Poincaré's Philosophy of Relativity and Geometrical Intuition
by Andreas Kamlah, Universität Osnabrück
Abstract
Many interpretations of Poincaré have not sufficiently taken into account the historical situation in which he formulated his theses which sometimes sound a bit odd. In the present paper I want to make clear that Poincaré's theory of mathematical and scientific knowledge was an adequate answer to the problems of his time. I want, however, to restrict myself to a special aspect of his epistemology, his philosophy of relativity and of geometric intuition. This theory becomes relevant in two respects.
On the one hand for many philosophers and scientists of that time physical knowledge in some way meant to have a picture or visual representation of the world, for example to know what galaxies or atoms look like. Thus the world was constructed in the visual space. A typical example is the epistemology of Kant which had become very popular at the end of the 19th century. The picture account of knowledge, however, was a serious obstacle for any kind of relativization of simultaneity and of geometrical shape. Poincaré was one of the first scientists who understood that mental pictures do not map reality in a way that they give an exclusively true picture of it not allowing for alternatives. He already knew that pictures have the character of signs and that different ways of picturing the world are only different languages representing it. For this purpose Poincaré had to destroy the illusion that we could have a "direct intuition" of a physical state of affairs which was not directly accessible to sense perception. 
On the other hand the majority of philosophers at the end of the 19th century believed in pure intuition of Euclidean geometry and Newtonian space-time theory. This creed was closely connected to the picture account of physical knowledge. If knowledge is given by pictures in Euclidean space and Newtonian space-time, classical space-time theory is a necessary presupposition of any kind of physical knowledge. Now one has to admit that for many mathematicians of Poincaré's time Kant's pure intuition was already discredited by leading in many cases to wrong results. But a conclusive refutation of pure intuition was not possible unless the illusion of alleged pure intuition was destroyed. Poincaré does this by his beautiful theory of geometric intuition which explained the efficiency of this faculty in cases where it does its job in a legitimate way, and which also explained the mistakes which we make by its illegitimate use when we for example represent in our mind two distant events taking place at exactly the same time. 
If we have understood Poincaré's discussion of visual representation and of pure intuition it finally becomes clear that some of his statements which were interpreted by A. Grünbaum as "geochronometrical conventionalism" have to be read in a completely different way as contributions to the refutation of pure intuition, and that they thus obtain a much more natural interpretation.
In the final section I will discuss two different meanings of "relativity" in Poincaré's writings, the confounding of which has led to much confusion. We will not obtain a consistent interpretation of Poincaré's philosophy, unless we carefully distinguish between these two different meanings.

1. Introduction
The aim of this paper is to contribute to an understanding of the conceptual revolution in science at the end of the 19th and the beginning of the 20th century. The aspect of this revolution which seems to be the most important is the replacement of the mentalistic conception of truth by a lingualist one. Formerly truth had been an agreement between reality and judgements consisting of representations. Later it became an agreement between statements (consisting of signs) and reality.
In order to make this step, one had to realize that ideas or representations are signs, too, which need a semantic interpretation. They are not directly related to their references. Poincaré says "we have no direct intuition" (intuition directe) of geometrical objects. If our ideas or representations have become ordinary signs like symbols drawn on paper, there is no motive any more for considering them important for knowledge. We obtain a simpler relation between the world and our knowledge of it, if we replace ideas by symbols and judgements by statements. And we are thus led to the modern conception of truth.
The insight that representations have no direct relation to their references (in the real and in possible worlds) appears in Poincaré's philosophy as "principle of relativity" which is always connected with the statement that we have no "direct intuition" of things. Poincaré has to defend this thesis against the Neo-Kantians and against those realists for which knowledge means the possibility to visualize the real world.

2. The Problem of Interpreting Poincaré
Poincaré's writings offer a lot of difficulties to the reader of today. Poincaré in the first place was not a philosopher but rather a mathematician. Therefore he did not feel obliged to compare his theses with those of others (there are, however, some exceptions as for instance his debates with Russell and Leroy). He did not use much philosophical terminology and tried to explain his ideas using pictures, examples, and ordinary language. His writings frequently have the character of talks or articles addressed to a mixed audience of scientists, philosophers, and educated laymen. According to the nature of such speech acts we are not entitled to expect a very systematic order of his ideas. In a monograph one will refer to preceding chapters reminding the reader of terms introduced earlier or one will draw parallels to ideas already discussed before. A talk or semipopular article has always to be self-contained. In such a text one will avoid doubts or complicated deliberations. Thus in such a case an author will tend to be more straightforward, more simple, more radical than in a written monograph.
One has also to keep in mind that in Poincaré's books, which are collections of lectures and articles, the terminology was not made uniform before publication, and differs considerably from one chapter to another. A most drastic example for such a discrepancy will be discussed in section 9. He uses two completely different concepts of relativity in Part 2 and Part 3 of Science et méthode without explaining that to the reader. 
All these features are present in Poincaré's writings. And in addition, the reading becomes even harder due to Poincaré very early becoming a kind of oracle who was invited to tell mankind the future of its science and mathematics (as for instance at Paris 1900, see SH, chapter 9 and 10, and at St. Louis 1904, see VS, chapter 7 to 9). Since Poincaré was permanently bursting with mathematical and physical ideas, he obviously had not much time to explain his ideas to people who had difficulties understanding them. His talks were published and quickly translated into other European languages, first of all into English and German. 
We have to take into account all peculiarities of Poincaré's style of communicating his philosophical ideas to the public, if we want to understand him. The greatest mistake to be made would be to pick out isolated theses from his books and treat them in the same way as theories of philosophers, label them with suitable "isms" and put them into a doxographic scheme. We have always to leave his utterances within the context of his time and its characteristic historical situation, keep in mind to whom he was addressing his talks or essays, and what he believed his audiences or readerships did not yet know and needed to learn. Poincaré did not waste his time with treating special subjects, he mostly talked about the nature of mathematics and physics both of which were considered to be in a fruitful crisis on the eve of dramatic changes. Thus Poincaré was a kind of herald of twentieth century science and mathematics.
