08 juillet 2009
Poisson kernels and Fisher information metric
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doi:10.1016/j.difgeo.2007.11.027
Copyright © 2007 Elsevier B.V. All rights reserved.
Fisher information metric and Poisson kernels
Mitsuhiro Itoha, 
, 
and Yuichi Shishidob, 1,
aInstitute of Mathematics, University of Tsukuba, Tsukuba-city, Ibaraki, Japan, 305-8571
bGraduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba-city, Ibaraki, Japan, 305-8571
Received 23 November 2006;
revised 1 February 2007.
Communicated by O. Kowalski.
Available online 31 December 2007.
Abstract
A complete Riemannian manifold X with negative curvature satisfying −b2
KX−a2<0 for some constants a,b, is naturally mapped in the space of probability measures on the ideal boundary ∂X by assigning the Poisson kernels. We show that this map is embedding and the pull-back metric of the Fisher information metric by this embedding coincides with the original metric of X up to constant provided X is a rank one symmetric space of non-compact type. Furthermore, we give a geometric meaning of the embedding.
Keywords: Fisher information metric; Poisson kernel; Rank one symmetric space of non-compact type; Minimal embedding
Mathematical subject codes: 53C35; 53C42; 58B20; 58J32
Article Outline
- 1. Introduction
- 2. Probability measures of positive density function and the Fisher information metric
- 3. Poisson kernels for a complete manifold with negative curvature
- 4. Isometries of X and the space of probability measures on ∂X
- 5. Proofs of theorems
- Acknowledgements
- References
1. Introduction
A Poisson kernel appears in the classical theory of Dirichlet problem for the Euclidean space. As is well known, the Poisson kernel for the n-dimensional unit ball Bn
Rn is given byNow, let X be a simply connected, complete n-dimensional Riemannian manifold with sectional curvature KX satisfying −b2KX−a2<0 for constants a,b.
Then, X is equipped with the ideal boundary ∂X, the space of all oriented geodesic rays up to asymptotic equivalence, which is identified with the space of unit vectors tangent to X at a reference point xo. So ∂X is identified with an (n−1)-dimensional sphere. We can therefore consider the Dirichlet problem for this ideal boundary. Like the Euclidean space case, on X a Poisson kernel Φ(x,θ) can be defined and moreover any solution of the Dirichlet problem can be written in terms of the Poisson integral. See the detailed argument given by Schoen and Yau in [15].
It is a fundamental and important fact that the Poisson kernels Φ(x,θ) are regarded as probability density functions on ∂X, that is,
X. This implies that there is an embedding from the space of Poisson kernels to the space Consider a compact, connected oriented Riemannian manifold M. The space of probability measures whose density function is positive on M has a structure of infinite dimensional manifold. See for this [17] and [19]. On this manifold
a Riemannian metric g which we call the Fisher information metric is defined as
Even though is infinite dimensional, this metric g possesses fine geometric properties. For instance, any orientation preserving diffeomorphism of M induces an isometry of
and moreover g is a metric of positive constant sectional curvature.
The Fisher information metric is a natural extension of the Fisher information matrix. The notion of Fisher information matrix is derived from the theory of statistical inference. The Fisher information matrix determines a Riemannian structure on a parametrized space of probability measures. Study of geometry of
with the Riemannian structure, which we call information geometry, contributes greatly to statistical inference. See [1] for details of information geometry.
We are now able to define a map φ from X to in terms of the Poisson kernels:
Remark that Douady and Earle defined the same map for real hyperbolic spaces and discussed the barycenter map associated with this map. See [6] for the detail. Also Besson et al defined in [3] a similar map to the space of L2-functions on ∂X in order to develop their studies.
The aim of this paper is to investigate geometry of the pull-back metric φ*g of the Fisher information metric g by this map φ.
When X is a rank one symmetric space of non-compact type, the map φ turns out to be an embedding, since the Poisson kernels are expressed as an exponential function of the Busemann functions (see the detailed argument in Section 3). We can then make use of an induced action of isometries of X on which plays an important role in our investigation.
In fact, the action of I(X), the isometry group of X, on ∂X is naturally defined by the fact that the asymptotical relation on geodesic rays is preserved under the isometric action. Further, we get an action of I(X) on by using Bourdon's argument about the Jacobian of the isometric action on the ideal boundary, given in [4], as
I(X) and xo is the reference point of X.
Therefore we obtain in Proposition 4.2 in Section 4 that the φ is equivariant with respect to the actions of I(X) on X and on , namely
Another important fact is that the action of I(X) on is isometric with respect to the Fisher information metric, as given in Proposition 4.3.
By making use of these facts together with the homogeneity of our manifold X we obtain the following
Theorem A
Let (X,h) be an n-dimensional rank one symmetric space of non-compact type. Let φ*g be the pull-back metric of the Fisher information metric via the embedding φ.
Then φ*g is proportional to the metric h. More explicitly,
This means that the embedding φ is isometric up to constant factor. Note that ρ2h is invariant even if we change conformally the metric h into λh by a constant λ. The following theorem asserts that this isometric embedding is minimal.
Theorem B
Suppose that (X,h) be a rank one symmetric space of non-compact type. Then, the mean curvature vector of the submanifold φ(X) in vanishes identically, that is, the φ is minimal.
The facts upon which this theorem depends crucially are that the Levi-Civita connection of the Fisher information metric has an explicit form as is seen in Section 2 and also that the Poisson kernels of rank one symmetric space of non-compact type are in exponential form with exponent of the Busemann functions. Therefore, we consider, conversely, whether this exponential expression of the Poisson kernels characterizes rank one symmetric spaces of non-compact type. We have, in fact, by using the argument of Besson et al. given in [3], the following characterization, though under an additional assumption.
Theorem C
Let (X,h) be a simply connected, complete, n-dimensional Riemannian manifold with sectional curvature satisfying −b2KX−a2<0 for some constants a,b. Suppose n
3 and that (X,h) admits a compact quotient.
If the Poisson kernels Φ(x,θ) for X are given by
It is still an interesting, open question whether the above theorem holds without a compact quotient assumption.
As Proposition 3.5 in Section 3 indicates, we can characterize the exponential expression of the Poisson kernels Φ(x,θ) given as Φ(x,θ)=exp(−cBθ(x)), by means of the mean curvature of horospheres in X geometrically. We have then
Corollary D
Let (X,h) be a simply connected, complete, n-dimensional Riemannian manifold with sectional curvature satisfying −b2KX−a2<0 for some constants a,b. Suppose that n3 and (X,h) has a compact quotient. If the mean curvature of every horosphere of X is constant c and this constant c takes the same value for all horospheres, then (X,h) is a rank one symmetric space of non-compact type.
In Section 2, we consider the space of probability measures and define the Fisher information metric. In Section 3, we give the Poisson kernels on each manifold with negative curvature and characterize a rank one symmetric space in terms of Poisson kernels (Theorem C and Corollary D). We show, in Section 4, that the space of probability measures carries the action of the isometry group of X.
2. Probability measures of positive density function and the Fisher information metric
Let M be a compact, oriented, n-dimensional Riemannian manifold and let dv be the canonical Riemannian volume element of unit volume. Over M we consider the space of smooth probability measures whose density function is everywhere positive;
We define the Fisher information metric g on by
On the Levi-Civita connection
for the Fisher information metric g at is given by
For geometrically interesting facts concerning the Fisher information metric, refer to [8], [18] and [19]. For example, T. Friedrich showed in [8] that the Riemannian manifold is a space of positive constant curvature. Furthermore, an orientation preserving diffeomorphism on M acts on
by pull-back. Then the action is isometric with respect to the Fisher information metric. This fact plays an important role in studying the space of probability measures.
3. Poisson kernels for a complete manifold with negative curvature
Let X be a simply connected, complete, n-dimensional Riemannian manifold with sectional curvature satisfying −b2KX−a2<0 for constants a,b.
The ideal boundary ∂X is defined as equivalent classes of geodesics. Here two geodesics of unit speed c1 and c2 are asymptotic equivalent if d(c1(t),c2(t)) is a bounded function in t, t0, where d is the Riemannian distance function on X. Fixing x0X, we can identify ∂X with the unit sphere of Tx0X. Therefore the ideal boundary ∂X is regarded as the standard sphere Sn−1. A natural topology called the cone topology is defined on . This topology gives a compactification of X. See [2] and [15] for details about the ideal boundary.
Let Δ be the Laplace–Beltrami operator of X. Now we consider the Dirichlet problem for X with respect to ∂X. The existence and uniqueness of a solution to this problem is given in the following theorem;
Theorem 3.1
(See [2] and [15].) Let X be a simply connected, complete, Riemannian manifold whose sectional curvature satisfies −b2KX−a2<0. For any ψC0(∂X), there exists a unique function such that
Fix x0X as a base point. Let dθ be the normalized Riemannian volume form on ∂X=Sn−1 defined by
Definition 3.2
(See [2] and [15].) Let θ be an element of ∂X. A continuous function ; x
Φθ(x) is called the Poisson kernel normalized at x0 for θ∂X if it is harmonic in X and satisfies the following properties;
1. Φθ(x)>0 for all xX, 2. Φθ(x0)=1,
3. limx→θΦθ(x)=+∞,
4. θ′
∂X, θ′≠θ
limx→θ′Φθ(x)=0.
