## 1. Biography



Jules Henri Poincare was born on April 29, 1854 in Nancy in the
Lorraine region of France. His father was professor of Hygiene in the
School of Medicine at the University of Nancy. His cousin Raymond was
to become the President of the Republic of France during the period
1913-1920 and his younger sister Aline married the philosopher
Emile Boutroux. Henri was a precocious student who rose immediately to
the top of his class, excelling in both science and letters. At age
13, his teacher told his mother that "Henri will become a
mathematician ... I would say a great mathematician"
(Bellivier 1956: 78). During the Franco-Prussian war in 1870 the
Germans occupied Nancy and the Poincare family was obliged to
billet the secretary of the civil commissar of Nancy, with whom Henri
would have a round of conversation each night after dinner in order to
improve his German (see Rollet 2012).



In 1871 Poincare passed the exams in letters with the grade
"good" and that in science with the grade
"fair." He received a zero in mathematics for answering a
different question from the one that was asked, apparently having
misunderstood the question. He then took the preparatory classes in
mathematics and was first in his class, also first in the academic
competition and first in the national competition (concours
general) in elementary mathematics. In Paris, he entered
the Ecole Polytechnique in 1873 and after graduating in 1875
(second in his class having apparently lost points for his inability
to draw), entered the Ecole des Mines. An extensive
correspondence with his mother has been preserved from this period of
study in Paris (Rollet 2017). After brief service as a mine inspector,
he submitted a dissertation on partial differential equations and was
then hired to teach the course on differential and integral calculus
at the University of Caen. In 1880 he submitted a paper solving a
problem in the theory of differential equations to the competition for
the grand prize in mathematics of the Academy of Sciences in Paris.
For the first time he made use of non-Euclidean geometry, which was
seen by most of his contemporaries as purely speculative.



Poincare married Louise Poulain d'Andecy on April 20,
1881 and soon thereafter joined the faculty of sciences at the
University of Paris. In 1886 he succeeded G. Lippmann in the chair of
mathematical physics and probability. In 1896 he was given the chair
of mathematical astronomy and celestial mechanics, in 1902 he was
named professor of theoretical electricity at the Bureau de Poste et
Telegraph and in 1904 professor of general astronomy at
the Ecole Polytechnique. Poincare also worked at the
French Bureau des Longitudes from 1893 (Galison 2003).



In 1889 Poincare won a prize from the King of Sweden for a
question posed by Weierstrass on the stability of the solar system,
that is, the three-body (or \(n\)-body) problem in
classical mechanics. Despite a mathematical error that he discovered
at the last moment (after questions raised by the Swedish
mathematician Lars Edvard Phragmen) and frantically corrected,
the work was important for its use of topology and as a founding
document in chaos theory, for Poincare showed that in general,
the stability of such systems cannot be demonstrated. It is also in
this context that he proved his famous recurrence theorem.



Poincare joined the French Academy of Sciences in 1887 and
became its president in 1906. He was elected to the Academie
Francaise in 1908 on the basis of his three books on
philosophical and general problems in science. He also became a
corresponding member of many international scientific organizations.
His travel included a trip to America to give a lecture at the St.
Louis World's Fair in 1904. Poincare's extensive
correspondence demonstrates the extent of his connection to the
scientific community of his time (Nabonnand 1999; Walter 2007, 2016).
He also co-signed a report that played an important role in the
rehabilitation of Dreyfus (Rollet 1999).



Poincare is also famous for his 1904 conjecture concerning the
topology of three-dimensional spheres which remained one of the major
unsolved problems in mathematics until the Russian mathematician
Grigori Perelman succeeded in demonstrating it nearly one hundred
years later. Poincare lectured on contemporary mathematical
physics for years and was very much abreast of current developments.
Popularizer of science, pedagogue, man of letters, he became an icon
of the popular press (see Rollet 2017, XLVI). In all he published over
five hundred scientific papers and over thirty books. He died July 17,
1912 in Paris from a blood clot to the brain which was a complication
of surgery, survived by his wife and their four children (see Rollet
2000, Boutroux 2012, Ginoux and Gerini 2012).


## 2. The Structure of Science



Poincare sets out a hierarchical view of the sciences in
*Science and Hypothesis* (1902), although he does not
explicitly use this terminology. In his view the special sciences
presuppose physics, which presupposes geometry, which in turn
presupposes arithmetic. Poincare treats topics in a serial
order--first arithmetic, then geometry, then physics, etc. In the
introduction, Poincare says that "Such is the conclusion
we will reach, but to get there, we must first review the sequence of
sciences, from arithmetic and geometry to mechanics and experimental
physics" (Poincare 1902, 2017, 2). The hierarchy
interpretation has recently come under criticism in an important pair
of articles (Dunlop 2016 and 2017). We believe that clarifying what is
meant by hierarchy will resolve this debate (see Stump 2017 and
Folina 2019 for further discussion). What is the relationship between
arithmetic, geometry and the rest of the sciences? Michael Friedman
originated the claim that Poincare presents the sciences in a
hierarchy, and in doing so makes clear that what he means by hierarchy
is that each lower level must first be in place in order to be able to
proceed to the next level:



> 
> 
> But the point perhaps can be made even more clearly if we consider
> Newton's theory of universal gravitation. For Newton's
> *Principia* had already shown clearly how we can empirically
> discover the law of universal gravitation--on the presupposition,
> that is, of the Newtonian laws of motion and Euclidean geometry.
> Without these presuppositions, however, we certainly would not have
> been able to discover the law of gravitation. And the same example
> also shows clearly how each level in the hierarchy of sciences
> presupposes all of the preceding levels: we would have no laws of
> motion if we did not presuppose spatial geometry, no geometry if we
> did not presuppose the theory of mathematical magnitude, and of course
> no mathematics at all if we did not presuppose arithmetic (Friedman
> 1999, 76).
> 
> 
> 



Note that Friedman mentions the theory of mathematical magnitude as
well. There is a step between arithmetic and geometry, namely the
theory of mathematical magnitude, which is precisely the topic of
Chapter 2 of *Science and Hypothesis*. The theory of
mathematical magnitude presupposes the possibility of indefinite
repetition, and therefore something like mathematical induction, which
Poincare presents as central to arithmetic in Chapter 1 of
*Science and Hypothesis*.



There is direct evidence that Poincare thought that mathematics
must be in place before we can do any empirical science. For example,
in the opening of Chapter 12 of *Science and Hypothesis*, he
remarks that: "The purpose of mathematical theories is not to
reveal the true nature of things. Such a claim would be unreasonable.
Their only goal is to coordinate the physical laws that experiment
reveals to us, but that we could not even state without the help of
mathematics" (Poincare 1902, 2017, 143). Dunlop, by
contrast, seems to have something much stronger in mind with the idea
of the hierarchy, which is perhaps why she calls the thesis the
dependency hierarchy interpretation. She seems to take the
interpretation to be a thesis of reductionism, that geometry reduces
to arithmetic, or at least, the intuitive reasoning used in arithmetic
is all that we need in geometry.



> 
> 
> Commentators have suggested that the intuition that grounds the use of
> induction in arithmetic also underlies the conception of a continuum,
> that the consistency of geometrical axioms must be proved through
> arithmetical induction, and that arithmetical induction licenses the
> supposition that certain operations form a group (Dunlop 2016,
> 274).
> 
> 
> 



Dunlop is absolutely correct in saying that this is not
Poincare's view. Rather, he says that we need arithmetic
and further things in order to work in geometry. All that we take
Poincare to be saying is that we still need to add, subtract,
multiply, and divide when we are doing geometry, but we will be doing
more than what we do in arithmetic. The reason that we need a theory
of magnitude before we can do geometry is because we measure things.
The reason we need geometry in classical mechanics is because we are
doing things like tracking the elliptical orbits of the planets. Thus,
we also agree with Dunlop when she says that
"Poincare's views of arithmetic and geometry cannot
be reconciled by supposing geometry to be founded on arithmetical
operations or principles" (Dunlop 2016, 306). We need
arithmetic, plus group theory, plus the conventions in order to have
metric geometry. There may be some authors who made reductionist
claims, but it is not what we take from Friedman's original
presentation. Furthermore, the reductionist interpretation cannot be
right when we get to the physical sciences, where there is clearly
more content in the higher-level sciences--empirical content that
could not possibly be reduced to mathematics.