Many of Poincaré's drastic and very misleading utterances will become much more plausible and perspicuous in the light of the circumstances under which they were made, and I think that in their own context they will indeed teach us a lot about the revolution which took place in the heads of contemporary mathematicians, physicists, and philosophers, and in the whole European society. We should consider Poincaré's utterances as speech acts which can only be understood when the conditions are known under which they are performed.
What was the message which scientifically educated mankind needed at the turn of the centuries? What was it that they were longing to be taught? It was essentially a new relation of knowledge to reality and a new understanding of what knowledge is at all, a second Copernican revolution, getting rid of a world view in which everything is neatly described by a suitable picture. 

3. Iconic realism.
From 1840 on, and simultaneous to the beginning of the industrial revolution in central Europe, there was a strong tendency to empirical research and connected with it a tendency to materialism. Nature was demythologized; the immortal soul was driven out of the body of man. His body became subject of conservation of energy as any piece of organic  or inorganic matter. 20 years later all living creatures were explained by Darwin in a natural way; there was no place left any more for divine creation. It was natural in this situation to consider human understanding as an instrument or organ developed in the course of billions of years by natural selection. Thus also the a priori principles of human understanding had to lose their supernatural character and to become a result of evolution like his liver or his lungs. They also had to lose their necessity and to become something like accumulated experience of thousands of generations, and that means experience after all. For E. Mach, at least, Darwin's theory was one of his strongest motives for empiricism.
The enormous success of natural sciences was very much in favour of a materialist realism according to which knowledge pictures the world, represents it in the mind by means of images. In terms of Peirce, who calls these images icons, one could say that these realists have an "iconic" conception of knowledge. To know is for them to have the true representation of something by means of mental images or "icons" in one's mind. Therefore I want to call those realists iconic realists. This realism has its roots in the Cartesian world view of the 17th century. For Galileo, Descartes and Locke it was quite clear that physical knowledge consists in a geometrical and mechanical model of the objectively existing world. A judgement is true if it says something about the model which pictures the world. The argument that we perceive the world only by our sense organs and that these will bias our perceptions is answered by Descartes with a recourse to God. He who sees the world as it really is guarantees that what we represent or perceive clearly and distinctly is true. Locke's argument is different: since space and time are common to many senses, our spatial knowledge is exempt from the disturbing influences of the senses. Descartes' and Locke's arguments certainly are much more sophisticated than I have sketched them here. But they have in common that human knowledge of nature is essentially something like having - at least in parts - a true mental picture of the world, a kind of mental photograph, or to see the world with God's eye. Therefore one could say that the Cartesian realists, too, already had an "iconic" conception of knowledge. And we may also say that they claimed knowledge to be intimately connected with a direct intuition of its objects, if we mean by this term either the ability to decide if a given image or icon truly represents a certain object or this true image itself. The interpretation of the concept of direct intuition in the writings of Poincaré will be our main issue in this talk. 
The iconic realists of the 19th century differed from Descartes by their atheism or at least by the peripheral role played by religion in epistemology. Nevertheless the image theory of knowledge was retained. Poincaré himself referred to this position, when he wrote:
"But if our senses were sufficiently keen to show us all the details of the bodies which the physicist studies, the spectacle thus disclosed would scarcely differ from the one the astronomer contemplates. There also we should see material points, separated from one another by intervals, enormous in comparison with their dimensions, and describing orbits according to regular laws. These infinitesimal stars are the atoms."
One sees here clearly that Poincaré in characterizing a widespread position of the 19th century talks about a represented "spectacle", and about mental "seeing". He uses terms characteristic for the iconic conception of knowledge.
There might have been many of those iconic realists under the natural scientists of Poincaré's time. And I think that every schoolboy (or schoolgirl) or even freshman has an iconic conception of knowledge in this broad sense. Before we started to learn modern physics, knowledge was for us to have in our mind images or icons of the world. I still can remember the difficulties which I had with the theory of special relativity, to understand that a body can be a sphere in one inertial system and an ellipsoid in the other. My difficulties were just of the kind already described. I believed in a true and absolute picture of the world.

4. Critical realism and positivism.
While in the mid 19th century crude materialism was at its peak, later critical realism and positivism became more powerful. This was partly due to the rise of sense physiology and psychology. One became aware of the fact that the relation between reality and human knowledge was much more complicated than the materialists tended to believe. This can be studied in the writings of von Helmholtz and Mach, the one being a critical realist, the other being a positivist. But both were unable to describe the relation between scientific theories and phenomena in a satisfactory way. Von Helmholtz only talked about theories in most general terms. He said that we make them in such a way that they can predict phenomena or tell us how we can produce at will phenomena which we need, and that was not very much. Mach however tried to reduce physical theories to relations between phenomena and to ban all theoretical entities like atoms from science.
Beyond that both Helmholtz and Mach were unable to offer a satisfactory theory of mathematics (nor could J. S. Mill) which was acceptable for anybody really familiar with the mathematics of that time.
These two gaps in the epistemology of the time had to be filled by some ingenious mind. That was the task to by done at the end of the century.

5. Neo-Kantianism.
The actual development of philosophy partly went back to Kant. The failure of positivism in its treatment of scientific theories became a strong argument in favour of a revival of Kant's philosophy even if evolutionary biology suggested biological and psychological contingency of a priori principles. But finally the way back to Kant could not have been a real solution. In the time after Darwin pure intuition and pure understanding had to be considered as something biologically contingent after all.
In order to understand this situation more thoroughly  let us first look at science and mathematics as it appeared at the end of the eighteenth century in the eyes of Kant.
