M Cunche

Task Force (IETF) V. Roca Request for Comments: 6816 INRIA Category: Standards Track

This document describes a fully specified simple Forward Error Correction (FEC) scheme for Low-Density Parity Check (LDPC) Staircase codes that can be used to protect media streams along the lines defined by FECFRAME. These codes have many interesting properties: they are systematic codes, they perform close to ideal codes in many use-cases, and they also feature very high encoding and decoding throughputs. LDPC-Staircase codes are therefore a good solution to protect a single high bitrate source flow or to protect globally several mid-rate flows within a single FECFRAME instance. They are also a good solution whenever the processing load of a software encoder or decoder must be kept to a minimum.

Introduction

The use of Forward Error Correction (FEC) codes is a classic solution to improve the reliability of unicast, multicast, and broadcast Content Delivery Protocols (CDPs) and applications [START_REF] Luby | The Use of Forward Error Correction (FEC) in Reliable Multicast[END_REF]. "Forward Error Correction (FEC) Framework" [START_REF] Watson | Forward Error Correction (FEC) Framework[END_REF] describes a generic framework to use FEC schemes with media delivery applications and, for instance, with real-time streaming media applications based on the RTP real-time protocol. Similarly, "Forward Error Correction (FEC) Building Block" [START_REF] Watson | Forward Error Correction (FEC) Building Block[END_REF] describes a generic framework to use FEC schemes with objects (e.g., files) delivery applications based on either the Asynchronous Layered Coding (ALC) [START_REF] Luby | Asynchronous Layered Coding (ALC) Protocol Instantiation[END_REF] or the NACK-Oriented Reliable Multicast (NORM) [START_REF] Adamson | NACK-Oriented Reliable Multicast (NORM) Transport Protocol[END_REF] protocols.

More specifically, the [START_REF] Luby | Raptor Forward Error Correction Scheme for Object Delivery[END_REF] (Raptor) and [START_REF] Roca | Low Density Parity Check (LDPC) Staircase and Triangle Forward Error Correction (FEC) Schemes[END_REF] (LDPC-Staircase and LDPC-Triangle) FEC schemes introduce erasure codes based on sparse parity check matrices for object delivery protocols like ALC and NORM. Similarly, "Reed-Solomon Forward Error Correction (FEC) Schemes" [START_REF] Lacan | Reed-Solomon Forward Error Correction (FEC) Schemes[END_REF] introduces Reed-Solomon codes based on Vandermonde matrices for the same object delivery protocols. All these codes are systematic codes, meaning that the k source symbols are part of the n encoding symbols. Additionally, the Reed-Solomon FEC codes belong to the class of Maximum Distance Separable (MDS) codes that are optimal in terms of erasure recovery capabilities. It means that a receiver can recover the k source symbols from any set of exactly k encoding symbols out of n. This is not the case with either Raptor or LDPC-Staircase codes, and these codes require a certain number of encoding symbols in excess to k. However, this number is small in practice when an appropriate decoding scheme is used at the receiver [START_REF] Cunche | Optimizing the Error Recovery Capabilities of LDPC-Staircase Codes Featuring a Gaussian Elimination Decoding Scheme[END_REF]. Another key difference is the high encoding/decoding complexity of Reed-Solomon codecs compared to Raptor or LDPC-Staircase codes. A difference of one or more orders of magnitude in terms of encoding/decoding speed exists between the Reed-Solomon and LDPC-Staircase software codecs [START_REF] Cunche | Optimizing the Error Recovery Capabilities of LDPC-Staircase Codes Featuring a Gaussian Elimination Decoding Scheme[END_REF] [START_REF] Cunche | High performances AL-FEC codes for the erasure channel : variation around LDPC codes[END_REF]. Finally, Raptor and LDPC-Staircase codes are large block FEC codes, in the sense of [START_REF] Luby | The Use of Forward Error Correction (FEC) in Reliable Multicast[END_REF], since they can efficiently deal with a large number of source symbols.

The present document focuses on LDPC-Staircase codes that belong to the well-known class of "Low Density Parity Check" codes. Because of their key features, these codes are a good solution in many situations, as detailed in Section 7. This document inherits from [START_REF] Roca | Low Density Parity Check (LDPC) Staircase and Triangle Forward Error Correction (FEC) Schemes[END_REF], Section 6 "Full Specification of the LDPC-Staircase Scheme", the specifications of the core LDPC-Staircase codes, and from Section 5.7 "Pseudo-Random Number Generator", the specifications of the PRNG used by these codes. Therefore, this document specifies only the information specific to Roca, et al.

Standards Track [Page 4]

the FECFRAME context and refers to [START_REF] Roca | Low Density Parity Check (LDPC) Staircase and Triangle Forward Error Correction (FEC) Schemes[END_REF] for the core specifications of the codes. To that purpose, the present document introduces:

o the Fully Specified FEC Scheme with FEC Encoding ID 7 that specifies a simple way of using LDPC-Staircase codes in order to protect arbitrary Application Data Unit (ADU) flows.

Therefore Sections 4 and 5 (except Section 5.7, see above) of [START_REF] Roca | Low Density Parity Check (LDPC) Staircase and Triangle Forward Error Correction (FEC) Schemes[END_REF], that define [START_REF] Watson | Forward Error Correction (FEC) Building Block[END_REF] specific Formats and Procedures, are not considered and are replaced by FECFRAME specific Formats and Procedures.

Finally, publicly available reference implementations of these codes are available [LDPC-codec] [LDPC-codec-OpenFEC].

Terminology

The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in RFC 2119 [RFC2119].

Definitions Notations and Abbreviations

Definitions

This document uses the following terms and definitions. Those in the list below are FEC scheme specific and are in line with [START_REF] Watson | Forward Error Correction (FEC) Building Block[END_REF]:

Source symbol: unit of data used during the encoding process. In this specification, there is always one source symbol per ADU.

Encoding symbol: unit of data generated by the encoding process. With systematic codes, source symbols are part of the encoding symbols.

Repair symbol: encoding symbol that is not a source symbol.

Code rate: the k/n ratio, i.e., the ratio between the number of source symbols and the number of encoding symbols. By definition, the code rate is such that: 0 < code rate <= 1. A code rate close to 1 indicates that a small number of repair symbols have been produced during the encoding process.

Systematic code: FEC code in which the source symbols are part of the encoding symbols. The LDPC-Staircase codes introduced in this document are systematic. ADU Information (ADUI): a unit of data constituted by the ADU and the associated Flow ID, Length, and Padding fields (Section 4.3). This is the unit of data that is used as source symbol.

FEC Framework Configuration Information (FFCI): information that controls the operation of the FEC Framework. The FFCI enables the synchronization of the FECFRAME sender and receiver instances.

FEC Source Packet: at a sender (respectively, at a receiver) a payload submitted to (respectively, received from) the transport protocol containing an ADU along with an optional Explicit Source FEC Payload ID.

FEC Repair Packet: at a sender (respectively, at a receiver) a payload submitted to (respectively, received from) the transport protocol containing one repair symbol along with a Repair FEC Payload ID and possibly an RTP header.

The above terminology is illustrated in Figure 1 --------------------- --------------------- Note that the use of the terminology "maximum source block size" and "maximum ADU block size" depends on the point of view that is adopted (FEC scheme versus FECFRAME instance). However, in this document, both refer to the same value since Section 4.1 requires there be exactly one source symbol per ADU. We now detail each of these aspects.

+----------------------+ | Application | +----------------------+ | | (1) Application Data Units (ADUs) | v +----------------------+ +----------------+ | FEC Framework | | | | |-------------------------->| FEC Scheme | |(2) Construct source |(3) Source Block | | | blocks | |(4) FEC Encoding| |(6) Construct FEC |<--------------------------| | | source and repair | | | | packets |(5) Explicit Source FEC | | +-
+ | Transport Layer | | (e.g., UDP) | +----------------------+ Figure 1:
The maximum source block size in symbols, max_k, depends on several parameters: the code rate (CR) and the Encoding Symbol ID (ESI) field length in the Explicit Source/Repair FEC Payload ID (16 bits), as well as possible internal codec limitations. More specifically, max_k cannot be larger than the following values, derived from the ESI field size limitation, for a given code rate: Additionally, a codec can impose other limitations on the maximum source block size, for instance, because of a limited working memory size. This decision MUST be clarified at implementation time, when the target use-case is known. This results in a max2_k limitation.