According to [2] and [15], the solution of the Dirichlet problem for X can be written as the Poisson integral formula
We remark here that for every point xX the Poisson kernel Φ(x,θ) is a probability density function on ∂X=Sn−1. In fact, assuming ψ=1 on ∂X we have from (5) and (6)
When X is a rank one symmetric space of non-compact type, we have the following relation between the Poisson kernel and the Busemann function;
Here, the Busemann function Bθ is defined by
We can easily show that the Busemann function Bθ(x) satisfies the following:
- 1. Bθ(x0)=0,
2. limx→θBθ(x)=−∞,
3. limx→θ′Bθ(x)=+∞, θ≠θ′.
If X is a rank one symmetric space of non-compact type, we have the realizations of X by the Poincaré models. For example, when X is a real hyperbolic space Hn(R) or a complex hyperbolic space Hn(C), we have
The Poisson kernels for a real hyperbolic space or a complex hyperbolic space are given by
is the canonical Hermitian inner product in Cn. We refer to [13] for the Poisson kernels for the hyperbolic plane and also to [9] for the complex hyperbolic space.
From the properties of the Busemann functions together with (8), it is clear that the above map φ is injective. Therefore we obtain
Lemma 3.3
When X is a rank one symmetric space of non-compact type, the map φ is an embedding.
Now, we characterize a rank one symmetric space of non-compact type by Poisson kernels. To obtain our theorem we make use of a fact which is showed by G. Besson, G. Courtois and S. Gallot;
Theorem 3.4
(See [3].) Let X be a compact manifold with negative curvature and be the universal covering of X. Suppose that
is asymptotic harmonic, i.e., the mean curvature of any horosphere in
is constant. Then
is a rank one symmetric space.
Proof of Theorem C
Let e1,…,en be an orthonormal basis for TxX such that eiej=0 for any i,j at xX. Now, we calculate the Laplacian of the function xΦ(x,θ). Then we have from the assumption of the theorem
gradBθ(x)=1 (see [16]), we obtainThe gradient vector −gradBθ(x) is a unit inward normal vector on the horosphere including θ and x. Remark that the second fundamental form Π(V,W) with respect to the normal vector −gradBθ(x) of a horosphere in X is given by
Π(V,W)=−
V(gradBθ(x)),W
=−HessBθ(x)(V,W),
, is the Riemannian metric on X and is the Levi-Civita connection of X (see [7]). Thus, ΔBθ(x)=−c implies that the mean curvature of any horosphere in X is constant c/n. Therefore X is a rank one symmetric space of non-compact type by Theorem 3.4. Moreover, from the uniqueness theorem of the Poisson kernel (Theorem 2.8 in [15]), the constant c is the volume entropy. □
We obtain naturally the following result from the calculation in the above proof.
Proposition 3.5
Let X be a simply connected, complete, n-dimensional Riemannian manifold whose sectional curvature satisfies −b2KX−a2<0 for some constants a,b. If all the horospheres in X have same constant mean curvature, then the Poisson kernels of X can be written as exponential functions whose exponents are the Busemann functions.
Proof
Let c/n be constant mean curvature of horospheres in X. Then, we have ΔBθ(x)=−c from (11). Define Φ(x,θ) by Φ(x,θ)=exp(−cBθ(x)). Since the function Φ(x,θ) satisfies the equation (10), Φ(x,θ) is harmonic in X.
We can see easily that Φ(x,θ) is continuous in and also satisfies the conditions of Definition 3.2 because of the properties of the Busemann function. The uniqueness theorem of the Poisson kernels implies that
It is also interesting to consider the following problem; how do the mappings behave in the case of supXiKXi→0 or infXiKXi→−∞?
4. Isometries of X and the space of probability measures on ∂X
Let be the embedding. Since the space
carries the Fisher information metric g, we are interested in geometry of the pull-back metric φ*g of the metric g via φ.
In order to investigate the metric φ*g, we rely on the equivariant property of the map φ.
We denote by I(X) the group of isometries of a Riemannian manifold (X,h). Let γI(X). Then γ induces naturally an action on the space ∂X. The following proposition shows that the Jacobian of the action of γI(X) on ∂X yields the Poisson kernel.
Proposition 4.1
(See [3] and [4].) The pull-back of the normalized Riemannian volume form by γ
I(X) is
The action of I(X) on ∂X induces naturally an action on the space which we denote by the same symbol γ as
I(X) and Φ(γ(x),θ)=Φ(x,γ−1(θ))Φ(γ(x0),θ).
Proposition 4.2
The embedding φ is I(X)-equivariant, namely for all γI(X)
Remark
In [3] Besson, Courtois and Gallot use systematically the notion of Γ-equivariant immersion of a rank-one symmetric space Y of non-compact type into the unit sphere in the L2-space L2(∂Y) over the ideal boundary ∂Y, where Γ is the discrete subgroup of I(Y). By applying such immersions they obtained characterizations of a rank one symmetric spaces of non-compact type in terms of the volume entropy. It is not vague to point out that the framework of our study is similar to theirs, whereas they use the L2-metric, not Fisher information metric.
Now we consider the Fisher information metric g on , and we will show that the action of I(X) on
preserves g.
Let Ω(∂X) be the space of smooth (n−1)-forms on ∂X. We define the action of γI(X) on Ω(∂X) by
I(X) on
Proposition 4.3
The action of I(X) on is isometric with respect to the Fisher information metric, that is, any γ
I(X) satisfies
g(dγ(σ1),dγ(σ2))γ(μ)=g(σ1,σ2)μ,
at any
Proof
Write and let
be two tangent vectors. Let γ
I(X). Since the differential map of γ is given by (16), we have
5. Proofs of theorems
Let (X,h) be a rank one symmetric space of non-compact type of dimension n, and x0 be the origin of X. X is two-point homogeneous so that I(X)x0 acts transitively on the unit sphere Ux0X in Tx0X (see [12] p. 355), where I(X)x0={γI(X)|γx0=x0} be the isotropy subgroup of I(X) at x0. Therefore, from Propositions 4.2 and 4.3 and Propositions 4.2 and 4.3 it is sufficient to consider at the single point x0=oX,
Proof of Theorem A
Let uToX be an unit vector with respect to the original metric h. From Φ(x,θ)=exp(−ρBθ(x)), the formula of the differential map of the embedding φ is given by
Then we have
Remark that dBθ(c′(0))=−1. For any unit vector uToX, we can take an orthonormal basis e1,…,en such that u=e1, then we have
Proof of Theorem B
Let e1,…,en be the basis for ToX. Since , we have
.
By the formula (4) of the Levi-Civita connection of the Fisher information metric g we obtain
dφo(ei)dφo(ej) vanishes, we complete the proof of Theorem B. □ Acknowledgements
The second author is grateful to Professor Takashi Kurose and Professor Hiroshi Matsuzoe for their valuable suggestions and encouragement. The authors would also like to thank to a referee for useful comments.
References
[1] S. Amari and H. Nagaoka, Methods of Information Geometry, Amer. Math. Soc. Translations of Mathematical Monographs, Oxford University Press (2000).
[2] M.T. Anderson and R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. Math. 121 (1985), pp. 429–461. MathSciNet | Full Text via CrossRef
[3] G. Besson, G. Courtois and S. Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictement négative, GAFA 5 (1995), pp. 731–799. MathSciNet | View Record in Scopus | Cited By in Scopus (97)
[4] M. Bourdon, Structure conforme au bord et flot géodésique d'un CAT(-1)-espace, Enseign. Math. 41 (1995), pp. 63–102. MathSciNet
[5] R. Courant and D. Hilbert, Methods of Mathematical Physics, Interscience, New York (1962).
[6] E. Douady and C. Earle, Conformally natural extension of homeomorphisms of the circle, Acta. Math. 157 (1986), pp. 23–48.
[7] P. Eberlein, Geometry of Nonpositively Curved Manifolds, The University of Chicago Press, Chicago (1996).
[8] T. Friedrich, Die Fisher Information und symplektische Strukturen, Math. Nachr. 153 (1991), pp. 273–296. MathSciNet | Full Text via CrossRef
[9] C. Robin Graham, The Dirichlet problem for the Bergman Laplacian I, Comm. Partial Differential Equations 8 (1983), pp. 433–476.
[10] Y. Guivarc'h, L. Ji and J.C. Taylor, Compactifications of Symmetric Spaces, Progress in Mathematics, Birkhäuser (1998).
[11] E. Heintze and H.C. Im Hof, Geometry of horospheres, J. Differential Geom. 12 (1977), pp. 481–491.
[12] S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press (1962).
[13] S. Helgason, Groups and Geometric Analysis, Academic Press (1984).
[14] H. Izeki, S. Nayatani, Entropy and rigidity of locally symmetric space of rank one (An introduction of the work of Besson, Courtois and Gallot), in manuscript (in Japanese), 1995.
[15] R. Schoen and S.T. Yau, Lectures on Differential Geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, International Press (1994).
[16] T. Sakai, Riemannian Geometry, Amer. Math. Soc. Translations of Mathematical Monographs, Oxford University Press (1996).
[17] Y. Shishido, Geometry of infinite dimensional statistical manifolds, master thesis, 2004.
[18] Y. Shishido, Strong symplectic structures on spaces of probability measures with positive density function, Proc. Japan Acad. 81 (2005), pp. 134–136. Full Text via CrossRef | View Record in Scopus | Cited By in Scopus (1)
[19] Y. Shishido, A note on the Poisson structures of the space of probability measures, Math. Nachr. 280 (2007), pp. 1827–1833. Full Text via CrossRef | View Record in Scopus | Cited By in Scopus (1)
Corresponding author.
