational because it is freed from any call to intuition. It is based on a number of axioms or postulates that are looked at, not as intuitive truths, but as disguised definitions. These axioms are divided into five groups. For four of these groups, I had the opportunity to say to what extent it is legitimate to look at them as containing only disguised definitions.
I would like to stress one of these groups here, the second, that of the “axioms of order”. To make clear what it is, I will quote one. If on any line the point C is between A and B, and the point D between A and C, the point D will be between A and B. For Hilbert, this is not an intuitive truth, we agree that in some cases C is between A and B, but we do not know not what it means, nor do we know what a point or a line is. We may, according to our conventions, use this expression between to designate any relation between three points, provided that this relation satisfies the axioms of the order. These axioms thus appear to us as the definition of the word between.
We can then use these axioms, provided we have demonstrated that they are not contradictory, and we can build on them a geometry where we will not need figures and that could be understood from a man who would have neither sight nor touch, nor muscular sense, no sense, and which would be reduced to a pure understanding.
Yes, this man would understand perhaps, in the sense that he would see that the propositions logically deduce from each other; but the assemblage of these propositions would seem to him artificial and baroque, and he would not see why it would have been preferred to a crowd of other possible assemblies.
If we do not experience the same astonishment, it is because the axioms are not, in fact, for us simple definitions, arbitrary conventions, but well justified conventions. For the axioms of the other groups, I wish that they are justified because they are the ones which best agree with certain experimental facts which are familiar to us and which are, therefore, most convenient to us; for the axioms of order, it seems to me that there is something more, that they are true intuitive propositions, related to Analysis Situs; we see that the fact for a point C to be between two other points of a line, is related to the way of cutting a one-dimensional continuum by means of cuts formed of impassable points.
But then a question arises; these truths, such as the axioms of order, are revealed to us by intuition; but is it the intuition of space itself, or of the intuition of the mathematical or physical continuum in general? What might tend to lean towards the first solution is that we easily reason on space and much more difficult on more complicated continuums, on continuums with more than three dimensions not likely to be represented in space.
And if this first solution were adopted, all this discussion would become useless; we would attribute space to three dimensions simply because the three-dimensional continuum would be the only one of which we would have a clear intuition.
But there is an Analysis Situs with more than three dimensions; I do not say that it is an easy science, I have devoted too much effort not to be aware of the difficulties encountered; but finally this science is possible and it does not rest exclusively on analysis; we cannot cultivate it fruitfully without constant calls to intuition. There is, then, an intuition of continuities with more than three dimensions and if it requires a more sustained attention than ordinary geometrical intuition, it is undoubtedly a matter of habit, and also the effect of the rapidly increasing complication of properties of continuums when the number of dimensions increases. Do not we see students in high schools who are strong in flat geometry and who “do not see in space”? It is not that they lack the intuition of three-dimensional space, but they are not used to using it and they need an effort for it. And besides, to represent a figure in space, does it not happen to us all to represent successively the various possible perspectives of this figure?
I will conclude that we all have in us the intuition of the continuum of any number of dimensions, because we have the faculty of constructing a physical and mathematical continuum; that this faculty preexists in us in all experience because without it, experience properly so called would be impossible and would be reduced to crude sensations, unfit for any organization, that this intuition is only the consciousness that we have of this faculty. However, this faculty could be exercised in various senses; it could allow us to build a four-space, just as much as a three-dimensional space. It is the outside world, it is the experience that determines us to exercise it in one direction rather than the other.