Then, max_k is given by: max_k = min(max1_k, max2_k)

Note that this calculation is only required at the encoder (sender), since the actual k parameter (k <= max_k) is communicated to the decoder (receiver) through the Explicit Source/Repair FEC Payload ID.

The source ADU flows can have real-time constraints. When there are multiple flows, with different real-time constraints, let us consider the most stringent constraints (see [START_REF] Watson | Forward Error Correction (FEC) Framework[END_REF], Section 10.2, item 6, for recommendations when several flows are globally protected). In that case the maximum number of ADUs of an ADU block must not exceed a certain threshold since it directly impacts the decoding delay.

The larger the ADU block size, the longer a decoder may have to wait until it has received a sufficient number of encoding symbols for decoding to succeed, and therefore the larger the decoding delay.

When the target use-case is known, these real-time constraints result in an upper bound to the ADU block size, max_rt.

For instance, if the use-case specifies a maximum decoding latency, l, and if each source ADU covers a duration d of a continuous media (we assume here the simple case of a constant bit rate ADU flow), then the ADU block size must not exceed:

max_rt = floor(l / d)
After encoding, this block will produce a set of at most n = max_rt / CR encoding symbols. These n encoding symbols will have to be sent at a rate of n / l packets per second. For instance, with d = 10 ms, l = 1 s, max_rt = 100 ADUs.

If we take into account all these constraints, we find: max_B = min(max_k, max_rt)

This max_B parameter is an upper bound to the number of ADUs that can constitute an ADU block. In its most general form, FECFRAME and the LDPC-Staircase FEC Scheme are meant to protect a set of independent flows. Since the flows have no relationship to one another, the ADU size of each flow can potentially vary significantly. Even in the special case of a single flow, the ADU sizes can largely vary (e.g., the various frames of a Group of Pictures (GOP) of an H.264 flow). This diversity must be addressed since the LDPC-Staircase FEC Scheme requires a constant encoding symbol size (E parameter) per source block. Since this specification requires that there be only one source symbol per ADU, E must be large enough to contain all the ADUs of an ADU block along with their prepended 3 bytes (see below).

In situations where E is determined per source block (default, specified by the FFCI/FSSI with S = 0, Section 5.1.1.2), E is equal to the size of the largest ADU of this source block plus three (for the prepended 3 bytes, see below). In this case, upon receiving the first FEC repair packet for this source block, since this packet MUST contain a single repair symbol (Section 5.1.3), a receiver determines the E parameter used for this source block.

In situations where E is fixed (specified by the FFCI/FSSI with S = 1, Section 5.1.1.2), then E must be greater or equal to the size of the largest ADU of this source block plus three (for the prepended 3 bytes, see below). If this is not the case, an error is returned. How to handle this error is use-case specific (e.g., a larger E parameter may be communicated to the receivers in an updated FFCI message, using an appropriate mechanism) and is not considered by this specification.

The ADU block is always encoded as a single source block. There are a total of B <= max_B ADUs in this ADU block. For the ADU i, with 0 <= i <= B-1, 3 bytes are prepended (Figure 2 -------------------------------------------------------------> +----+----+-----------------------+------------------------------ ---+----+----------+------------+------------------------------ ---+----+----------+------------------------------------------- ---+----+------+----------------------------------------------- ---+----+------+-----------------------------------------------+ \_______________________________ _______________________________/ \/ simple FEC encoding +----------------------------------------------------------------+ ---------------------------------------------------------------+ --------------------------------------------------------------- The FEC Framework Configuration Information (or FFCI) includes information that MUST be communicated between the sender and receiver(s). More specifically, it enables the synchronization of the FECFRAME sender and receiver instances. It includes both mandatory elements and scheme-specific elements, as detailed below.

+ |F[0]|L[0]| ADU[0] | Pad[0] | +-
+ |F[1]|L[1]| ADU[1] | Pad[1] | +-
+ |F[2]|L[2]| ADU[2] | +-
+ |F[3]|L[3]|ADU[3]| Pad[3] | +-
| Repair 4 | +-
. . . . +----------------------------------------------------------------+ | Repair 7 | +-

Mandatory Information

o FEC Encoding ID: the value assigned to this fully specified FEC scheme MUST be 7, as assigned by IANA (Section 8).

When SDP is used to communicate the FFCI, this FEC Encoding ID is carried in the 'encoding-id' parameter.

FEC Scheme-Specific Information

The FEC Scheme-Specific Information (FSSI) includes elements that are specific to the present FEC scheme. More precisely:

o PRNG seed (seed): a non-negative 32-bit integer used as the seed of the Pseudo-Random Number Generator, as defined in [START_REF] Roca | Low Density Parity Check (LDPC) Staircase and Triangle Forward Error Correction (FEC) Schemes[END_REF].

o o N1m3 parameter (n1m3): 3-bit field. 0 1 2 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 -------------------------------+ | Transport Header | +- -------------------------------+ | ADU | +--------------------------------+ | Explicit Source FEC Payload ID | +- ------------------------------- o Source Block Length (k) (16-bit field): this field provides the number of source symbols for this source block, i.e., the k parameter. 0 1 2 3 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 -------------------------------+ | Transport Header | +- -------------------------------+ | Repair FEC Payload ID | +- -------------------------------+ | Repair Symbol | +- ------------------------------- o Source Block Length (k) (16-bit field): this field provides the number of source symbols for this source block, i.e., the k parameter.

+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | PRNG seed (seed) | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | Encoding Symbol Length (E) |S| resvd | n1m3| +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | Source Block Number (SBN) | Encoding Symbol ID (ESI) | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | Source Block Length (k) | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

o Number of Encoding Symbols (n) (16-bit field): this field provides the number of encoding symbols for this source block, i.e., the n parameter. 0 1 2 3 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+- +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ o The FEC repair packet creation MUST follow the procedures specified in Section 5.1.3.

+-+-+ | Source Block Number (SBN) | Encoding Symbol ID (ESI) | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | Source Block Length (k) | Number Encoding Symbols (n) | +-

FEC Code Specification

The present document inherits from [START_REF] Roca | Low Density Parity Check (LDPC) Staircase and Triangle Forward Error Correction (FEC) Schemes[END_REF] the specification of the core LDPC-Staircase codes for a packet erasure transmission channel (see Section 1).

Because of the requirement to have exactly one encoding symbol per group, i.e., because G MUST be equal to 1 (Section 4.1), several parts of [START_REF] Roca | Low Density Parity Check (LDPC) Staircase and Triangle Forward Error Correction (FEC) Schemes[END_REF] are not of use. In particular, this is the case of Section 5.6, "Identifying the G Symbols of an Encoding Symbol Group".

Security Considerations

The FEC Framework document [START_REF] Watson | Forward Error Correction (FEC) Framework[END_REF] provides a comprehensive analysis of security considerations applicable to FEC schemes. Therefore, the present section follows the security considerations section of [START_REF] Watson | Forward Error Correction (FEC) Framework[END_REF] and only discusses topics that are specific to the use of LDPC-Staircase codes. to the way this solution is applied (e.g., Is encryption applied before or after FEC protection? Is it within the end-system or in a middlebox?), to the operational constraints (e.g., performing FEC decoding in a protected environment may be complicated or even impossible) and to the threat model.

Content Corruption

The LDPC-Staircase FEC Scheme specified in this document does not change the recommendations of [START_REF] Watson | Forward Error Correction (FEC) Framework[END_REF]. To summarize, it is RECOMMENDED that one of the solutions mentioned in [START_REF] Watson | Forward Error Correction (FEC) Framework[END_REF] be used on both the FEC source and repair packets.