1 Present address: Meikei High School, Inarimae, Tsukuba, Ibaraki, Japan, 305-8502.
a Clifford-Finslerian unification
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doi:10.1016/j.chaos.2008.09.004
Copyright © 2008 Elsevier Ltd All rights reserved.
A Clifford–Finslerian physical unification and fractal dynamics
Fred Y. Ye, a,
aDepartment of Information Resources Management and Center for Interdisciplinary Studies, Zhejiang University, Hangzhou 310027, China
Accepted 8 September 2008.
Available online 15 October 2008.
Abstract
A Clifford–Finslerian physical unification is proposed based on Clifford–Finslerian mathematical structures and three physical principles. In the Clifford–Finslerian mathematical structure, spontaneous symmetry breaking is automatically embedded in fractal branches. With the action principle, connection principle and computation principle, physics can be unified, in which the Riemman–Einstein system and Euclid–Newton system are naturally included when quaternion are reduced to complex and real phases.
Article Outline
- 1. Introduction
- 2. Mathematical methods
- 2.1. Algebraic structure
- 2.2. Analytical structure
- 2.3. Geometrical structure
- 3. Physical principles
- 3.1. The first physical principle
- 3.2. The second physical principle
- 3.3. The third physical principle
- 4. Quaternionic Finslerian physics
- 5. Finslerian metric and fractal dynamics
- 6. Discussion
- 7. Conclusion
- Acknowledgements
- References
1. Introduction
Some scientists believe that super-symmetry and string theory are biased towards mathematics rather than physics [1] and [2] and so do I. Let us go back to physical tradition and recall experimental and empirical foundations and the main theories of physics. Although various string theories and M theory have shown us many beautiful situations in theoretical physics, they are only mathematical images. According to our knowledge, real physics that relies on experiments ended at three theories: (1) General relativity, (2) Standard model of cosmos (big-bang), and (3) Standard model of particle physics [3]. Thus, these theories are our starting point for further consideration.Since the 1980s, some mathematicians and physicists have probed into quaternionic physics [4] and [5] and Clifford–Finsler structures [6] and [7], while others developed string theory or M theory [8], [9] and [10], loop quantum gravity [11] and [12], spinor and twistor theory [13] and [14], causal dynamical triangulations [15] and [16] and E-infinity unification [17], [18] and [19]. In this paper, some basic principles [20] are developed for new ideas of Clifford–Finslerian physical unification and fractal dynamics.
2. Mathematical methods
A quaternion consists of two complex or four real factors. In mathematics, a quaternion has both Cayley–Dickson construction [21] and scalar–vector construction [22].
2.1. Algebraic structure
According to the Cayley–Dickson construction, we can construct logically complex or quaternion with real number or function X, in which a complex is expressed by two real parameters or functions and a quaternion by four real parameters or functions.
For real X, we define its conjugate . And two real X1 and X2 multiply as
For complex Z = (X, X′) with Re(Z) = X and Im(Z) = X′, we have its conjugate
For quaternion Q = (Z,Z′) with dual complex Z and Z’, we have its conjugate
And according to scalar–vector construction, a quaternion Q = q0 + q1i + q2j + q3k = (φ, A) with scalar q0 = φ and vector (q1, q2, q3) = A has its conjugate
Q1Q2=(φ1,A1)(φ2,A2)=(φ1φ2-A1·A2,φ1A2+A1φ2+A1
A2),
is wedge product and a b = -b a.
As (5) and (7) co-exist, we have
2.2. Analytical structure
When we consider mathematical analysis, the differential operator d and integral operator ∫ should be defined as a whole operator:
With same reason, the integral operator ∫ acts on differential form ω in a Cayley–Dickson construction or scalar–vector construction as different branches
All the algebraic and analytical branches formulate fractal branches of the quaternionic world.
2.3. Geometrical structure
According to Chern’s analysis [23], there is a Chern connection form in Finsler bundle p*TM → PTM as the unique solution of structural equations
ωj ωk is a Cartan tensor. The Finsler structure becomes Riemann structure if A = 0. And the curvature of the Chern connection isThe Clifford–Finslerian structure with above fractal branches can be a general mathematical structure for physics.
3. Physical principles
For formulating a physical system, we need some basic physical principles with some basic physical quantities. Based on physical tradition, time t and space S are needed, so are energy E and momentum p, which just construct two scalar–vector quaternion pairs X = (t, −S) and F = (E, −p). While scalar time t is one dimension, vector space S naturally is three dimensions.
In synthesized traditional and modern physics, the following three principles should be introduced and maintained.
3.1. The first physical principle
The first physical principle is the action principle, which links dynamic mechanism and symmetry as
The first physical principle originates from the Lagrange–Hamilton principle and the Nöther theorem and determines the kinematical and dynamic mechanism of a physical system, including Newton system, Einstein system, and Yang–Mills system.
3.2. The second physical principle
The second physical principle is the connection principle, which links physical potential and mathematical connection, physical field strength and mathematical curvature together, with the following mathematical structure:
The second physical principle originates from kinematics, Newton’s 2nd law and Einstein‘s equivalence principle, which determines the kinematical characteristics and the dynamic structure of a physical system, which covers Newtonian theory, Einstein’s general relativity, and Yang–Mills gauge field theory. While every potential has its relative connection and each connection determines its potential, every field strength links its relative curvature and each curvature determines its field strength. If a relation between connection ω and time–space X with ω = dX, and energy E (scalar) and momentum p (vector) constructing with F = (E, −p) = dP become physical laws, matching curvature tensor Ω = d2X, a physics will be established in the time–space.
The above two principles link physical structures with mathematical structures together.
3.3. The third physical principle
The third physical principle is computation principle, which links inner geometric structure to outer topological index together as
The third physical principle originates from Gauss–Bonnet and Stokes theorems, which are mathematical laws, with which one can probe into inner structure with surface parameters so that they construct a computation system for physics.
In these three principles, each consists of both mathematical and physical structures. While the mathematical algebra–geometry structures are respectively real-Euclid, complex-Riemann, quaternion-Finsler, the physical systems may become Newton system, Einstein system and so on. We can apply the above three principles to any physical system as a general approach framework of physics.
4. Quaternionic Finslerian physics
Applying the above mathematical structures and physical principles to any physical system, we see the following quaternionnic Finslerian physics in quaternionnic Finslerian time–space X = (t, S).
Cayley–Dickson branch → scalar–vector branch
F=(FZ,FZ′)=k2Ω=k2(dω-ωω)=k2(∂,)2(t,S)((1-(t,S)).
In a real-Euclidean structure, (18), (19), (20) and (21) embed Newtonian mechanics and Maxwellian equations. In a complex-Riemannian structure, (18), (19), (20) and (21) support Einstein’s general relativity and Dirac’s theory, when energy–momentum tensor (E, p) is applied. And in a quaternion-Finslerian structure, (18), (19), (20) and (21) provide a new system. As a quaternion-Finslerian structure contains double complex-Riemannian structures, it constructs two branches of physics in complex-Riemannian time–space. And as one complex-Riemannian structure includes two real-Euclidean structures, a quaternion-Finslerian structure can be divided into four real-Euclidean structures, which means that four branches co-exist and may match four basic interactions.
When the quaternion-Finslerian structure is reduced to the complex-Riemannian structure, its algebra is complex (it can be divided as two real parts) and its geometry is Riemannian. And when it is reduced further to the real-Euclidean structure, its algebra is real and its geometry is Euclidean.
5. Finslerian metric and fractal dynamics
If we want to measure the geometry of space–time, we need a key element, which is metric ds of time–space. In a Clifford–Finslerian structure, the metric ds should apply the Lorentz–Cao transformation [24], which is
In time–space X = (t, S), the Lorentz–Cao invariance remains as
In (22), we see that there is a fractal factor with a typical catastrophic pattern like symmetry breaking
And (21) embeds a mechanism of fractal dynamics with the following geodesic line:
When it is Riemann space, (26) reduces into Riemann metric
When Einstein’s view of space as warped was confirmed, we knew that Riemann’s geometry fits Einstein’s physics, or Einstein’s physics is based on Riemann’s geometry. While we logically expand algebraic structures from real, complex to quaternion, we have to develop geometric structures from Euclid, Riemann to Finsler. The developments will introduce some new approaches to the world and some new ideas for physics.
6. Discussion
In the above Clifford–Finslerian unification, we see three constants k1, k2 and k3. The three constants may be sources of basic physical constants. If minimum quantum space is the unit of reduced Planck constant , we can have
. If we accept special relativity, we know max(dS/dt) = c, where c means the velocity of light. So, we have min(dt/dS) = 1/c. And if minimum two order derivative of time–space is gravitational constant g, we have min(d2S/dt2) = g. That is summed up as
If we think that time–space is always the combination of minimum quantum units and all constants relate to the minimum quantum units, we can introduce the physical constants by
In the new system, we can maintain experimental foundations and theoretical traditions of physics, while general relativity and standard models can be saved. Beyond standard models, string theory and M theory have developed so far from phenomena, as well as other unified theories [25], [26], [27] and [28]. And this is a new investigation.
7. Conclusion
The above physical unification is based on Clifford–Finslerian mathematical structures and three physical principles. The first physical principle is the action principle, which links dynamic mechanism and symmetry, and determines the kinematical and dynamic mechanism of a physical system. The second physical principle is the connection principle, which links physical potential and mathematical connection, and physical field strength and mathematical curvature and determines the kinematical characteristics and dynamic structure of a physical system. The third physical principle is the computation principle, which links inner geometric structure to outer topological index together and constructs a computation. These three principles can be applied to physical systems with real, complex and quaternion algebraic phases and Euclidean, Riemmanian, Finslerian geometric structures, which produce Newton’s approach, Einstein’s approach, and new approaches to the world and new physics.
