P G Grinevich

R G Novikov
email: novikov@cmap.polytechnique.fr

Moutard transform approach to generalized analytic functions with contour poles *

. In particular, we show that generalized analytic functions with the simplest contour poles can be Moutard transformed to the regular ones, at least, locally. In addition, the later Moutard-type transforms are locally invertible.

Introduction

We study the equations

∂ z ψ = u ψ in D, (1)
∂ z ψ + = -ū ψ+ in D, (2)
where D is an open domain in C, u = u(z) is a given function in D, ∂ z = ∂/∂ z.

The functions ψ = ψ(z) satisfying equation [START_REF] Arkad'ev | Singular solutions of the KdV equation and the inverse scattering method[END_REF] are known as generalized analytic functions in D, equation ( 2) is known as the conjugate equation to [START_REF] Arkad'ev | Singular solutions of the KdV equation and the inverse scattering method[END_REF]; see [START_REF] Vekua | Generalized Analytic Functions[END_REF]. In the present article the notation f = f (z) does not mean that f is holomorphic.

The classical theory of generalized analytic functions is presented in [START_REF] Bers | Theory of pseudo-analytic functions[END_REF], [START_REF] Vekua | Generalized Analytic Functions[END_REF]. In addition, very recently in [START_REF] Grinevich | Moutard transform for generalized analytic functions[END_REF], [START_REF] Grinevich | Generalized analytic functions, Moutard-type transforms and holomorphic maps[END_REF] it was shown that a new progress in this theory is possible by involving ideas of Moutard-type transforms going back to [START_REF] Moutard | Sur la construction des équations de la forme 1 z ∂ 2 z ∂x∂y = λ(x, y) qui admettenent une intégrale générale explicite[END_REF]. Actually, ideas of Moutard-type transform were developed and successfully used in the soliton theory in dimension 2+1, in the spectral theory in dimension 2 and in the differential geometry; see [START_REF] Doliwa | Integrable lattices and their sublattices: From the discrete Moutard (discrete Cauchy-Riemann) 4-point equation to the self-adjoint 5-point scheme[END_REF], [START_REF] Matveev | Darboux transformations and solitons[END_REF], [START_REF] Nimmo | Superposition principles associated with the Moutard transformation: an integrable discretization of a 2+1dimensional sine-Gordon system[END_REF], [START_REF] Novikov | Two-dimensional von Neumann-Wigner potentials with a multiple positive eigenvalue[END_REF]- [START_REF] Taimanov | On the Moutard transformation and its applications to spectral theory and Soliton equations[END_REF] and references therein.

We recall that in our case the Moutard-type transforms assign in quadratures to a given coefficient u and fixed solutions f j , f + j , j = 1, . . . , N, of equations ( 1), [START_REF] Beals | The spectral problem for the Davey-Stewartson and Ishimori hierarchies[END_REF], and all formal solutions ψ, ψ + of (1), ( 2) a new coefficient ũ and new related formal solutions ψ, ψ+ for these generalized analytic function equations; see [START_REF] Grinevich | Moutard transform for generalized analytic functions[END_REF], [START_REF] Grinevich | Generalized analytic functions, Moutard-type transforms and holomorphic maps[END_REF]. In turn, the construction of [START_REF] Grinevich | Moutard transform for generalized analytic functions[END_REF] was stimulated by recent articles by I.A. Taimanov [START_REF] Taimanov | Blowing up solutions of the modified Novikov-Veselov equation and minimal surfaces[END_REF], [START_REF] Taimanov | The Moutard transformation of two-dimensional Dirac operators and Möbius geometry[END_REF] on the Moutard-type transforms for the Dirac operators in the framework of the soliton theory in dimension 2 + 1.

In the classical theory of generalized analytic functions it is usually assumed that

u ∈ L p (D), p > 2, if D is bounded, (3) u ∈ L p,2 (C), p > 2, if D = C, (4)
where L p,ν (C) denotes complex-valued functions u such that u ∈ L p (D 1 ), u ν ∈ L p (D 1 ), where

u ν (z) = 1 |z| ν u 1 z , (5)
D 1 = {z ∈ C : |z| ≤ 1};
see [START_REF] Vekua | Generalized Analytic Functions[END_REF].

On the other hand, one of the most important applications of the generalized analytic functions theory is associated with the inverse scattering in two dimensions, see [START_REF] Beals | The spectral problem for the Davey-Stewartson and Ishimori hierarchies[END_REF], [START_REF] Fokas | On the solvability of the N-wave, Davey-Stewartson and Kadomtsev-Petviashvili equations[END_REF]- [START_REF] Grinevich | Faddeev eigenfunctions for multipoint potentials[END_REF], [START_REF] Grinevich | Two-dimensional "inverse scattering problem" for negative energies and generalized-analytic functions. 1. Energies below the ground state[END_REF], [START_REF] Kazeykina | A large-time asymptotics for the solution of the Cauchy problem for the Novikov-Veselov equation at negative energy with non-singular scattering data[END_REF], [START_REF] Lakshtanov | A global Riemann-Hilbert problem for two-dimensional inverse scattering at fixed energy[END_REF], [START_REF] Novikov | The inverse scattering problem on a fixed energy level for the two-dimensional Schrödinger operator[END_REF] and references therein. In addition, already in [START_REF] Grinevich | Two-dimensional "inverse scattering problem" for negative energies and generalized-analytic functions. 1. Energies below the ground state[END_REF] it was shown that, for the case of the two-dimensional Schrödinger equation, not only regular generalized analytic functions, where u satisfies (3) or ( 4), but generalized analytic functions with contour poles, also naturally arise. However, in the latter case, assumptions (3), [START_REF] Crum | Associated Sturm-Liouville systems[END_REF] are not valid at all. It is quite likely, that the classical methods of generalized analytic functions dot not admit appropriate generalizations for this case. In particular, the problem of proper solving the generalized analytic function equation ( 1) for u with contour poles was remaining well-known, but open.

It is in order to precise that in the framework of inverse scattering for the Schrödinger equation in two dimensions the generalized analytic function equation [START_REF] Arkad'ev | Singular solutions of the KdV equation and the inverse scattering method[END_REF] with contour poles arises for the case when

u = u -1 ∆ , ψ = ψ -1 ∆ , ( 6
)
where ∆ is real-valued and u -1 , ψ -1 , ∆ are quite regular (e.g. real-analytic) functions on D, and the pole contours are the zeroes of ∆. In this setting ψ = ψ(x, z, E) are the Faddeev exponentially growing solutions of the Schrödinger equation in x ∈ R 2 at fixed energy E ∈ R, u is a particular case of the Faddeev generalized scattering data, ∆ is the modified Fredholm determinant for the Lipman-Schwinger-Faddeev integral equation for ψ, and z is a fixed-energy spectral parameter; see [START_REF] Faddeev | Inverse problem of quantum scattering theory. II[END_REF], [START_REF] Henkin | The ∂-equation in the multidimensional inverse scattering problem[END_REF], [START_REF] Grinevich | Two-dimensional "inverse scattering problem" for negative energies and generalized-analytic functions. 1. Energies below the ground state[END_REF] and references therein.

Proceeding from the aforementioned inverse scattering motivation, we consider the generalized analytic function equation [START_REF] Arkad'ev | Singular solutions of the KdV equation and the inverse scattering method[END_REF] with contour poles for the case when u is of the form as in [START_REF] Duistermaat | Differential equations in the spectral parameter[END_REF], and when this equation has sufficiently many (more or less as in the regular case) local solutions ψ of the form as in [START_REF] Duistermaat | Differential equations in the spectral parameter[END_REF]. Adopting the terminology of [START_REF] Grinevich | Spectrally meromorphic operators and nonlinear systems[END_REF] we say that in this case equation ( 1) is of meromorphic class. Note also that equation (1) may be of meromorphic class only if principal terms of u near pole contours satisfy solvability conditions; for simple contour poles these conditions were found in [START_REF] Grinevich | Two-dimensional "inverse scattering problem" for negative energies and generalized-analytic functions. 1. Energies below the ground state[END_REF].

Actually, our recent works [START_REF] Grinevich | Moutard transform for generalized analytic functions[END_REF], [START_REF] Grinevich | Generalized analytic functions, Moutard-type transforms and holomorphic maps[END_REF] were motivated considerably by the aforementioned open problem of proper solving the generalized analytic function equation [START_REF] Arkad'ev | Singular solutions of the KdV equation and the inverse scattering method[END_REF] with contour poles for the meromorphic case. In particular, in [START_REF] Grinevich | Moutard transform for generalized analytic functions[END_REF] we give examples of coefficients u when equation ( 1) is of meromorphic class and can be efficiently studied using Moutard-type transforms. In addition, we were stimulated by results of [START_REF] Arkad'ev | Singular solutions of the KdV equation and the inverse scattering method[END_REF], [START_REF] Crum | Associated Sturm-Liouville systems[END_REF], [START_REF] Grinevich | Singular soliton operators and indefinite metrics[END_REF], [START_REF] Novikov | The Moutard transformation and twodimensional multipoint delta-shaped potentials[END_REF], [START_REF] Taimanov | On the Moutard transformation and its applications to spectral theory and Soliton equations[END_REF], [START_REF] Taimanov | Blowing up solutions of the modified Novikov-Veselov equation and minimal surfaces[END_REF] on efficient applications of Darboux-Crum and Moutard-type transforms to studies of some important linear ODE's and PDS's with singular coefficients.

The results of the present work can be summarized as follows:

• We give composition and inversion formulas for the simple Moutardtype transforms for the equations ( 1), [START_REF] Beals | The spectral problem for the Davey-Stewartson and Ishimori hierarchies[END_REF]; see Theorems 1, 2 in Subsection 2.2;

• We show that any equation (1) of meromorphic class with a simple contour pole can be transformed to a regular one in a neighborhood of the pole contour via an appropriate simple Moutard-type transform; see Theorem 3 in Subsection 3.4.

2 Moutard-type transforms for generalized analytic functions

Basic construction

Let

f j = f j (z) and f + j = f + j (z), j = 1, . . . , N, (7)
denote a set of fixed solutions of equations ( 1) and (2), respectively. Let ψ, ψ + be arbitrary solutions of (1), [START_REF] Beals | The spectral problem for the Davey-Stewartson and Ishimori hierarchies[END_REF]. We define an imaginaryvalued potential ω ψ,ψ + = ω ψ,ψ + (z) such that

∂ z ω ψ,ψ+ = ψψ + , ∂ z ω ψ,ψ + = -ψψ + in D. (8)
We recall that this definition is self-consistent, at least, under the assumption that D is simply-connected. The integration constant is imaginary-valued and may depend on concrete situation.