Kant himself was no scientist or mathematician, even if he had some important scientific ideas. His account of sciences, however, contains much of the commonly accepted view of his contemporaries. Otherwise his philosophy would not have had the chance to spread all over Europe as it did in the 19th century. At the centre of Kant's epistemology we find the distinction between "concept" (Begriff) and "intuition" (Anschauung). In order to have a meaning a scientific or mathematical term has to designate a concept which by its turn has either to refer to intuitions, or to be traced back to Kant's categories, as for instance the concepts of substance and force. Thus at the basis of mathematical or scientific knowledge we will always come across intuitions, in the first instance pictures of single objects or events, but also representations of non-visual impressions. These intuitions, however, are not necessarily possible sense impressions. They can also be mental pictures of objects which are too small to be seen by the naked eye (Kant 1787, p. 273). Kant considered the limits of human sense organs as more or less arbitrary and was also ready to admit to "phenomena" of a kind not accessible to them. The paradigm of mathematics for Kant was geometry, and geometry was about visual geometrical figures. Thus mathematics had its objects, it was about "pure intuitions".  Physics differed from mathematics only in the fact that matter was its object. But this matter had to be described in mathematical terms, i. e. by geometrical figures, geometrical forms. A physical state of affairs was considered as understood if and only if it could be visually described, constructed in pure intuition (making use of some additional concepts which were derived from the categories).
We have to assume that for many of Poincaré's readers or listeners mathematics and physics still went just this way. This is at least clear for the majority of German (and I suppose also for many French) philosophers of that time. When in 1870 Helmholtz presented his ingenious account of visualization of Lobachevskian geometry (Helmholtz 1962, p. 242, 1966, p. 62), he was understood by hardly anybody. The general reaction to his famous talk was only: Helmholtz had constructed strange objects in Euclidean pure intuition (see for example Natorp 1910, p. 309f.). Euclidean geometry was in an absolutely unchallengeable position. It was the grammar of our scientific imagination and thus the grammar of any scientific description whatsoever.
We get a completely wrong impression of the history of mathematics and physics in the 19th century if we learn it only as a history of winners and not of continuous struggle against stubborn ideas deeply implanted in human minds. The elite of less than a hundred leading mathematicians and scientists might have been ahead of the others. But these people were not the typical expected readers of Poincaré's books but rather those who by Toretti in his monograph From Riemann to Poincaré (1978, on p. 285ff.) are called "Boeotians" alluding to Gauß who was afraid of them in 1829 in his famous letter to Bessel (see Becker 1964, p. 178). To be a "Boeotian" in Toretti's sense at that time was normal, even at the end of the century. (In ancient times Boeotians by the Athenians were considered being rather backward.)
We read in our history of mathematics that non-Euclidean geometry was discovered by ca. 1830. But what many books on the history of science do not tell us, is that before 1865 this discovery was only noticed by few experts (see Volkert 1993, p. 8). For a Kantian the works of Bolyai and Lobatchevky could only be curiosities, since it was clear from the start that their concepts would remain empty forever and without any object in pure intuition. When Riemann (see Riemann 1959) and Helmholtz presented their interpretations of the new geometries, this did not change much, except for some mathematicians like F. Klein or S. Lie. Even G. Frege was firmly committed to pure intuition like most of his philosophical colleagues:
"The craziest feverish dreams, the boldest inventions of legends and poets, which make animals talk, stop stars in their revolution, turn stones into men and men into trees, and teach how one can pull oneself by ones own tuft out of a swamp, they are yet as far as they remain visual bound to the axioms of geometry."
How can we explain the tenacity of apriorism in geometry? As long as judgements were considered as connections of ideas, as we find it in nearly all logic textbooks of the 18th and the 19th century, and ideas considered as related to intuitions as Kant liked to point out at many places in his works, physical knowledge as a whole had to refer to intuitions and these had to conform to Euclidean geometry, since that geometry was so to speak the grammar of geometrical imagination. This was expressed nicely in the Logic of J. v. Kries, a distinguished physiologist and philosopher:
"A world picture [Weltbild] as we conceived as ultimate goal of scientific progress, has to apply a conceptual material which is in some way accessible to us, given in our mental life. Thus we come back to the question how it is possible that for a such world picture [Weltbild] the content of mathematics has an obligatory validity." 
He then talks about the mechanical description of the world by interacting point particles, and continues: 
"It seems to be clear that as long as we follow that approach, and as long as scientific knowledge rests upon the conceptions of space and time, we are bound to the nature of these conceptions. Spatial and temporal characterizations [Bestimmungen], too, which we use in our world picture [Weltbild], have to conform to the concepts given within these conceptions, as for instance to equality etc. Also for them the internal relations and connections are obligatorily fixed, which result from the nature of the conceptions of space and time. In this sense the contents of mathematics  ....  can be considered obligatory for experience , i. e. for a scientific world picture [Weltbild]" 
Nothing seems to be more plausible than the idea that science is bound to the nature of our thinking. Von Kries and most of the other "Boeotians", however, made the mistake not to scrutinize every single axiom of geometry if it really is a consequence of the very nature of our reasoning. The persuasion of our old habits was to powerful.
According to von Kries and the other Boeotians a physicist has always to try to explain unexpected new phenomena within the frame of Euclidean geometry. Thus if a large triangle seems to have a sum of angles not equal to 180°, he should conclude that its legs are no straight lines and not that Euclidean geometry is violated. If for example a triangle realized by light rays, as the famous triangle Hoher Hagen, Brocken, Inselsberg surveyed by Gauß, should not have yielded 180°, one should have concluded that the light rays were diffracted by the atmosphere or something else. This argument was used by W. Wundt, C. Sigwart, Lotze, J. von Kries, P. Natorp, by nearly everybody (see Kamlah 1993, p. 10, 19, 24). Very few philosophers or scientists of that time would have considered it deceptive. We can therefore say that already the Neo-Kantians and the adherents of pure intuition in geometry used a conventionalist strategy for their defence, which certainly must have influenced Poincaré. He had only to add that not only Euclidean axioms could have been saved in this way but also any other geometry. Therefore we find the favourite argument of the Boeotians also in Poincaré's writings where it, however, does not play the same role:
"If, therefore, we were to discover negative parallaxes, or to prove that all parallaxes are higher than a certain limit, we should have a choice between two conclusions: we could give up Euclidean geometry, or modify the laws of optics, and suppose that light is not rigorously propagated in a straight line. It is needless to add that every one would look upon this solution as the more advantageous. Euclidean geometry, therefore, has nothing to fear from fresh experiments." 