Poincare discusses the sciences in a sequence, starting with
arithmetic. Mathematical induction is essential in arithmetic, because
only by using it can we make assertions about all numbers.
Poincare considers mathematical induction to be a genuine
synthetic a priori. He next considers magnitude, which requires
arithmetic, but goes further. Likewise, geometry extends our knowledge
still further, but requires the theory of magnitude to make
measurements, and arithmetic to combine numbers. Poincare then
considers classical mechanics, which again extends our knowledge while
relying on the mathematics that came before it. Finally, he considers
theories of physics, where we have genuine empirical results, but
based on the mathematics, hypotheses and conventions that came before.
Thus the sciences are laid out like expanding concentric circles, with
new content being added to the base at each level. The lower levels
are necessary preconditions for higher levels, but they are not
sufficient.



Poincare makes hypotheses central to his analysis of science,
distinguishing four kinds, which are actually given in two lists of
three each (Poincare 1902: 24 and 166-167; 2017: 1 and
109-110). Combining the two lists, these are:


1. Verifiable hypotheses
2. Indifferent (or neutral) hypotheses
3. Natural hypotheses
4. Apparent hypotheses



By verifiable hypotheses, Poincare means general statements
that have been confirmed by experiment. These are the backbone of
natural science and can be seen non-controversially to be compatible
with standard accounts of, say, the Hypothetical-Deductive method.
However, it is important for Poincare that this is not all of
science, as he is arguing for a view of science that is between
empiricism and rationalism.



Indifferent hypotheses are ontological in nature and are, for example,
mechanical models of underlying mechanism. As Poincare
emphasizes, these can frequently be exchanged without sacrificing
empirical accuracy. Take, for example, the propagation of heat, where
"through the effects of the averages and thanks to the
medium's symmetry, all the differences even out" and we
need not inquire how each molecule radiates. Such hypotheses are mere
tools useful for thought but "unverifiable" and
"useless" as such (Poincare 1902: 169; 2017:
111).



Natural hypotheses are necessary conditions for science but are
experimentally inaccessible:



> 
> 
> It is difficult not to suppose that the influence of very distant
> bodies is altogether negligible, that small motions obey a linear law,
> that the effect is a continuous function of its cause. I would say the
> same of the conditions imposed by symmetry. All these hypotheses form,
> so to speak, the foundation common to all the theories of mathematical
> physics. (Poincare 1902: 166; 2017: 109)
> 
> 
> 



Finally, apparent hypotheses are definitions or conventions rather
than actual claims about the world. They therefore may not be
considered to be hypotheses at all, although they are, however, often
mistaken for hypotheses. Poincare argues that (metric) geometry
is the hypothesis most prone to such confusion. Although he presents
arithmetic in *Science and Hypothesis* without mentioning
hypotheses explicitly, we can see a role for them in his analysis here
as well, which we will present in the order that he follows.



Central to Poincare's analysis of arithmetic is the
principle of recurrence, or mathematical induction, which he
considered to be a synthetic *a priori* principle that
"would then seem so self-evident that we would not be able to
conceive of the opposite proposition" (Poincare 1902: 74;
2017: 42). The principle of mathematical induction is necessary to
prove general statements about numbers and is thus a necessary tool in
mathematics. Therefore the principle of mathematical induction is
similar to a natural hypothesis or at least plays a similar role to
that of a natural hypothesis, i. e. a mathematical structure, which is
presupposed and "more fundamental than any particular systems of
natural numbers" (Folina 2020: 293). Of course, there are
important differences between mathematical induction and the principle
of induction used in the natural sciences. The principle of
mathematical induction gives certainty, while the inductive
generalizations in the natural sciences give only probability (1902:
27; 2017: 3). The principle of mathematical induction is based on
"the affirmation of the power of a mind that knows itself
capable of conceiving the indefinite repetition of a particular action
as soon as this action is possible" (1902: 41-42; 2017:
15), so that there can be no uncertainty. It can be argued that
Poincare is justified in including his analysis of arithmetic
in *Science and Hypothesis,* that is, in his analysis of
empirical science, given that there is a striking similarity between
the role played by the principle of mathematical induction and the
principle of induction in natural science, which "is as
difficult to justify [...] as it is to get along without
it" (1905b: 176; 1913b: 345).



Poincare argues that (metric) geometry is neither *a
priori* nor empirical, but rather conventional. It is important to
note, however, that these conventions are guided by experience
according to Poincare, rather than being totally arbitrary, as
some later writers take conventions to be. This connotation of the
word 'convention' has led to many misrepresentations of
Poincare's views. The existence of consistent alternative
geometries shows that the geometry of space cannot be determined *a
priori*. The fact that we experience material bodies and their
relations, rather than space, leads Poincare to argue that the
geometry of space cannot be determined empirically either. Thus,
statements about the geometry of space are apparent hypotheses or
conventions that appear to be making descriptive claims, however they
are actually closer to definitions, though they cannot be reduced to
being merely linguistic.



Poincare conceives of our perception of objects as a
combination of motor and visual impressions that are at first
non-spatial. In order to classify these sensations, which only appear
because of the movement of our bodies, Poincare says we
introduce "sensible space" as a category "of our
understanding" that is not physical space and does not even have
a metric, meaning that "it involves no idea of
measurement" (Poincare 1898). Sensible space is a
necessary condition for our understanding of our place in the world,
that is, for our understanding of the objects around us, their
relations and how they move, but it is not a form of our sensibility,
since individual sensations can exist without it (Poincare
1898: 3). Thus sensible space can be seen as an example of a natural
hypothesis. We introduce a metric to this space in order to measure
distance by using the language of group theory, that is, the
mathematical study of abstract algebraic structures. Groups serve as a
tool of reasoning about representations of muscular sensations. We can
choose a group (or sub-group) which corresponds to the geometry of
Euclid, Lobachevskii or Riemann to describe space. Poincare
proposes a criterion of choice between the three possible geometries,
positing



> 
> 
> the existence of an *invariant* sub-group, of which all the
> displacements are interchangeable and which is formed of all
> translations. (1898: 21)
> 
> 
> 



Only Euclidean geometry meets this criterion, so the conventional
choice of Euclidean geometry is determined by the simplicity of
interchangeability that "translated into analytical language
[...] means that there are fewer terms in the equations"
(Poincare 1898: 43). The empirical underdetermination of the
groups has a purely contrastive (arbitrary) character if the groups
are isolated from the background beliefs furnished by our mind
(Stanford 2009). "Holistic" considerations (commodity,
simplicity) overcome the empirical equivalence between different
geometries and give more evidence for Euclidean geometry. In contrast
to geometry, no choices between alternatives are given in
arithmetic.



We have thus far discussed arithmetic and geometry and examined *a
priori* principles, natural hypotheses, and apparent hypotheses.
In discussion of natural science Poincare presents his views on
verifiable hypotheses, which are characterized by mechanical laws that
are arrived at through experiment:



> 
> 
> What then is a good experiment? It is one that informs us of something
> besides an isolated fact, one that allows us to make predictions; that
> is, that allows us to generalize. For without generalization, we
> cannot make predictions. (Poincare 1902: 158; 2017: 104)
> 
> 
> 



The hypotheses that arise from experiment are verifiable hypotheses.
The process of generalization presupposes a belief in the unity and
simplicity of nature (see 1902: 161 ff.; 2017: 105 ff.), although it
is important to remember that for Poincare experience is always
a complex phenomenon that is theory laden. Any good experiment in
physics can be generalized in an infinite number of ways, depending on
our preconceived ideas, and every generalization results in a
hypothesis. While he emphasizes the creative role of the scientist,
Poincare does not go so far as to say that natural laws are
simple definitions, as Le Roy proposed (1901: 143).