The Clifford–Finslerian physical unification is a new idea based on Clifford–Finslerian mathematical structures and three physical principles. In the Clifford–Finslerian mathematical structure, spontaneous symmetry breaking is automatically embedded in fractal branches. With the action principle, connection principle and computation principle, physics can be unified, in which the Riemman–Einstein system and Euclid–Newton system are naturally included when quaternion are reduced to complex and real phases. While formula (15), (16) and (17) construct an abstract framework, the formula (20), (21), (22), (23) and (26) will provide the concrete physics when they are applied. Although this is only a preliminary idea, we hope that deeper development will be fruitful.
Acknowledgement
Thank Ms. Regina P. Entorf at Wittenberg University in the US for her assistance with English wording.
References
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[4] S.L. Alder, Quarternionic quantum field theory, Commun Math Phys 104 (1986), pp. 611–656. Full Text via CrossRef | View Record in Scopus | Cited By in Scopus (13)
S.L. Alder, Quarternionic quantum mechanics and quantum fields, Oxford University Press, Oxford (1995).
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[6] Vacaru S. Nonholonomic clifford structures and noncommutative Riemann–Finsler geometry. arXiv: math.DG/0408121, 2004 or open access at http://www.mathem.pub.ro/dgds/mono/va-t.pdf [chapter 15].
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[11] A. Ashtekar and J. Lewandowski, Background independent quantum gravity: a status report, Classical Quant Grav 21 (2004), pp. R53–R152. Full Text via CrossRef | View Record in Scopus | Cited By in Scopus (299)
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[18] M.S. El Naschie, Quantum gravity, clifford algebras, fuzzy set theory and the fundamental constants of nature, Chaos, Solitons and Fractals 20 (2004), pp. 437–450. Article | PDF (348 K) | View Record in Scopus | Cited By in Scopus (52)
[19] M.S. El Naschie, Quantum golden field theory – ten theorems and various conjectures, Chaos, Solitons and Fractals 36 (2008), pp. 1121–1125. Article | PDF (119 K) | View Record in Scopus | Cited By in Scopus (7)
[20] Ye FY, Approaches to the natural world based on three principles of mathematical physics. Analysis and creation, 2nd ed. Beijing: China Society Press; 2008. p. 175–84. Open access at http://www.paper.edu.cn, 200802-2.
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30 juin 2009
from phenomenology to phenomenotechnique
From phenomenology to phenomenotechnique: the role of early twentieth-century physics in Gaston Bachelard’s philosophy
Cristina Chimissoa,
aDepartment of Philosophy, Faculty of Arts, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK
Available online 29 August 2008.
Abstract
Bachelard regarded the scientific changes that took place in the early twentieth century as the beginning of a new era, not only for science, but also for philosophy. For him, the theory of relativity and quantum mechanics had shown that a new philosophical ontology and a new epistemology were required. I show that the type of philosophy with which he was more closely associated, in particular that of Léon Brunschvicg, offered to him a crucial starting point. Brunschvicg never considered scientific objects as independent of the mind, and as a consequence questions such as the existence of particles independently of the mind, theory or apparatus, were absent from his philosophy, which was rather aimed at analyzing the mind critically, and above all historically. Bachelard accepted the fundamental ideas of Brunschvicg’s philosophy; however, his own reading of contemporary science enabled him to go beyond it, as shown by his emphasis on the social production of knowledge, and by his removal of the distinction between ideas and technologically produced objects of knowledge. For him, modern science teaches philosophy that knowledge is not a phenomenology but rather a ‘phenomenotechnique’. I argue that Bachelard’s view that philosophy ‘should follow science’ stems from moral considerations.
Keywords: Gaston Bachelard; Léon Brunschvicg; Phenomenotechnique; Chosisme; Theory of relativity; Quantum mechanics
Article Outline
- 1. Introduction
- 2. At the roots of Bachelard’s third way: beyond the traditional philosophical dichotomies
- 3. Noumena and Bibliomena : the creativity of human knowledge
- 4. If they are not things, then what are they? The ontology of microphysics
- 5. From phenomenology to phenomenotechnique
- 6. Why should philosophy follow science?
- References
1. Introduction
The era of the new scientific mind could be very precisely dated from 1905, when Einstein’s relativity came along and deformed primordial concepts that we thought were fixed forever. From then on, reason multiplied its objections, dissociating fundamental ideas and then making new connections between them, trying out the boldest of abstractions. Over a period of twenty-five years, ideas appear that signal an amazing intellectual maturity, any one of which would suffice to shed lustre on the century. Among these are quantum mechanics, Louis de Broglie’s wave mechanics, Heisenberg’s physics of matrices, Dirac’s mechanics, abstract mechanics, and doubtless there will soon be abstract physics which will order all the possibilities of experience. (Bachelard, 1993 [1938], p. 19)
There is no doubt that Gaston Bachelard regarded the dramatic developments that took place in early twentieth-century physics as the beginning of a new era, not only for science itself, but also for philosophy. The new physics had shaken common sense assumptions about time, space, causality and indeed the very bases of philosophical ontology. Far from trying to reconcile contemporary physics with traditional philosophical concepts, not to mention common sense, Bachelard thought that the former should lead the way to a revolution in our way of conceiving the world and of reasoning. It was contemporary science that for Bachelard had completed the rupture between scientific knowledge and common knowledge, and in so doing had started the ‘fourth’ period of philosophy — the contemporary period — following the ancient period, the Middle Ages and the Modern era (Bachelard, 1986 [1949], p. 102). He remarked that ‘science in effects creates philosophy’ (Bachelard, 1991 [1934], p. 7), and that reason itself ‘should obey science’. However, he argued, science changes across time, and as a consequence reason should obey ‘the most highly evolved science, science in the process of evolution’ (Bachelard, 1988 [1940], p. 144). The philosophical concepts of time, movement and causality were for him based on nineteenth-century science, and they did not reflect the impressive ‘rupture’ that science had experienced at the beginning of the twentieth century. Indeed, a large part of Bachelard’s polemics with some contemporary philosophers rested on his perception that they based their theories on concepts that had been superseded in the sciences. For him the philosophy of physics lead to a rejection of current epistemological doctrines, such as positivism, pragmatism and realism, which in his words become ‘positivist agnosticism’, ‘pragmatic tolerance’ and ‘traditional philosophical realism’ (Bachelard, 1991 [1934], p. 9). These doctrines were in his view still rooted in pre-twentieth-century science and rationality, and still linked to an immediate, intuitive and common sense way of looking at the world.
The philosophical meaning that Bachelard attached to contemporary physics did not necessarily coincide with that attributed to it by many scientists, or by many contemporary philosophers. However, I shall argue that the philosophical tradition with which Bachelard was most closely associated lent solid bases for a philosophy which would embrace the counter-intuitive character of the new physics. This philosophical tradition, especially in the form given to it by Bachelard’s professor and mentor Léon Brunschvicg, was particularly attentive to modern science, and particularly receptive to the philosophical significance of its innovations. This openness to scientific novelty was based on a historicist conception of knowledge. I shall explain. Brunschvicg saw as the aim of his philosophy that of analysing the mind, and to investigate the ways in which knowledge develops. His analysis of knowledge, and science in particular, was done a posteriori: rather than developing an epistemology starting from principles and rules, he aimed at extracting these principles and rules from scientific and philosophical texts. He believed that the structure of knowledge is different in different times, indeed that knowledge develops and progresses by changing, sometimes radically, its central concepts and aims. He therefore needed to study history of science and history of philosophy as comprehensively as possible, and to investigate texts from different epochs. He regarded contemporary science as the most advanced type of knowledge, and believed that its analysis would reveal how the contemporary mind works and what its principles and values are.
Quantum mechanics and the theory of relativity presented many problems for philosophy, as they appeared to violate traditional philosophical conceptions, for instance of time, space, causality and substance, and the distinction between the subject and object of knowledge. However, Bachelard found in his philosophical training an approach that allowed him to regard what appeared as ambiguities or contradictions to other philosophers as the foundations of a new philosophy. In particular, the conception, present in Brunschvicg’s philosophy, of the object as mind-dependent meant that the question of the existence of microparticles as independent of the mind never arose, and that the fact that observations of microphysical systems necessarily involved an interference by the observer was not problematic. It was from his own observation of contemporary physics, however, that Bachelard conceptualised one of the most interesting aspects of his philosophy: the erosion of the boundary between the theoretical and the technical part of science. His original concept of ‘phenomenotechnique’ results from his reading of the practices of contemporary physics, and supports his revision of traditional philosophical views concerning the existence and essence of things. I shall discuss Bachelard’s interpretation of modern physics in the context of his philosophical background and questions. In the concluding part, I shall suggest the reasons why Bachelard thought that philosophy should follow science.