In order to face the arguments of the Neo-Kantians Poincaré had to shake the confidence of his readers or hearers in the reliability of the power of visual imagination. He had to show that the relation of mathematical and physical terms to their meanings or references is much more complicated than they believed.
But it was not so easy to attack Kantianism. Many of Poincaré's arguments against it must have been absolutely unable to convince any Neo-Kantian of his time. Poincaré tells us a fairy tale about intelligent beings living on a surface of the sphere (SH, p. 67 and 75). These beings would come to a rough empirical knowledge of spherical geometry and would therefore not accept Euclidean geometry. And that's why Euclidean geometry cannot be true a priori. The Kantians must have considered this argument absolutely ridiculous. Nobody can predict what two-dimensional intelligent beings are disposed of to do or not to do. Therefore such a story proves nothing (see for example Krause 1878, p. 41ff.). And even if we knew anything about these "flatmen", we were not entitled to draw any conclusions on real three dimensional men. In the writings of the Kantians we frequently find the opinion that the enemies of the apriori had no better arguments than this silly story about flatmen or "Beltrami's bugs". Thus Kantianism was in a reasonably firm position at Poincaré's time and even used his ideas for strengthening its own arguments.
It may be considered a shortcoming of this article that German authors are treated as examples of 19th century philosophy rather than French ones. The actual influences on Poincaré are still unknown to me. If we can believe Toretti (1978), the situation in France was not so much different from that in Germany. But it certainly deserves a detailed study. We have to find out the actual ideas which influenced Poincaré. Therefore my paper should rather be considered a kind of zeroth approximation to the solution to be found.

6. The origin of Poincaré's conventionalism.
Let us now come back to Poincaré. He was in the 1880th a young mathematician who was familiar with the most advanced mathematical ideas of his time. Mathematics was on the way to becoming the science of structures. At that time mathematicians were not yet completely aware of what they were doing, but they were already concerned with groups, manifolds and the like, and became used to talking about geometries in a way that it did not matter any more if these referred to the one or the other of its models. By handling mathematics the identity criteria of mathematical theories gradually became structural. This can easily be seen in Poincaré's famous paper on geometry of 1887.
In this situation Poincaré was ready to give mathematics the new role it had to play in the performance of science. If one reads the last two pages of the mentioned paper, one sees immediately that his geometrical conventionalism was motivated by the most obvious argument, that if Euclidean geometry were true, Lobatchevskian geometry had to be false. But since both are the study of two different groups, there cannot be one true and the other false, if they describe these groups in a mathematically correct way:
"... Geometry is nothing but the study of a group, and in this sense one could say that truth of Euclidean geometry is not incompatible with that of the Geometry of Lobatchevski, since the existence of a group is not incompatible with that of another group."
The paragraphs which follow in the article are to be read as an abstract of the first chapters of La science et l'hypothèse. They contain Poincaré's most important philosophical theses in a nutshell.
It seems that Poincaré dealt with geometry just in the same way as the present day structuralists à la Sneed deal with scientific theories in general (Sneed 1971, Stegmüller 1973). Was Poincaré, therefore, the first proponent of Sneed's and Stegmüller's "non-statement view"(Stegmüller 1975, p. 81), according to which theories are species of structure, and as such can neither be true nor false? One has to be careful, however. Poincaré did not talk about structures but rather about continuous groups, and it is not clear from the start that he did not mean by "groups" just transformations of real space. 
This starting point of Poincaré's conventionalism has been noticed by many students of his philosophy, for instance by J. Giedymin (Giedymin 1977) and A. Coffa (Coffa 1986). Therefore I do not want to discuss this point here in further detail. 
In the following years Poincaré worked out his conventionalism. In its original formulation his conventionalism is still open to the objection that it is just one way to look at geometry but that one is not forced to conceive geometry as one of many possible structures. One might have objected that instead of studying arbitrary geometries as separate mathematical objects one could also have asked for the geometry of the real world. He therefore added some other claims to the original ones, tried to make them plausible and to defend them against the apriorists on the one side and against the empiricists on the other. Poincaré never changed his central thesis that geometry is neither true nor false, and therefore neither empirical nor synthetic a priori. But later he argued for it in a different way. His arguments now concerned the role played by geometry in physical knowledge.

7. Poincaré's thesis of relativity of space and space-time.
Poincaré's conventionalism contains at least two main theses:
1. Geometry and general principles of physics are disguised definitions (SH English, p. 50; SH French, p. 76)
2. Space and space-time are relative.
Certainly there is a relationship between these two theses. But logically they are not identical, and we have to keep that in mind. My concern here will be relativity in the broad sense in which Poincaré uses this word. This relativity is usually looked at as an alternative to primitive empiricism, and in the present literature on Poincaré the relation between conventionalism and crude empiricism is discussed extensively. Therefore I do not want to deal here with this issue. I am more interested in Poincaré's relation to apriorism on the one hand and to iconic realism on the other. 
First I think that many of Poincaré's arguments in favour of relativity are directed against the Neo-Kantians and others for whom geometry and some fundamental principles are true though being no empirical truths, and I believe also that some formulations of Poincaré which sound obscure become perfectly understandable if one reads them as directed against the apriorists. Secondly I believe that iconic realism is another important target of Poincaré's discussion. We cannot in all cases say how far Poincaré's argument aims at one or the other of these epistemologies. In many instances they will hit both of them. This will become especially clear for the claim that "space is amorphous".
Let us first look at the essay "The Relativity of Space" (SM English, p. 93 - 116; SM French, p. 95 - 122):
The article is divided into sections. 