Not all laws are directly accessible to experiment. Generalization in
physical science has to take the mathematical form of a differential
equation in order to grasp complexity and to unify science:



> 
> 
> It is not sufficient for each elementary phenomenon to obey simple
> laws, all those to be combined must obey the same law as well.
> (Poincare 1902: 171; 2017: 113).
> 
> 
> 



The law of inertia, for example, obtains only with respect to the
pre-existing category of differential equations. These differential
equations are, in the final analysis, the physical laws:



> 
> 
> Newton has shown that a law is only a necessary relation between the
> present state of the world and its immediately subsequent state. All
> the laws discovered since are nothing else; they are in sum,
> differential equations. (Poincare 1905b: 119; 1913b: 292)
> 
> 
> 



In geometry, Poincare supposes the pre-existent category of a
group. In mechanics, the categorical tools are the principle of
(empirical) induction and differential equations. So,
Poincare's position represents a hierarchy of elements
that, as in Kant, are presupposed as necessary conditions of the
possibility of the science on the next level. The scheme of
generalization in mechanics appears as follows: Experience provides
complex phenomena, i.e., "a confused mass of facts" which
we reduce into a number of elementary phenomena by means of
experiment. From these elementary phenomena, we generalize (e.g., via
the empirical induction principle) to elementary facts and move, by
means of differential equations, to laws and verifiable hypotheses
whose number should be kept as small as possible (Poincare
1902: 168-171; 2017: 110-113). Finally, laws can be
elevated by decree to the status of principles, thus allowing the
basic principles of mechanics to become conventions; not arbitrary
conventions however, since "They would be arbitrary [only] if we
were to lose sight of the experiments that led the founders of
mechanics to adopt the laws and which, as imperfect as these
experiments may be, suffice to justify them" (1902: 128; 2017:
82). For Poincare the *principles* of mechanics
(principle of inertia, law of acceleration, principle of relative
motion, etc.) are not hypotheses but rather conventions
(Poincare 1902: 151; 2017: 99), although they are of course of
experimental origin. These conventions do not constitute a choice
between physical alternatives, but rather must be interpreted as
defining the basic terms of a theory. Immune to experimental
disconfirmation without being *a priori* in the classical
sense, they are necessary preconditions to the sciences further up in
the hierarchy (Poincare 1902: chap. VI and VII).



In the physical sciences proper, such as optics and electrodynamics,
the scene changes entirely and principles seem no longer to share the
conventional character of the geometrical postulates.
Poincare's "indifferent hypotheses" are those
that the analyst assumes at the beginning of calculations and that are
neither true nor false, but rather play roles as occupants in a
structure. Indifferent hypotheses differ from geometrical conventions
in one very important sense. While geometry rests on a conventional
decree, the structures of physics (mechanics included) do not
systematically rest on indifferent hypotheses; rather, these
hypotheses have only a psychological and pedagogic function as mental
constructs that are in a sense superfluous, since they can change
while leaving the experimental results intact. The basis of
Poincare's structural realism can be seen in his idea
that the content is superfluous and that it is the relations expressed
in natural laws that remain through time that are essential to
science.



Here then is the methodological analogy and link between the principle
of induction, conventions in the sense of apparent hypotheses and
verifiable hypotheses. All are guided by experience without being
(entirely) experimental, and all involve presupposed categories,
natural hypotheses or a given structure (Folina 2020). Verifiable
hypotheses employ conventional elements in the generalization process
and may presuppose the (empirical) principle of induction. Geometrical
conventions are apparent hypotheses guided by the experience of
muscular sensations and involve the category of groups. Both these
sorts of hypotheses, verifiable and apparent, employed in physics and
geometry, are in Poincare's hierarchy of sciences framed
on the one hand by *a priori* principles and natural hypotheses
that are tools both for mathematicians and experimenters, and on the
other hand, by indifferent hypotheses and physical principles. The
indifferent hypotheses are conventional (in the ordinary sense)
stipulations of ontological entities--mechanical models that
increase our theoretical insight, while the physical principles are
either verifiable hypotheses elevated by decree beyond the crucible of
experiment or well-founded with respect to the network of actual
science.


## 3. Philosophy of mathematics


### 3.1 Logic and foundations: intuition and predicativity



Concerning logic and foundations of mathematics,
Poincare's position is governed by two theses:


1. Logical inferences alone are epistemically inadequate to express
the essential structure of a genuine mathematical reasoning in view of
its understandability (see Poincare 1908: 159; 1913b:
452).
2. As a consequence of the logical antinomies, one should avoid any
impredicative concept formation.



Historically, both theses are directed broadly against the founders of
modern logic and set theory such as Cantor, Peano, Frege, Russell,
Zermelo, and Hilbert. On closer inspection, the historical situation
is more complicated. With respect to Georg Cantor, for example, we
must distinguish his work as a logical-set theoretical method useful
in treating mathematical problems especially in the theory of
functions on the one hand, and on the other hand, set theory as a tool
for foundations or for considerations on infinite cardinality. Through
Gosta Mittag-Leffler, Poincare was familiar with
Cantor's work of the first type and uses it in his theory of
Fuchsian functions in 1882 (Poincare 1882: 1167), even having
personal contact with Cantor (see Gray 1991: 22; Cantor 1991: 188; and
Decaillot 2008: 281). However, Poincare rejected the
later foundational work of Cantor, saying that



> 
> 
> There is no actual infinity, the Cantorians have forgotten that, and
> they have fallen into contradiction. It is true that Cantorism
> rendered services, but that was when it was applied to a real problem
> whose terms were clearly defined, and we could walk safely.
> Logisticians as Cantorians have forgotten. (Poincare 1908:
> 212-213; 1913b: 484)
> 
> 
> 



While it has been stressed by all commentators that
Poincare's concerns with the foundations of mathematics
are very different from those of the logicians, there is no unanimity
on the focus of the dispute. Emile Boutroux and Warren Goldfarb claim
that Poincare is condemning the "logical" point of
view for not providing psychological conviction (Boutroux 1914;
Goldfarb 1985), while Detlefsen and Heinzmann argue for a difference
at the epistemological level rather than a psychological one
(Detlefsen 1992, 1993; Heinzmann 1995). In a famous response to
Russell, Poincare refuses to consider logic or epistemology
independently from psychology (Poincare 1909: 482). It would be
false, however, to believe that he thus conflates logic and
psychology. Indeed, Poincare does not use the term psychology
in the modern sense, but rather in all situations where he wants to
emphasize the dimension of understanding, in particular its historical
elements, which he contrasts with the mere correct logical exposition
of the result. It is therefore not surprising that, according to
Poincare, mathematics requires intuition, interpreted as an
element of understanding, not only in the context of discovery, but
equally in the context of justification. As we have already seen, in
arithmetic pure intuition is necessary to justify the complete (or
mathematical) induction principle. It should also be noted that the
term "intuition" is quite ambiguous, a fact that is well
known and explicitly discussed by Poincare himself. In *The
Value of Science*, he distinguished three kinds of intuition: an
appeal to sense and to imagination, generalization by induction, and
intuition of pure number--whence comes the axiom of induction in
mathematics. The first two kinds cannot give us certainty, but, he
says, "who would seriously doubt the third, who would doubt
arithmetic?" (Poincare 1905b: 33; 1913b: 216).



From 1905, Poincare therefore opposed the logicist thesis that
claims to be able to prove all mathematical truths without recourse to
intuition, once the principles of logic are admitted. Poincare
suspected that the logicians were actually making an equivocal use of
the term logic and that they no longer sought the former kind, but
rather a "new logic" containing principles of synthetic
proof or formations of non-logical concept. He was evidently right
given that not only is the modern logic of predicates richer than the
traditional (syllogistic) logic, but in order to carry out the
logicist project, one is even led to expand logic again by certain
set-theoretic postulates of existence.



As Detlefsen points out, Poincare's most interesting
argument does not concern the status of the basic laws of the logical
systems, but rather the "place of logical inference in
mathematical proof" (Detlefsen 1992: 349). According to
Poincare, today's admitted validity of a logical, i.e.,
topic-neutral, inference from a proposition \(p\) to a
proposition \(q\) (in Tarski's conception: every
model of \(p\) is a model of \(q)\) is not enough to
sustain epistemic growth of *mathematical* knowledge, which
requires rather that "\(p\) be seen
*mathematically* to imply \(q\) (i.e., that \(p\)
and \(q\) be seen as united by a
common *mathematical* structure)" (Detlefsen 1993: 272).
Poincare insists on the non-invariance of mathematical
reasoning with respect to its content and advances, so to speak, a
local conception of reasoning (for example, the mathematical induction
principle for arithmetical reasoning). According to this view, a gap
in a proof is no longer a logical gap, but a gap in the mathematical
understanding (Detlefsen 1992: 360, 366). If
"formalization" and a universal calculus are, then, no
longer the criterion of mathematical rigor, how shall rigor be
maintained? Poincare's response mentions in particular
the condensing of the content of the premises to the content of the
conclusion by means of intuition. Rigor is obtained by combining Ernst
Mach's principle of the economy of thought with the concept of
harmony, that is, thanks to "the fortunate inventions of
language" that introduce an order-structure, the complexity of a
domain of objects is made more harmonious by the introduction of an
invariance (see Poincare 1908: 23-30; 1913b:
371-375). Thus, in mathematical reasoning, rigor is placed in
the context of a "mathematical architecture" whose
simplest expression is found in mathematical induction.