As in [START_REF] Grinevich | Moutard transform for generalized analytic functions[END_REF], [START_REF] Grinevich | Generalized analytic functions, Moutard-type transforms and holomorphic maps[END_REF] we consider imaginary-valued potentials ω f j ,f + k , j, k = 1, . . . , N, and we set

Ω =      ω f 1 ,f + 1 ω f 2 ,f + 1 . . . ω f N ,f + 1 ω f 1 ,f + 2 ω f 2 ,f + 2 . . . ω f N ,f + 2 . . . . . . . . . . . . ω f 1 ,f + N ω f 2 ,f + N . . . ω f N ,f + N      . (9)
Following [START_REF] Grinevich | Moutard transform for generalized analytic functions[END_REF], for equations (1), (2) we consider the Moutard-type transform

u M / / ũ , (10)
{ψ, ψ + } M / / { ψ, ψ+ } , (11)
M = M u,f,f + = M u,f 1 ,...,f N ,f + 1 ,...,f + N ( 12
)
defined as follows:

ũ = u + f 1 . . . f N Ω -1    f + 1 . . . f + N    , (13)
ψ = ψ -f 1 . . . f N Ω -1    ω ψ,f + 1 . . . ω ψ,f + N    , (14)
ψ+ = ψ + -f + 1 . . . f + N (Ω -1 ) t    ω f 1 ,ψ + . . . ω f N ,ψ +    , (15)
where ψ, ψ + are formal solutions to equations ( 1), (2), respectively, ω ψ,f + j and ω f j ,ψ + are defined as in [START_REF] Fokas | On the solvability of the N-wave, Davey-Stewartson and Kadomtsev-Petviashvili equations[END_REF]; t in (15) stands for the matrix transposition.

Due to results of [START_REF] Grinevich | Moutard transform for generalized analytic functions[END_REF], the transformed functions ψ, ψ+ solve the transformed generalized-analytic function equations:

∂ z ψ = ũ ψ in D, (16)
∂ z ψ+ = -ũ ψ+ in D. (17)

Composition and inversion of simple Moutard transforms

We say that the Moutard-type transforms ( 10)-( 15) are simple if N = 1. In this case:

ψ = ψ -f 1 ω ψ,f + 1 ω f 1 ,f + 1 , (18)
ψ+ = ψ + -f + 1 ω f 1 ,ψ + ω f 1 ,f + 1 , (19) ũ
= u + f 1 f + 1 ω f 1 ,f + 1 . ( 20
)
Proposition 1 For a simple Moutard transform ( 18)-( 20) the following formula holds:

ω ψ, ψ+ = ω ψ,ψ + ω f 1 ,f + 1 -ω ψ,f + 1 ω f 1 ,ψ + ω f 1 ,f + 1 + c ψ, ψ+ , (21)
where ω ψ, ψ+ is defined according to [START_REF] Fokas | On the solvability of the N-wave, Davey-Stewartson and Kadomtsev-Petviashvili equations[END_REF], and c ψ, ψ+ is an imaginary-valued integration constant.

Proposition 1 was announced in [START_REF] Grinevich | Generalized analytic functions, Moutard-type transforms and holomorphic maps[END_REF] and is proved in Section 4 of the present work.

Let f 1 , f 2 and f + 1 , f + 2 be some fixed solutions of equations ( 1) and ( 2), respectively, with given u. Let

u M / / ũ , (22)
ψ, ψ + M / / ψ, ψ+ , (23)
M = M 2 • M 1 , (24)
where 1. M 1 is the simple Moutard transform for equations ( 1), ( 2) with coefficient u, generated by f 1 , f + 1 and given by formulas ( 18)-(20);

2. M 2 is the simple Moutard transform for equations ( 16), [START_REF] Grinevich | Spectrally meromorphic operators and nonlinear systems[END_REF] with coefficient ũ = M 1 u, given by:

ψ = ψ -f2 ω ψ, f + 2 ω f2 , f + 2 , (25)
ψ+ = ψ+ -f + 2 ω f2 , ψ+ ω f2 , f + 2 , (26) ũ
= ũ + f2 f + 2 ω f2 , f + 2 , ( 27
)
where we assume that:

• ψ = M 1 ψ, ψ+ = M 1 ψ + , f2 = M 1 f 2 , f + 2 = M 1 f + 2 ;
• ω ψ, f + 2 , ω f2 , ψ+ and ω f2 , f + 2 are given by ( 21) with c ψ, f + 2 , c f2 , ψ+ and c f2 , f + 2 equal to zero. 3. We assume that

ω f 1 ,f + 1 = 0, ω f2 , f + 2 = 0. Theorem 1 Let f 1 , f 2 and f + 1 , f + 2
be some fixed solutions of equations ( 1) and ( 2), respectively, with given u, and let M be defined as in ( 22)- [START_REF] Novikov | The inverse scattering problem on a fixed energy level for the two-dimensional Schrödinger operator[END_REF]. Then M coincides with the Moutard transform given by formulas ( 10)- [START_REF] Grinevich | Two-dimensional "inverse scattering problem" for negative energies and generalized-analytic functions. 1. Energies below the ground state[END_REF] for N = 2 and generated by the initial functions

f 1 , f 2 , f + 1 , f + 2 .
Schematically, the result of Theorem 1 can be also presented as follows:

ψ {f 1 ,f + 1 } M 1 / / {f 1 ,f 2 ,f + 1 ,f + 2 } 6 6 ψ { f2 , f + 2 } M 2 / / ψ ( 28
)
Theorem 1 is proved in Section 4.

Next, in order to inverse simple Moutard transforms, we consider:

f 2 ≡ 0, f + 2 ≡ 0, ω f 2 ,f + 1 = i, ω f 1 ,f + 2 = i; (29) f2 = M 1 f 2 = f = -i f 1 ω f 1 ,f + 1 , f + 2 = M 1 f + 2 = f + = -i f + 1 ω f 1 ,f + 1 . (30)
In particular, f and f + defined in [START_REF] Vekua | Generalized Analytic Functions[END_REF] satisfy equations ( 16) and ( 17), respectively, with the coefficient ũ given by [START_REF] Lakshtanov | A global Riemann-Hilbert problem for two-dimensional inverse scattering at fixed energy[END_REF].

Theorem 2 Let M 1 be a simple Moutard transform defined as in ( 18)- [START_REF] Lakshtanov | A global Riemann-Hilbert problem for two-dimensional inverse scattering at fixed energy[END_REF], where ω f 1 ,f + 1 = 0. Let M 2 be the simple Moutard transform for equations ( 16), [START_REF] Grinevich | Spectrally meromorphic operators and nonlinear systems[END_REF] with coefficient ũ = M 1 u, given by ( 25)- [START_REF] Taimanov | Blowing up solutions of the modified Novikov-Veselov equation and minimal surfaces[END_REF], where ψ = M 1 ψ, ψ+ = M 1 ψ + , f2 = f , f + 2 = f + , where f and f + are defined in [START_REF] Vekua | Generalized Analytic Functions[END_REF]. Then: 1. The potentials ω ψ, f + 2 , ω f2 , ψ+ , ω f2 , f + 2 can be chosen as follows:

ω ψ, f + 2 = -i ω ψ,f + 1 ω f 1 ,f + 1 , ω f2 , ψ+ = -i ω f 1 ,ψ + ω f 1 ,f + 1 , ω f2 , f + 2 = 1 ω f 1 ,f + 1 ; (31) 2. Under assumptions (31), the composition M = M 2 •M 1 is the identical transformation.
Thereom 2 is proved in Section 4.

3 Removing the simplest contour pole singularity

Real analytic pole contour

We consider a real analytic curve Γ ⊂ D:

Γ = {z(τ ) : τ ∈]it 1 , it 2 [}, t 1 , t 2 ∈ R, (32)
where:

• z(τ ) ∈ D for τ ∈]it 1 , it 2 [, • z is (complex-valued) real-analytic on ]it 1 , it 2 [, • z(τ ′ 1 ) = z(τ ′ 2 ) for τ ′ 1 = τ ′ 2 , • dz(it)/dt = 0 for for t ∈]t 1 , t 2 [.
Here D is the domain in (1), [START_REF] Beals | The spectral problem for the Davey-Stewartson and Ishimori hierarchies[END_REF].

Let

T = T a,b,ε = {τ ∈ C : a < Im τ < b, |Re τ | < ε}, (33)
where a, b, ε ∈ R, ε > 0.

As a corollary of our assumptions, z in (32) could be continued to a holomorphic bijection Z:

Z : T a,b,ε → D a,b,ε , τ → z(τ ), (34) Z -1 : D a,b,ε → T a,b,ε , z → τ (z), for some a, b, ε such that t 1 < a < b < t 2 , ε > 0, where D a,b,ε ⊂ D.
Actually, we consider Γ of (32) as a pole contour for equations (1), (2).

Holomorphic change of variables

In the domain D a,b,ε of (34) we rewrite equations ( 1), (2) in variable τ ∈ T a,b,ε . In this connection, following [START_REF] Grinevich | Generalized analytic functions, Moutard-type transforms and holomorphic maps[END_REF] we consider

u * (τ ) = u(z(τ )) ∂z(τ ) ∂τ , (35)
ψ * (τ ) = ψ(z(τ )) ∂z(τ ) ∂τ , ψ + * (τ ) = ψ + (z(τ )) ∂z(τ ) ∂τ , (36)
where u(z), ψ(z), ψ + (z) are the same that in equations ( 1), (2), z(τ ) is the same that in (34), τ ∈ T a,b,ε . Then (see [START_REF] Grinevich | Generalized analytic functions, Moutard-type transforms and holomorphic maps[END_REF]):

∂ τ ψ * = u * ψ * in T a,b,ε , (37)
∂ τ ψ + * = -ū * ψ+ * in T a,b,ε . (38)
In addition, we have (see [START_REF] Grinevich | Generalized analytic functions, Moutard-type transforms and holomorphic maps[END_REF]):

M u * ,f * ,f + * • Z -1 = Z -1 • M u,f,f + , (39)
where:

• Z -1 is considered as a map of the conjugate pairs of equations ( 1), ( 2) in D a,b,ε into the conjugate pairs of equations ( 37), (38) in T a,b,ε and is defined according to (35), (36);

• M u,f,f + for (1), ( 2) in D a,b,ε and M u * ,f * ,f + * for (37), (38) in T a,b,ε are defined according to formulas ( 10)- [START_REF] Grinevich | Two-dimensional "inverse scattering problem" for negative energies and generalized-analytic functions. 1. Energies below the ground state[END_REF], where u * , f * = {f 1, * , . . . , f N, * }, f + * = {f + 1, * , . . . , f + N, * } are defined according to (35), (36), and

ω ψ * ,ψ + * (τ ) = ω ψ,ψ + (z(τ )), (40)
for all involved potentials.

In the framework of the Moutard transform approach, using the commutativity relation (39) we reduce local studies of equations ( 1), [START_REF] Beals | The spectral problem for the Davey-Stewartson and Ishimori hierarchies[END_REF] with contour pole at Γ in (32) to the case of contour pole at the straight line

Γ * = {τ ∈ T a,b,ε : Re τ = 0}, (41)
where T a,b,ε is defined as in (33), (34).