1. In the first section the essay starts after an introductory passage with Delboeuf's claim that we would not notice anything if all things in the world became 1000 times bigger overnight than they had been before.
2. Then Poincaré discusses the Lorentz contraction and argues that it also would remain undiscovered for the same reasons.
3. At the end of the first section we find a sentence which seems to support A. Grünbaum's well known geochronometrical conventionalism (Grünbaum 1963, p. 27f.): 
"Space is in reality amorphous, and it is only the things that are in it that give it a form."
But already the next sentence falsifies Grünbaum's interpretation:
"What are we to think, then, of that direct intuition we have of a straight line or of distance? We have so little the intuition of distance in itself that, in a single night, as we have said, a distance could become a thousand times greater without our being able to perceive it, if all other distances had undergone the same alteration."
We wonder what Poincaré may mean with the expression "intuition directe". The second section of the article will make this clear. It starts with the question:
"If this intuition of distance, of direction, of the straight line, if, in a word, this direct intuition of space does not exist, whence comes it that we imagine to have it?" 
The answer to this question may teach us its sense. It consists in a theory of visual intuition which Poincaré develops in the second and the following sections.
If Grünbaum was right with his interpretation according to which Poincaré wants to tell us that space is in some sense ontologically amorphous, we would never understand why Poincaré continues the article with a theory of visual intuition acquired by individual and evolutionary experience. 
What then could he have meant? Who had claimed to have a direct intuition of geometrical or physical objects? There may be two possible answers. On the one side there were iconic realists of the kind of Galileo, Descartes or Locke, on the other side the Kantians. 
Did Poincaré address himself to the iconic realists? In the essay "The Relativity of Space" Poincaré tells us a story of intelligent beings bred in a world "B" with other kinds of rigid bodies: 
"But if one day a window were to open for us upon the universe B, we should feel contempt for them, and we should say, 'These wretched people imagine that they have made a geometry, but what they so name is only a grotesque image of ours; their straight lines are twisted, their circles hunchbacked, and their spheres have capricious inequalities. ' We should have no suspicion that they were saying the same of us, and that no one will ever know which is right."
Immediately after this passage comes the one I have already quoted:
"Space is in reality amorphous, and it is only the things that are in it that give it a form." 
Poincaré wants to tell us that we can also have different mental pictures of the world which are no more and no less true than those we presently have of it.
Let us continue with the Kantians. Also if we make ourselves clear that "intuition directe" is what the apriorists believe to have from geometrical objects in physical space, the connection between the first and the following sections of the article becomes perspicuous. Thus the "amorphness" of space (see Grünbaum 1963, p. 10-11) is just the fact that a mental picture of a geometrical object can neither be matched with something real directly, i. e. without any further specification, nor with an object in the world of phenomena which according to Kant we construct in a unique way by using a priori principles and self-evident Euclidean geometry.
Are these quotations sufficient for our interpretation? Let us look for other references too! In his essay on time, which already seven years before Einstein's famous paper contained ideas today commonly attributed to Einstein, Poincaré asked what might have been the reason for the difficulty to accept the relativity of simultaneity. He wrote:
"We should like to represent to ourselves the external universe, and only by so doing could we feel that we understood it. We know we never can attain this representation: our weakness is too great. But at least we desire the ability to conceive an infinite intelligence for which this representation would be possible, a sort of great consciousness which would see all, and which should classify all in its time, as we classify, in our time, the little we see."
Poincaré's "infinite intelligence" reminds us of Laplace's famous "intelligence" on the first pages of the introduction of his Essay (Laplace 1886) who is able to know all data of the world at a given time. But Poincaré's intelligence sees the world, it is designed as a sort of great consciousness which would see all
This intelligence is a substitute for Galileo's, Descartes' and Leibniz' God who perceives the world by direct intuition. Immediately after the quoted paragraph Poincaré himself compares his "intelligence" with a god, a strange kind of god, however, who perceives at once only events of one single time slice in space-time, containing present events, and has in the same way as human beings access to the past events only by his memory and not by direct perception. This time slice then defines simultaneity. But this "demigod" (demi-dieux) or "infinite intelligence" is only a fata morgana, it does not exist for us, and therefore a definition of simultaneity of this kind is empty and useless.
I think that the quoted passages make it quite clear that iconic realism is one of the targets of Poincaré's argumentation. But what about the Kantians?
If a mathematician had told Kant that he had no direct intuition of geometrical objects, say triangles or straight lines, Kant would have answered: "Then you cannot be a mathematician, since mathematics is about intuitions. Please change your profession, become a butcher or do something else useful for your fellow citizens!"
Kant and the Kantians were no iconic realists, who wanted to give a picture of a world existing in itself. But on another level the human mind constructs a physical world in Euclidean space and as far as this model world conforms to experience and synthetic principles a priori it is nevertheless the true phenomenal world (including also "phenomena" which are too small or too big to be perceived by our sense organs). Thus Kant and his followers, too, had an iconic conception of knowledge, as I have already mentioned (in section 5).
Did Poincaré direct his arguments also against the Kantians? I have already pointed out above that the article on relativity of Space continues in section III with a theory of geometrical intuition. If the sections III to V have something to do with the preceding ones, one has to assume that they are written in order to show that a direct intuition of geometrical objects does not exist. The Kantians and the Neokantians (except those of the Marburg school, see Natorp 1910, p. 276f.), however, still believed that geometry is given to us by pure intuition. Therefore Poincaré had to dismantle the Kantian theory of intuition. He did it by developing his own theory according to which intuition is nothing more than experience accumulated by the individual and by the race. Intuition, therefore, is not able to teach us more than our sense impressions do.

8. Poincaré's theory of geometrical intuition
It is hard to imagine a knowledge which does not rest at least partly on intuition. Single empirical perceptions do not lead themselves to generalizations without some additional principles whatever these may be. 