Regarding the antinomies discovered at the turn of the century--
such as Richard's or Russell's paradoxes--they are,
according to Poincare, the consequence of an abusive use of
intuition with respect to abstract entities that arises from the
adoption of conceptual realism (Platonism). In a definition of the
form \(\exists y \forall x ((x \in y) \leftrightarrow D[x])\), the
existence of the set \(y\) is intuitively postulated
without any restriction for the predicate \(D[x]\). For an
anti-Platonist like Poincare was from 1906 on, intuition is not
a mode for representing an object, but rather concerns our capacity to
follow a rule. We see this quite clearly in Poincare's
discussion of mathematical induction.



We know that Poincare hoped to avoid the known antinomies by
limiting himself to predicative definitions (see Heinzmann 1985). It
was Russell who introduced the terms "predicative" and
"non-predicative" to highlight the difference between two
kinds of propositional functions, those that determine and those that
do not determine a class. He called the first
"predicative" and the second
"non-predicative." Poincare attributed the mistake
of non-predicative definitions to a vicious circle: a definition is
"non-predicative" if the *definiens* contains or
relates to a totality of which the *definiendum* is a part. He
formulated a principle to avoid circularity, which leads directly to
Russell's famous vicious circle principle:



> 
> 
> I recognize further this element of truth in Poincare's
> objection to totality, that whatever in any way concerns *all*
> of *any* or *some* (undetermined) of the members of a
> class must not be itself one of the members of a class. (Russell 1973
> [1906]: 198)
> 
> 
> 



Russell succeeded in developing a theory that respects his principle,
the ramified type theory. In Poincare, we find nothing that is
comparable, for he believed that the principle guarded against the
definitional mistakes that he denounced. He did not seriously take up
the question and only formulated successively different variants of
the principle when he learned from Ernst Zermelo that the proof,
proposed by Argand and Cauchy, of the fundamental theorem of algebra
makes an appeal precisely to the rejected, non-predicative
definitions. Hence, only when the measures taken against the logical
antinomies affected "real mathematics" did he try to
modify the principle. The discussion between Poincare, Russell,
Peano, and Zermelo on the measures to take lasted for six years. The
difficulty consisted in the obligation to formulate a principle that
is neither too restrictive for the important results in analysis, nor
too liberal with respect to the formations of concept to be excluded
by reason of Poincare's philosophical position. The
absence of a truly convincing solution shows that the analysis of the
concept of predicativity with the help of informal methods appeared to
have reached its limits.



In 1912, the year of Poincare's death, Luitzen Egbertus
Jan Brouwer in his inaugural lecture at Amsterdam characterized two
opposing orientations in mathematics: intuitionism (in large part
French) and formalism (in large part German). The two orientations
were represented by the greatest mathematicians of the antagonistic
Franco-German pairing: Henri Poincare and David Hilbert
(Brouwer 1912: 124-125). According to the historians, the
modernists are thos eat the beginning of the 20th century who
supported an algebraization of the contents up to and including the
language of signs, while the conservative tendency finds the
justification of mathematical knowledge before language, that is, in
the intellect. The polemics of the period persist today, even though
its political metaphor (conservatism/modernism) has been reversed.
Since the death of Bourbaki, it is fashionable to emphasize the
intuitive aspects of mathematics. Yet, it is remarkable that Brouwer
himself criticizes Poincare's conflation of "the
language of mathematics [and] the true mathematical
construction" (Brouwer 1907: 176). Brouwer is quite right to
emphasize that Poincare does not separate language and
mathematics. Poincare ascribes to language--but not to
logic--an essential role in mathematical reasoning. In this
sense, we may say that Poincare is, with respect to Brouwer, a
semi-intuitionist who is located squarely between the two fronts of
the dispute. For him, mathematical reasoning possesses and must
conserve an intuitive content that defies the formal contents. In
summary, the similarities and differences between Poincare and
Brouwer can be shown as follows:


* Poincare and Brouwer share the conviction that intuition is
a guarantee of the certainty inherent in mathematics.
* They share the conviction that complete induction is the
"mathematical reasoning par excellence."
* In contrast with Poincare, Brouwer considers intuition as
the unique basis of mathematical construction.
* Their positions differ with respect to the relation that binds
intuition and language.
* The involvement of paradoxes is attributed by Brouwer to the
application of logical laws to a linguistic structure that can never
be transformed into proper mathematics. According to Brouwer,
formalism is useless; according to Poincare, the Platonistic
philosophy that 
accompanies it must be corrected.
* Poincare rejects an actual infinity; Brouwer accepts it
provided that it can be confined to an intuitive construction.


### 3.2 Geometry: conventions, intuition and aesthetics



Poincare was strongly influenced by and attuned to the French
philosophical scene, which refers to the Kantian tradition to show the
limits of methodological positivism. Members of the so-called
*Critique de la science* movement (Benrubi 1928), which
includes members of the "Boutroux Circle" (Nye 1979)
advocate a mixture of positivism and Neo-Kantianism. They criticize
both Comte's determinism and Kant's static view of the
mind's structure. According to Poincare, mathematics
requires intuition not only in the context of discovery but also in
the context of justification, especially in arithmetic and logic. So
it is not surprising that Poincare was discussed in German
philosophical circles in the Kantian tradition, for example by Ernst
Cassirer (1910), Ilse Rosenthal-Schneider (1914), who is a disciple of
Alois Rhiel, and Moritz Schlick (1918). Thus, Poincare was
taken to be one of the Neo-Kantian precursors of logical positivism.
However, unlike Kant, the arithmetical "pure intuition"
Poincare introduces is intellectual in character and
Poincare does not in the least solve the problem of the unity
of spontaneity and receptivity by the introduction of a pure
sensibility. Rather, he changes the terms of the Kantian contrast,
emphasizing the balance between exactness, that which is common to
many minds and is transmissible by discourse, and subjective
acquaintance, which is intransmissible sensation. He expresses this
difficult balance in a formula well known due to its popularization by
Einstein:



> 
> 
> what they [mathematics] have gained in rigor, they have lost in
> objectivity. It is by distancing themselves from reality that they
> acquired this perfect purity. (Poincare 1908: 131; 1913b:
> 435)
> 
> 
> 



Because of his concern for the application of mathematics,
Poincare did not confine himself to the perfect purity of
mathematics.



The existence of relatively consistent non-Euclidean geometries leads
him to the view that geometric propositions cannot be determined by
*a priori* intuitions and that it is necessary to study
"the structural relations between Euclidean and non-Euclidean
geometry" (Nye 1979: 111). Poincare established a famous
dictionary translating concepts and propositions of
Lobachevskii's geometry into Euclidean geometry in order to
create a model to show the relative consistency of
Lobachevskii's geometry (Poincare 1891: 771; cf. 1902:
68; 2017: 37). Concerning pure *and* applied geometry,
Poincare holds the modernist-sounding views that we have no
pre-axiomatic understanding of geometric primitives, that rigor
demands that we eliminate appeals to intuition in geometry, and that
(metric) geometry is neither true nor false, not only because we
cannot *verify* which one is true, but also because we cannot
empirically determine the metric geometry of space 'in
principle.' When we speak of space, all we can talk about is the
relations of physical bodies:



> 
> 
> [B]ecause of the relativity and passivity of space, they will not
> depend on the absolute position and orientation of the system. In
> other words, the state of the bodies and their mutual distances at a
> given instant will depend only on the state of these same bodies and
> their mutual distances at the initial instant, but will in no way
> depend on the initial absolute position and initial absolute
> orientation of the system. For brevity's sake, this is what I
> might call *the law of relativity.* [...] [S]uch a
> statement is independent of any interpretation of the experiments. If
> the law is true in the Euclidean interpretation, it will also be true
> in the non-Euclidean interpretation (Poincare 1902:
> 98-99; 2017: 61-62).
> 
> 
> 



Although distinguishing in 1903 the n-dimensional
"representative space", which is a physical continuum,
from the geometrical space, which is a mathematical continuum
(Poincare 1905b: chap. IV), Poincare always held a
relationalist view of space that goes further than current space-time
theories allow (Stump 1989; Walter 2010). 