Remark 1 We recall that, in view of formulas (35), (36), the generalized analytic functions ψ, ψ + of ( 1), ( 2) can be treated as spinors, i.e. differential forms of type 1 2 , 0 , and u can be treated as differential form of type 1 2 , 1 2 ; see [START_REF] Grinevich | Generalized analytic functions, Moutard-type transforms and holomorphic maps[END_REF]. These forms can be written as:

u = u(z) √ dzdz, ψ = ψ(z) √ dz, ψ + = ψ + (z) √ dz. ( 42
)

Constraints on the meromorphic class coefficients at singularity

We consider equations (37), (38) in T = T a,b,ε defined by (33) for the case of simplest pole at Γ * defined by (41). We write τ = x + iy, τ = x -iy.

We assume that

u * (τ ) = e 2iφ(y) +∞ j=-n r j (y) x j in T , (43)
where φ, r j are quite regular functions on the interval ]a, b[≈ Γ * , and φ, r -n are real-valued, n ∈ N. For this case we consider solutions ψ * , ψ + * of (37), (38) near Γ * of the following form:

ψ * = +∞ j=-n ′ α j (y) x j , ψ + * = +∞ j=-n ′′ α + j (y) x j , n ′ , n ′′ ∈ N, (44)
where α j , α + j are quite regular on ]a, b[ and α -n ′ , α + -n ′′ are not-zero almost everywhere at ]a, b[. Note that in this section we consider u * , ψ * , ψ + * in formulas ( 43), (44) as formal power series in variable x.

Lemma 1 Assume that equation (37) with coefficient u * as in (43) has, at least, one solution ψ * of the form as in (44). Then n = 1 and |r -1 (y)| ≡ n ′ /2 in (43).

Lemma 1 follows from formal substitutions of (43), (44) into (37), (38). In the present article we restrict ourselves to the simplest (but generic) case when n ′ = n ′′ = 1. In this case, without loss of generality, we can assume that r -1 = -1/2, adding π/2 to the phase φ, if necessary.

Then, as a corollary of results of [START_REF] Grinevich | Two-dimensional "inverse scattering problem" for negative energies and generalized-analytic functions. 1. Energies below the ground state[END_REF], equation (37) is of meromorphic class, at least formally, if and only if:

Re r 0 (y) ≡ 0, y ∈]a, b[, (45)
Im r 1 (y) = 1 2

d 2 φ(y) dy 2 , y ∈]a, b[. (46)
For completeness of exposition, this result is proved in Section 5.

Here, belonging of equation (37) to meromorphic class means that equation (37) has local solutions ψ * near Γ * ⊂ T of the form (44) with n ′ = 1 parametrised by two real-values functions (one complex-valued function) on Γ * , i.e., roughly speaking, equation (37) has as many local solutions ψ * near Γ * as in the regular case.

Actually, equation (37) and formulas (43), (44) for u * , ψ * with n = 1,

n ′ = 1, r -1 = -1/2 imply that Im e -iφ(y) α -1 (y) = 0, y ∈]a, b[. (47)
In addition, under conditions (45), ( 46), the solutions ψ * of (37), (44) are parametrised by β -1 (y) and Im β 1 (y) on ]a, b[, where α j (y) = e iφ(y) β j (y); see Section 5. Finally, one can see that the meromorphic class conditions (45), (46) for equation (37) imply the related meromorphic class conditions for the conjugate equation (38).

Moutard transform to the regular case

We consider equations (37), (38) for the case when

u * = e 2iφ(y) - 1 2x + r 0 (y) + r 1 (y)x + O x 2 in T ∪ ∂T , (48)
where

φ is real-valued, φ ∈ C 2 ([a, b]) , u * + e 2iφ(y) 2x ∈ C 1 (T ∪ ∂T ) , (49)
r 0 , r 1 satisfy (45), (46), T is defined by (33). We assume that equations (37), (38) have some solutions f * , f + * such that

f * = e iφ(y) β -1 (y) x + β 0 (y) + O(x) , β -1 is real-valued, β -1 > 0, β -1 ∈ C 1 ([a, b]) , (50)
f * - e iφ(y) β -1 (y) x ∈ C 1 (T ∪ ∂T ) , f + * = e -i(φ(y)+π/2) β + -1 (y) x + β + 0 (y) + O(x) , β + -1 is real-valued, β + -1 > 0, β + -1 ∈ C 1 ([a, b]) , (51)
f + * - e -i(φ(y)+π/2) β + -1 (y) x ∈ C 1 (T ∪ ∂T ) ,
where

∂ x O(x) = O(1), ∂ y O(x) = O(x).
Note that from point of view of formal considerations of Subsection 3.3 such solutions f * , f + * always exist.

Theorem 3 Let u * satisfy (48), (49) and equations ( 37), (38) have some solutions f * , f + * satisfying (50), (51). Let ω f * ,f + * be some potential defined according to [START_REF] Fokas | On the solvability of the N-wave, Davey-Stewartson and Kadomtsev-Petviashvili equations[END_REF]. Then

ũ * = M u * ,f * ,f + * u * = O(1) in T a,b,δ ∪ ∂T a,b,δ (52)
for some δ ∈]0, ε[, where M u * ,f * ,f + * is defined according to formulas ( 10)-( 15) for N = 1.

Theorem 3 is proved in Section 6.

The point is that that the results of Theorems 2, 3 and the commutativity formula (39) reduce local studies of equations ( 1), ( 2) near the simplest contour pole singularity to the regular case.

Proofs of Proposition 1 and Thereoms 1-2 4.1 Proof of Proposition 1

Let ω ψ, ψ+ be given by [START_REF] Matveev | Darboux transformations and solitons[END_REF]. Then it is sufficient to show that

∂ z ω ψ, ψ+ = ψ ψ+ , ∂ z ω ψ, ψ+ = -ψ ψ+ . ( 53
)
Using the definitions of ω ψ,ψ

+ , ω ψ,f + 1 , ω f 1 ,ψ + , ω f 1 ,f + 1
, and ψ, ψ+ we have:

∂ z ω ψ, ψ+ = ∂ z ω ψ,ψ + ω f 1 ,f + 1 -ω ψ,f + 1 ω f 1 ,ψ + ω f 1 ,f + 1 + c ψ, ψ+ = = ∂ z ω ψ,ψ + -∂ z ω ψ,f + 1 ω f 1 ,ψ + ω f 1 ,f + 1 -∂ z ω f 1 ,ψ + ω ψ,f + 1 ω f 1 ,f + 1 + ∂ z ω f 1 ,f + 1 ω f 1 ,ψ + ω f 1 ,f + 1 ω ψ,f + 1 ω f 1 ,f + 1 = = ψψ + -ψf + 1 ω f 1 ,ψ + ω f 1 ,f + 1 -f 1 ψ + ω ψ,f + 1 ω f 1 ,f + 1 + ψψ + ω f 1 ,ψ + ω f 1 ,f + 1 ω ψ,f + 1 ω f 1 ,f + 1 = = ψ -f 1 ω ψ,f + 1 ω f 1 ,f + 1 ψ + -f + 1 ω f 1 ,ψ + ω f 1 ,f + 1 = ψ ψ+ ; ∂ z ω ψ, ψ+ = ∂ z ω ψ,ψ + ω f 1 ,f + 1 -ω ψ,f + 1 ω f 1 ,ψ + ω f 1 ,f + 1 + c ψ, ψ+ = = ∂ z ω ψ,ψ + -∂ z ω ψ,f + 1 ω f 1 ,ψ + ω f 1 ,f + 1 -∂ z ω f 1 ,ψ + ω ψ,f + 1 ω f 1 ,f + 1 + ∂ z ω f 1 ,f + 1 ω f 1 ,ψ + ω f 1 ,f + 1 ω ψ,f + 1 ω f 1 ,f + 1 = = -ψψ + + ψf + 1 ω f 1 ,ψ + ω f 1 ,f + 1 + f 1 ψ + ω ψ,f + 1 ω f 1 ,f + 1 -ψψ + ω f 1 ,ψ + ω f 1 ,f + 1 ω ψ,f + 1 ω f 1 ,f + 1 = = -ψ -f 1 ω ψ,f + 1 ω f 1 ,f + 1 ψ + -f + 1 ω f 1 ,ψ + ω f 1 ,f + 1 = -ψ ψ+ .
Thus, the proof of Proposition 1 is completed.

Proof of Theorem 1

Due to formula [START_REF] Matveev | Darboux transformations and solitons[END_REF] for ω f2 , f + 2 with c f2 , f + 2 = 0 and due to the assumptions that ω f 1 ,f + 1 = 0, ω f2 , f + 2 = 0, we have:

det Ω = ω f 1 ,f + 1 ω f2 , f + 2 = 0, ( 54
)
where Ω is defined according to [START_REF] Grinevich | The scattering transform for the two-dimensional Schrödinger operator with a potential that decreases at infinity at fixed nonzero energy[END_REF] for N = 2. Due to the definition of M according to ( 22)-(24), using [START_REF] Matveev | Darboux transformations and solitons[END_REF] for ω ψ, ψ+ = ω f2 , f + 2 , ω ψ, f + 2 with c ψ, ψ+ = 0 and using (54), we obtain:

ψ = ψ -f2 ω ψ, f + 2 ω f2 , f + 2 = ψ -f 1 ω ψ,f + 1 ω f 1 ,f + 1 - -f 2 -f 1 ω f 2 ,f + 1 ω f 1 ,f + 1 ω ψ,f + 2 ω f 1 ,f + 1 -ω ψ,f + 1 ω f 1 ,f + 2 ω f 1 ,f + 1 ω f 2 ,f + 2 ω f 1 ,f + 1 -ω f 2 ,f + 1 ω f 1 ,f + 2 ω f 1 ,f + 1 -1 = = ψ -f 1 1 ω f 1 ,f + 1 ω f 2 ,f + 2 -ω f 2 ,f + 1 ω f 1 ,f + 2 × × ω ψ,f + 1 ω f 2 ,f + 2 - ω ψ,f + 1 ω f 2 ,f + 1 ω f 1 ,f + 2 ω f 1 ,f + 1 -ω ψ,f + 2 ω f 2 ,f + 1 + ω f 2 ,f + 1 ω ψ,f + 1 ω f 1 ,f + 2 ω f 1 ,f + 1 - -f 2 1 ω f 1 ,f + 1 ω f 2 ,f + 2 -ω f 2 ,f + 1 ω f 1 ,f + 2 ω ψ,f + 2 ω f 1 ,f + 1 -ω ψ,f + 1 ω f 1 ,f + 2 = = ψ -f 1 f 2 1 ω f 1 ,f + 1 ω f 2 ,f + 2 -ω f 2 ,f + 1 ω f 1 ,f + 2 ω ψ,f + 1 ω f 2 ,f + 2 -ω ψ,f + 2 ω f 2 ,f + 1 ω ψ,f + 2 ω f 1 ,f + 1 -ω ψ,f + 1 ω f 1 ,f + 2 .
(55) Taking into account that

Ω = ω f 1 ,f + 1 ω f 2 ,f + 1 ω f 1 ,f + 2 ω f 2 ,f + 2 , Ω -1 = 1 ω f 1 ,f + 1 ω f 2 ,f + 2 -ω f 2 ,f + 1 ω f 1 ,f + 2 ω f 2 ,f + 2 -ω f 2 ,f + 1 -ω f 1 ,f + 2 ω f 1 ,f + 1 , Ω -1 ω ψ,f + 1 ω ψ,f + 2 = 1 ω f 1 ,f + 1 ω f 2 ,f + 2 -ω f 2 ,f + 1 ω f 1 ,f + 2 ω ψ,f + 1 ω f 2 ,f + 2 -ω ψ,f + 2 ω f 2 ,f + 1 ω ψ,f + 2 ω f 1 ,f + 1 -ω ψ,f + 1 ω f 1 ,f + 2 , formula (55)
can be rewritten as:

ψ = ψ -f 1 f 2 Ω -1 ω ψ,f + 1 ω ψ,f + 2 . (56)
One can see that (56) coincides with formula [START_REF] Grinevich | Generalized analytic functions, Moutard-type transforms and holomorphic maps[END_REF] for N = 2. The computations for ψ+ are similar.