Attacks against the FEC Parameters

The FEC scheme specified in this document defines parameters that can be the basis of several attacks. More specifically, the following parameters of the FFCI may be modified by an attacker (Section 5.1.1.2):

o FEC Encoding ID: changing this parameter leads the receiver to consider a different FEC scheme, which enables an attacker to create a Denial of Service (DoS).

o Encoding symbol length (E): setting this E parameter to a value smaller than the valid one enables an attacker to create a DoS since the repair symbols and certain source symbols will be larger than E, which is an incoherency for the receiver. Setting this E parameter to a value larger than the valid one has similar impacts when S=1 since the received repair symbol size will be smaller than expected. Contrarily, it will not lead to any incoherency when S=0 since the actual symbol length value for the block is determined by the size of any received repair symbol, as long as this value is smaller than E. However, setting this E parameter to a larger value may have impacts on receivers that pre-allocate memory space in advance to store incoming symbols.

o Strict (S) flag: flipping this S flag from 0 to 1 (i.e., E is now considered as a strict value) enables an attacker to mislead the receiver if the actual symbol size varies over different source blocks. Flipping this S flag from 1 to 0 has no major consequences unless the receiver requires to have a fixed E value (e.g., because the receiver pre-allocates memory space).

o N1 minus 3 (n1m3): changing this parameter leads the receiver to consider a different code, which enables an attacker to create a DoS. The LDPC-Staircase FEC Scheme specified in this document does not change the recommendations of [START_REF] Watson | Forward Error Correction (FEC) Framework[END_REF] concerning the use of the IPsec/ESP security protocol as a mandatory to implement (but not mandatory to use) security scheme. This is well suited to situations where the only insecure domain is the one over which the FEC Framework operates.

Operations and Management Considerations

The FEC Framework document [START_REF] Watson | Forward Error Correction (FEC) Framework[END_REF] provides a comprehensive analysis of operations and management considerations applicable to FEC schemes. Therefore, the present section only discusses topics that are specific to the use of LDPC-Staircase codes as specified in this document. rate, enables a successful decoding with a probability 0.5), and an overhead of 1.46% (i.e., receiving 15 symbols in addition to k, which corresponds to a 32.36% channel loss rate) is sufficient to reduce the probability that decoding fails down to 8.2*10^^-5. This is why these codes are a good solution to protect a single high bitrate source flow as in [START_REF] Matsuzono | Performance Analysis of a High-Performance Real-Time Application with Several AL-FEC Schemes[END_REF] or to protect globally several midrate source flows within a single FECFRAME instance: in both cases, the source block size can be assumed to be equal to a few hundred (or more) source symbols.

LDPC-Staircase codes are also a good solution whenever the processing load at a software encoder or decoder must be kept to a minimum. This is true when the decoder uses an IT decoding algorithm, an ML algorithm (we use a Gaussian Elimination as the ML algorithm) when carefully implemented, or a mixture of both techniques, which is the recommended solution

[Cunche08][CunchePHD10][LDPC-codec-OpenFEC].
Let us consider the same conditions as above (k=1024 source symbols, CR=2/3, N1=7, G=1), with encoding symbols of size 1024 bytes. With an Intel Xeon 5120/1.86 GHz workstation running Linux/64 bits, the average decoding speed is between 1.78 Gbps (overhead of 2 symbols in addition to k, corresponding to a very bad channel with a 33.20% loss rate, close to the theoretical decoding limit, where ML decoding is required) and 3.91 Gbps (corresponding to a good channel with a 5% loss rate only, where IT decoding is sufficient). Under the same conditions, on a Samsung Galaxy SII smartphone (GT-I9100P model, featuring an ARM Cortex-A9/1.2 GHz processor and running Android 2.3.4), the decoding speed is between 397 Mbps (bad channel with a 33.20% loss rate, close to the theoretical decoding limit) and 813 Mbps (good channel with a 5% loss rate only).

As the source block size decreases, the erasure recovery capabilities of LDPC codes in general also decrease. In the case of LDPC-Staircase codes, in order to limit this phenomenon, it is recommended to use a value of the N1 parameter at least equal to 7 (e.g., experiments carried out in [START_REF] Matsuzono | Performance Analysis of a High-Performance Real-Time Application with Several AL-FEC Schemes[END_REF] use N1=7 if k=170 symbols, and N1=5 otherwise). For instance, independently of FECFRAME, with a source block of size k=256 symbols, CR=2/3 (i.e., 128 repair symbols are added), N1=7, and G=1, the average overhead amounts to 0.706% (i.e., receiving 1.8 symbols in addition to k enables a successful decoding with a probability 0.5), and an overhead of 5.86% (i.e., receiving 15 symbols in addition to k) is sufficient to reduce the decoding failure probability to 5.9*10^^-5.

The processing load also decreases with the source block size. For instance, under these conditions (k=256 source symbols, CR=2/3, N1=7, and G=1), with encoding symbols of size 1024 bytes, on a Samsung Galaxy SII smartphone, the decoding speed is between 518 Mbps (bad channel) and 863 Mbps (good channel with a 5% loss rate only). With very small source blocks (e.g., a few tens of symbols), using for instance Reed-Solomon codes [SIMPLE_RS] or 2D parity check codes may be more appropriate.

The way the FEC repair packets are transmitted is of high importance.

A good strategy, that works well for any kind of channel loss model, consists in sending FEC repair packets in random order (rather than in sequence) while FEC source packets are sent first and in sequence. Sending all packets in a random order is another possibility, but it requires that all repair symbols for a source block be produced first, which adds some extra delay at a sender.

IANA Considerations

This document registers one value in the "FEC Framework (FECFRAME) FEC Encoding IDs" registry [START_REF] Watson | Forward Error Correction (FEC) Framework[END_REF] as follows:

o 7 refers to the Simple LDPC-Staircase FEC Scheme for Arbitrary Packet Flows, as defined in Section 5 of this document.

  FEC source and repair packets v +-

  max1_k = 2^^(16 -ceil(Log2(1/CR))) Some common max1_k values are: o CR == 1 (no repair symbol): max1_k = 2^^16 = 65536 symbols o 1/2 <= CR < 1: max1_k = 2^^15 = 32,768 symbols o 1/4 <= CR < 1/2: max1_k = 2^^14 = 16,

Figure 3 :

 3 Figure 3: FSSI Encoding Format 5.1.2. Explicit Source FEC Payload ID A FEC source packet MUST contain an Explicit Source FEC Payload ID that is appended to the end of the packet as illustrated in Figure 4.

+Figure 4 :

 4 Figure 4: Structure of a FEC Source Packet with the Explicit Source FEC Payload ID More precisely, the Explicit Source FEC Payload ID is composed of the following fields (Figure 5): o Source Block Number (SBN) (16-bit field): this field identifies the source block to which this FEC source packet belongs.o Encoding Symbol ID (ESI) (16-bit field): this field identifies the source symbol contained in this FEC source packet. This value is such that 0 <= ESI <= k -1 for source symbols.

Figure 5 :

 5 Figure 5: Source FEC Payload ID Encoding Format

+Figure 6 :

 6 Figure 6: Structure of a FEC Repair Packet with the Repair Payload ID More precisely, the Repair FEC Payload ID is composed of the following fields (Figure 7): o Source Block Number (SBN) (16-bit field): this field identifies the source block to which the FEC repair packet belongs.o Encoding Symbol ID (ESI) (16-bit field): this field identifies the repair symbol contained in this FEC repair packet. This value is such that k <= ESI <= n -1 for repair symbols.

o

  Figure 7: Repair FEC Payload ID Encoding Format

Roca

  Then, zero padding is added to ADU i (if needed) in field Pad[i], for alignment purposes up to a size of exactly E bytes. The data unit resulting from the ADU i and the F[i], L[i], and Pad[i] fields is called ADU Information (or ADUI). Each ADUI contributes to exactly one source symbol of the source block.

	RFC 6816	Simple LDPC-Staircase FEC Scheme	December 2012
		Encoding Symbol Length (E)
	< -
	Roca, et al.	Standards Track	[Page 11]

): o The first byte, F[i] (Flow ID), contains the integer identifier associated to the source ADU flow to which this ADU belongs. It is assumed that a single byte is sufficient, or said differently, that no more than 256 flows will be protected by a single instance of FECFRAME. o The following two bytes, L[i] (Length), contain the length of this ADU, in network byte order (i.e., big endian). This length is for the ADU itself and does not include the F[i], L[i], or Pad[i] fields.

  Therefore, it is RECOMMENDED that security measures be taken to guarantee the FFCI integrity, as specified in[START_REF] Watson | Forward Error Correction (FEC) Framework[END_REF]. How to achieve this depends on the way the FFCI is communicated from the sender to the receiver, which is not specified in this document.