2. At the roots of Bachelard’s third way: beyond the traditional philosophical dichotomies
For Bachelard traditional philosophical doctrines, such as rationalism and empiricism, idealism and materialism, and realism and positivism had had their time. It is not by chance that one of his books on modern science is entitled Rational materialism: for him the traditional dichotomy between rationalism and materialism had been made obsolete by the advancement of the sciences. In his view, the alternatives to which philosophy had grown used had been dramatically exploded by modern physics. It is in his discussion of microphysics that he developed his critique of the traditional philosophical distinctions between theory and experiment, scientific object and experimental apparatus, and even between substance and attributes. Teresa Castelão-Lawless has argued that Bachelard’s lack of distinction between theory and experiment was radically different from the mainstream philosophy of science of the period. She is right, especially when she clarifies that by mainstream philosophy of science she means the Vienna Circle’s and Karl Popper’s (Castelão-Lawless, 1995, p. 11). However, the tradition of French philosophy to which Bachelard belonged made available to him the intellectual tools not only to reject such distinction, but also to expel it from the set of philosophical problems. Starting from the standpoints of this tradition, Bachelard was in a position to develop an original philosophy by reflecting on modern physics, as I shall show.
The most important exponent of this philosophical tradition was Léon Brunschvicg, professor of history of modern philosophy, and, according to many, one of the most influential philosophers of the first half of the twentieth century in France (Deschoux, 1969, p. 5).1 Bachelard, who was a student of Brunschvicg, has been presented as the scholar who freely developed Brunschvicg’s philosophy (Wahl, 1962, p. 114), and his ‘view of the relation of science and philosophy’ as ‘most directly’ derived from Brunschvicg (Gutting, 2001, pp. 85–86).2 Many of Brunschvicg’s works look at first sight as histories of science. For instance, Les étapes de la philosophie mathématique (Brunschvicg, 1912) is chronologically very thorough, starting from chapters on the origin of the concept of number to proceed to Pythagoreanism, Euclid, Descartes on geometry, and from Zeno to Kepler, Pascal, Fermat up to Lobachevsky, Riemann and Cauchy. The same historical approach can be seen in books like L’expérience humaine et la causalité physique (Brunschvicg, 1922) and others dealing with history of philosophy and thought (e.g., [Brunschvicg, 1927] and [Brunschvicg, 1934]). However, when asked by George Sarton to collaborate more closely with Isis, he replied that he was not a historian of science, but rather a philosopher who studied the mind.3
Brunschvicg’s aim was to examine the ways in which human beings had thought in the past and think in the present, in particular when faced with mathematical problems and the study of nature. Two aspects of his approach are interesting here: one is his detailed analysis of the history of scientific doctrines, and the other that he never judged these doctrines in terms of their success in representing the physical world. As far as the latter characteristic of his work is concerned, it is a consequence of his epistemology and of his concept of truth. Brunschvicg judged as naı¨ve and ‘common sense’ (in the French pejorative meaning) the belief that there is a dichotomy between ideas on the one hand and mind-independent things on the other. If these independent things exist, he argued, we cannot say anything about them (Brunschvicg, 1921, pp. 47–48). Similarly, the idea that truth is something independent of us, and external to us, is for him a naı¨ve conception. In his view, truth has its source in the mind, for it is the mind that establishes intelligible relationships between phenomena (ibid., pp. 86–87). He followed Kant in defining truth as the result of the mutual relation between the mind and experience; and he stressed that this relation is between the human mind and human experience. It is easy to see that Brunschvicg’s philosophy is a type of idealism, as he himself admits (see, for example, Brunschvicg, 1923, p. 170), and of the Kantian variety. He is indeed normally classed as a neo-Kantian, and this is right in many ways, first of all because his philosophy is aimed at critically examining knowledge and at assessing the capabilities of the mind.
There is an important aspect of his philosophy, however, which departs from Kant’s inspiration, although for him his disagreement with Kant was his way of being ‘more faithful than Kant to the spirit of critical idealism’.4 For Brunschvicg, Kant had been wrong to characterize his table of categories as universally valid, and more in general to present the conditions of knowledge as ahistorical. He argued that Kant claimed to have described the ‘intellectus archetypus’, the ideal form of intellect, but in fact he had only described the form of Newtonian science ([Brunschvicg, 1922] and [Brunschvicg et al., 1923]). The a prioris of Transcendental aesthetics, that is absolute space and time, no longer correspond, for Brunschvicg, to the bases of contemporary ‘speculation’ (Brunschvicg, 1922, p. 458), by which he meant contemporary physics and geometry. Many philosophers, he believed, were still trapped in a Newtonian world, including Henri Bergson, who, when criticising science, in fact targeted classic physics, ignoring the revolutions of the twentieth-century (ibid., p. 591).5 His criticism of Kant explains why he needed to analyse the history of thought. If there are no categories given once and for all, and if ways of thinking change, it is not possible to reach general conclusions about the mind by only analysing it in one particular time and place. He claimed that history is for the philosopher what the laboratory is for the scientist (Brunschvicg, 1923, p. 162): in history the mind can be analysed in its diverse ways of applying itself to various questions.
Brunschvicg’s belief in the ‘dynamism and plasticity of intellectual functions’ (Brunschvicg, 1922, p. 550) and his neo-Kantian idealism bear consequences for his conception of the physical world. If the physical world (studied by science) is dependent on the mind, and the mind is not fixed, then the physical world will also change; indeed Brunschvicg concluded that ‘nature … will never be a given’ (ibid., p. 609). In his view, the mind and nature interact and change as a result of their interaction. He argued that Kant separated the mind and the content of experience because he had a ‘static and schematic’ conception of both (ibid., p. 608). However, the theory of relativity for Brunschvicg had overcome this separation. As he put it to Einstein in a seminar, in the Kantian world there is a ‘container’ (space and time) and a content (matter and force), but in the Einsteinian world, this distinction is no longer valid, for neither space nor time are any longer conceived as ‘empty and homogeneous’ receptacles of reality (Brunschvicg in Einstein, 1922, p. 100).6 For him, the forms of experience that Kant believed to be a priori have been shown to be neither fixed nor distinguishable from the content of experience.
Brunschvicg’s ‘fluid mind’ is a philosophical version of the more famous concept of ‘mentality’, which Lucien Lévy-Bruhl had used and popularised in his books Le fonctions mentales dans les sociétés inférieures (Lévy-Bruhl, 1910) and La mentalité primitive (Lévy-Bruhl, 1922). The links between the two historians of philosophy were rather strong: Brunschvicg succeeded Lévy-Bruhl in the Sorbonne chair of history of modern philosophy, and it seems that Lévy-Bruhl played a role in the choice of his successor (see Febvre, 1997, p. 271). Brunschvicg also used Lévy-Bruhl’s work, especially in his own analysis of ‘pre-historic’ science ([Brunschvicg, 1921] and [Brunschvicg, 1934]). The belief that there are different ways of thinking across time was very widespread in France in the first half of the twentieth-century, although in different forms. Brunschvicg and Lévy-Bruhl shared this view with the historians of mentalities, including Lucien Febvre, and with numerous historians of science, such as Alexandre Koyré, Abel Rey and Hélène Metzger-Bruhl. These historians of science investigated past texts and doctrines in order to unveil the mentality which informed them. By doing this, they often aimed to show that past doctrines, which may seem absurd from the point of view of modern science, were internally logical, but followed a different logic and exhibited different values from modern science. Just to cite an example from many Metzger described the way of thinking of seventeenth-century ‘philosophers of metal’ as analogical; in other words, they saw the world as a network of analogies. As a consequence, from their point of view, it made perfect sense to think that lead could be transformed into gold, as lead is to gold what a child is to an adult: in both cases, the former is an imperfect embodiment of the latter and can develop into its/his perfect form (Metzger, 1969 [1923], p. 108).7
For the philosopher Brunschvicg, the aim was not just to unveil one particular mentality, but rather to draw general conclusions about the mind. However, since he shared the historicism of these other scholars, he could carry out his study of the mind only through historical research. For him the ‘philosopher’s mission’ is to follow ‘the indefinite progress of rationality and objectivity, in their indissoluble link’ (Brunschvicg, 1922, p. 595). In other words, the Brunschvicgian philosopher studies the history of knowledge, and observes its ‘progress’ through the ages, which involves changes in logic, principles, categories and change in what counts as an object of knowledge.
3. Noumena and Bibliomena : the creativity of human knowledge
Brunschvicg’s philosophy can be broadly regarded as Bachelard’s starting point, as far as his questions, perspective and priorities are concerned. Like Brunschvicg, Bachelard studied the mind by observing it ‘at work’, which in practice for him meant to study the history of natural philosophy, science, philosophy, literature and also popular beliefs. Bachelard studied scientific knowledge as a whole; its components—theory and experiment, mathematics and experimental apparatus, scientists and scientific objects—for him cannot be understood in isolation, but they have rather to be analysed in their dialectic relationships. He acknowledged Brunschvicg’s legacy when rejecting the conception of the mind and object as absolute and fixed, and subscribing instead to the thesis of the mutual relativity of the two (Bachelard, 1986 [1949], p. 9). In Bachelard’s hands, this thesis developed to become one of the dialectics that form the very motor of the evolution of knowledge. The dialectic between subject and object is for him particularly evident in science. Indeed, he argued that philosophy of science ‘is … essentially … a philosophy of the mutual transformation of man and things’ (Bachelard, 1951, p. 3). He regarded scientific objects as the result of a process of transformation—which he often called ‘rectification’ and ‘rationalisation’—operated by human knowledge. The scientific ‘fact’, he argued, is fictitious, or ‘made up’ (Bachelard, 1970a [1931–1932], p. 12); indeed for him contemporary science is fictitious (Bachelard, 1951, pp. 3–6).