In additions, due to the definition of M according to ( 22)-(24), using (21) for ω f2 , f + 2 with c f2 , f + 2 = 0 and using (54), we obtain:

ũ = ũ + f2 f + 2 ω f2 , f + 2 = u + f 1 f + 1 ω f 1 ,f + 1 + f2 f + 2 ω f2 , f + 2 = = u+ f 1 f + 1 ω f 1 ,f + 1 + f 2 -f 1 ω f 2 ,f + 1 ω f 1 ,f + 1 f + 2 -f + 1 ω f 1 ,f + 2 ω f 1 ,f + 1 ω f 1 ,f + 1 ω f 1 ,f + 1 ω f 2 ,f + 2 -ω f 2 ,f + 1 ω f 1 ,f + 2 = = u + f 1 f + 1 ω f 2 ,f + 2 -f 2 f + 1 ω f 1 ,f + 2 -f 1 f + 2 ω f 2 ,f + 1 + f 2 f + 2 ω f 1 ,f + 1 ω f 1 ,f + 1 ω f 2 ,f + 2 -ω f 2 ,f + 1 ω f 1 ,f + 2 = = u + 1 ω f 1 ,f + 1 ω f 2 ,f + 2 -ω f 2 ,f + 1 ω f 1 ,f + 2 f 1 f 2 ω f 2 ,f + 2 -ω f 2 ,f + 1 -ω f 1 ,f + 2 ω f 1 ,f + 1 f + 1 f + 2 . ( 57
)
One can see that (57) coincides with formula (13) for N = 2. This completes the proof of Theorem 1.

Proof of Thereom 2

First, we check that ω f2 , ψ+ , ω ψ, f + 2 , ω f2 , f + 2 defined in (31) are the potentials for the pairs { f , ψ+ }, { ψ, f + }, { f , f + }:

∂ z ω ψ, f + 2 = -i∂ z ω ψ,f + 1 ω f 1 ,f + 1 = -i ψf + 1 ω f 1 ,f + 1 + i ω ψ,f + 1 ω 2 f 1 ,f + 1 f 1 f + 1 = = ψ -f 1 ω ψ,f + 1 ω f 1 ,f + 1 -if + 1 ω f 1 ,f + 1 = ψ f + , ∂ z ω ψ, f + 2 = -i∂ z ω ψ,f + 1 ω f 1 ,f + 1 = i ψf + 1 ω f 1 ,f + 1 -i ω ψ,f + 1 ω 2 f 1 ,f + 1 f 1 f + 1 = = -ψ -f 1 ω ψ,f + 1 ω f 1 ,f + 1 -if + 1 ω f 1 ,f + 1 = -ψ f + ; (58)
the calcualtions for ω f2 , ψ+ are analogous to the calculations for ω ψ, f + 2 ;

∂ z ω f2 , f + 2 = ∂ z 1 ω f 1 ,f + 1 = - ∂ z ω f 1 ,f + 1 ω 2 f 1 ,f + 1 = - f 1 f + 1 ω 2 f 1 ,f + 1 = f f + , ∂ z ω f2 , f + 2 = ∂ z 1 ω f 1 ,f + 1 = - ∂ z ω f 1 ,f + 1 ω 2 f 1 ,f + 1 = f 1 f + 1 ω 2 f 1 + 1 = -f β+ . (59)
Here, we used also that all potentials ω ψ,ψ + are pure imaginary. Second, we calculate the transform M 2 • M 1 :

ψ = ψ - f ω ψ, f + 2 ω f2 , f + 2 = ψ -f 1 ω ψ,f + 1 ω f 1 ,f + 1 --i f 1 ω f 1 ,f + 1 -i ω ψ,f + 1 ω f 1 ,f + 1 1 ω f 1 ,f + 1 = = ψ -f 1 ω ψ,f + 1 ω f 1 ,f + 1 + f 1 ω ψ,f + 1 ω f 1 ,f + 1 = ψ;
the calcualtions for ψ+ are analogous to the calcualations for ψ;

ũ = ũ + f f + ω f2 , f + 2 = u + f 1 f + 1 ω f 1 ,f + 1 + f f + ω f2 , f + 2 = u + f 1 f + 1 ω f 1 ,f + 1 + -i f 1 ω f 1 ,f + 1 -i f + 1 ω f 1 ,f + 1 1 ω f 1 ,f + 1 = = u + f 1 f + 1 ω f 1 ,f + 1 - f 1 f + 1 ω f 1 ,f + 1 = u. (60)
This completes the proof of Thereom 2.

Proof of meromorphic class conditions

We consider equation (37) and formulas (43), (44) for u * , ψ * , where n = 1, r -1 = -1/2, n ′ = 1. In this case formulas (43), (44) can be written as:

u * = e 2iφ(y) - 1 2x + r o (y) + r 1 (y)x + r 2 (y)x 2 + . . . , (61)
ψ * = e iφ(y) β -1 (y) x + β 0 (y) + β 1 (y)x + β 2 (y)x 2 + . . . . (62)
We substitute (61), ( 62) into (37), and we use that

2∂ τ ψ * = e iφ(y) - β -1 (y) x 2 + iβ -1 (y) -φ ′ (y)β -1 (y) x + (63)
+ +∞ k=0 [iβ ′ k (y) -φ ′ (y)β k (y) + (k + 1)β k+1 (y)] x k , 2u * ψ * = e iφ(y) - β -1 (y) x 2 + -β 0 (y) + 2r 0 (y)β -1 (y) x + (64) + +∞ k=0 -β k+1 (y) + 2r k+1 (y)β -1 (y) + 2 k l=0 r l (y)β k-l (y) x k .
¿From this point and till the end of the proof ′ denotes ∂ y .

Collecting the terms at x k , k = -2, -1, 0, 1, 2, . . ., we obtain:

β -1 (y) =β -1 (y) for k = -2, ( 65
)
β 0 (y) = -iβ ′ -1 (y) + φ ′ (y)β -1 (y) + 2r 0 (y)β -1 (y) for k = -1, (66) β 1 (y) + β 1 (y) = -iβ ′ 0 (y) + φ ′ (y)β 0 (y) + 2r 1 (y)β -1 (y)+ (67)
+ 2r 0 (y)β 0 (y) for k = 0,

β k+1 (y) + (k + 1)β k+1 (y) = -iβ ′ k (y) + φ ′ (y)β k (y)+ (68) +2r k+1 (y)β -1 (y) + 2 k l=0 r l (y)β k-l (y) for k ≥ 1.
One can see that: relation (65) coincides with (47); relation (66) can be considered as a formula for finding β 0 ; relations (68) can be considered as recursion relations for finding β j , j ≥ 2. In addition, the real part of (67) can be considered as a formula for finding Re β 1 , whereas the imaginary part of (67) can be rewritten as

Im -iβ ′ 0 (y) + φ ′ (y)β 0 (y) + 2r 1 (y)β -1 (y) + 2r 0 (y)β 0 (y) = 0. (69)
Actually, relations (65), (66), (69) yield the solvability constraints on φ, r 0 , r 1 . Substituting (65), (66) into (69) we obtain:

Im iφ ′ (y)β ′ -1 (y) + ((φ ′ (y)) 2 + 2φ ′ (y)r 0 (y))β -1 (y) + 2r 1 (y)β -1 (y)--2ir 0 (y)β ′ -1 (y) + (2r 0 (y)φ ′ (y) + 4(r 0 (y)) 2 )β -1 (y) + β ′′ -1 (y)-

-i(φ ′ (y) + 2r 0 (y))β ′ -1 (y) -i(φ ′′ (y) + 2r ′ 0 (y))β -1 (y) = 0.

In turn, (70) can be rewritten as:

Im β ′′ -1 (y) -2i[r 0 (y) + r 0 (y)]β ′ -1 (y) + [(φ ′ (y)) 2 + (71) + 2φ ′ (y)(r 0 (y) + r 0 (y)) + 4(r 0 (y)) 2 + 2r 1 (y) -iφ ′′ (y) -2ir ′ 0 (y)]β -1 (y) = 0.

In addition, taking into account that φ, β -1 are real-valued, we simplify (71) as follows:

Im -2i[r 0 (y)+r 0 (y)]β ′ -1 (y)+[4(r 0 (y)) 2 +2r 1 (y)-iφ ′′ (y)-2ir ′ 0 (y)]β -1 (y) = 0. (72)

One can see that (72) is fulfilled for all sufficiently regular real-valued β -1 if and only if (45), (46) are fulfilled. Finally, using (65)-(68) under conditions (45), (46), one can see that all ψ * of (62) satisfying (37) with u * of 61) are parametrised by β -1 and Im β 1 .

This completes the proof.

6 Proof of Thereom 3

Substituting (48), ( 50), ( 51) into (37), (38) we obtain:

β
As in Section 5 we assume that ′ denotes ∂ y . Using (50), ( 51) and ( 73), (74) we obtain:

f * f + * =e -iπ/
for some δ ∈]0, ε[. Note also that f * f + * = ie 2iφ(y) β -1 (y)β + -1 (y)

x 2 + O 1 x in T ∪ ∂T . (80)
Finally, due to ( 20), ( 48), ( 79

Theorem 3 is proved.

  0 (y) = i(β -1 (y)) ′ + (φ ′ (y) -2r 0 (y))β -1 (y), y ∈ [a, b],(73)β + 0 (y) = i(β + -1 (y)) ′ + (-φ ′ (y) + 2r 0 (y))β + -1 (y), y ∈ [a, b].