	RFC 6816	Simple LDPC-Staircase FEC Scheme	December 2012
	Similarly, attacks are possible against the Explicit Source FEC
	Payload ID and Repair FEC Payload ID: by modifying the Source Block
	Number (SBN), or the Encoding Symbol ID (ESI), or the Source Block
	Length (k), or the Number Encoding Symbols (n), an attacker can
	easily corrupt the block identified by the SBN. Other consequences,
	that are use-case and/or CDP dependent, may also happen. It is
	therefore RECOMMENDED that security measures be taken to guarantee
	the FEC source and repair packets as stated in [RFC6363].
	6.3. When Several Source Flows Are to Be Protected Together
	The LDPC-Staircase FEC Scheme specified in this document does not
	change the recommendations of [RFC6363].
	6.4. Baseline Secure FEC Framework Operation
	Roca, et al.	Standards Track	[Page 18]

Simple LDPC-Staircase FEC SchemeDecember 2012

Acknowledgments

The authors want to thank K. Matsuzono, J. Detchart, and H. Asaeda for their contributions in evaluating the use of LDPC-Staircase codes in the context of FECFRAME [Matsuzono10].

Authors' Addresses


ON EXCESSIVE MATHEMATIZATION, SYMPTOMS, DIAGNOSIS, AND PHILOSOPHICAL BASES FOR REAL WORLD KNOWLEDGE

Several articles on the misuse of mathematics in economics have already appeared in this journal. They all denounce this excess and list numerous weaknesses of liberal economics and theoretical economics that are due to, or at least related to, too much math.

This subject is worthy of further comment because it seems to me that these articles have mostly described symptoms, albeit a great many symptoms, but have barely begun to diagnose the causes and have given no hint of the kind of knowledge that would enable us to escape this noman's-land of using a little math but not too much.

The most recent contribution, by Michael Hudson (RWER No. 54), focuses on the important issues that escape mathematical models, such as the structural and historical evolution of societies, prevention of crises, psychological phenomena, long-term thinking. It emphasizes the normative nature of marginal analysis and equilibrium models, and denounces rough quantifications such as GNP and the staggering increase in debt. He acknowledges Marx's openness to the big issues in society that are currently excluded from political debate by an economic philosophy that tries to impress its opponents with sophisticated mathematics. These questions are analyzed thoroughly. On several occasions, however, one feels that the criticism is that the math is being misused and should be developed in some other direction (e.g. a statistical analysis of the financial tendencies that polarize wealth and income, or a study of the positive feedback mechanisms, etc.). This leaves a certain dissatisfaction -on a philosophical level -a feeling that the problem of excess math has not been addressed in all its aspects.

My thesis is that economics adds its own particular difficulties to these issues (because of its status as "conseiller du prince", and because through teaching it gives useful professional skills, etc.) and that things become clearer when we step back and frame the question in terms of knowledge in general. As the reader will see, this enables us to trace, with great epistemological force, the direction of a different type of knowledge. This allows us to escape from the addiction of mathematization while building a better quality knowledge.

We will take in a number of examples in economics and finance, but the fact remains that economics has many distinctive characteristics, as several authors have noted, which tend to prevent a reasoned consideration of its social function. Consequently there remain several points that will need to be developed further.

A. The contribution of mathematics to knowledge: some history and preliminary remarks.

Since the beginnings of civilization mathematics has been associated with most forms of knowledge. Early examples are Archimedes's work in engineering and, from the same era, The Nine Chapters about land measures and economy in China1 . Few areas have not been influenced in some way by mathematics. From this long and multi-faceted history we extract some key features.

I. The Baconian program served by mathematics. It is in Il Saggiatore (The Assayer) in 1623 that Galileo posits that the universe is written in the language of mathematics. This  as it is called, became the foundation of all Western science. This clarifies Francis Bacon's program, which asserts that man has a Promethean perspective, because he is subject to God choosing to share his power. He can conquer, dominate and transform nature. Galileo tells us how he can know and understand it. In fact later in his work -as Alexander Koyre has clearly shown2 -Galileo proceeds essentially by thought experiments following mathematical reasoning, not by experiments providing data for subsequent modeling.

He believed that mathematics was a sufficient sign of the essence of God in nature that nature would reveal its secrets purely by geometric and algebraic deductions. Over a century later, Kant built his philosophy around the explicit idea that mathematics, although not based in sensory experience (a priori judgments), nevertheless teaches about the world (synthetic judgments). Subsequently mathematics has gradually yielded the philosophical throne of synthetic a priori judgments, but without ever losing the prestige of a natural fertility. In the early 19th century there was a separation with mathematics on one side, taking a modern and rigorous turn in the writings of Gauss, Cauchy and Bolzano, and philosophy on the other side, which, with Hegel's Logic, had no mathematical element. But then the emergence of non-Euclidean geometries and crises in the foundations of mathematics gave rise to a plurality of views about mathematics and its role in the development of scientific knowledge. At the end of the 19th and 20th centuries, with the development of physics that became the focus of epistemology, mathematics is, with variations depending on the authors, mainly considered as a servant of the natural sciences; we refer to this as its ancillary role.

II. The appearance of mathematics in economics. Sociology, as introduced by Auguste Comte, takes a non-mathematical road, except through the use of statistics, particularly by Durkheim. Subsequently it acquired its own methodological bases with Max Weber in the early 20th century. Economics, on the other hand, was mathematized as early as the mid 19th century with Jules Dupuit and Augustin Cournot, without really using statistics. Prior to this, economics presented itself as a kind of philosophy of accounting operations. After Dupuit and Cournot economics was full of talk of derivatives, equations and integrals. How did math come to be accepted into the very heart of this social science?

To answer this we follow the path of Jules Dupuit (1804-1865). A civil engineer, he realized that one can do better than simply fixing a single price for the tolls on a bridge since, whatever the price, some users will find it too expensive, while others would happily pay an even higher toll. He is the inventor of what today is called market segmentation. Having a good mathematical training he had the intuition that with a single price one cannot recover all of the integral of the curve that quantifies the willingness to pay; one can only recover that of a truncated curve. This idea of an integral is quite clear in his articles.

Yet we must note that this "willingness to pay" is a poorly defined concept. It depends on many factors, the weather, time of day, seasons, and a thousand social and economic causes. It seems impossible to measure. A collection of experiments measuring traffic against toll level would not provide a curve but a cloud of points. It also depends on the tolls levied on other crossings, and on whether users collude and sell their rights of crossing etc.

In the early 19th century, this concept was debated under the name "utility". Dupuit pursued the belief that the mathematical phenomenon that he had discovered would help to clarify the concept. He postulated the existence of this quantity as a property of the commodity being exchanged and its price, which is shared according to the benefits of the seller/manufacturer and the consumer. "Political economics," he wrote [as opposed to social economics], "should measure the utility of an object by the sacrifice that each consumer is prepared to make in order to acquire it" and he took the still famous example of a bridge: "[the utility of a toll bridge] can be separated into two main parts: 1) the lost utility, which corresponds to those crossings that would have occurred if the toll were abolished but which do not take place with the current charge, and 2) the utility produced, which corresponds to the crossings which do take place. This latter splits into two further parts: a) utility for the producer, i.e., the money raised by the toll, and b) utility for the consumer, i.e., the excess value of the service over the price it costs." 3 Dupuit explains: "[In a shop we see] the fine, the very fine, the super fine, the extra fine, which, though from the same barrel and showing no difference other than the superlative of the label, are sold at very different prices"4 and this changes the optimization of public taxes: "So when the bridge is built and the State establishes a tariff, it stops caring about production costs. It charges less for a heavy cart which wears out the bridge more, than for a carriage with good suspension. Why two different prices for the same service? Because the poor do not value the crossing as highly as the rich, and raising the tariff would only prevent them from using the bridge." He explains: "The goal is always the same: to charge for the service rendered, not what it costs, but what the buyer thinks its value to be." 5 Dupuit fully realizes that, being defined by thought experiments, this notion of utility is difficult to measure. He acknowledges that it is abstract. "It may be objected that the calculation for which we have given the formula is based on data that no statistics can provide, thus we will never be able to express precisely the utility provided by a machine, by a road, by any work ..." But he advances the famous argument, which has been repeated endlessly by neo-classicists ever since, that economic science is only an approximation. It is this argument that led to all the ambiguities in the passage from descriptive to normative and to the performativity of discourse, and which opened economics up to all the mathematical refinements imaginable.

Dupuit starts from a mathematical property and uses it to account for the psychological, and it is interesting to compare his approach with that of Condorcet, who, at the end of the preceding century, proposed a different kind of mathematization of the social.