Bachelard regarded the view that science is about ‘things’ that exist independently of our knowledge so bizarre, that, instead of refuting it philosophically, he decided to psychoanalyse those holding it. He clearly thought that the attachment shown by human beings to the mind-independent reality of the objects of their knowledge could only be explained psychologically. For him realism is an innate, instinctive attitude towards the world, which has more to do with our desires, and in particular desire of possession, than with any scientific approach. Realism is for him a psychological complex rather than a philosophical position. Realists, says Bachelard, suffer from the Harpagon complex: they are misers who want to ‘possess’ the ‘riches of reality’ (Bachelard, 1993 [1938], Ch. 7).8 Realists want to possess substances, that is, what is permanent in things—what things ‘really’ are, below the surface of their attributes, of what is transient, and, one could add, of what is dependent on our perception of them. In La formation de l’esprit scientifique, in which he presents the theory of realism as a psychological complex, Bachelard principally relies on the analysis of alchemic texts. When applied to microphysics, though, his approach gives results as well. Early atomic models such as J. J. Thomson’s, represent the atom as ‘containing a large number of smaller bodies’.9 The history of the atom seems to show what for Bachelard is the journey of human knowledge: in its first stages it was chosiste (‘thingist’), that is to say it interpreted its objects as things similar to those ordinarily perceived in everyday life. However, atomic theory progressively became less akin to anything representable in everyday images, or even every-day concepts. For those, who, unlike Bachelard, think that what is familiar counts as more ‘real’ than what contradicts our everyday experience, it is hard to say that microparticles are ‘real’. It is interesting that Werner Heisenberg, in contesting the idea that waves in configuration space could be considered ‘real’, argued that ‘real’ comes from the Latin word res, ‘which means ‘thing’; but things are in the ordinary three-dimensional space, not in an abstract configuration space’ (Heisenberg, 1958, p. 116). For Bachelard, this argument would have been an example of chosisme, because it states that for something to be real it must be a ‘thing’, and a thing which is similar to those of our everyday experience, in this case, situated in a three-dimensional space. This attitude for him only leaves the alternative between ‘traditional realism’ (if scientific objects are seen as things in the world) or ‘positivist agnosticism’ (if scientific objects are not seen as things in the world).
For Bachelard a ‘corpuscle’10 is not ‘a miniature of a common object’; it is rather a noumenon (Bachelard, 1951, p. 96). Bachelard’s use of the term noumenon I believe is partly descriptive and partly polemical. On the one hand, he simply says that subatomic particles are not, indeed in principle they cannot be, objects of our senses, or phenomena. They are rather the product of our nous, of our intellect. He argues that the fact that particles are thought of mathematically, ‘is the mark of an … objective existence’, and concludes with the ‘formula’ cogitatur ergo est: something exists by virtue of being (mathematically) thought of (Bachelard, 1970a, p. 18). By arguing this, however, Bachelard goes against the fundamental principles of Kant’s conception of noumenon. For Kant, the pure concepts of the intellect can only have an empirical use; in other words, they can only be applied to experience. For instance, for Kant human beings apply the concepts of unity and totality to the objects of their perceptions, but unity and totality cannot be the direct object of human knowledge: they are the form of our thought, rather than its content. The content of human knowledge is given by our senses, in other words, we know phenomena. However, Kant argues, the very fact that we denote certain objects as phenomena, implies a distinction between our perception of things and things in themselves. In other words, things-in-themselves, and in general whatever is not the object of our senses, are simply thought of by the intellect, they are noumena. However, in the Critique of pure reason, Kant denied that noumena can be objects of our knowledge: they are rather negative concepts, in the sense that they indicate the limits of our intellect, or what our intellect cannot know.11
Electrons cannot be perceived by our senses, but this for Bachelard does not make them less real, or less important objects of scientific knowledge. For him, they certainly are more rational than the objects that we easily perceive. For instance we can see and feel fire. However, he argued that fire is no longer an object of chemical knowledge (Bachelard, 1949 [1938], Ch. 5), no matter how clearly we perceive it. Indeed, for Bachelard, we do not experience fire (or any other object) ‘directly’, despite what we might believe; in fact, we load the experience we have of it with our desires, in particular sexual desires, and thoughts of destruction, death and renewal (Bachelard, 1949 [1938]). For Bachelard all human knowledge or opinion carries ‘the human mark’, but this human mark can be provided either by imagination and subjectivity, as in the case of fire, or by rationality, as in the case of scientific objects. However, the form of our rationality cannot be decided once and for all, as Kant intended. Indeed, for Bachelard the ‘form’ envisaged by Kant was only the analysis of scientific knowledge as it was in a particular historical moment. Scientific knowledge has changed, and so, for Bachelard, should philosophy.
As seen above, Brunschvicg, despite presenting himself as a Kantian, already challenged the distinction between form and content of human knowledge, and thought that modern physics had exploded this distinction. Bachelard fully accepted the rejection of this distinction, which he placed at the core of his epistemology. His own reading of history of science fully confirmed it. For him the objects of human knowledge are not given once and for all, but rather they are constructed in ways that change in time, as demonstrated by the difference between the objects of early twentieth-century physics and those of Newtonian physics. Scientific objects are for him constructions of historically-situated knowledge. As a result of his view, he had no qualms in asserting that the electron has a firmer existence than the moon. His proof of this is that more books have been written about the electrons than about the moon: the electron, he argued with some humour, is a bibliomenon, it exists in books; and he adds that this type of existence is ‘so human, so solidly human!’ (Bachelard, 1951, pp. 6–7). For him, truth does not result from finding out how things are independently of the mind; such an aim would be for him a pointless and irrational pursuit; indeed a matter for psychoanalysis.
4. If they are not things, then what are they? The ontology of microphysics
Bachelard thought that modern physics had eliminated the philosophical distinctions not only between ‘form’ and ‘content’ of knowledge, but also between subject and object, and between substance and attributes. Corpuscles are for Bachelard the best illustration of the power of physical knowledge to shake philosophical concepts and categories. He argued that the atom has a ‘new ontological status’ (Bachelard, 1951, p. 76), and he listed and discussed the characteristics that make corpuscles irreconcilable with traditional philosophical ontology. Not only are they not bodies, but they cannot be described as substances. First of all, Bachelard explained that corpuscles do not have assignable absolute dimensions, or shape. He quoted the physicist and mathematician Hermann Weyl who wrote that the dimensions attributed to the electron radium should be interpreted as ‘the distance at which two electrons get closer to each other with a speed comparable to the speed of light’ (ibid., p. 77). The corpuscle, Bachelard concluded, is defined as ‘a power of opposition’; in other words the corpuscle is investigated through relations rather than its own essence. With his usual humour, elsewhere Bachelard wrote that in microphysics ‘in the beginning is the Relation’ (Bachelard, 1970a, p. 19). He provided other reasons why corpuscles cannot be substances in the traditional sense. Their individuality is not stable; two corpuscles that go through a narrow region may not be distinguishable (Bachelard, 1951, p. 81); corpuscles, finally, can be annihilated. Bachelard summarised his view by saying that in ‘contemporary corpuscular philosophy’ an ‘ontology of corpuscles’ is combined with an ‘ontology of corpuscular transformation’, and referred to a page of Heisenberg’s Two lectures (ibid., p. 83). This is the page where Heisenberg claims that:
the experiments prove directly that there are transitions between nucleons and mesons . . . mesons, electrons and neutrinos; protons, electrons and light quanta, etc. Therefore one can go from any particle to any other particle at least by intermediate steps, and it is thereby very probable that in a collision process of sufficiently large energy particles of any type can be created. (Heisenberg, 1949, p. 13)
This possibility of annihilation, and of transformation of one particle into another, for Bachelard represents the defeat of chosisme (or ‘thingism’) (Bachelard, 1951, p. 82): corpuscles have nothing to do with the ‘things’ of our everyday experience. A further blow to chosisme, and indeed to philosophical ontology and epistemology, is for him the very specific sense in which it can be said that corpuscles are in a certain point in space. When we say that something (material) exists, we imply that it exists in a specific place (at one given time). It is certainly the case that we perceive things in space. Kant expressed it very clearly by saying that space, with time, is the a priori of sensation: we cannot perceive anything if not in space. However, it is not the case that particles, although not perceivable by our senses, can still be regarded as occupying a space in the same way as the other objects of our experience do. The localisation of subatomic particles presents all sorts of constraints that make it very problematic to imagine them being situated in space just as a chair or a table would. These difficulties lead Heisenberg to argue that it is not possible to talk about the position of an electron independently of the specific experiments used to determine it. In his own words:
if one wants to be clear about what is to be understood by the words ’position of the object’ for example the electron … then one must specify definite experiments with whose help one plans to measure the ’position of the electron’; otherwise this word has no meaning. (Heisenberg, 1983 [1927], p. 64)
Such restrictions did not lead Bachelard to become sceptical about the objective value of the results microphysics, but rather to argue in favour of a revision of the idea of existence as situated in space. He claimed that:
the localisation of the corpuscle … is dependent on such restrictions that the function of the situated existence no longer has an absolute value’. (Bachelard, 1951, p. 80)
It is easy to see how Bachelard’s presentation of sub-atomic particles is inconsistent with the traditional philosophical concept of substance. It is not possible to indicate what the unchangeable structure of a corpuscle’s being is, as we should according to Aristotle’s sense of substance. Similarly, it would be difficult to say of a particle what Descartes said of substance, that is to say that it is ‘a thing that exists in such a way as to depend on no other thing for its existence’ (Descartes, 1985 [1644], Pt. I, p. 51). Moreover, the traditional philosophical distinction, of Aristotelian derivation, between substance and attributes no longer holds. Bachelard pointed out that we must not regard the electron as a small body which is negatively charged, as we would following classical physics, whereby distinguishing between the electron as substance, and its charge as attribute, in the same way that we would say that a man is tall. For him, electrons and protons exhibit a ‘total synthesis of substance and attribute’ (Bachelard, 1951, p. 77).