2 β - 1 ( 1 )- 1 Remark 2

 1112 in T ∪ ∂T . (75) Next, equation[START_REF] Fokas | On the solvability of the N-wave, Davey-Stewartson and Kadomtsev-Petviashvili equations[END_REF] can be rewritten as:∂ x ω ψ,ψ + = 2i Im ψψ + , ∂ y ω ψ,ψ + = 2i Re ψψ + . (76)As a corollary of (75), (76) and the property that β 1 , β + 1 are real-valued, we have:∂ x ω f * ,f + * = -2iβ -1 (y)β + -1 (y) x 2 + O(1), ∂ y ω f * ,f + * = 2i β -1 (y)β + -1 (y) (y)β + -1 (y) + O(1) in T ∪ ∂T . (78) Note that res x=0 ∂ x (ω f * ,f + * ) = 0. for any fixed y = y 0 .Using also the strict positivity of β -1 β + -1 we obtain:1 ω f * ,f + * = -ix 2β -1 (y)β + -1 (y) + O x 2 in T a,b,δ ∪ ∂T a,b,δ ,

  ), (80):ũ * = u * + f * f + * ω f * ,f + * = e 2iφ(y) -1 2x + r 0 (y) + O(x) + e 2iφ(y) 1 2x + O(1) = = O(1) in T a,b,δ ∪ ∂T a,b,δ .

* This work was started during the visits of the first author to the Centre de The first author was partially supported by the Russian Foundation for Basic Research, grant 13-01-12469 ofi-m2, by the program "Leading scientific schools" (grant NSh-4833.2014.1), by the program "Fundamental problems of nonlinear dynamics".


Hourya Benis-Sinaceur

Marco Panza
email: marco.panza@univ-paris1.fr

Gabriel Sandu
email: sandu@mappi.helsinki.fi

Frege's and Dedekind's Logicisms

, p. 41). Though ascribing to Russell the role of "chief proponent" of it, Carnap also adds that "Frege was the first to espouse this view" (ibid.).

Still, strictly speaking, Frege never argued for such a thesis. At most, he argued that arithmetic and real analysis are part of logic. But, also if it is so restricted, this thesis renders his view only very roughly. For what makes Frege's view distinctive is the way the inclusion relation between these mathematical theories and logic is conceived. And once this way is made clear, it becomes also clear that this relation does not depend, for him, on the mere possibility of a reduction of the former to the latter. Frege's point is, indeed, less that of showing how coming back from arithmetic and real analysis to logic, than that of developing logic enough so as to find natural and real numbers within it, and, then, show that what arithmetic and real analysis deal with is logical in nature.

,

p. 99) 1 . Frege's point seems, then, that "arithmetic is a branch [Zweig] of pure logic" ( [START_REF] Frege | [END_REF], Einleitung, pp. 1 and 3; [START_REF] Frege | [END_REF], pp. 1 1 and 3 1 ) because the former results from an appropriate development (but not an extension), of the latter, that is, because "the simplest laws of cardinal number [Anzahl ] 2 [. . . ][are] derived by logical means alone" ( [START_REF] Frege | [END_REF], Einleitung, p. 1; [START_REF] Frege | [END_REF], p. 1 1 ).

The crucial point of Frege's arithmetical logicism consists in fixing these logical means. This, in turn, entails identifying appropriate logical laws (or axioms, in modern terminology), deductive rules, and definitions from and according to which arithmetical truths follow. The purpose of the first and second parts of the Grundgesetze ( [START_REF] Frege | [END_REF], § § I.1-II.54) 3 is precisely that of fixing these means and using them for deriving these truths.

In Frege's mind, what ensures the logical nature of these laws, rules and definitions is that the laws and rules are appropriately general, while the definitions are explicit and have recourse, in their definiendum, only to linguistic tools already introduced by previous analogous definitions, or directly belonging to the language in which the laws are stated. The appropriate generality of the laws and rules is, in turn, ensured by the fact that all they concern is (the values of) a small number of basic functions, defined by merely appealing to two basic objects, the True and the False (whose existence is taken for granted), and to the totalities of objects and of first-and second-level one-argument functions (so as to avoid to appeal to each of these totalities of functions for defining a function belonging to it). In other terms, these laws and rules are general because they merely pertain to (the values) of some basic functions, which are defined by relying on no device used for selecting some specific portions or elements of these totalities other than the True and the False. Now, defining these basic functions results in fixing a language to be used to form either names of values of these functions or of whatever other functions resulting from appropriately composing them, or general marks apt to "indeterminately indicate" [unbestimmt andeuten]" ( [START_REF] Frege | [END_REF], § § I.1, I.8, I.17; [START_REF] Frege | [END_REF], pp. 5 1 , 11 1 , 31-32 1 ) these values. It follows, that, for Frege, the boundaries of logic are established by fixing a functional formal language, a small number of basic truths stated in this language, and a 1 Here and later, time to time, both in the present foreword and in the three following chapters, we feel free to slightly modify the English translations we quote, for sake of faithfulness to the original. 2 We agree with Ebert and Rossberg in translating Frege's term 'Anzahl', when used in a technical context, with 'cardinal number', by conserving 'number' for his term 'Zahl' (cf. [START_REF] Frege | [END_REF], "Translators' Introduction", p. xvi). A reason for using 'cardinal number', rather than 'natural number' is that Frege explicitly distinguishes (both in Grundlagen and in Grundgesetze) endlich Anzahlen from unendliche ones, namely finite cardinal numbers from infinite ones, among the latter of which he pays particular attention to the Anzahl Endlos, the cardinal number belonging to the concept endliche Anzahl (cf. [Frege, 1884], § § 84-86 and [START_REF] Frege | [END_REF], Vorwort, p. 5, and § § I.122-157). Notice, moreover, that, in Grundlagen, Frege also uses twice ( § § 19 and 43) the term 'natürliche Zahl' (to be mandatorily translated with 'natural number')-in the latter case, merely in a quote from Schröder, but in the former by speaking on his own behalf-and many times ( § § 76-79, 81-84, 104, and 108) the term 'natürlichen Zahlenreihe' (to be mandatorily translated with 'series of natural numbers' or 'natural numbers series'). Though in Grundgesetze ( § § I. 43-46, I.66, I.88, I.100, I.104, etc.), this last term is replaced with 'Anzahlenreihe' (to be translated with 'series of cardinal numbers' or 'cardinal numbers series'), it seems, then, that Frege takes a natural number to be a finite cardinal one.

3 We shall come back later on the third part.

small number of rules used to draw truths stated in this language from other such truths.

To put it shortly, when he speaks of logic, Frege is referring to a well-identified and (in his mind) appropriately established formal system, and when he claims that arithmetic is a branch of logic, he is implying that arithmetical truths are nothing but theorems of this system.

As we shall see pretty soon, this is not as trivial as it could appear at first glance. But it is still compatible with a conception of arithmetical logicism as a reductionistic program. To see what makes Frege's arithmetical logicism much more than that, one has to consider another distinctive and essential aspect of it. This depends on Frege's considering that values of functions (of whatever level) are objects, and that "objects stand opposed to functions", to the effect that "everything that is not a function" is an object ( [START_REF] Frege | [END_REF], § § I.2; [START_REF] Frege | [END_REF], p. 7 1 ). Insofar as functions are, for him, unsaturated, this entails that cardinal, and, a fortiori, natural numbers could not but be objects, for him. It follows, that Frege's arithmetical logicism involves the thesis that natural numbers are objects, namely logical objects-objects whose intrinsic nature is made manifest by explicit definitions stated in the language of logic-, and arithmetical truths are truths about these objects. But, insofar as it seems quite clear that natural numbers cannot be the True and the False, arguing for this thesis requires admitting that the language of logic is enough for defining some objects other than the True and the False.

The problem arises, then: how can such other objects be defined through this language, provided that it merely results from defining the basic functions of logic, and this is done by merely appealing to the True, the False, and to the totalities of objects and first-and second-level one-argument functions? The answer depends (and could not but depend) on Frege's countenance, among his basic functions, of a function having values other than the True an the False. This is the case of the value-ranges function: a second-level oneargument function taking first-level one-argument functions (without any restriction), and giving value-ranges. Still, given the mentioned restrictions on the way basic functions are defined, taking such a function as a basic one entails renouncing to definite it explicitly, and, then, admitting of an implicit definition for it. Frege's infamous Basic Law V provides such a non-explicit definition: it implicitly defines value-ranges by stating an identity condition for value-ranges of first-level one-argument functions, that is, by asserting, as it is wellknown, that the value-range of a first-level one-argument function Φ (ξ) is the same as that of a first-level one-argument function Ψ (ξ) if and only if the value of Φ (ξ) is the same as that of Ψ (ξ) for whatever argument, which in Frege's formal language is expressed thus: (ἐf (ε) = ἀg(α)) = ( a f (a) = g (a)), where 'f ' and 'g' are marks used to indeterminately indicate first-level one-argument functions.

Frege was perfectly aware that, by admitting of such an implicit definition, he was derogating from the strict criterion of logicality that any other ingredient of his system meets. In the Vorwort of the Grundgesetze he recognises, indeed, that "a dispute" concerning the logical nature of this system "can arise [. . . ] only concerning [. . . ] Basic Law of value-ranges (V)" ( [START_REF] Frege | [END_REF], Vorwort, p. VII; [START_REF] Frege | [END_REF], p. VII 1 ). Still, according to Frege, without this Law, and without value-ranges, there could not be other logical objects but the True and the False, and arithmetical logicism would, then, not be viable. This is what he openly claims in his tentative reply to Russell's paradox: "[. . . ] even now I do not see how arithmetic can be founded scientifically, how the numbers can be apprehended as logical objects and brought under consideration, if it is not-at least conditionally-permissible to pass from a concept to its extension" ( [START_REF] Frege | [END_REF], Nachwort, p. 253; [START_REF] Frege | [END_REF], p. 253 2 )4 . Hence, for Frege, calling Basic Law V in question was not just his "approach to a foundation in particular, but rather the very possibility of any a logical foundation of arithmetic" (ibid.). For, Frege seems to argue, if the value-range function is to be dismissed, what other logical function having other values than the True and the False is permissible? And if no such function might be permissible, how can natural numbers be logical objects? And if natural numbers are not logical objects, how can arithmetic be a branch of logic?