Condorcet, a great mathematician, aimed to use the calculus of probabilities to understand the propagation and sharing of a "reason to believe", a concept somewhat similar to that of utility but based on the truth or fallacy of judgments6 . He pursued this program at length, making, along the way, the great discovery of the "paradox of the vote of an assembly". But he did not think that it would be possible to go so far as to calculate peoples' behavior.

"On the use of language of geometry, the amount of universal commodity, that of a particular commodity, these can be approximated by numbers, but the urge to buy and sell cannot be calculated. Yet the changes in price depend on this moral quantity which, in turn, depends on opinions and passions. It's a beautiful idea to try to calculate everything, but look at the greatest mathematicians of Europe, the likes of d'Alembert and Lagrange. They seek to understand the motion of three attracting bodies: they assume that these bodies are point masses, or are very nearly spherical, and yet this issue, despite being limited by a hundred conditions that make calculation easier, has occupied them for twenty years without an answer. The effect of the forces acting on the head of the dullest shopkeeper is much more difficult to calculate." 7 Condorcet's approach starts from the psychological, the reason to believe, and attempts a mathematization of sociality by the calculus of probabilities. His epistemology is an extension of that of Laplace: we cannot determine everything -principles, laws of forces and their way of acting -only the calculation of probability is relevant. It is an approach with an a priori limitation of science. Condorcet had to spell out all his assumptions -independence or correlation of opinions etc. -before doing calculations.

Dupuit, on the other hand, can immediately perform calculations, and does so in his articles, he constructs concepts which interpret price curves (assumed to be obtained). His concepts require very strong assumptions of independence, but he leaves the details of these hypotheses to be spelt out and improved later.

These features -the independence of agents presented as approximation, the progression from prices and quantities to concepts and then, during the 19th century, production function, and problem-solving by local differentiation -these will be the backbone of the neo-classical theory with Stanley Jevons, Carl Menger, Léon Walras (general equilibrium), von Böhm-Bawerk, Vilfredo Pareto (theory of optimum), Irving Fisher, etc. creating an evocative and highly flexible language that is still in use today.

III. Advanced mathematization of finance. This is a very recent and well-known phenomenon, whose history I have recounted elsewhere8 . I will simply explain how an apparently very clever mathematization of risk, helped lead financiers away from safe practices and facilitated the emergence of the subprime crisis 9 .

The crisis has occured in an era when finance is thoroughly mathematized, as a result of the "Black-Scholes revolution". A rediscovery of the work of Bachelier and the use of Brownian motion in modeling, and developments of stochastic calculus after the Second World War, particularly the work of K. Itô (1915-2008), provided a mathematical language (that of semimartingales) in which the non-arbitrage principle could be expressed under broad assumptions that were suitable for operational cases. Methods for pricing and hedging options were thus provided by partial differential equations. The simplest case is when volatility is constant, but it is clear to everyone that these methods are largely perfectible, a point which is epistemologically essential.

This led to three historical phenomena: the development of derivatives markets in the U.S. first, then Japan and Europe, a transformation of professional profiles in banks and a call for new mathematical skills, and an enhanced political role for finance which was felt during the construction of the European Union and then in the globalization movement.

From the hedging of (European or American) options on stocks and currencies, the mathematical formalization then spread to more delicate issues: rate models. In particular, the bond market and the term structure of interest rates. The Cox-Ingersoll-Ross and Heath-Jarrow-Morton models allow the non-arbitrage principle to be applied here. Furthermore the theory can make use of infinite-dimensional models that must be simplified and calibrated to the current data. These model the behavior of agents over five, ten or twenty years and are therefore highly uncertain, this uncertainty being expressed in the language of probability theory.

But the most ambitious level of mathematization goes even further and deals with securitization of debts and risk assessments. Putting risks on the market is a priori a good idea, in the sense that it is better not to put all your eggs in one basket. But this assumes that the players (banks, insurance companies) can assess the risks.

This gave rise to a mathematical innovation worth mentioning here. It was noted that to estimate the risk of a portfolio of contingent claims, the classical method known as "value at risk," based on a criterion of the form (level of losses, probability of this level), entailed some logical difficulties. It has been shown that any criterion satisfying the desired consistency was of a particular mathematical form called a "coherent risk measure"10 . We emphasize that these tools allow calculations for complex portfolios assuming known probability of rare events, i.e., the tails of probability distributions which have great influence on the results. These methods, in other words, yield a quantification based on unknowns.

In the credit-risk market financial institutions have mathematical tools to estimate risks on reassembled portfolios for the purpose of exchanging them and improving the situation of each individual with respect to their own utility function and their aversion to risk. It has often been stressed in the commentaries on the crisis that the new tools of these markets especially CDO and CDS (credit default swaps) did not encourage operators to exercise caution. That is correct. The changes in the way agents dealt with risk when protected by insurance, termed "moral hazard" by the Anglo-Saxons, surely had a role in making the "soufflé" of the crisis rise. But equally important is the fact that it was wrong to think that the risk was "in the portfolio". The risk is interpretative in nature and just as "the beauty of the Parthenon is not found in the dust of the Parthenon", so these mathematical tools do not see the global economic interpretations related to the decline in U.S. household savings etc.

IV. The quantification of uncertainty is a removal of meaning. From an epistemological point of view this fundamental fact needs to be stressed. It is the significance of the event that creates the risk. The probabilistic representation of risk is classically a pair of mathematical quantities: 1) a probability law that governs the states that can arise, 2) a random variable, i.e., a function that maps each state to the damage, that is to say the cost (counted algebraically if there are also benefits). This representation by a pair of quantities is a mathematical model both too simple and too ideal for thinking about risk. It is too ideal because we are almost never in a situation where this model is well informed. We do not know the tails of probability distributions because they concern rare events for which there is insufficient data. We do not know what correlations occur to assess the damage and we do not have a full description of what can happen. Moreover the model is too simplistic because it removes the reasons that make us interested in the events as if their translation into costs could be done automatically and objectively.

The true purpose of risk analysis is to move forward with a little foresight in organizing facts and social practices. It may be the risk that a child be knocked down while crossing the street, the risk that the air of Paris be toxic, that the failure of one business will cause that of others, etc. The intellectual operation of probabilizing a situation is fundamentally one of removing meaning. It is largely problematic for all matters concerning human behavior. Risk analysis necessarily involves understanding interpretations.

It is the meaning of the event that creates the risk. As an example, suppose a particular type of cancer is found in a certain proportion of the Swiss population. This proportion is then used to estimate the risk. If it is subsequently found that most of the people with this cancer had consumed cannabis twenty years ago, say, then all cannabis users become potential patients. The risk is much higher; the meaning of the event has changed. Reducing risk to a probability distribution of sums of money amounts to trusting mathematization as an approximation, as if it were describing a physical reality, whereas it is actually a question of meaning whose subjectivity permeates every interaction between the agents. This epistemological point is extremely important. They are interpretations, and hence meanings, that are replaced by numbers.

Recently there have been significant improvements in financial analysis, especially with the so-called coherent risk measures. All these methods for making decisions in the face of uncertainty have the innate defect of assuming the interpretative process to be closed. Yet, on the contrary, new interpretations are constantly emerging. Once a new reading is made, new risks are created, but perceived only by those who understand it. If in 2006, nobody had seen the growth of house prices and the decline of household savings in the United States as a phenomenon open to several interpretations, the corresponding risk would not have been perceived. Mathematization of risk conceals these difficulties behind assumptions about the tails of probability distributions. It is not enough to say that those are poorly known. They are by nature provisional and changeable according to the interpretative knowledge that agents bring from their understanding of economic phenomena.

V. In liberal economics, every quantification opens a possible extension to the market. There are numerous examples. The most recent is the quantification of research work. Up until the end of the last century, the quality of researchers was seen in terms of idiosyncratic talents that could only be truly appreciated by researchers themselves experienced in the same type of activities. Putting in place all the machinery of publication indices and journal citations has profoundly disrupted the working relations in the profession. I will say no more. The result has been the emergence of an international market for students, teachers and researchers, with Universities being faced with a new logic where their financial budgets determine what league of intellectual athletes they can afford.