Poincaré, too, accepted intuitions as a source of knowledge in some fields of science, in logics and arithmetic at least. In Le valeur de la science he discussed four examples of intuition:
"Compare these four axioms: (1) Two quantities equal to a third are equal to one another; (2) if a theorem is true of the number 1 and if we can prove that it is true of n + 1 if true for n, then it will be true of all whole numbers; (3) if on a straight the point C is between A and B and the point D between A and C, then the point D will be between A and B; (4) through a given point there is not more than one parallel to a given straight.
All four are attributed to intuition, and yet the first is the enunciation of one of the rules of formal logic; the second is a real synthetic a priori judgement, it is the foundation of rigorous mathematical induction; the third is an appeal to the imagination; the fourth is a disguised definition."
The first example belongs to what Euclid called "axioms" (in distinction to his "postulates" and "definitions") and we today "analytical statements about the meaning of concepts". The second example is important. Poincaré considered the principle of complete induction indispensable for mathematics. It rests on a direct intuition of numbers. Since numbers and their infinite series are products of the human mind, we can have a complete direct intuition of them:
"Why then is ... [the principle of complete induction] imposed upon us with such an irresistible weight of evidence? It is because it is only the affirmation of the power of the mind which knows it can conceive of the indefinite repetition of the same act, when the act is once possible. The mind has a direct intuition of this power, and experiment can only be for it an opportunity of using it, and thereby of becoming conscious of it."
This is important for the considerations in this article, since direct intuition is according to Poincaré what we do not normally have of geometrical objects. While Kant treated arithmetic and geometry on the same footing, for Poincaré both have a radically different epistemological status. Let us now come to the last two examples.
If one studies the literature on geometrical intuition, one will observe that most expositions do not contain a detailed psychological analysis of the way in which intuition works. Frequently only some examples are discussed and geometrical intuition is compared to other intuitive constraints of imagination, for instance to the impossibility to imagine a red patch which is blue at the same time. It sounds plausible that the sense impressions of our sense organs enter the brain in a certain organized form, that for instance four points on a straight line in the visual field always have to satisfy the axioms of the betweenness relation, for example:

A       D                     C            B
If on a straight line  the point C is between A and B and the point D between A and C, then the point D will be between A and B.
This was Poincaré's third example. He himself accepted this kind of intuition as legitimate. He called the example just given "un appel à l'imagination". I want to call intuition of this kind synchronic intuition, since it deals with visual pictures at one given moment of time. But is all geometrical intuition of this kind? This question seems not to have been raised frequently. To have some examples in which geometrical intuition seems to be successful does not entitle us to found the whole of Euclidean geometry on intuition. And indeed a moment's reflection will teach us that the successful examples of pure intuition solely concern the subjective visual space at one moment. Only synchronic intuition can be a priori. From synchronic intuition we have to distinguish diachronic intuition which deals with processes extended in time. Intuition, generally speaking, can never yield a priori relations between visual impressions of two different times to each other. Any sense impression can follow any other in time. That was Hume's famous argument. Therefore diachronic intuition has to be empirical. No a priori insight can prevent a straight line from becoming a circle in the next second. Therefore a sequence of sense impressions in time in any case is empirical.
There are many instances for diachronic intuition. Poincaré's fourth example is not a case of immediate diachronic intuition. Therefore I prefer a more direct one. We expect that a car which approaches us changes its apparent visual shape in a certain way, or that the perspective of the things surrounding us changes in a certain way if we move ourselves. But in any case while synchronic intuition may be a priori, diachronic intuition is empirical. The mistake of the Neo-Kantians and of many psychologists of those times was that they did not make a difference between synchronic and diachronic intuition and considered instances of the former as a guarantee for the efficiency of pure intuition altogether. 
Now it is clear that synchronic intuition cannot be sufficient for a geometry as part of natural science or even practical life. In science and in our daily life we do not deal with imagined geometrical figures but rather with figures drawn on paper or with objects having certain shapes in physical space. We move in space and look at those figures or shapes from different sides hoping them to remain the same in different situations. Therefore synchronic intuition cannot give us what we need for our scientific or practical aims. To deal with figures and shapes in the objective world we need diachronic intuition. Helmholtz has characterized this kind of intuition as follows:
"By the much abused expression 'to represent' or 'to be able to think how something happens' I understand - and I do not see how anything else can be understood by it without loss of meaning - the power of imagining the whole series of sensible impressions that would be had in such a case." 
If we read Poincaré, we will notice that he, too, deals with diachronic intuition only. He emphasizes [original in italics]:
"None of our sensations, if isolated, could have brought us the concept of space; we are brought to it solely by studying the laws by which those sensations succeed one another." 
Now there is still a third way how geometrical intuition works which is very important in science. We do not confine ourselves to pictures which represent possible sense impressions, but we draw pictures and construct models of atoms and of planetary systems, of the interior of the earth cutting the earth into two halves like a melon. In short: we use our imagination for many purposes different from the anticipation of future perceptions, and in doing so we rely on the axioms of Euclidean geometry according to which objects can be enlarged or reduced in size at will. In a Lobatchevskian world these operations would be illegitimate. I want to call this third kind of intuition model constructing intuition.
Poincaré sometimes talks also about this model constructing intuition:
"Now if I wish to pass to the great space, which is no longer to serve for my individual use only, but in which I can lodge the universe, I shall arrive at it by an act of imagination. I shall imagine what a giant would experience who could reach the planets i a few steps, or, if we prefer, what I should feel myself in the presence of a world in miniature, in which these planets would be replaced by little balls, while on one of these little balls there would move a Lilliputian that I should call myself. But this act of imagination would be impossible for me if I had not previously constructed my restricted space and my extended space for my personal use."
Another passage has already been quoted in which Poincaré invokes a demigod for the imagination of absolute simultaneity.
It is clear that model constructing intuition is an important mental tool for scientific research and engineering. But it rests upon experience like diachronic intuition and to use it for a proof of geometrical axioms from pure intuition would be illegitimate.