We know today that an alternative route for arithmetical logicism-allegedly understood, if not in the same way, at least in a way close to Frege's-has been suggested ( [Wright, 1983], [Hale and Wright, 2001]). Still, it is clear that also this route depends on the admission of a basic function, namely the cardinal-number function, which, while being taken to be a logical function, is required to have as its possible values some particular objects whose existence is not a necessary condition for the admissibility of the relevant system of logic. This is, in Frege's original terminology, a second-level one-argument function, like Frege's value-range one. And it is, like this latter function again, defined by a principle, namely Hume's principle, working as an axiom of the relevant system, and taken as an implicit definition. But, differently from Frege's value-range function, the cardinal-number function is not second-level insofar as its arguments are taken to be first-level functions. These arguments are rather taken to be concepts no more intended as functions from the totality of objects to the True and the False, but rather as the items designated by monadic first-order predicates5 . The cardinal-number functions is, thus, a total function, like the value-range one, only insofar as a previous restriction is, so to say, incorporated in the logical system that its definition depends on: a restriction that makes the predicate variables of this system range only over concepts, rather than over items so generally conceived as to render the larger variety of Frege's first-level one-argument functions. This goes together with the fact that the values of the cardinal-number function are ipso facto cardinal numbers, rather then more general items among which cardinal, and, more specifically, natural numbers, are selected with the help of appropriate explicit definitions (which might suggest that this function is not general enough to count as logical in Frege's sense).

It is not our purpose, here, to discuss neologicism. Touching on it is only a way to emphasise the main difficulty of Frege's logicism, by showing that, mutatis mutandis, it is still a crucial difficulty for its modern consistent version. This is the difficulty of fixing objects to be identified with natural numbers by having recourse only to means recognised as logical.

The way we have presented this difficulty hides a decisive aspect of it, however. This aspect only appears when it is made clear that, for Frege (as well as for the neologicists), nothing could be taken to be an object if it were not also taken to exist (in the only rightful sense in which anything can be taken to exist, both for him and for them), and no statement could be taken to be a truth if the singular terms and the first-order quantified variable included in it (if any) were not respectively taken to be names of, or to vary over existing individuals. The difficulty does not only consist, then, in defining natural numbers by having recourse only to means recognised as logical, but in doing it so as to ensure that these numbers exist, that is, that the (non-atomic) term that provides the definiendum of the explicit definition of each of them denotes an existing individual, and the (nonatomic) formula that provides the definiendum of the explicit definition of the property of being a natural number is satisfied by some (namely a countable infinity) of existing individuals (which means, in Frege's formalism, that the explicit definition of the first-level concept natural number designates a function whose value is the True for some, namely for countably many, arguments).

There is no room here for discussing the reason for the neologicists to claim that their definitions comply with this condition. What is relevant is that, for Frege, no independent existence proof is needed for this purpose, since, for him, the relevant explicit definitions are so shaped as to ensure by themselves that this condition obtains. In other words, according to Frege, his explicit definition of each natural number directly exhibits an object to be identified with this number, while his definition of the property of being a natural number makes directly manifest that there are these numbers and which objects they are. This means that, according to Frege, these explicit definitions make directly manifest that "there are logical objects" and that "the objects of arithmetic [i. e. the natural numbers] are such" ( [START_REF] Frege | [END_REF], § II.147; [START_REF] Frege | [END_REF], p. 149 2 ).

For real numbers, Frege does not seem to have thought that something like this would have been achievable, instead. Since, though he closed his informal exposition of the way he was planning to define these numbers by claiming that in this way he would have succeeded "in defining the real number purely arithmetically or logically as a ratio of magnitudes that are demonstrably there" ( [START_REF] Frege | [END_REF], § II.164; [START_REF] Frege | [END_REF], p. 162 2 ), his plan explicitly calls for an existence proof of domains of magnitudes going far beyond the simple inspection of the definition of these domains, and consisting, rather, in the independent exhibition of a particular domain of magnitudes generated from natural numbers. Frege actually realised only a part of his plan: in the third part of the Grundgesetze ( [START_REF] Frege | [END_REF], § § II.55-II.245), after having discussed and questioned several (informal) definitions of natural numbers (ibid. II.55-II.155) and having exposed his plan (ibid. II.155-II.164), he proceeds to formally defining domains of magnitudes (ibid. II.165-II.245) and to prove some crucial properties of them, by leaving to a never appeared third volume of his treatise the accomplishment of the remaining part of the plan, including the existence proof of such domains, and the definition of real numbers as ratios over them.

Let D(ξ) be the first-level concept of domains of magnitudes, namely the concept under which an object falls if and only if it is a domain of magnitudes, which means that D(s) is the True if and only if s is such a domain. Frege's formal definition of domains of magnitudes consists in stating an identity like 'D(s) = D (s)', where 'D (s)' stands for an appropriate (non-atomic) formula (ibid. II.173-174 and II.197)6 . In modern terminology, this means that domains of magnitudes are explicitly defined as the objects that satisfy this formula. And this formula is such that s satisfies it (which means, in Frege's terminology, that D (s) is the True) if and only if s is the extension of another first-level concept M (ξ), under which an object falls, in turn, if and only if it is the extension of a first-level binary relation that, if taken together with all the other extensions of a first-level binary relation that fall under this very concept, forms a certain structure7 . This means that domains of magnitudes are explicitly defined as extensions of first-level concepts under which the extensions of some first-level binary relations that form, when taken all together, a certain structure fall.

This definition is stated within the same functional formal language in which natural numbers are defined. Still, Frege openly claims (ibid. II.164) that it does not ensure that there are objects that stand to each other in some binary relations whose extensions, when taken all together, meet the relevant structural condition, and are many enough for the ratios over them could be identified with the real numbers. And, he argues, if there were not such objects, real numbers could not be defined as ratios over domains of magnitudes. The existence proof of domains of magnitudes envisaged by Frege should have consisted in showing how, by starting from natural numbers and by appropriately operating on them, one can get enough-i.e continuous many-other suitable objects. It is not necessary to enter the details of the way Frege planned to conduct this proof, in order to understand that he could not have imagined that the relevant objects could be directly exhibited by explicit definitions, as he held to have done for natural numbers. This, together with the fact that he held that his definition of domains of magnitudes does not secure, by itself, the existence of appropriate such domains, is enough for concluding that real numbers could not have been taken by Frege as logical objects in the same sense as natural numbers were taken to be logical objects by him. Hence, his logicism about real numbers, once completely expounded in agreement with his plan, could not have appeared similar in nature to his arithmetical logicism.

These short and quite general remarks should be enough to make clear that Frege's logicism is quite complex a thesis, or better that it consists of two distinct quite complex theses, respectively pertaining to natural and real numbers, that are only very partially and broadly rendered by the simple claim that arithmetic and real analysis are part of logic. It follows that it is not enough for someone to be credited with the same foundational program as Frege's that he made this same claim. Only a careful comparative scrutiny of way this claim is justified and explained could allow one to evaluate whether this claim is an expression of logicism in the same sense as Frege's.

A case in point is that of Dedekind. From the very beginning of the Vorwort to the first edition of Was sind und was sollen die Zahlen?, he explicitly identifies the "simplest science" with "that part [Theil ] of logic which deals with the theory of numbers [Lehre von den Zahlen]", and refers to it as to "arithmetic (algebra, analysis) [Arithmetik (Algebra, Analysis)]" ( [START_REF] Dedekind | [END_REF], p. VII; [START_REF] Dedekind | [END_REF], p. 14). There is little doubt that what Dedekind is meaning here with 'theory of numbers' is much more than the theory of natural numbers, and also includes real analysis. This is not only suggested by the parenthesis following the term 'Arithmetik', but also by the possessive pronouns 'its [ihr ]' in what one finds some lines below: "it is only through the purely logical construction of the science of numbers and in its acquiring the continuous number-realm that we are prepared accurately to investigate our notions of space and time" (ibid.)8 . It seems then quite clear that Dedekind here is endorsing the claim that both arithmetic and real analysis are parts of logic. Still, the way this claim is justified with respect to arithmetic, as well as and the tacit extension of it to real analysis, delineate a quite different conception than Frege's.

Dedekind's main point is that "the number-concept [Zahlbegriff ] [is] entirely independent of the conceptions or intuitions of space and time", being rather "an immediate result from the laws of thought", since "what is done in [looking for] the number of a set [Zahl der Menge] or the number of some things [Anzahl von Dingen]" depends on "the ability of the mind to relate [beziehen] things to things, to let a thing correspond [entsprechen] to a thing, or to represent [abzubilden] a thing by a thing, an ability without which no thinking is possible" ( [START_REF] Dedekind | [END_REF], p. VIII; [START_REF] Dedekind | [END_REF], p. 14).

The reference, here, is to the crucial role played, in Dedekind's definition of natural numbers, by the notion of a "mapping [Abbildung]. In his terminology: a "thing [Ding]" is "any object of our thought [Gegenstand unseres Denkens]" ( [START_REF] Dedekind | [END_REF], §. 1; [START_REF] Dedekind | [END_REF], p. 21); a "system [System]" is that which "different things [. . . ] constitute [bilden]" when they are "considered from a common point of view [unter einem gemeinsamen Gesichtspuncte aufgefasst]" and are, then, "associated in the mind [im Geiste zusammengestellt]" ( [START_REF] Dedekind | [END_REF], §. 2; [START_REF] Dedekind | [END_REF], p. 21); the "elements [Elemente]" of a system are the things that constitute such a system (ibid.); and a "mapping [Abbildung][. . . ] of a system S [. . . ][is] a law [Gesetz ] according to which a determinate thing pertains to [gehört zu] every determinate element [. . . ] of S" ( [START_REF] Dedekind | [END_REF], §. 21; [START_REF] Dedekind | [END_REF], p. 24).

His point is, then, that if these notions are appropriately used together, they provide enough conceptual tools for defining the natural numbers and setting up a theory of them. The way this is done in Dedekind's treatise is, however, essentially informal. The foregoing explanations are everything Dedekind's exposition relies on in order to make these notions operate and carry out the required definition of natural numbers and the corresponding theorems. Hence, Dedekind's theory of natural numbers is in no way embedded, like Frege's, within a well-identified formal system. If it is logic, or a part of logic, it is, then, not because it is part of such a system, but because of the intellectual abilities these notions and our handling them hinge on. In other terms, what is taken to be logical, in Dedekind's arithmetical logicism, is not a formal system and its ingredients, but some notions, informally explained, and our intellectual ability to handle them. Moreover, the relation of being part of, which relates arithmetic to logic, is not conceived as a relation of inclusion of a system into a system, but rather as a relation depending on the sufficiency of this ability for realising the relevant task.

Concerning real numbers, things are even clearer. Since, Dedekind's extension of his arithmetical logicism to real analysis merely depends on his remark that the "creation [Schöpfungen] of [. . . ] negative, fractional, irrational [. . . ] numbers is always accomplished by reduction to the earlier concepts [. . . ] without the introduction of foreign conceptions", as thoroughly shown, for irrational numbers, in Stetigkeit und Irrationale Zahlen, and suggested in section III of this very treatise, for the other numbers ( [START_REF] Dedekind | [END_REF], p. X; [START_REF] Dedekind | [END_REF], p.15; [Dedekind, 1872]). The point, here, is not only that the definition of irrational numbers offered in Stetigkeit is as informal as that of natural numbers offered in Was sind, and that in the former treatise Dedekind advances no thesis assimilable to some sort of logicism, but also, and above all, that Dedekind seems to consider useless to show explicitly how the logical intellectual ability the theory of natural numbers depends on is also enough to pass from these numbers to real ones. All that is relevant, for him, is a generic appeal to the possibility of a conceptual reduction. There is nothing, then, like what justifies Frege's logicism for real numbers, namely a further development of the same formal system in which the definition of natural numbers is embedded, resulting in an independent formal definition of the former numbers.