Another example, one which is more serious in its long-term consequences, is biodiversity. Mathematization here is based on separating species into two categories. On the one hand are the "remarkable" species, those officially considered as threatened. For these species we calculate the cost of conservation much as for historical monuments. On the other hand for the "ordinary" species we calculate the ecological service they provide, from prokaryotes (bacteria) to eukaryotes (higher species) by standard methods of cost-benefit analysis. One can then buy and sell any part of nature or exchange it against goods or services already quantified by the economy.

B. When and how is there excessive mathematization?

We now examine the particular type of inefficiency and problem that suggests a diagnosis of excessive mathematization.

VI. We only realize after the fact. The recent financial crisis is quite illustrative in this regard. While the crisis had not yet occurred -except in the eyes of some non-orthodox observers as there always are -every agent and every financial institution believed that they should estimate the risk of their portfolios (comprised of complex products such as credit derivatives) by the methods best suited to the very mathematical nature of these products. Coherent risk measures make assumptions on the tails of laws but enable one to handle multiple scenarios. The weak point is that they omit scenarios based on global interpretations where the value of each portfolio cannot be calculated by considering the others as ne varietur.

Once the crisis had started, and after the resultant upheavals, what happened was the result of political forces: on one hand a strong current of opinion emerged urging the adoption of regulatory measures in order to avoid future crises or at least limit their damage, on the other hand most financial workers felt that all that was needed was to take into account the interpretation that had been neglected, to improve, in other words, the global readings of risky situations by strengthening the role of rating agencies in particular. The latter have now been warned, and have learnt to keep in mind the previously neglected facts (resistance to "stress" of the various institutions, etc.). For public opinion we are back where we started, with the same tools with the same defects. 11 VII. Calculations conceal ignorance. This is obvious for financial risks. Because we do not know how to quantify counterparty risks, or those related to market liquidity, and much less those which are due to human error or to changes in the law, very precise calculations are mixed with crude estimates hoping that they will have no appreciable impact on the outcome. Applying sophisticated calculations, such as coherent risk measures, to complex portfolios supposes that the risks are expressed perfectly in the ontology of the objects considered at the outset. In other words it adds a second level: one ignores one's ignorance. This affects the market (organized or OTC) in credits and their derivatives. By the market, portfolios acquire a value where everyone trusts everybody else's calculations though they are no better. This leads to an instability that may be called "methodological moral hazard" which is the belief that mathematics is able to capture new interpretations if the calculations are done by everyone. This kind of instability is worse than in conventional markets in assets and their options because the timescales are much longer (tens of years instead of tens of months) and the punishment of economic reality comes much more slowly.

VIII. The ancillary role of mathematics as servant is confused with that of the subjects being served.

The previous idea can be generalized to all situations of mathematized knowledge. Let us take the case of physics. It is obviously helpful to physics when the mathematics used by physicists is improved. There is a real fertility there which has been particularly emphasized by Gaston Bachelard. But it works with the same interpretations as the served science. We are in the syntactic part of normal science in Kuhn's sense. Although Bachelard, with his usual talent, shows that mathematics can suggest questions for physicists, it is impossible to get genuinely new interpretations of phenomena occuring in the domain of the master discipline in this way. Mathematization is an essential component in the phenomenon of scientific crisis as described by Thomas Kuhn.

IX. That a theoretical representation be perfectible does not mean it is the only way to deal with reality and does not guarantee that it is capable of taking into account every aspect of the situation in question. By theoretical representation I mean a semi-artificial language using mathematics, as in physics or modeling. The fundamental point is that perfectibility gives the illusion of completeness. Ptolemy's geocentric planetary system provides a good example: the excess of mathematization lies in cycles and hypocycles that can be added at will. The original system was improved by Tycho Brahe and is infinitely perfectible, and the excess only became apparent after the new interpretation given by Copernicus. The only flaw in Ptolemy's system is that it has no place for this new interpretation. Yet the new interpretation was much less precise, at least initially, when Copernicus was proposing heliocentric circles. But this is astronomy not planar geometry, and the new reading acquires legitimacy from the fact that it too could be a starting point for improvements; it also has room for possible enhancements. Galileo cannot depart from this new interpretation because he recognized in Jupiter and its satellites a Copernican system. Nevertheless, having, at that pre-Newtonian time, only a kinematic description of phenomena, he has no compelling argument against the geocentric system. He was accused during his trial of basing his position on "beliefs" that are not in the sacred texts. It is a case of one interpretation against another, a situation cleverly analysed by Augustin Cournot12 . The position of Cardinal Robert Bellarmine is that faith has a monopoly of beliefs and that science must remain a means of describing what is allowed in God's creation.

X. There is confusion between creativity of the representation and creativity of the world. Within a system of thought, especially one that is perfectible, one cannot see a reason to escape the system. This is related to Quine's remarks on ontological commitment and on the near impossibility of talking about things we either don't know about or deny the existence of. Quine emphasizes our strong tendency to "talk and think about objects" 13 both in ordinary language and in physical or economic theories where agents and objects are subject to certain relationships. "It is hard to say how else there is to talk, not because our objectifying pattern is an invariable trait of human nature, but because we are bound to adapt any alien pattern to our own in the very process of understanding or translating the alien sentences." 14 Quine also takes into account the ontological conflicts in order to clarify them. The novelty of the famous article "On What There Is" 15 is the proposal of a definition of ontological commitment which in principle applies quite generally. In fact these fine arguments inspired by mathematical logic are based on the use of logical quantifiers and are quite abstract, and they do not focus on the emergence of new objects. A more concrete historical example is very illuminating: the abandonment of the natural scale in music. The octave, fifth and other basic musical intervals correspond initially to the division of a vibrating string into simple fractions, one-half for an octave, two-thirds for the fifth, three-fourths for the fourth, etc. This is a strict mathematization of the harmony that is actually perceived by the ear through sound frequencies. If we move from fifth to fifth by iterating the operation of taking two-thirds of the length, then we find that twelve fifths are approximately seven octaves. Hence, translating these divisions back onto the original octave yields the twelve intervals of the so-called Pythagorean scale. It is approximate since 12 fifths are not exactly 7 octaves, but it is very close to the mathematics of vibrating strings, which is the natural (and scientific) basis of sound. It took more than twenty centuries before the natural scale and its improvements were abandoned and the so-called "even-tempered" scale, which gives exactly the same role to all intervals, was adopted. The instruments built on the even-tempered scale do not give preference to a particular key, but they do not respect fully the laws of vibrating strings. The creativity of the musicians has won over that of mathematics in music. The victory is in fact not total, because of some harmonics that are heard as dissonance, etc. But the point to emphasize here is that the idealized world of mathematics has been put to one side in favour of a world based on practice.

C. Why normal science and jolts of revolutions? Why orthodox economics and crises?

Things seem to move like tectonic plates, in jolts. Why is this? How can we implement a production of knowledge that goes beyond the Kuhnian epistemology? XI. As Kuhn thought, normal science is very close to the Popperian vision. Only the modalities of its functioning are seen with a more social emphasis on paradigms as shared understandings of scientific communities. The real difference with Popper is that the disorder that precedes a crisis is more complex than simply encountering a decisive experiment that could refute the theory: there are also attempts to negotiate with the forms of interpretations. Usually the plasticity of the paradigms allows the acceptance of new facts or events in the theory. Kuhn takes the example of a child learning to distinguish ducks, swans and geese in a zoo, with his father playing the role of experimental verdict. He stresses the importance of slightly fuzzy categories whose vagueness is not mathematically quantified 16 . But in certain historical situations, the various ways of arranging things lead to choices that are too artificial (properties of the ether, for example), which gives rise to the search for and the legitimization of more radical interpretative changes.

XII. But most mathematization situations are not Popperian. Economic theories are not likely to be refuted by any observations of facts. The social environment is constantly changing and is never the same twice. Specialized models with predictive aims are probabilistic and cannot be falsified by a single event. More generally, mathematizations useful for studying changes in the environment (pollution, climate change) are always open to several competing models, each based on a different perspective (extrapolation from ice cores or CO2 emmissions), each perfectible as new data become available. The simplest generic example is that of modeling the flow of a river for flood forecasting. Families of models based on Gaussian ARMA factoring in 1) the water depth, 2) the flow rate, 3) the logarithm of the depth and 4) the logarithm of flow, are each infinitely perfectible if new measured data are available yet they do not give the same probabilities of reaching a certain level 17 . This does not mean that these models are useless, far from it. It just shows that it is not because reality is plural that it is not scientific. In fact, for one type of phenomenon, the data are always finite in number and a finite number of points can be matched either by polynomials or by combinations of real exponentials or trigonometric functions etc. If you think about the immense range of subjects opened up by modeling, then you quickly become convinced that it is the Popperian cases that are the exception. For a theory to be Popperian it must have a fixed number of parameters, each fixed numerically. It is hard to think of any apart from gravitation and some physical theories. Probabilistic theories never fall into this category because an infinite number of events is needed to determine a probability distribution.