In summary, the difficulties that subatomic particles create for philosophical ontology are so great that it is not really possible for traditional Cartesian–Kantian philosophy to answer questions such as ‘what is an electron?’. The duality of representation of subatomic particles as waves and as corpuscles is perhaps the most striking example of the inability to answer the question of what the ‘essence’ of a particle is. Bachelard followed in particular the works of Louis de Broglie and Werner Heisenberg on this problem, and, as did de Broglie after 1928,12 he accepted Bohr’s thesis of complementarity. In other words, Bachelard did not see anything peculiar about describing the same phenomena as wave and as corpuscle, that is to say in two different ways which are incompatible with each other. His conclusion is that we must accept that the concept of substance is irrational, and in general that ‘old’ philosophical doctrines that cannot be reconciled with the new achievements of microphysics should be abandoned. It is clear that for Bachelard it is science that dictates what is rational. If modern science is incompatible, as it is, with some long-held philosophical beliefs, for him there is no doubt: philosophy should rethink its own concepts in the light of the latest development of science.
Bachelard’s epistemology is normative: it is aimed to judge what counts as rational knowledge and what does not. This is why he could write books explaining why alchemy could not possibly be rational knowledge: because he applied a norm to all investigations of nature, and this norm was current science. Alchemy, just as any inquiry prior to the beginning of the nineteenth century, for him fell short of the parameters of rationality posed by current science. This norm of rationality, however, is not given once and for all; it changes with the development of science itself, and with the contextual change of minds and objects. This is why Bachelard criticised philosophers for relying, in his eyes, on concepts that belonged to a previous scientific era.13 However, not only philosophers, but also scientists were not as ready as Bachelard to accept the philosophical implications of quantum mechanics. Without going into a detailed discussion of the different points of view held by the leading physicists of the time, it is worth mentioning that some of their positions could broadly fall into the positions that Bachelard described as superseded by modern physics. A good example of this is Einstein. I have mentioned above that Brunschvicg told Einstein that his theory of relativity had gone beyond the oppositions of Kantian philosophy, but that Einstein did not really accept to have revolutionized philosophy, and vaguely reasserted his Kantian epistemology. Analogously, Bachelard thought that contemporary physics underpinned the negation of the classical Cartesian opposition between an unchangeable subject and an unchangeable object. Of course this does not mean that Einstein should see it that way. He started his essay Maxwell’s influence on the evolution of the idea of physical reality with the statement that: ‘the belief in an external world independent of the perceiving subject is the basis of all natural sciences’ (Einstein, quoted in Holton, 1973, p. 241). In their famous paper on the (in)completeness of quantum mechanics, Albert Einstein, Boris Podolsky and Nathan Rosen stressed once again their fundamentally realist epistemology based on a distinction between subject and object:
If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity. (Einstein et al., 1983 [1935], p. 138)
For Bachelard, as for the philosophical tradition to which he belonged, truth was not a matter of correspondence between the intellect and the world, or between our theories and physical reality. The existence of subatomic particles, or any other object, independently of the mind was not a question which interested him or that indeed made any sense in his approach. However, his rejection of what he calls ‘traditional philosophical realism’ did not translate for him into ‘positivist agnosticism’: he really believed that what science achieved was the truth—though what counted as true did change in history—and not just truth about ‘phenomena’. For him, no agnostic suspension of judgement on how the world ‘really is’ is necessary, nor it makes sense in his philosophy. As to why science rather than another form of inquiry or human endeavour should have been singled out as having this very special status is something that Bachelard cannot resolve by saying that it provides a true description of the world. I will discuss this crucial point in the concluding part of this article. Now a few words must be spent on how modern physics for Bachelard has realized that quantum leap from a phenomenological to a phenomenotechnical approach.
5. From phenomenology to phenomenotechnique
In the 1980s the philosophers Ian Hacking and Bas van Fraassen engaged in a discussion as to whether we ‘really’ see through a microscope. In brief, the former has claimed that we do, and the latter that we do not ([Hacking, 1985] and [van Fraassen, 1985]). Van Fraassen has represented the empiricist point of view: for him, the human body, or rather its senses, are the measure of all things. He has therefore argued that we do see through a telescope, because, at least in principle, we could go and observe, say, the moons of Jupiter (van Fraassen, 1980, pp. 13–19). By contrast, since our unaided senses cannot see in principle what we ‘see’ with the help of an electronic or acoustic microscope, he has concluded that we cannot say that we see through these devices. For him, to put it roughly, if we cannot observe something, we have no basis to believe in its existence. Hacking, on the other hand, has argued that we do have reasons to believe in the existence of the so-called theoretical entities (which include Bachelard’s corpuscles) because we can manipulate them. As the subtitle of his Representing and intervening suggests: ‘if you can spray them, then they are real’ (Hacking, 1983).
This debate gives us the measure of how different Bachelard’s chosen philosophical background and its values were from those of large part of recent English-language philosophy of science. For Bachelard, an object does not gain any special status by virtue of being observable: scientific objects for him are not the objects of our perception. As I mentioned above, he thought that the objects of our untrained perceptions are loaded with our fantasies and desires. For instance, he investigated the imagery associated with water in its different forms, as fresh water and sea water, as streams, ponds and lakes. Our diverse experience of water for him cannot really be dissociated from ideas and images of purity, motherhood or death (Bachelard, 1942). Indeed, in his view, scientific knowledge must overcome the obstacles posed to the rationalisation of nature by the rich imagery that populates our experience of the world. To see or not to see through a microscope is not really the problem. But the difference between the questions that Bachelard on the one hand and van Fraassen and Hacking on the other pose is more profound than this. Bachelard, as we shall see, attached great importance to the human ability of manipulating objects. This may seem rather close to Hacking’s stance in Representing and intervening. However, it is not.14 The whole debate between van Fraassen and Hacking is based on the question of what warrants our belief in the existence of something that we cannot observe. In other words, their respective arguments are based on an opposition between subject and object. As a consequence, the question of the mind-independent existence of sub-atomic particles is regarded as an important one, while it is just not one of Bachelard’s philosophical questions. His aim is rather to observe how scientific knowledge, theories and instruments are created by historically situated research. For Bachelard, to say that a scientific object exists independently of science would have been meaningless, for scientific objects are created by science.
On one point, however, Bachelard would have agreed with Hacking: the modern scientist does not observe the world, but rather manipulates it. Bachelard argued that science is not a phenomenology of the world. By phenomenology, Bachelard loosely intended a doctrine aimed at knowing the ‘external world’ by privileging what is ‘felt, perceived, indeed imagined’ (Bachelard, 1951, p. 2). Bachelard’s theory of knowledge, based on his interpretation of modern physics, counters what normally are considered to be the fundamental characteristics of phenomenology, such as its aim at description and its lack of presuppositions. For Bachelard, in modern science it is impossible to bracket presuppositions and previous knowledge; in fact for him scientific knowledge develops by revisiting, opposing and rectifying previous knowledge, and by overcoming irrational beliefs. Without previous knowledge and irrational beliefs for Bachelard there would be no knowledge at all. This is why he claimed that modern science’s rationalism never begins, but always re-starts, ‘rectifies’, ‘regularises’ and ‘normalises’ previous conceptions (cf. Bachelard, 1986 [1949], pp. 112, 122–123). Knowledge always springs out of a dialectic engagement with its past. For Bachelard, only errors are immediate, and they must be revised and rectified rationally.
Bachelard directly criticised Edmund Husserl, the founder of phenomenology as a philosophical doctrine, for conceiving of knowledge as ‘reception’ of ‘data’ by the mind. He commented that Husserl’s ‘dualism’ between mind and data is not ‘close enough’ or ‘systematic enough’ (Bachelard, 1986 [1949], p. 43). In other words, it is not dialectic, as there is no mutual intervention of the two elements, which do not change each other. For Bachelard, the mind does not ‘receive’ the object, but rather judges it and makes it rational. What, however, for Bachelard really distinguishes modern science from phenomenology, is that scientists do not simply observe or directly capture essences, but rather technically manipulate and indeed create the objects of their knowledge. Bachelard pointed out that the only possible study of corpuscles is technical, that is to say it is carried out by using experimental apparatus; in his own words, ‘of all corpuscles of modern physics, one can only do a phenomenotechnical study’ (Bachelard, 1951, p. 92). He continued by saying that in phenomenotechnique, no phenomenon appears naturally, no phenomenon is a given.15 I have mentioned above that Bachelard called corpuscles noumena and even bibliomena. Here we see how Bachelard stressed the technical character of these objects; in modern physics, he argued, ‘ontology [is] conditioned by technical experience’ (Bachelard, 1951, p. 82). Indeed, for him a concept becomes scientific insofar as it becomes technical (Bachelard, 1993 [1938], p. 61). The structure of nature itself has a human and technical character: Bachelard argued that the ‘true order’ of nature is that which we technically put into nature itself (Bachelard, 1991 [1934], p. 111). Modern physics, and quantum mechanics in particular, here inspire and support Bachelard’s view that experimentation has nothing to do with the direct observation of nature: the concepts itself of observation, he pointed out, is thrown into doubt in certain domains of quantum mechanics (Bachelard, 1986 [1949], p. 43). In quantum mechanics, the measuring apparatus inevitably interferes with the system, as Niels Bohr explained on several occasions, for instance by saying that ‘the procedure of measurements has an essential influence on the conditions on which the very definition of the physical quantities in question rests’ (Bohr, 1983 [1935], p. 144). This inextricability of apparatus and object was easy to accept for Bachelard precisely because he thought that the scientific object could not be independent of our knowledge (theoretical or technical) and could not be grasped as it is ‘in itself’. The impossibility of drawing a clear line between object and apparatus comes therefore as an extension of the thesis that no sharp distinction can be drawn between the subject and the object of knowledge.