But as essential as these differences might appear, they are far from being the only ones. Another, possibly even more essential one, already shines through Dedekind's speaking of creation of negative, fractional and irrational numbers by reduction to earlier concepts. It depends on Frege's and Dedekind's respective conceptions of the very nature of natural and real numbers. For Frege, they are objects, that is, individuals existing as such, and they are logical not insofar as their existence depends on logic, but rather insofar logic is enough for defining them (and, at least in the case of natural numbers, for ensuring their existence). This is not at all Dedekind's view. For him, they are, rather, "free creations [frei Schöpfungen] of the human mind" to be used for "apprehending more easily and more sharply the difference of things" ( [START_REF] Dedekind | [END_REF], pp. VII-VIII; [START_REF] Dedekind | [END_REF], p.14), and these creations are logical insofar as logic, understood as we have said above, is enough for fixing their concept, which is all what is needed to create them. Since, for Dedekind, the only sense in which a number can be said to be an object depends of its being a thing, namely, as we have already said, an object of our thought, and "a thing is completely determined [vollständig bestimmt] by all that can be affirmed or thought concerning it" ( [START_REF] Dedekind | [END_REF], §. 1; [START_REF] Dedekind | [END_REF], p. 21). The more difficult task of Frege's logicism, consisting in offering a justification of the existence of natural and real numbers is, then, simply dismissed by Dedekind, which merely implies that it is nonsensical.

Though the first edition of Was sind appeared four years after Frege's Grundlagen, no mention of the latter is made in the former. A short mention is made, instead, in the Vorwort of the second edition ( [START_REF] Dedekind | [END_REF], p. XVII; [START_REF] Dedekind | [END_REF], p. 19). Though declaring of having become acquainted with Frege's treatise only after the publication of his own, and bringing to the reader's mind the difference of his and Frege's "view on the essence of number [Wesen der Zahl ]", Dedekind emphasises Frege's standing "upon the same ground" with him. Had be been aware of the way Frege's view is spelt out in the Grundgesetze, whose first volume just appeared in the same year as the second edition of Dedekind's treatise, the latter would have probably advanced a different judgement, since the main differences among the two approaches are much more evident when Was sind is compared with Grundgesetze (as we have done above), rather than with the Grundlagen, where Frege's logicism is, indeed, merely sketched out informally.

It was, then, up to Frege to emphasise the differences of his and Dedekind's approaches ( [START_REF] Frege | [END_REF], Vorwort, p. VII-VIII; [START_REF] Frege | [END_REF], pp. VII 1 -VIII 1 ):

My purpose demands some divergences from what is common in mathematics. This will be especially striking if one compares Mr Dedekind's essay, Was sind und was sollen die Zahlen?, the most thorough study I have seen in recent times concerning the foundations of arithmetic. It pursues, in much less space, the laws of arithmetic to a much higher level than here. This concision is achieved, of course, only because much is not in fact proven at all. Often, Mr Dedekind merely states that a proof follows from such and such statement; he uses dots, as in 'M (A, B, C . . .)'; nowhere in his essay do we find a list of the logical or other laws he takes as basic; and even if it were there, one would have no chance to verify whether in fact no other laws were used, since, for this, the proofs would have to be not merely indicated but carried out gaplessly. Mr Dedekind too is of the opinion that the theory of numbers is a part of logic; but his essay barely contributes to the confirmation of this opinion since his use of the expressions 'system', 'a thing belongs to a thing' are neither customary in logic nor reducible to something acknowledged as logical.

If the most part of this quote keeps our attention to the first difference we have remarked above, namely the fact that Dedekind's presentation is informal, to the effect that his logicism is in no way the thesis that the relevant truths are theorems of a formal system taken as logic, the last remark points in a different direction: what makes Dedekind's alleged logic not be logic at all is its involving set-theoretic notions.

As Frege makes clear in the Einleitung of his treatise, by coming back to the same point, the complaint, here, is not merely with Dedekind's using the term 'system' with the "same intention [Absisht]" with which others were using, in the same years, the term 'set [Menge]' ( [START_REF] Frege | [END_REF], Einleitung, p. 1; [START_REF] Frege | [END_REF], p. 1 1 ), or with his considering systems to be things to which other things belong9 . After all, Frege himself will largely use a set-theoretic term as 'class [Klasse]' in the second volume of the Grundgesetze, and will have no reticence to say that a class "contains [umfassen]" objects, and that an object "belongs [angehört]" to a class ( [START_REF] Frege | [END_REF], § II.164 and II.173, for example). Still, for Frege, a class is the "extension of a concept" ( [START_REF] Frege | [END_REF], § II.99 and II.161), that is, a value-range. This is exactly the point. For Dedekind, he argues, "every element [. . . ] of a system [. . . ] can itself be regarded as a system", and, "since in this case element and system coincide, it here becomes very perspicuous that, according to Dedekind, the elements are what properly makes out the system [die Elemente den eigentlichen Bestand des Systemes ausmachen]" ( [START_REF] Frege | [END_REF], Einleitung, p. 2; [START_REF] Frege | [END_REF], p. 2 1 ), which is, in fact, inappropriate. Since, "every time a system has to be specified", even Dedekind, despite his "lack of insight", cannot but mention "the properties a thing must have in order to belong to the system, i. e. he defines a concept in virtue of its characteristic marks", and so makes it appear that "it is the characteristic marks that make out the concept, rather than the objects falling under it" ( [START_REF] Frege | [END_REF], Einleitung, p. 3; [START_REF] Frege | [END_REF], p. 3 1 ; cf. also the quote from Frege's undated letter to Peano quoted in § ?? of chapter ??, below). In other terms: if specifying a system reduces to defining a concept, what makes out the system cannot but be the same as what makes out the concept, and this is not given by the extension (or value-range) of this concept, but by what makes an object be part of such an extension. So, it is not Dedekind's use of set-theoretic notions that Frege is questioning, but his taking them as primitive notions, his not clearly submitting them to the more general notion of a concept, or, even, to the still more general notion of a function. Logic, Frege seems to mean, can include consideration of sets, but only insofar this is part of a more general study of the objects-concepts relations, of the falling of objects under concepts: dealing with the notions of a set as a primitive notion, and merely conceiving of a set as an aggregate of things is ipso facto departing from logic. This is, possibly, the deeper difference between Frege's and Dedekind's logicism, and it sheds new light also on the differences between Dedekind's notion of a mapping and Frege's notion of a function. For the former, a mapping is something that "relates things to things" or makes "a thing correspond to a thing", or a thing "represent" a thing, and only appears, then, when appropriate things, which we independently take to be there as such, are related. For the latter, a function is that which an object is a value of, and it appears any time we refer to a certain object, through the way we refer to it, with the only exception of the primitive reference to the True and the False, which is necessary for defining the basic functions of logic. This exception, however, does not make our reference to the True and the False is always independent of any function. Logic is, rather, for Frege, the study of our referring to the True and the False as values of particular sorts of functions, namely concepts and relations, and it is our referring to the True or to the False as the value of a certain concept or relation, for a certain object or a certain pair of objects as arguments, that makes this object fall or not fall under this concept, or the objects composing this pair are or are not related by this relation. The notions of a concept and a relation, rather than those of a system and its mappings are then, as Frege explicitly says, "the foundation stones [Grundsteine] on which [. . . ][he] build [. . . ][his] construction" ( [START_REF] Frege | [END_REF], Einleitung, p. 3; [START_REF] Frege | [END_REF], p. p. 3 1 ).

We have, thus, arrived at the two basic, strictly connected, topics the present book is devoted to, namely the way Dedekind's logicism differs from Frege's, and the conception of a function Frege's logicism is grounded on.

The first chapter, by Hourya Benis Sinaceur, is specifically devoted to the former topic, and goes much further than we could have done here in accounting for the specific features of Dedekind's logicism, and of his conception of logic, in connection, of course, with his conception of systems and mappings.

The second chapter, by Marco Panza, provides a historically situated account of Frege's notion of a function, by insisting on the intensional nature of this notion, and then questioning a widespread tendency to ascribe to Frege a Platonist view about functions, and, more specifically, concepts and relations.

This question is related to a lively discussion that took place after the publication, in 1992, of a paper by Jakko Hintikka and Gabriel Sandu ([Hintikka and Sandu, 1992]), where the ascription to Frege of a notion assimilable to the modern, essentially extensional and set-theoretic notion of an arbitrary function is questioned. In the third chapter, Gabriel Sandu comes back on this discussion, by offering new arguments for his views, based on the consideration of Ramsey's reaction to Russell's logicism.

The intimate connection among these topics should be clear from the previous considerations. But it becomes even more evident when it is observed that the two features of Frege's logicism that are mainly responsible for the difference between this logicism and Dedekind's, namely Frege's conceiving of logic as a formal system, and his grounding it on a general notion of a function-as opposed, respectively, to Dedekind's taking as logical some informal notions and our intellectual ability to handle them, and to his conceiving of systems as aggregates of things that we take to be there as such-, are strictly connected, in turns. For Frege's notion of a function is fashioned as to provide the ground on which this logical system is erected, that is, it is, essentially, a linguistic notion. While Frege conceives of objects as existing individuals, he does not assign existence to functions, as such. He rather conceives them as matrices to be used to form objects-names, which is precisely what, for him, makes a function appear through the way we refer to an object, as we have said above.

This has a crucial consequence on the way the totalities of objects and functions, and, then, an arbitrary function, are conceived by him. The question is largely discussed in Panza's and Sandu's chapters, but a short remark is in order, here as well.

True, Frege was aware of Cantor's distinction among different sorts of infinity. He entitled § § 84-86 of Grundlagen "Infinite cardinal numbers [Unendliche Anzahlen]"10 (in the plural), by explicitly referring to Cantor's very recently appeared Grundlagen einer allgemeinen Mannigfaltigkeitslehre ( [START_REF] Cantor | [END_REF]). Though, in them, he specifically deals, in fact, only with "the cardinal number which belongs to the concept finite cardinal number ", he significantly denotes it with '∞ 1 ', by suggesting that other infinite cardinal numbers follow, namely ∞ 2 , ∞ 3 , etc. After having come back to this same number in the first volume of Grundgesetze, this time by calling it 'Endlos' 11 and denoting it with ' ' ( [START_REF] Frege | [END_REF], Vorwort, p. 5, and § §.122-157), he remarks, in the second volume (ibid., II.164), that his envisaged definition of real numbers as ratios of magnitudes could not be carried our without having available an infinite "class of objects" having an infinity greater than Endlos, namely that of the cardinal number of the concept class of finite cardinal numbers . Still, he seems to have realised neither that the objects that such a class would contain could not all have a (finite) name (even in a languages including countably infinite atomic names), nor that they would have allowed to form enough subclasses as to assign an extension to an even greater infinity of unnameable concepts, and, more in general, a value-range to an even greater infinity of unnameable functions. In other terms, he does not seem to have realised that allowing for uncountable classes of objects goes together with admitting unnameable objects, including the possible value-ranges of unnameable functions, and, then, also unnameable functions, in extension.