This remark also applies equally well to normal science in the sense of Kuhn. It is an extremely restrictive view of knowledge. Let us be more precise.

XIII. It is the monism required at each step that causes the jolts.

Where does the new interpretation that is characteristic of a scientific revolution come from? It can only come from differences in the subject community. In other words, the jolts come from the absolute will that the community accept only one truth. Yet this is one particular vision of knowledge and social organization of science. If we accept instead that "reality" is also, and indeed primarily, people, groups, with their abilities, their habits, their psychology, and their means of interacting with their environment, we see that the only way to capture, or at least to take some account of, the innovation in the world is to make space for the instances where new representations are constructed : users' associations, professional groups, consulting experts, victims of unforeseen circumstances, etc. As Funtowicz and Ravetz have thoroughly analyzed, this route leads to a better quality of knowledge, more reliable and in which we can have more confidence 18 .

It is a pluralistic knowledge, but that is not to say that it is relativistic. This distinction is crucial. Specifically, as soon as one demands a certain level of rigor and consistency, one is limited to a small number of different approaches, just as the major political ideas concern a limited number of parties in multiparty parliamentary systems. To say that departing from the monism of unique truth leads one into relativism is the coarse argument of dominant representations, which the jolts of scientific crises regularly refute.

Nevertheless, if the implementation of such pluralistic knowledge is progressing well in some areas such as climate change or the protection of sensitive areas (despite clashes with political power, which are nothing new), it presents particular difficulties for economics. With globalization, knowledge about economic exchanges has a strong tendency to monism. One would think, however, that the growing environmental problems should lead us to greater tolerance in the implementation of specific economic experiments and their running as a condition of better support for natural equilibriums.

D. Interpretative pluralism is not destructive of knowledge; it is a better type of knowledge.

We now propose to examine more thoroughly the features of that better quality and what role mathematics can play. This will necessitate a step back from science as it is currently most often understood and practiced. Beyond the concept of "confined research" introduced by Michel Callon 19 , it appears that what is at stake is the conquering character of the Baconian program and the masculine virtues connected with them.

For convenience we shall use the term challenge-science to describe the view, held until recently by most scientists, that sees knowledge as a challenge to nature. It challenges nature to a duel. The honor in the game is to respect the assumptions that govern the rules for experiments. This includes Popperian science and Kuhn's normal science. In fact it is very old; the induction principle advocated by many philosophers and scientists to account for knowledge is similar in nature. Put simply, Popper proposes an induction articulated on a theory. Instead of accepting the thesis that knowledge is essentially philosophical in its ability to spot a pattern and extrapolate itan idea championed simultaneously (in 1843) by John Stuart Mill and by Augustin Cournot who finely analysed it -thus drawing from a large number of results, or a large number of circumstances, a prospective law that is to be evaluated, Popper strengthens the criterion by requiring that we move from observed facts to a representation with the dress of a theory, that is to say, based on a mathematical syntax like mechanics as formulated by Lagrange or Hamilton. Historically, it is indisputable that during the whole period where industrialization had not yet complexified technology too much, science was practiced with little experimentation and as many challenges were presented to colleagues as to nature. The discoveries at the time of Pascal, Fermat and Father Mersenne were often announced as puzzles, whose answer was known only to the finder, to challenge the wit of contemporaries 20 .

In these early years of the 21st century, a new awareness, unique in the history of man, is happening. Endless continual growth is impossible, and even if the limit is not yet reached, the current pace is so destructive that it must be drastically curbed21 . It is becoming less and less clear that using challenge-science vis-à-vis the environment, with new technical devices and a progressive mathematization to calculate the economic optimum by cost-benefit analyses in the context of democracy and liberal economy, can overcome the global challenges : arable land, species, climate change, pollution of soil and water, etc.. New options for production and consumption (e.g. use oriented product service systems, etc.) and for democratic structures (new bicameralism22 ) are probably essential. But, more fundamentally, we must also consider the question of what kind of knowledge. The epistemological question of how knowledge is produced also arises.

XIV. What logical status can the new knowledge have? Is there "room" for anything else? What are the characteristics of forms of knowledge that are not falsifiable theories -are there any? They would eventually be forgotten but they are innumerable. Included in this field are all useful discoveries that form the logical category complementary to that of refutable hypotheses. The vast majority of knowledge about animal, mineral and vegetable, and a great deal of technical expertise, is of this type.

In this class we find most of the chemistry that has long been viewed as pre-scientific when compared with physics. The great chemist Henry Le Chatelier in the early twentieth century says: "These two sciences have a similar purpose, they both study phenomena that result in transformations of energy, i.e., mechanical, calorific, electrical or chemical power. In teaching physics one refers only to the laws of natural phenomena: the laws of Mariotte, Gay-Lussac, Ohm, Joule, Descartes, Carnot, etc.. [...] In chemistry, on the other hand, there is an endless list of small particular facts [...] the material thus accumulated will be very useful for the subsequent establishment of science but they do not yet constitute it in any way" 23 . Why such a disgrace? Is it justified in terms of services rendered?

This class also contains most medical and environmental knowledge. Long before Popper, Claude Bernard wrote the following about medicine: "in science you can make two kinds of discoveries. Some are predicted by theory; these suppose two conditions: a very advanced science, e.g., physics, and simplicity of the phenomena. The other kind are unexpected: they appear unexpectedly in the experiment, not as corollaries of the theory and devoted to confirm it, but always outside of it and therefore contrary to it." 24 More generally, outside the challenge-science category lies all the knowledge about how the world is, what features make it the way we find it, and not another that follows the same laws. This is not inconsistent with general knowledge in Aristotle style, but these innumerable and fortuitous data, that reflect what life and history have made, are essential for nature and the society. Besides, without them challenge-science is nothing. Computers can help us to store them but they do not reduce to dimensions or coordinates. They are interpretative like the new paradigms that Kuhnian revolutions bring. We must therefore accept that some are complementary -plural answers to the same question, differing accounts written in different styles and emphasizing different points.

XV. A knowledge whose social function is not prediction but caution and care. We have to make a place for stories, testimonies, for what makes our current understanding of the world in all its diversity. They are the basis for the uses and values that give meaning to representations, even scientific ones.

With regard to mathematics, there is no reason to exclude it, we need it here too. But symbols may be used more freely than in axiomatized theories. It is perfectly legitimate to reveal a phenomenon, to represent a trend or a natural evolution using existing scientific languages from the established sciences or from engineering which are semi-artificial languages with partial mathematization. For managing natural equilibriums of life and for working on collective decisions of social groups, it is necessary to allow various representations and even different rationalities to coexist. The use of mathematics as thought patterns, for the linguistic value of symbols and combinations thereof, is useful and desirable. They are not reserved for expressing the truths of challenge-science.

XVI. The main tool of a better quality science is critical and contradictory modeling. The models are able first to take into account the distinctive features of situations and to apply proven knowledge to them and secondly to translate, by the ordinary language which forms the internal cement and the external context, an interpretation of the complexity into what we are interested in.

If they are not to be seen as low level or amateur challenge-science, it is essential that models be always viewed as a facet of a plurality. Firstly, they must be validated by data with the same rigor as usually required by scientists. This validation is not a test of truth, but simply a process of eliminating the unlikely. Secondly they must be recognized as a social expression, i.e., a form of communication from an agent (be that a group, association, company, territorial entity, etc.) to an audience in order to contribute to a decision and therefore subject to criticism by other models. Knowledge is no longer formed exclusively by a struggle between theory and nature but by a contest between models. This process obviously requires a specific organizational context, just as challenge-science requires cautious experimental protocols. The "rules" are not currently codified, but the experiments are underway at international level for the IPCC and in the public debates, citizen juries etc., in a kind of applied living epistemology still under development.

To critique a model is difficult. The quantitative arguments are linked together, everything is connected. It is a huge task to draw out all the implicit assumptions of a model. Even though we know that every model is arbitrary in some aspects, we do not see this arbitrariness explicitly. When discussing one model, our thinking remains stuck in a rut. The best way is to build another model from scratch -the options are much clearer then.