Since Poincare so profoundly modifies the traditional view of
geometry as the *a priori* science of space, authors such as
Schlick, Carnap, and Ernest Nagel (1939, 1979) attribute to him a
leading role in the development of the formal conception of
mathematics. Alberto Coffa extends Nagel's analysis by claiming
that there was a crisis in the interpretation of primitive geometrical
terms in the nineteenth century that led to the development of formal
axiomatic systems (Coffa 1986). References to a central role played by
Poincare can also be found in Joan Richards' survey
(Richards 1994) and in Jeremy Gray's account of the development
of modernism (Gray 2008). The modernist character of
Poincare's approach is also shown by the fact that he
must be ranked with Duhem among those forerunners of logical
empiricism who survive Quine's criticism of the movement, not
primarily for his holistic thesis concerning theory-confirmation, more
so for his conception of geometrical conventions as a kind of hybrid
analytic-synthetic expressions "guided" by experience and
for his ontological "relationalism" which in certain
aspects comes close to Quine's doctrine. Both attempt to account
philosophically for the incompleteness of (scientific) objects.



Poincare argues that



> 
> 
> contrary to the naive dogmatists' view, that which science
> captures are not the things themselves, but simply relationships
> between them. Beyond these relations, there is no knowable reality
> (1902: 25; 2017: 2).
> 
> 
> 



Thus, he rejects metaphysical realism:



> 
> 
> No, beyond doubt a reality completely independent of the mind which
> conceives it, sees or feels it, is an impossibility. A world as
> exterior as that, even if it existed, would for us be forever
> inaccessible (Poincare 1905b: 23; 1913b: 209)
> 
> 
> 



Although Quine's argument ("no entity without
identity") finds a different motivation in a logical context
quite distant from that of Poincare's, the result is
similar. For Quine, mathematical objects are "known only by
their laws" and any other properties that they may have are
irrelevant (Quine 1969: 44).



Poincare's geometrical conventions and his ontological
relationalism are nothing more than two different aspects of his
structural approach, as can be seen by considering two parallel
debates at the turn of the 20th century on the nature of
geometry--one between Henri Poincare and Bertrand Russell
and the other between David Hilbert and Gottlob Frege. Poincare
and Hilbert argue for a new conception of geometrical systems, while
Frege and Russell defend a traditional way of looking at these
systems. The debate takes up the questions: Which geometry is true?
What is the nature of definitions and axioms? To deal with the
multiplicity of geometries, questions about their truth, and the
meaning of their primitive terms, Poincare and Hilbert propose
a radically new view of geometry. They hold that all that we can say
about the meaning of 'point', 'straight line',
'distance', etc. is that which we have stated in the
axioms or principles of the system, and that geometry is not a set of
truths about some previously known objects. Thus Poincare
formulates a new view of geometric theories, that geometry does not
express true or false propositions and that there are no special
objects which geometry studies. Rather, geometry is just a system of
relations that can be applied to many kinds of objects.



Poincare argues against Russell's view of definition,
claiming that the meaning of primitive terms can be fixed by the
geometrical system in which we are working. Fixing the meaning of
primitive terms by the system is the only way to proceed,
Poincare argues, because, if we were to take a primitive term
out of the context of a system, it would lose all meaning:



> 
> 
> If one wants to isolate a term and exclude its relations with other
> terms, nothing will remain. This term will not only become
> indefinable, it will become *devoid of meaning*.
> (Poincare 1900: 78, Poincare's emphasis)
> 
> 
> 



As Poincare explains in a section entitled "Form and
Matter" of his Monist article of 1898, the idea is that in
geometry the properties of the primitives or of the objects to which
geometrical relations are applied are not important at all, as far as
geometry is concerned. The set of relations that hold between the
primitives constitute the form, not the matter, of geometric objects,
and these relations are what are studied. The form constitutes the
relational aspect of Poincare's structuralism.



By recognizing Poincare's heritage as a structural point
of view, Schlick insists on the conventional aspect of his
structuralism. He sees in Poincare's conventions a third
type of definition between the implicit definitions of the axiomatic
method and the concrete or ostensive definitions of physical objects.
These conventions are, as Friedman observes "crucial for an
understanding of how we achieve a coordination between concepts and
empirical reality in the mathematical exact sciences" (2007:
100). As Schlick explains:



> 
> 
> To define a concept implicitly is to determine it by means of its
> relations to other concepts. But to apply such a concept to reality is
> to choose, out of the infinite wealth of relations in the world, a
> certain group or complex and to embrace as a unit by designating it
> with a name. By suitable choice it is always possible under certain
> circumstances to obtain an unambiguous designation of the real by
> means of the concept. Conceptual definitions and coordinations that
> come into being in this fashion we call conventions (using this term
> in the narrower sense, because in the broader sense, of course, all
> definitions are agreements). It was Henri Poincare who
> introduced the term convention in this narrower sense into natural
> philosophy; and one of the most important tasks of that discipline is
> to investigate the nature and meaning of the various conventions found
> in natural science (Schlick 1925, 91-92).
> 
> 
> 



According to Schlick, Poincare's conventions combine
conceptual definitions and coordinations. Given the fact that both
expressions are completely foreign to Poincare's style,
how should one understand Schlick's claim? In fact,
Schlick's expressions designate two different aspects of
Poincare's structural approach, which pay attention to
the fact that Poincare is not an *ante rem*
structuralist (the *ante rem* structuralist takes the pattern
to exist independently of any systems that exemplify it). At first
glance he seems to be an ontological realist with respect to
relational universals (but not with respect to individuals
(*relata*)), despite his predicativist attitude. As Elie Zahar
emphasizes, such a position is not naturally compatible with classical
semantics (logic) because this latter "seems unable to interpret
relations except through their *relata*": one cannot
confirm \(R(a,b)\) without determining



> 
> 
> the referents of both, '\(a\)' and
> '\(b\)' in order then to ascertain whether
> \(\lt a,b\gt\) belongs to the referent of '\(R\)'.
> (Zahar 2001: 38)
> 
> 
> 



An answer to this difficulty can be found by examining four steps of
Poincare's psycho-physiological reconstruction of the
genesis of geometry. In his early articles, Poincare argues
that geometry concerns only the relations expressed in the axioms, not
some inherent features of the primitives:



> 
> 
> What we call geometry is nothing but the study of formal properties of
> a certain continuous group; so we may say, space is a group.
> (Poincare 1898: 41)
> 
> 
> 



The first step of Poincare's construction of geometrical
space via groups proceeds from the observable fact that a set of
impressions can be modified in two distinct ways: on the one hand
without our feeling muscular sensations, and on the other, by a
voluntary motor action accompanied by muscular sensations. So, as in
Carnap's *Aufbau*, the starting point here is the
definition (guided by experience) of two two-place relations: an
external change \(a\) (with '\(x a
y\)' for '\(x\) changes in \(y\)
without muscular sensation') and an internal
change \(S\) (with '\(x S y\)'
for '\(x\) changes in \(y\)
accompanied by muscular sensations'). In the second step, he
introduces a classification of external changes, some of which can be
brought back to the starting point by an internal change (changes of
position), while others (changes of state) cannot. One presupposes "by
convention" or better by decision, that the internal compensation
for external changes is exact and not approximate. In the third step,
Poincare defines the equivalence class of changes of position
through an identity condition with respect to the compensation by
internal changes and calls it a displacement:


1. Two internal changes have to be considered identical iff they
have induced the same muscular sensations. (Poincare 1905b: 79;
1913b: 258 ff.).
2. \( \dfrac{\alpha, \beta \text{ external}}{S \text{ internal}} \Bigg\} \text{ changes}\)
  

\(\alpha \sim \beta\) *iff* \(\exists S (S\alpha \doteq S\beta \doteq I)\)
  

This means that two external changes are equivalent iff they possess a
common character (i.e., to be canceled by \(S\)).
3. \(S = S'\) iff \(\exists \alpha(\alpha S \doteq \alpha S \doteq I)\)
4. If "\(\sim\)" and "\(\approx\)" are equivalent
relations, then each equivalence class of the changes of position is a
*displacement*, so we can recognize that two displacements are
identical.



The fourth step and Poincare's main result is that each
set of displacement classes (external & internal) forms a group in
the mathematical sense. The general concept of a group is a form of
our understanding that "pre-exists in our mind" (1902:107;
2017: 56 and 68; 1898: 42-43) or is an *ante rem*
structure, which the specific displacement-structure \((=\)
transformation group) exemplifies. Thus, the genesis of geometry is
based on an epistemological process founded on previous
classifications, carried out as a relationship between a structure as
norm of invariance and conventionally adapted systems as
instantiations or exemplifications of these norms.