By this interpretation of Poincaré's chapter "The relativity of space" I believe to have shown that for Poincaré relativity of space means that there is no legitimate representation of spatial objects by mental pictures without semantical specification, no "direct intuition", and that Poincaré in order to refute both iconic realism and Kantian iconic semantics had to destroy the illusion of pure intuition.
We may still be not yet convinced and doubt that our interpretation is correct, and therefore we may also look for additional evidence for it. In 1898 Couturat reviewed Russells dissertation (Russell 1956) which had just appeared, and praised it as a comeback of philosophy into the theory of space and time and as a victory for apriorism (Couturat 1898). This caused Poincaré to react and he, too, reviewed Russells dissertation (Poincaré 1899). The review led to a reply by Russell (Russell 1899) and an answer by Poincaré (Poincaré 1900). Russell required in his reply that science had to start with clear and simple concepts like for instance that of straight line or distance which had to be the basis for the definition of more complicated ones. And Poincaré answered that we had no direct intuition  of straight lines and the like:
"Both parties [the Kantians and the adherents of Russell] claim to have a direct intuition of distance, i. e. of a certain something appearing to them as enlightened by a vivid insight [éclairé d'une vive lumière]". 
For the Neo-Kantians the world could only be understood, if it was constructed in Euclidean space. The frame into which this construction was put, was a space accessible to pure intuition. Poincaré could not believe in the "vive lumière" which enables us to analyze Euclidean geometrical figures.

9. Two meanings of "relativity" in Poincaré's writings
In the introduction I have already mentioned the difficulties which we face when we try to understand Poincaré. Some of these can be removed by interpretation, others remain. I want to mention here his use of two different concepts of rela-tivity. In order to make this clear, let us start with Poincaré's dictum that the group of motions of rigid bodies can be chosen by convention. 
"... only, from among all possible groups, we must choose one that will be the standard, so to speak, to which we shall refer natural phenomena." 
If a spatial group is considered a rule for the measurement of space, it cannot become inapplicable to the world, but only more or less convenient. Poincaré originally believed that the degree of convenience of a geometrical group used as a rule is only determined by its agreement with inexact immediate experiences which we make with rigid bodies, when we move in space, while a physical principle though being a convention, too, can become more or less convenient by the results of physical experiments (SH, p. 152 and 153). This is all very well known, and I need not comment upon this point in detail.
Therefore Poincaré could also say: "La géométrie euclidienne n'a donc rien à craindre d'experiences nouvelles." (SH, p. 96.) This untenable position of Poincaré in his La science et l'hypothèse has been widely criticized, and he himself must have noticed somewhat later that it could not be maintained. Let us try to understand what caused him finally to reject it. In this context we will also be confronted with two different meanings of "relativity".
Helmholtz had already discovered that the empirical content of a physical geometry is contained in the characteristics of the group of motion of rigid bodies (Helmholtz 1886). A rigid body is physically described by a stable solution of the equations valid for matter. If one transports the body to another place or gives it a different spatial orientation, it will be described by another solution of these equations which is obtained from the first by an invariance transformation of the laws. Thus geometry is given by the spatial invariance transformations of the laws of nature.
Helmholtz, however, did not say just this explicitly. In his famous talk on "the origin and significance of geometrical axioms" from 1870, after having discussed the temptation to consider the group of spatial motions a transcendental condition of knowledge, he continued:
"But if to the geometrical axioms we add propositions relating to the mechanical properties of natural bodies, were it only the axiom of inertia, or the single proposition that the mechanical and physical properties of bodies and their mutual reactions are, other circumstances remaining the same, independent of place [my italics], such a system of propositions has a real import which can be confirmed or refuted by experience, but just for the same reason can also be gained by experience" (Helmholtz 1962, p. 245)
Obviously Helmholtz knew quite well that the group of motions of rigid bodies is also the group of reproduction of identical physical processes in general, "that the mechanical and physical properties of bodies and their mutual reactions are, other circumstances remaining the same, independent of place". Instead of a rigid rod we could use equally well a standing wave or something else and we would always see that the same experimental arrangement is possible at any place in space and in any direction, spatial distances (in a metric which is also invariant under the same group) remaining the same. To Helmholtz it was also clear how a Lagrangian has to be formulated for a Lobatchevskian space, he referred to two articles of Lipschitz, where this was shown (1962, p. 238).
It is true when Poincaré said: "..geometry is nothing else than the study of a group", and again: "The object of geometry is the study of a particular 'group' ". But he should have added: "The true physical geometry is the study of the invariance group of physics."
Some time after the publication of La science et l'hypothèse  Poincaré must have changed his mind and realized that not only immediate experience but also scientific experiments can turn a convenient geometrical group into an inconvenient one. When Poincaré studied Lorentz' theory he was confronted with the indistinguishability of inertial systems, with the principle of relativity. In 1904, in his famous talk at St Louis he discussed the question if this principle is conventional or empirical:
"Let us pass to the principle of relativity: this not only is confirmed by daily experience, not only is it a necessary consequence of the hypothesis of central forces, but it is irresistibly imposed upon our good sense, and yet it also is assailed."
If Poincaré was consistent with his own epistemology the principle of relativity should be conventional in any case. the question is only if we have to consider it as a physical or as a geometrical principle. We could take it to be a geometrical, since it constitutes an important aspect of space-time geometry. That the principle is confirmed by every day experience would be no argument against it's conventionality. Empirical confirmation by "rough facts" of every day experience would only make the principle more convenient ("com-mode").  But then he says "and yet it is also assailed." He refers to the difficulty of maintaining the principle if classical mechanics is combined with Maxwell's electrodynamics. If the principle had been geometrical, it could according to Poincaré's position in La science et l'hypothèse not have been affected by any physical consideration. Only immediate experiences could make it more or less convenient, that would be all. 
But if the principle is considered to be physical, it can become less convenient not only by unfavourable immediate or every day experience but also by the results of physical experiments and by the hypotheses to which we are led by them. Therefore it was important for Poincaré what happened to the principle of relativity. Theoreticians were ready to introduce the ether for formally saving the principle, but experimentalists proved to be more faithful to it and to test it by experiment, which turned out in favour of it:
"..the theorists, to put in accord their other general views, would not have spared it; but experiment has been stubborn in confirming it."