A symptom of his unawareness is his using Latin letters to form universal statements. Within Frege's system, a particular statement is formed by the name of a value of a concept or relation (which is a name of a truth-value), preceded by a special sign of assertion, namely ' '. Such a statement is taken to assert, then, that the truth-value named by this name is the True. A general statement is formed in the same way, except for the replacement of the name of a value of a concept or relation with a Roman object-marker, namely an expression involving Roman letters ( [START_REF] Frege | [END_REF], §. 26; [START_REF] Frege | [END_REF], p. 44). And it is taken to assert that the object-name that is obtained from such an object-marker, by replacing within it each Roman letter for objects with whatever object-name (which actually refers to an object), and any Roman letter for functions with whatever functionname (of the appropriate sort), is a name of the True ( [START_REF] Frege | [END_REF], § § I.5, I.8, I.17 and I.19). So, to give a simple example, the general statement ' There is no doubt that a notion of a function leaving no room for the idea of an extensionally arbitrary function, and a notion of a set based on such a notion of function could not have provided a basis for the development of mathematics and mathematical logic that followed the pioneering works of Frege, Dedekind, and Cantor, among others. Hence, despite the lack of intelligibility that the idea of an extensionally arbitrary function brings with it, and despite the promises of clarification that alternative notions, intensional, in nature, could generate or have generated, it is a fact that a foundational program based on these last notions could hardly have been suitable for accounting for this development, even if it had presented no other internal difficulty. Still, the question that the present book is devoted to is not whether Frege's or Dedekind's different forms of logicism were based on suitable conceptions. Our aim is rather that of keeping the reader's attention on some crucial aspects of these forms of logicism, so as to contribute to a better understanding of their nature, their motivations, and their mutual differences.

  s system ([START_REF] Frege | [END_REF], § I.50), is taken to assert that the truth-value named by the object-name got from 'f (b) f (a) a = b', by replacing 'a' and 'b' with whatever object-names (which actually refer to an object), and 'f (ξ)' with whatever name of a one-argument first-level function, is the True. It is clear that Frege could have not conceived of a general statement in this way if he had not admitted that any possible object and function can somehow be named 12 .

Remember that a concept is, in Frege's terminology, a one-argument function whose values are either the True or the False, and its extension is nothing but its value-range.

This entails that taking the cardinal-number function as a second-level function is imprecise, strictly speaking: this is, rather, a second-order function.

As a matter of fact, this formula is not openly written by Frege, but it is easily deducible by other formulas which he openly writes.

Remember that for Frege a binary first-level relation is a first-level two-arguments function whose values are either the True or the False.

This is also confirmed by the reference, which Dedekind makes few lines below, to his supplement XI to Lejeune Dirichlet's Vorlesugen über Zahlentheorie, which is devoted to the theory of finite algebraic numbers ([Dirichlet, 1879], pp. 434-626, esp. p. 470, footnote).

As a matter of fact, in Was sind, one finds only one occurrence, in § 34, of the verb 'to belong to [gehören zu]' or its cognates used in this sense; possibly Frege's was here referring to Stetigkeit, where this use is much more frequent, namely in § § IV-V.

Cf. footnote 2, above.

Cf. again footnote 2, above.

Unless he were admitting, instead, that the relevant generality is merely a linguistic one, that is, one that concerns only objects and function that can be somehow named, which is quite implausible, indeed.


Arash Maskooki

Viktor Toldov

Laurent Clavier

Valeria Loscrí

Nathalie Mitton

Competition: Channel Exploration/Exploitation Based on a Thompson Sampling Approach in a Radio Cognitive Environment

Machine learning approaches have been extensively applied in interference mitigation and cognitive radio devices. In this work, we model the spectrum selection process as a multi-arm bandit problem and apply Thompson sampling, a fast and efficient algorithm, to find the best channel in the shortest time interval. The learning algorithm will work on top of a network layer to efficiently route the event information to the sink.In this work we address the problem on two layers: channel decision layer and network layer. Channel decision layer formulates the channel selection as a multiarm bandit problem. Thomson sampling [START_REF] Chapelle | An empirical evaluation of thompson sampling[END_REF] is a classical solution in Bandit problems. We adapt Thompson sampling to our context and implement it in hardware in the channel selection layer. The network layer uses the results of this channel selection to route data on the specified channel.

We formulate channel selection and exploration dilemma as a multi-arm bandit problem. Then, we describe an efficient and simple learning algorithm for the channel selection process.

In a multi-arm bandit problem, an agent tries to obtain as much reward as possible by playing the most rewarding arm among N arms. However, each arm rewards randomly upon being played according to an unknown distribution. Hence, the objective is to minimize exploration to find the most rewarding arm. A policy A is an algorithm that defines the actions of the agent usually based on the previous observations. We assume n j to be the number of times j th arm has been played after n steps and µ j to be the expected reward of j th arm. In other words, channel j is found available in average µ j n j times in n j measurements. µ j is associated with the statistics of other networks.

To keep track of the other networks statistics, an expiration time can be defined to trigger the search for a new channel. Another criterion to trigger the search for a new channel is to define a threshold for the number of consecutive unsuccessful channels access after which the user will search for a new channel.

Thompson sampling [START_REF] Chapelle | An empirical evaluation of thompson sampling[END_REF] is best understood in Bayasian context. Assume we observed S j , the observation vector, after accessing channel j, n j times. Assuming Bernoulli distribution for each access trial with parameter µ j , the parametric likelihood function for observation vector S j is as follows,

p j (S j |µ j ) = µ j t j (1 -µ j ) n j -t j , (1)
where t j is the number of successful transmissions on j th channel in n trials. Without loss of generality, we use Beta distribution as the prior for the distribution of parameter µ j . This is because Beta distribution is conjugate prior for the likelihood function in Equation ( 1) which simplifies the derivations [START_REF] Diaconis | Conjugate priors for exponential families[END_REF]. Using Bayes rule we can write,

p j (µ j |S j ) = p j (S j |µ j ) Γ(α+β) Γ(α)Γ(β) µ j α-1 (1 -µ j ) β-1 p j (S j ) , (2)
where,

Γ(α) = ∞ 0 x α-1 e -x dx (3)
and α and β are the shape parameters of the Beta distribution; as we assume no prior information on µ j we initialize α = β = 1 which yields uniform distribution. Substituting (1) in (2) yields,

p j (µ j |S j ) = Γ(α+β) Γ(α)Γ(β) p j (S j ) µ j t j +α-1 (1 -µ j ) n j -t j +β-1 . (4)
α = t j + α and β = n jt j + β can re-write (4) as:

p j (µ j |S j ) = Cµ j α -1 (1 -µ j ) β -1 (5)
Substituting the normalizing factor C we obtain,

p j (µ j |S j ) = Γ(α + β ) Γ(α )Γ(β ) µ j α -1 (1 -µ j ) β -1 , (6)
which is the beta distribution with parameters α and β ,

p j (µ j |S j ) = beta(α , β ). (7)
Thompson sampling Channel selection algorithm is described in Algorithm 1.

j: channel index n: total number of channel access s j : current state of the channel j t j : number of successful transmissions so far x j : empirical mean of the channel j states, α: a priori (beta distribution) model parameter β: a priori (beta distribution) model parameter α : a posteriori (beta distribution) model parameter

α j = t j + α j (8)
β : a posteriori (beta distribution) model parameter

β j = n j -t j + β j (9)
TRANSMIT(): Packet transmission function Initialization:

1: for all j do 2:

if channel j is busy then 3: update t j , n j , α j and β j

s j = 0 4:

21: end while

As there is no prior information about the channels, we set α = 1 and β = 1 which yields uniform distribution in [0, 1].

we implemented the in TelosB nodes to measure its performance in real-time. In our setup, we used 3 pairs of laptops occupying 3 orthogonal Wi-Fi channels, 1, 6 and 11 overlapping with standard 802.15.4 channels, 12, 17 and 22 respectively. The traffic was generated using "Distributed Internet Traffic Generator" [START_REF] Botta | A tool for the generation of realistic network workload for emerging networking scenarios[END_REF] in single flow mode with packet size 500 bytes which is the average packet size on the Internet [START_REF] Thompson | Wide-area internet traffic patterns and characteristics[END_REF]. Two TelosB nodes were programmed in Contiki operating system one with a learning algorithm and one as oracle fixed on the best channel. To generate samples of Beta distribution used in Thompson sampling algorithm we used "GEN SEQUENCE" open-source library [START_REF] Karplus | Online library[END_REF].

To monitor the availability rate of the channel we programmed one TelosB node as monitor which just sampled the channel. The availability rate obtained as the average number of samples the channel is detected available. The RSSI sensitivity of TelosB node was set to -40dbm. This relatively high threshold was set to suppress the RSSI received from other networks present in the building. With this sensitivity, the monitor node registers approximately 90% availability rate for the channels. The availability rate of channel 6 drops to approximately 40% when the traffic generator is activated at 2000pkt/sec and packet size of 500 bytes. The availability rate of the channel 1 drops to ap- prox. 60% when the traffic generator occupies the channel with 500pkt/sec. Channel 11 is left without traffic although the server and client were connected. The monitor shows approx. 90% availability on the channel. Note that the channel occupancy rate is affected by our traffic, other networks traffic and noise. However, it was roughly constant during the experiment at the given rates.

We programmed two Telosb nodes; one as an oracle which always operated on the channel with the best availability rate. The other node was programmed with the implementation of a learning algorithm to find and use the best channel. In our results, we considered an available channel as a successful transmission. In reality, the packet transmission can be disrupted in the middle of the transmission and cause the transmission to fail. However, the collision will affect the throughput results of all algorithms including the oracle the same way. Hence the comparison results would not be affected.

In each set of experiments, we performed 3 experiments where occupancy rate of the channels were permutated. We repeated each experiment 3 times. The relative throughput of each algorithm in each experiment is divided by the oracle performance of the best channel and then averaged over all the experiments for each algorithm. Figure 1 shows the performance of the algorithm. As seen in the figure, the learning algorithm finds the best channel and reaches about 99% of the throughput of the oracle in 0.8 seconds.

Results show that Thompson sampling formulation achieves high average throughput as it spends less time on exploring the channels and converges to the best channel fast.

  t j , n j , α j and β j 8: end for 9: while True do 10: for all j do 11: sample r j ∼ beta(α j , β j )

Figure 1 :

 1 Figure 1: Relative throughput on three accessible channels using Empirical real-time test-bed.