Where there is a choice between different possibilities, conventions
only become involved at a further step of the mathematical
construction where the properties of the transformation group are
studied and choices are made concerning distance (a metric). It
follows that the Euclidean axiom of distance is not
*exclusively* the result of a conventional (decisional) choice
in the linguistic sense, rather, it is a "disguised
definition" or an "apparent hypothesis."
Poincare uses the term "disguised definition"
before 1899, the year of Hilbert's famous *Foundations of
Geometry*, to express the fact that language apparently used
descriptively is not descriptive in actuality. Certain axioms appear
descriptive, but instead define the objectivity of a fact
(Poincare 1899: 274). Such a fact is found to be defined only
up to structure, although it clearly reflects the truth of certain
relations between sensations whose qualities remain unknowable, as
shown by Helmholtz and others (Poincare 1905b: 174.



Poincare's often quoted structuralist *Credo* that
in mathematics an entity "exists" means "that
its defintion does not imply a contradiction" (1902: 70; 2017:
38-39; 1905c: 819), must be seen in a non-Hilbertian light.
Poincare's geometric propositions (conventions), which
are neither true nor false, are not propositional schemata that can be
satisfied by a model. For reasons concerning in particular the
involvement of a vicious circle with respect to the principle of
complete induction, Poincare rejects the idea of ensuring
mathematical reliability with a Hilbert-style consistency proof
(Poincare 1906: 301; cf. Detlefsen 1986: 59 ff. for a critical
assessment). Even with respect to geometry, he takes a structural
position without, however, disengaging meaning and knowledge
completely from ostension. Nevertheless, Poincare begins his
alternative reliability construction only *apparently* with
sensations as ostensive contacts with the given, since he is not an
empiricist. Instead, in a move similar to Helmholtz's conception
of intuition as imagined sensible impressions, Poincare
introduces a representation of a two-place sensation relation, based
on the *imagination* of single sensations. The representation
of an *object* in sensible space means nothing other than the
deliberate and conscious representation of muscular sensations
*thought* necessary to reach the object (Poincare 1902:
82; 2017: 48-49). The sensation relations are then the
place-holder of the categories (forms) of sensible space and of
groups. Poincare's conventions in geometry are first of
all the tool that he uses to close (by decision) the gap between the
*exactness* of a structure and the *objectivity* of
sensation-relations based on an imagined ostensive contact
(*reflecting on sensations*). If this interpretation is right,
then Poincare's concept of structure is not the new
Hilbertian one that derives from the axiomatization of Euclidean
geometry, but rather constitutes a development of the traditional
algebraic concept of structure and concerns continuous groups on the
one hand and the type of iteration and complete induction on the other
hand. His epistemological stance has a strong affinity with
Schlick's *General Theory of Knowledge*. Given, on the
one hand, that "pure quality is impenetrable" and, on the
other hand, that knowledge is public--"no discourse, no
objectivity" (Poincare 1905b: 179; 1913b:
347-48)--how is it possible that although we cannot be sure
that our *experience* is the same, we can intersubjectively
*know* the same thing? The answer is found in the thesis that
"science is a system of relations" (Poincare 1905b:
181; 1913b: 349).



Contrary to *in re* structuralists, Poincare's
structures are not ontologically but rather epistemically dependent on
their instances. The faculty capable of creating the general concept
of a group or of indefinite iteration is the expression of a form of
our understanding "prior to all experience"
(Poincare 1898: 41). He defends a kind of constructive
epistemological Platonism or anti-nominalism (Linnebo 2011), in which
intuition is a cognitive faculty used to grasp from series of
imagined sensations or concrete iterations the general structure. This
is the solution that explains Folina's characterization of
Poincare, which at first sight seems inconsistent, as an
ante-rem structuralist and as an anti-realist with respect to
mathematical objects (Folina 2020). The structure is not itself a
position in a meta-structure, but the psycho-physiological procedure
is the way in which we know of its existence in our mind.
Poincare uses it as a metamathematical "instrument which
serves not to represents things to ourselves, but to reason"
upon the psycho-physiological genesis of *real* actions with
*imagined* sensations (Poincare 1898: 41). The genesis
of geometry is based on an epistemological process founded on previous
classifications, carried out as a relationship between a structure as
norm of invariance and "conventionally" adapted systems as
instantiations and exemplifications of these norms.



The exemplification of the structure by a large variety of systems
(imagined sensations, mathematical and physical facts, objects) having
some characters in common is an aesthetic operation, not a logical
one. The general group structure is given by intuition rather than by
an explicit identity criterion and the determination of its formal
properties expressed by the means of sub-groups is the result of a
mastery occasioned by concrete systems (samples). The mathematician
creates structural harmony as does an artist. In art, the structure
can be implicit, while in mathematics it is normally explicit, even if
it is qualitative. The number of their exemplifications, i.e., of
their applications, is the aesthetic element "uniting"
*both*, mathematical and artistic structures (Poincare
1905b: 104; 1913b: 280). There is for example nearly no mathematical
theory where the notion of group does not intervene by giving the same
name to different "things" (Poincare 1921: 137).
Poincare's geometric or algebraic conventions form a
conceptual construction that is irreducible to a combination of
*clearly distinguished* parts of structural elements and
aesthetic exemplifications. In this sense the result of the
psycho-physiological genesis of geometry is as a whole a system
understood as a *pragmatic* procedure.



The insight that geometry is the result of a genesis where the
description and the construction of objects are two aspects of the
same procedure leads to an alternative interpretation of Quine's
thesis of the incompleteness of mathematical objects and the ideas to
which they belong. The freedom that we have to choose a geometric
distance function is due neither to a general skepticism nor to a
purely verbal accommodation with respect to a set theoretic
development of the foundations of mathematics that itself possesses
ontological commitment, but rather to an ontological peculiarity of a
new sort (Quine 1986: 401). Structures (groups) are not patterns whose
positions are given entities. The genesis of the geometrical metric
structure is as the advent of relations linked to the concrete in an
*aesthetic analysis* of the general idea, rather than either
the creation of the concrete material from the general idea (a group)
or as the creation of the category (a group) from the concrete. What
Poincare calls "objective reality common to many thinking
beings" is exactly the "harmony expressed by mathematical
laws" (Poincare 1905b: 23; 1913b: 209). To move along
this predefined track of a reconciliation of aesthetic sensitivity and
cognition leads directly to Nelson Goodman (1969).



As Philippe Nabonnand has remarked, Poincare's early
presentation of geometrical space is as a whole in fact circular:



> 
> 
> in his 1898 paper, [Poincare] put forward a (mathematical)
> explanation of the three dimensions of space. He observed that the
> Euclidean group, selected after many conventions, can be seen as
> acting on a space of three, four or five dimensions. The choice of a
> three-dimensional space is justified by considerations of commodity.
> Unfortunately, Poincare's argument is viciously circular
> because the choice of the Euclidean group was grounded on Lie's
> classification of transformation-groups operating on \(\mathrm{R}^3\)
> (Heinzmann and Nabonnand 2008: 171).
> 
> 
> 



Nevertheless, Poincare noted his mistake and introduced in
1905b "representative space" -- a physical continuum
to which experience proves that it is more convenient to attribute
three dimensions -- and which may also imply the matching
three-dimensionality of "geometric space," which is
a mathematical continuum, for convenience (Poincare 1905b: 94;
1913b: 272 -- in order to justify his utilization of Lie's
classification. The consequence is that geometry is not independent of
any further mathematical space, the structure of which must be
presupposed as a primitive notion, contrary to the pragmatically
suggested group notion existing in our mind.


## 4. Philosophy of Physics



Concerning the epistemological status of mechanics, Poincare
positions himself, as in his discussion of geometry, as holding a
position between empiricism and a priorism (Poincare 1902: 111;
2017: 71). The principles of mechanics certainly have, according to
Poincare, an empirical origin, but they nonetheless surpass the
bounds of strict empiricism since the experience that leads us to an
experimental result rests on the repetition of a phenomenon and
therefore requires physical or empirical induction (Poincare
1902: 26; 2017: 3). Generalizations are either a hypothesis, a law, or
a confirmable thesis obtained by the fact that each result of
empirical physics can be generalized in different ways. Indeed, we are
obligated to simplify the experimental data and the criterion of
simplicity is relative to the analytical apparatus we employ, so
consequently, simplicity may be only be "apparent." Such
analysis of experimental data may even force us to adjust the
experiment. Reflecting on the generalization process shows that it
presupposes a belief in the unity and simplicity of nature
(Poincare 1902: 159-165; 2017: 105-108). However,
this is only the most general kind of presupposition that is required.
The hypothesis of the law of inertia, for example, cannot be obtained
except with respect to the preexisting category of the differential
equation, which thus replaces the physical induction that leads to a
simple experimental fact.