All attempts to measure the absolute velocity of the earth failed.
Somewhat later Poincaré himself discovered that the equations of electrodynamics are invariant under the transformations of the Lorentz group. He considered this invariance as an implication of the "principle of relativity" (Poincaré 1906, p. 495).
The Euclidean group, however, is a subgroup of what we today call the Poincaré group, the Lorentz group plus spatial translations. Therefore Poincaré's result should have changed his attitude towards geometry. In this moment he should have realized that geometry is nothing besides physics and that for geometrical conventions the same rules should be valid as for physical conventions. Both can either improve or turn out to become rather inconvenient in the light of the results of new physical experiments.
In 1907, two years after his famous paper on the dynamics of the electron, Poincaré used the expression "principle of relativity" in a completely different sense in the article "La relativité de l'espace" (ch. II,1 in VS), from which I have already taken some quotations in section 7 of this paper. 
Poincaré starts with the story of the man who does not notice that during the night when he is sleeping the world and all things in it become 1000 times larger than they were before. He would not notice anything. Evidently he cannot talk of relativity in the physical sense here, since spatial dilatations are no invariance transformations of the physical laws. This was, in a qualitative way, already stated by Galileo (Discorsi, 1st day, p. 571f.). If we would build a ship ten times as big as an existing one, but in the same proportions, with the boards, the beams, and the mast also ten times thicker than those of the original, the ship would probably break down. And Galileo was right. Everybody knows that big animals have thick legs (in proportion to their body) while small animals can have very thin legs like insects or spiders. A spider 1000 times bigger probably would be unable to carry its own weight, and so would the man in his bed.
What Poincaré did not tell us was that all physical constants have to be altered in a suitable way, if the change of the world during the night is really going to go unnoticed. The strange growth of anything in the world is then taken by Poincaré as an example to explain the Lorentz contraction and the fact that we do not become aware of it. But in the latter case we are dealing with a real invariance transformation, the Lorentz transformation, while the former example was not of that kind (and thus was no example of relativity). Poincaré thus had begun in 1907 with a very misleading discussion using the word "relativity" equivocal in two different senses, a discussion which kept philosophers occupied for seventy years. In Science et méthode he uses the term in part 2 in the sense in which we have discussed it above in section 7 and in part 4 of his book in the sense of a physical principle of invariance.
It was only six weeks before his death in a lecture given at London, that Poincaré made the distinction between "physical relativity" and "psychological relativity", the first being a physical principle, the second a "psychological" truth about the limitations of our sense organs and of our faculty of direct intuition. (Poincaré's label "psychological relativity" is more to the point than Reichenbach's "philosophical relativity", 1928. p. 206, since it is an empirical fact that we do not immediately perceive or have an immediate intuition of all physical facts.)
After this clarification Poincaré discussed the relation of the principle of relativity to geometry. He again talks about the relativity of space but now in a physical sense and says:
"This principle of physical relativity can serve to define space; it provides us, so to speak, with a new measuring instrument. 
Now space is not any more measured exclusively by rigid bodies but by any kind of mechanical systems:
"We will say: two figures are equal when the same mechanical system, far enough from the neighbouring systems to be considered as isolated and placed first in such a way that its different material points reproduce the first figure and again in such a way that they reproduce the second figure, behaves in the same manner."
These lines agree completely with the quoted passages from Helmholtz. Most probably by his occupation with the theory of Lorentz and the physics of fastly moving electrons Poincaré was led to a more empiricist understanding of geometry which resembles that of the famous Prussian scientist.
The last pages of the lecture are not easy to understand. I had to read it at least half a dozen of times, before I felt I understood them in the essential points. That is partly due to Poincaré's style of presentation. While he first seems to consider the physical principle of relativity a verifiable statement ("It is verifiable, and consequently it might not be verified."), this claim is later withdrawn. Certainly like all general principles of physics the principle of relativity is conventional. But, as I have already pointed out, this holds in the same way as physical principles are conventional: They can become less convenient ("commode") in the light of new experimental data than they seemed to have been before. 
Thus Poincaré at the end of his life - six weeks before his death at least - was much nearer to the later position of the logical empiricists than before. If he had lived some years longer he would have learnt from Einstein's theory of general relativity that space-time and physics are closely connected with each other and cannot be treated as separate subjects. Einstein had his new theory already conceived when Poincaré died, he only hesitated - for reasons which I do not want to discuss here - three further years to publish it (see Pais 1982. p. 223).
Thus fortunately, all important aspects of Poincaré's different concepts of relativity can be elucidated. Even if his formulations sometimes seem to be rather perplexing they can be incorporated into a consistent hole which contains many deep and penetrating insights worth being recognized even today.

10. Summary
It is generally known that Poincaré's conventionalism embraces at least two theses:
1. Geometrical axioms and some physical principles are disguised definitions and as such neither true nor false. (This thesis was not subject of my talk.)
2. Space and space-time are relative.
I hope to have made clear in this talk that the latter thesis is one about "direct intuition". We have
a) no direct empirical perception of geometrical objects,
b) no pure intuition of them which could enable us to ascribe geometrical forms in a unique way to perceived objects,
c) no possibility to identify figures in physical space which are "really" circles etc., not even with the aid of physical principles.
Besides that we have tried to discuss a second, a physical concept of relativity which appears in Poincaré's writings and is not to be confounded with the first. The acceptance of the physical principle of relativity also changed Poincaré's conception of geometry.
Thus we may conclude that Poincaré's conventionalism is much more complex and more profound than what was taken to be his opinion by logical empiricists or many American philosophers of science of our century. Even if we do not accept many of Poincaré's theses, we could only admire the depth of his thought.
Acknowledgement: I am indebted to Robert J. Murphy for checking my English style.

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