For Bachelard, modern physics shows that even an intervention on nature that is not informed by theoretical and technical knowledge is a dream of the past. This is why he described experimentation in modern science as ‘non-Baconian’ (Bachelard, 1970b, p. 45). It is only thanks to the apparatus that the objects of scientific theories come into being, so to speak; Bachelard claimed that the phenomena of contemporary scientific thought are phenomena of apparatus, created by the apparatus (Bachelard, 1951, p. 5). His stress on the technical side of science and the role of experimentation was an important development that made his philosophy different in character from Brunschvicg’s idealism. In Bachelard’s philosophy, the distinction between thinking and doing is eliminated: one cannot ‘observe’ nature without thinking and without theories, but one cannot have scientific theories and scientific objects without experimentation. For Bachelard, instruments are part of scientific knowledge just as theories, scientific objects and minds are. All these components are dependent on one another: there would be no scientific objects without theories or without instruments, without theorists or without technicians.
6. Why should philosophy follow science?
Bachelard thought that twentieth-century physics had revolutionized not only science but also philosophy. Indeed, he believed that science should revolutionize philosophy: his criticism of some philosophers for being still attached to nineteenth-century thinking implies that philosophy was lagging behind. But why should philosophy follow science? Or, in other words, what makes science preferable to other types of investigation? It should be clear from what I said so far, that Bachelard would not have answered these questions by saying that science is preferable because it provides a true picture of the world as it is independently of us. Scientific knowledge is for Bachelard the result of the dialectic interaction of minds, objects, theories and apparatus. But one can easily argue that so was alchemic knowledge. The objects of alchemy may be heavily shaped by a specific world-view, but so are the objects of science.
However for Bachelard there is a crucial difference between the way in which science and alchemy order their worlds: science employs rationality, while alchemy employed imagination and emotions. There is no doubt that for Bachelard rationality is superior to imagination, although it is so only in the social context. He wrote many books analysing and indeed celebrating imagination, but always in the private sphere, and he always thought the ‘diurnal and nocturnal man’, that is rational and oneiric life, should be kept separate (see for example, (Bachelard (1972b [1953]), p. 19 ). Rationality for him should regulate society, while imagination should enrich our private life. He commented that the alchemist ‘cannot communicate his dreams’. By contrast, scientific knowledge not only can be communicated, but for him it is always created socially, through the interaction between individuals, as well as between minds and objects, and between new knowledge and old knowledge. Modern science for Bachelard is not the ‘architectonic knowledge’ of philosophers, who believed that knowledge could be built little by little starting from its foundations. Scientific knowledge for him is ‘polemical’: it results from clashes between different views, and between present results and past failures ([Bachelard, 1950] and [Bachelard, 1986]). He regarded science as objective, but objective for him means intersubjective and social, rather than belonging to the object as opposed to the subject. Rather than the individualistic Cartesian cogito, Bachelard’s rationalism is based, in his expression, on the cogitamus, that is to say collective and dialectic thinking (Bachelard, 1986 [1949], pp. 56, 59–60). On the other hand, he pathologized what he called pre-scientific thought—that is any inquiry prior to the nineteenth century—and employed psychoanalysis in order to understand the roots of its ‘mentality’.16 For him the capital sin of ‘pre-scientific’ thought is that of applying to the social sphere what it should stay private: emotions, desires and images.
It is, however, almost a tautology to say that for Bachelard science is the model of knowledge because it is rational, for the norm of rationality for him is provided by current science. In other words, for him, it is science that dictates what counts as rational. The reasons why Bachelard had to assign such an important role to science are, I think, of two orders. The first is simply historical. Bachelard’s epistemology is historical, and results from the observation of the historical forms that knowledge has taken. In other words, for him humankind, in its history, has exercised rationality through science, and the role of epistemologists is to analyse these forms, by taking as their norm the historically most advanced form. The other reasons why Bachelard valued science above any other type of social activity belong to ethics. I believe that Bachelard regarded the overstepping of the boundaries between private and public life as morally undesirable. In pre-scientific knowledge, the relationships between people are not acceptable to him, because they are based on personal authority. This is marked even by the physical space: the apprentice in the art of alchemy would live in their master’s home, unlike the student of science who would learn their trade in public spaces. He denounced the ‘false values’ of ‘intimate thought’, which in his view is ‘lazy’, and exhorted his readers to get rid of those ‘strange problems’ that ‘laziness’ creates (Bachelard, 1951, p. 4).17
The type of rationalism based on modern science, by contrast, is for him ‘the consciousness of a rectified science, of a science which bears the mark of human action, of the well-considered, industrious, normalising action’ (Bachelard, 1986 [1949], p. 123; my italics). Science for Bachelard is the embodiment of change, of work and action, which he clearly valued more than inaction. He valued the transformation that the mind undergoes as a result of scientific work very highly, and believed that a philosophy inspired by scientific practices would have the same effect as science. In scientific activity, the subject, in his interaction with the object, not only ‘regularises’ the object, but ‘eliminates … irregular attitudes from his own intellectual behaviour’ (Bachelard, 1970b, p. 92). For Bachelard science should be chosen because in his view it enables human beings to overcome their selfish, individualist and emotional drives and to enter a world of objectivity and work. The history of science for him is a history of progressive socialization, and science’s aim at objectivity assures that ‘selfishness’ cannot have a place in it. He thought that the historian of science’s mission should be to show the ‘profoundly human value of current science’ (Bachelard, 1972a, p. 152). Although the application of science is one of the central concerns of his philosophy, he never discussed these applications outside the laboratory, for instance in the two world wars that took place in his lifetime, and the materialization of atomic theory in the atomic bomb. His analysis seems to be rather impermeable to the world outside books and laboratories, and until the end of his life he believed that science embodied human values which must be preserved and developed. Moral values were what made him choose science as guide to philosophy and social life.
Manuscript sources
12 juin 2009
Mathesis universalis
L'humanité de ce début du 21 ème siècle se trouve, pour la première fois de son histoire, face à l'Abîme, dans une situation de sursis qui se prolongera sans doute indéfiniment : depuis le 6 août 1945, nous sommes passés à deux doigts de la destruction totale, et ce plus d'une dizaine de fois .
Cet état de fait peut sembler terrible et écrasant, invitant à la fuite en avant et à l'autodestruction finale, d'une façon analogue à l'alcoolique qui a tellement peur de la rechute qu'il boit un verre pour ne plus avoir peur d'en boire un, justement, et consommer le désastre dans une jouissance rageuse et mortelle.
Mais il est aussi susceptible d'une autre interprétation : l'homme (c'est à dire nous tous, hommes et femmes, qui vivons ou vivrons sur cette planète après 1945) a conquis grâce à la science moderne née au 17 ème siècle européen des pouvoirs immenses sur la Nature, bien supérieurs à ceux de l'antique magie, propre aux sociétés primitives d'avant la science, des pouvoirs qui dans les anciennes mythologies auraient été réservés aux "dieux".
S'il ne veut pas périr , il lui faut être à la hauteur de ses pouvoirs et des responsabilités qui en dérivent.
Il lui faut donc se faire "Dieu", se déifier.
Mais qu'est ce que cela veut dire : se déifier ?
Dieu est Esprit, Raison, Logos : telle est l'unique leçon que nous retenons de l'Evangile.
Se déifier, cela signifie donc : élever sa pensée propre à la hauteur de la Pensée Infinie qui est Dieu.
Un tel acte de pensée , nous le nommons, en empruntant avec quelques raisons pensons nous ce terme à Descartes : Mathesis universalis.
L'homme se déifiant dans un processus infini d'acheminement de l'âme vers la Raison pure, n'est donc pas un "autre" que Dieu : nous sommes Dieu envisagé (s'envisageant) dans le temps.
Oui, nous sommes Dieu, mais nous sommes aussi le cobaye universel : cela nous donne en plus quelques droits...
Le Temps est la Mathesis universalis existant empiriquement.
Celle-ci ne doit pas être confondue avec la mathématique , ou la science , qui en est le résultat : elle est l'activité pure de pensée qui en est la condition de possibilité.
Et nous pensons ici que puisque la théorie des nombres (l'arithmétique) est la reine des disciplines mathématiques, et que la mathématique est la reine des sciences, alors c'est là, au coeur même de l'activité intellectuelle-spirituelle qui constitue le monde dans sa réalité ultime, que la Mathesis universalis comme acheminement vers l'Esprit doit être cherchée avant tout .
Les Nombres ne sont autres que les Idées de Platon.
Voici quelques blogs que j'ai créés , et où cette activité de pensée est développée à un rythme plus ou moins régulier:
http://mathesisuniversalis.multiply.com
http://mathesisuniversalis0.multiply.com/
http://mathesisuniversalis.blogg.org
http://principiatoposophica.blogg.org