Poincare distinguished empirical laws from conventional
principles. How does one move from "simple" empirical
laws, understood as *confirmable* hypotheses, to principles as
the result of a conventional decision to withdraw a confirmable
hypothesis from the judgment of experience? In mechanics he employs
the same methodological procedure as in geometry with regard to
classes of displacements and groups:



> 
> 
> When a law has received a sufficient confirmation from experiment, we
> may adopt two attitudes: Either we may leave this law in the fray; it
> will then remain subject to incessant revision, which without any
> doubt will end by demonstrating that it is only approximate. Or else
> we may elevate it into a principle by adopting conventions.
> (Poincare 1905b: 165-166; 1913b: 335)
> 
> 
> 



Thus, in outline, a generalization scheme in physics is presented as
follows: From the phenomenon we move by physical induction to the
experimental result and, thanks to differential equations, to the laws
and the general hypotheses which, by a common decision of the
scientific community, can be elevated finally to a principle.
Certainly this is only a general scheme and the concrete process of a
generalization could include many variations. The difference between
the geometric axiom as a "disguised definition" and the
physical law as a principle seems to define Poincare's
epistemological criterion that distinguishes geometry from mechanics
from the ontological point of view. In geometry, the convenient
conventions (or definitions) are chosen as a function of objects
(solid bodies, rays) which are not the objects of the geometry.
Furthermore, in geometry we presuppose the category of group, so
geometry requires a double conventional abstraction. In mechanics, the
conventions are useful and apply to physical objects (Poincare
1902: 152; 2017: 100). In this context we presuppose the categories of
empirical or physical induction and of differential equations as
necessary preconditions for doing science. Poincare's
approach therefore integrates specifically Kantian elements, the idea
that there are preconditions, while still completely eliminating the
transcendental basis of these components that we find in Kant's
thought.



Poincare's argument in favor of the necessary
transmission of the conventional character of geometry to physics is
based on an analogy, which shows another important relationship
between geometry and physics. The methodological function of
transformation groups with respect to the representative space
corresponds to the function of geometry with respect to physics. The
analogy runs as follows: Just as the law of displacements may
correspond only approximately to the law of groups and is consequently
considered to be the result of two component changes, the first being
a displacement, and the second a qualitative alteration, so
Poincare regards the physical "complex" relation
between two bodies \(A\) and \(B\) as
the result of two components. The first is regarded as a
"simple" geometrical principle, while the second is itself
composed of two "epistemological laws." The bodies \(A\)
and \(B\) are related to figures
\(A'\); and \(B'\) of the geometrical space so that

 

\[ R(A,B) \leftrightarrow R'(A',B') \wedge r\_A (A,A') \wedge r\_B (B,B') \]


\(R'\) being a "properly" geometrical proposition, and the
\(r\_i\) expressing the relationships between objects of the
representative space and the geometrical space, such as the relation
between solid bodies and motion invariants. Poincare declares
that by changing the relations \(r\_i\), the geometrical proposition
\(R'(A', B')\) could even serve to describe the relationship between
two different physical bodies (see Poincare 1905b:
166-67; 1913b: 336). Poincare thereby rules out any
application of geometry to physics based on a structural isomorphism
with a presupposed reality (Mette 1986: 75-80).



The conventional elements in physical theories were often
misunderstood by Poincare's contemporaries. So, for
example, his affirmation that neither of the two propositions
'The earth turns round' and 'The earth does not turn
round' is "more true than the other in the kinematic
sense," is not a rehabilitation of Ptolemy's system, but
the consequence of the fact that in physics, the empirical systematic
(not epistemic) under-determination of theories is limited by unifying
considerations: "A physical theory is by so much the more true,
as it puts in evidence more true relations." So, the 'The
earth turns round' has a richer content expressed by the
"flattening of the earth, the rotation of Foucault's
pendulum, the gyration of cyclones, the trade-winds, and who knows
what else?" (Poincare 1905b: 184-85; 1913b: 353).
However, in physics proper--that is, in optics and in
electrodynamics--the conventional elements seem to be further
weakened:



> 
> 
> We come across a different kind of hypothesis and fully perceive its
> productivity. Undoubtedly, the theories seem at first sight fragile
> and the history of science shows us how fleeting they are. However,
> they do not entirely die out and something remains of each of them. It
> is this something that we must try to sort out, for there and only
> there is true reality (Poincare 1902: 26; 2017:
> 3-30).
> 
> 
> 



In fact, Poincare's model of explanation, founded on a
minimum of well-confirmed hypotheses erected into principles and from
which any meaningful propositions can be deduced, is called into
question by Maxwell's approach. In view of the crisis in physics
that was brewing toward the end of the 19th century,
Poincare's general position will not work without posing
problems with respect to concrete principles. We will briefly mention
the most interesting case, that of the application of the principle of
relativity to electrodynamics, which is one of the most discussed
subjects by historians (e.g., Goldberg 1967; Paty 1993; Miller 1996a;
Zahar 2001; Darrigol 2004; Rouche 2008; Walter 2011, DiSalle
2014)--not least because Poincare and Einstein did not
cite each other on relativity despite the fact that Einstein read
Poincare's *Science and Hypothesis* before 1905
and Poincare wrote Einstein a recommendation letter sometime
after their only meeting at the first Solvay Congress in 1911 (Walter
2007: chap. 59.3). What is uncontroversial is that Poincare
discovered salient points of the special theory of relativity, such as
an operational definition of clock-synchronization to the first order
in \(v/c\), and a relativistic formula of the composition of speed,
the determination of the structure of the Lorentz-group.



Since the Maxwell-Lorentz equations for electromagnetics are not
Galileo-covariant, the classical principle of relativity according to
which measurement in one inertial reference frame can be converted to
another by Galilean transformation is called into question. Using in
his famous St. Louis lecture of 1904 the designation (physical)
"principle of relativity" (Poincare 1905a: 607),
which does not apply "to finite equations that are directly
observed, but to differential equations" (Poincare 1913a:
103; 1963: 19), Poincare reports that Lorentz introduces the
conjectures (i.e., the *ad hoc* hypotheses) of "local
time" and of "uniform contraction in the direction of
motion" in an attempt to save the principle in its application
to the electromagnetic domain (see Poincare 1905b: 132 ff.;
1913b: 305 ff.). It was indeed Poincare (1906) who made
Lorentz's theory fully compatible with the relativity principle.
We are thus naturally inclined, remarks Poincare, to admit the
postulate of relativity in every domain. In postulating the principle
as a "general law of Nature" (Poincare 1906: 495),
its extended form finds its origin in the Galilean principle, and its
motivation in its compatibility with a theory explaining why no
experiment is able to inform us of the earth's motion relative
to the ether. So, the principle is an element of a work in progress
containing several conjectures and there is the possibility that it
will lose its soundness (Poincare 1905b: 146; 1913b: 319).



In fact, there are now two principles: the old one, which was made
immune to revision by considering it as a *conventional
principle* and which serves to construct geometrical space via the
transformation group that is the mathematical expression of the
psychological principle of the free mobility of "rigid"
bodies, and the new one, where the invariance concerns differential
equations and where the geometry could now be rooted in the Lorentz
group at the price of accepting a four-dimensional spacetime with
indefinite metric (see Gray 2012: 111; Walter 2009). So, it seems that
Poincare's old conventional construction of space where
he prefers Euclidean geometry on the basis of considerations of
simplicity and commodity must be subsumed to a new convention on
space-time. Indeed, Poincare never abandons Galilean spacetime
but it is not even clear if before 1912 the principle of special
relativity with Lorentz covariance (instead of with Galilean
covariance) was confirmed well enough to be unassailable by experiment
and to make the principle with Galilean covariance unfruitful (Walter
2009). This situation is what Larry Sklar has called
"transient" underdetermination,



> 
> 
> that is, theories which are *not* empirically equivalent but
> are equally (or at least reasonably) well confirmed by all the
> evidence we happen to have in hand at the moment (see Stanford 2009
> and Sklar 1975: 380)
> 
> 
> 



while waiting to make a decision in face of new results.