To construct another model, the dualities introduced by the philosophy of science are relevant -they facilitate a dialectic setting for the occurence of what may be called co-truths. Let us consider a few examples.

Discrete / continuous. Much of the economic theory can be developed without individualizing agents or goods. Some scholars find it illuminating to derive global laws from a micro-economic individual rationality. When studying traffic, depending on the question we may use flow models or we may model each vehicle individually. Sometimes it is thought that discretization, spatial or temporal, simplifies the problems, with the recurrence rules being more elementary than differential equations and finite element algorithms reducing partial differential equations to simple algebra. But often the opposite happens: the discrete probabilities are sometimes intractable and some algorithms (such as Kalman), are best understood in continuous time.

Descriptive / explanatory. In 1970, two American authors, G. E. P. Box and G. M. Jenkins took methods invented by Wiener for signal processing and applied them to economic predictions. Treating annual series without any regard to their economic meaning, they sometimes obtained better predictions. This is the fundamental duality which we began with in this article. In the history of science, it often occurs in successive periods. The purely descriptive approach can be an advance when it frees us from certain loaded interpretations. On the other hand, explanations allow a reading to shed light on situations other than those already considered.

Quantitative / qualitative. The philosophical work of René Thom has brilliantly illustrated that mathematics provides representation tools that go far beyond the quantitative. A huge field of natural phenomena can be addressed qualitatively through a language adapted to the evolution of forms.

Deterministic / random. A huge number of modeling situations involve risks. The instinctive tendency of modelers is to probabilize the uncertainties -we have already discussed this tendency. This provides a very efficient syntax thanks to the stochastic calculus developed in the 20th century. But this, especially in the tails of laws, conceals ignorance. Uncertainty is sometimes better illustrated by some typical or extreme trajectories obtained from different scenarios.

Image / symbol. Let us take the example of dance. Dozens of notation systems have been developed by the choreographers to record ballets, either based on a limited vocabulary of successive steps (Feuillet system 1700) or more elaborate, noting the dancer's energy in each movement (Laban system 1927). The problem is one of modeling, with the usual constraints of relevance for the choreographer and dancers. But is this not a false problem since film and video can provide us with an almost perfect image of the ballet ? The image reproduces, it can provide the perfect illusion of reality, but it does not, by itself, allow choreographic creation. The notation systems have the immense superiority of enabling one to record a ballet that has never been danced.

Critiques of models cannot come from recipes or an a priori classification, especially since, as we have emphasized, their relevance depends on the social group that proposes them. The quality of the plural knowledge thus produced comes particularly from the things that it can draw out of reality but which challenge-science fails to see. Applied in good conditions of open democracy, it is likely to show hidden effects, unnoticed risks, possibly unsuspected solutions. Challenge-science instead, with the successive stages of its rockets, heads only in one direction.

Conclusion:

The problem is not that there is too much mathematics, but that it is used exclusively as a framework for theories that claim to univocal truth.

The propensity to mathematize more and more can occur in the development of a classical theoretical line of thought as much as one based on modeling, especially if one assigns a value of absolute truth to the interpretative framework we work in, so that syntactic developments will be seen as revealing reality. This occurs in modeling because the modelers tend to think that their models are reality. But faced with other models they are forced to acknowledge the scope of their approach. In contrast, in a Popperian conception, mathematization can be pursued without any restraint, until a crisis occurs. Our analysis of mathematization is an Ariadne's thread that opens up the philosophy of knowledge to a new and immense field of thought. It turns away from the jousts, catapults and knights-in-armor of the conquering knowledge, it takes a step back, whereupon challenge-science starts to look like a very particular way of understanding the world.

It is ultimately a choice between what is important and what is not. A river basin for example, may remain for centuries. But we are faced here with contradictory logics, politicians who want to develop jobs, farmers who want to irrigate, associations that want to respect the landscape, companies that want to build dams for electricity, etc. Often neither the economic interest nor the democratic vote, can overcome the basic dominance of selfishness. Maintaining the scenes of natural life involves intermediate languages between native speech and falsifiable science, languages which oppose but do not destroy each other, which, by their plurality, are open to the interpretation of data and the imagination of eventualities. About mathematics itself, there is no need to worry. Real mathematicians know what drives them: the pleasure of an intellectual game 25 . Maths does not need to be the framework for a grand and unique building of knowledge. On the contrary, freedom from applications and doctrines has always been maintained : non-Euclidean geometries, non-standard analysis, etc. Explorations off the beaten track are rewarded with the surprise of the treasures discovered there.

Nicolas Bouleau

April 2011

25 Cf. N. Bouleau, Dialogues autour de la création mathématique, in coll. with Laurent Schwartz, Gustave Choquet, Paul Malliavin, Paul André Meyer, David Nualart, Nicole El Karoui, Richard Gundy, Masatoshi Fukushima, Denis Feyel, Gabriel Mokobodzki, 1997, on line : http://www.enpc.fr/HomePages/bouleau/DialoguesInterferences.html

SHEN Kangshen The Nine Chapters on the Mathematical Art, Oxford 1999; K. Chemla et G. Shuchun, Les Neuf Chapitres, Dunod,

Cf. A. Koyré Etudes d'histoire de la pensée scientifique Gallimard 1973, and Galilée Dialogues et lettres choisies Hermann 1966.

J. Dupuit Annales des Ponts et Chaussées 1849.

Annales des Ponts et Chaussées 1944.

Ibid. Note that today's large online shops can charge "good" customers more than new customers, thanks to the information they receive from cookies. Good customers are those users attracted to this way of buying and can therefore be charged more for the service. The screen presented to the customer is not a public price tag, it depends on the user's IP number.

The reason to believe is what we call today the degree of certainty. Condorcet studied how it accumulates when we collect uncertain information or when members of an assembly vote.

Letter to P. Verri 1773.

Financial Markets and Martingales, Observations onScience and Speculation, Springer 1998.

For more détails cf. N. Bouleau "Finance et opinion" Esprit nov. 1998 and "Malaise dans la finance, malaise dans la mathématisation" Esprit fév. 2009, p37-50.

For details see N. Bouleau Mathématiques et risques financiers Odile Jacob 2010.

It is impossible to predict the next crises, but we can guess that they will revolve around the failure to take into account limits. Bounds, finiteness of the world, resources, raw materials, agricultural land, etc. are all ignored by economics. Anticipation of increasing scarcities in an uncertain environment may provide unpredictable instabilities.

cf. N. Bouleau Risk and Meaning Springer 2011, chap. II. Cournot's "philosophical probabilities".

"Speaking of objects" in Ontological Relativity and Other Essays, Columbia University Press, 1969

in From a Logical Point ofView (1953), Harpers & Row 1963.

T. Kuhn "Second thoughts on paradigms"(1974), in The Essential Tension, Univ. of Chicago Press 1977.

Auto-Regressive Moving Average processes are the simplest Gaussian models with linear recurrence used in theory of time series. Obviously to the above four families we could add linear alpha-stable models and models with quadratic recurrence (heteroscedasticity) etc. All these examples illustrate the relevance of the remarks of Quine on underdetermination of theories by experience.

S. O. Funtowicz and J. R. Ravetz "Three Types of Risk Assessment and the Emergence of Post-Normal Science" in Social Theory of Risk, Sh. Krimsky and D. Golding eds, Preager

 19 Cf. M. Callon, P. Lascoumes et Y. Barthe Agir dans un monde incertain, Essai sur la démocratie technique, Seuil 2001.

See Koyré, ibid, on the fact that Galileo never experienced the stone that falls from the mast of a moving ship, and on the conundrum in which he announces to his contemporaries his discovery of the phases of Venus.

Cf. D. Bourg and A. Papaux Pour une société sobre et désirable PUF, FNH, 2010.

Cf. B. Latour Politiques de la nature, Comment faire entrer les sciences en démocratie La découverte 1999, et D. Bourg et K. Whiteside Vers une démocratie écologique, Le citoyen, le savant et le politique Seuil 2010.

H. Le Chatelier, Leçons sur le carbone, la combustion, les lois chimiques, preface,Paris, 1908.

Cl. Bernard Leçons de physiologie expérimentale appliquée à la médecine faites au collège de France, Paris, 1885.