## 5. The Influence of Poincare



There is no doubt that Poincare's work has been very
influential both in the sciences and in philosophy. It was already
widely discussed at the time it was first presented -- not only
in France but also in Germany (e.g. Ferdinand von Lindemann (1904) and
Emil Meunier (1919))-- and his geometric conventionalism also
greatly influenced the logical empiricists, who were linked to
Poincare's follower Louis Rougier (Rougier 1920), one of
the organizers of the 1935 Paris congress on scientific philosophy.
Poincare's conventionalism was widely discussed again in
the post-WWII literature on spacetime and relativity theory in the
philosophy of science. His intervention in the philosophy of
mathematics has been a source of great inspiration to critics of
logicism and others. In the tradition of Kant and Mach, who excluded
"metaphysical" speculations from philosophy of science,
Poincare is also considered to be one of the founders of this
field as a discipline (Brenner 2003; Moulines 2006). Poincare
is considered the founder of structural realism and is thought to have
held a productive middle of the road position on scientific realism
(Worrall 1989). His discussion of scientific creativity has spawned a
literature of its own and became a standard account of how scientific
creativity works. His work on the stability of the solar system
influenced the development of chaos theory and the classical concept
of causality based on the fact that similar causes produce similar
effects. Poincare is one of the starting points for arguments
showing the compatibility of ontological determinism and
epistemological indeterminism. His famous recurrence theorem raised
questions for statistical mechanics and his work on probability theory
influenced the early work of Hans Reichenbach (Glymour and Eberhardt
2012).



Insofar as Poincare's philosophical views focus on the
comprehension and explanation of the activities and practice of
mathematicians and scientists, his approach seems more current today
than the approach of the logical empiricists, given that the
contemporary "practical turn" in the philosophy of science
pays less tribute to logic and the foundations of mathematics by
formal representation than did the canonical tradition of the 20th
century philosophy of science. His approach restores the depth to some
of the critical discussions that the Vienna Circle undertook in the
spirit of Poincare from the 1930s onward (Heinzmann 2016).
Though they are missing from standard histories in philosophy of
science, the Swiss philosopher Ferdinand Gonseth, the French
philosophers Jean Cavailles and Albert Lautman, the Italian
epistemologist Federico Enriques and, finally, Jean Piaget, all have
in common a focus on mathematical practice that rejects the logical
empiricists' strict distinction of analytic and synthetic
propositions.



Poincare is often invoked in the philosophy of mathematics
because of his discussions of the paradoxes and his diagnosis, which
became the vicious circle principle. According to
Poincare's pragmatic and anti-Platonist standpoint,
mathematical definitions and proof-procedures should be limited to
predicative ones. Herman Weyl (1918) was the first to propose a
predicative continuum, while later Lorenzen (1955, 1965) and,
especially, Kreisel (1960) and Feferman (1964) each tried to find
predicatively definable sets following Poincare's rough
sketch. Hintikka (2012) argues that logicians and mathematicians did
not know how to implement Poincare's insight because they
did not pay enough attention to the quantifier dependence and
independence and proposes a solution with IF-logic.
Poincare's critical remarks about the logicist project
and about the new logic in general have been evaluated many times and
continue to be discussed in the literature and to inspire those with
sympathy towards intuitionistic or Kantian points of view (Cassirer
1910; Dingler 1931; Folina 1992; Goldfarb 1988; Heinzmann 1995).
Although the influences on Wittgenstein's work are notoriously
difficult to interpret, there are clearly evident similarities to
Poincare's fundamental views (Rodych 2011).



Moritz Schlick is mainly responsible for bringing an interpretation of
Poincare's conventionalism to the Logical Empiricists. In
particular, he advocated Poincare's conventionalism (or
at least what he thought of as Poincare's
conventionalism) in letters to Hans Reichenbach in regards to
Reichenbach's 1920 book *The Theory of Relativity and A
Priori Knowledge* (Coffa 1991, Parrini 1998, Friedman 1999;
Oberdan 2009). Later, however, Schlick criticized generalized
conventionalism in his reprinted and widely-read article of 1935
"Are Natural Laws Conventions?" Reichenbach took up
geometric conventionalism in his *Philosophy of Space and Time*
(1928) and Carnap generalized conventions, taking them to be the
adoption of a formal language of science. Poincare was widely
cited and discussed in the heyday of the Logical Empiricists, though
the meaning of his conventionalism is often distorted, for example,
when Popper characterizes Poincare's conventions as
analytic judgments (Popper 2007: chap. VIII). Einstein's
interpretation of Poincare in his important article
"Geometry and Experience" is emblematic of the fate of his
geometric conventionalism in the philosophical literature. Einstein
says there that although current physics (i.e., the General Theory of
Relativity) is not in accord with Poincare's views, he is
correct "*sub specie aeterni*," that is in
principle in an ideal physics (Einstein 1921). Thus the specific
geometric conventionalism that Poincare advocated was opposed,
but a generalized form of conventionalism was frequently embraced.



Although Poincare's ideas are not the central focus of
the discussion of spacetime theories that blossomed in the 1960s and
1970s, his views are always in the background. Adolf Grunbaum
(1968) defended geometric conventionalism, albeit with his own unique
arguments rather than Poincare's, and critics of
Grunbaum and of conventionalism generally could not help but
direct some of their attention to Poincare who was rightly seen
as the preeminent precursor to all further work on geometric
conventionalism. Some of this literature does directly engage with
Poincare, notably Torretti (1978).



In 1989 John Worrall published a paper defending structural realism as
"the best of both worlds" in the debate over scientific
realism, drawing his inspiration from Poincare. The view came
to be widely discussed in the philosophy of science and spawned
further elaborations and differentiations into different types, such
as ontic and epistemic structural realism (Worrall 1989; Ladyman
2009). Poincare's epistemological thesis that all
statements depend, at least partially, on a conventional conceptual
framework is generalized in Kasimir Ajdukiewicz's radical
conventionalism (Ajdukiewicz 1978) and is related to the Quinean
notion that there are logically incompatible but empirically
equivalent theories (Quine 1990: chap. 41). (See Poincare
1905b: chap. X.3, which is embedded in the discussion about nominalism
with his former pupil Edouard Le Roy for the limits that
Poincare places on these ideas, and (de Paz 2021) for more on
the relation of Le Roy and Poincare.))



Poincare also spawned a large literature on scientific
creativity with his brief account of his discovery while stepping onto
an omnibus during an outing of the relation between Fuchsian (or
automorphic) functions and non-Euclidean geometries:



> 
> 
> At the moment when I put my foot on the step the idea came to me,
> without anything in my former thoughts seeming to have paved the way
> for it, that the transformation that I had used to define the Fuchsian
> functions were identical with those of non-Euclidean geometry. (1908:
> 49; 1913b: 387-388)
> 
> 
> 



The story has been very widely cited and his four-step account of how
one arrives at novel ideas has been adopted by those presenting the
development of creative ideas in analogy with Darwinian blind
variation, such as Donald Campbell (1987 [1960]: 99-100) and
Dean K. Simonton (1988: 27-33; 1995: 468-86; 1999:
32-4); see Kronfeldner for a critical survey of the literature
(2011: esp. 64-65). Poincare's creativity has also
been analyzed and compared to that of other scientists, especially
Einstein, by Arthur Miller (1984, 1992, 1996b, 1997).



Poincare's writings influenced artists (Henderson 1983),
especially the cubist movement which associated their artistic
approach to that occupied by non-Euclidean geometries in the history
of geometry. In "the mathematician of Cubism" Maurice
Princet closely studied the work of Poincare (Decimo
2006) and how he influenced Picasso among others, and Albert Gleizes
and Jean Metzinger (1912), the authors of the famous treatise
"Du Cubisme" that provides theory to the artistic
movement. Poincare's work combines the sensations
produced by movement with the sense of touch (as in *Science and
Hypothesis*) to create the fourth dimension. Maurits Cornelis
Escher uses a paving motif of the Poincare disk of the
hyperbolic plane in order to represent an infinite structure in a
finite space (see Dunham 2010 (Other Internet Resources), Figure 3, p.
3). Poincare also directly influenced James Joyce, who
experimented with non-linearity and topology in his depiction of
events in his literature (McMorran 2020).