FORMATTING POWER OF “MATHEMATICS IN A PACKAGE”:
A CHALLENGE FOR SOCIAL THEORISING?
Ole Skovsmose and Keiko Yasukawa
Abstract: In this paper we, firstly, want to give
an example of “mathematics in a package” to which we relate most of the
rest of our considerations. The package we will open and look into is a
encryption packages used for the secure transfer of files and emails over
the Internet. In particular, we will locate some of its mathematical content.
In this way we want to clarify the question “What is in the package”
Secondly, we want to emphasise that
this package operates in a way more “real” than clusters of mathematical
theorem and results normally do. Although the package represents knowledge,
and mathematical knowledge in particular, it does not operate simply as
a cluster of theoretical knowledge. It becomes relevant to ask: “Whose
package is it?” as it has become part of our social and economic reality.
Thirdly, we want to clarify the question “What
could be done by means of the package”. In the clarification we will
be more specific about the thesis of the formatting power of mathematics,
which condense the idea that mathematics operates almost everywhere,
and that this operation has a social significance. The aspects of the
formatting power which we will examine are: (1) By means of mathematics
it is possible to provide new space for socio-technological action. (2)
By means of mathematics it is possible to investigate details of a hypothetical
situation. (3) Mathematics becomes “locked in” in reality and becomes
inseparable from other aspects of society.
Fourthly, we want to discuss the question: “What
does these observations mean to social theorising?” In particular
we will consider what our observations related to the formatting power
of mathematics may have of implications for social theorising in general.
An understanding of how mathematics may be operating is not only of relevance
for the philosophy and the sociology of mathematics, but also for social
theorising in general. In particular, notions like “thrust”, “risk” and “reflexivity” must
include the study of mathematics based technological actions in order
to provide adequate interpretation of the propensities of the “informational
society”.
Introduction
People
wake up to a world where their level of health can be determined for them
by quantified comparators such as weight, height, blood cholesterol level,
and other measures of their physiological machinery. Furthermore, they
are able to “fix” some of their deficiencies by taking appropriate dosages
of vitamins and other drugs correlated mathematically to their particular
physiological indicators. For many working people across the industrialised
world, the working day is determined by some form of industrial award which
specifies the number of hours, if not which hours, they are expected to
work. Increasingly, the work done is translated
into performance or productivity measures, which in turn can be calculated
into performance based pay. The business risks that organisations decide
to take are based on a range of quantitative cost-benefit analyses. Out
of ones daily or fortnightly pay, a certain amount is set aside for superannuation
whose rate of contribution and returns are based on complex actuarial calculations.
Coming home from work, people sort through their mail, and perhaps they
will find that the rates are due on their property - perhaps there has
been an increase due to a higher valuation of their property. In addition
to this excessive quantification, which have a direct impact on ones life
and identity, there is the mathematically determined wider environment
in which people operate. Thus, models of the national economy determine “acceptable” levels
of unemployment, immigration quota, inflation rates, interest rates; assessment
of public risks, such as pollution levels, that are deemed acceptable or
otherwise through some “scientific” models. And so on.
Investigators of the social studies of
mathematics have examined society’s increased propensity for measuring
and quantifying social phenomena, reliance on mathematically based “intellectual
technologies” to replace human decision-making, and use of “inscription
devices” to abstract and categorise physical and non-physical features. Such studies focus on the processes or
effects of quantifying aspects of social and physical phenomena, and how
this may in turn mediate social interactions. This paper also examines manifestations
of mathematics in society. In particular we shall try to condense these
observations into a theses of the formatting power of mathematics. Furthermore,
we shall argue that these observations are essential to any social theorising.
What is of interest to us is the sharp
contrast between the impotence of mathematics claimed by the mathematician
G.H. Hardy and the central role that his very area of mathematics, Number
Theory, has played in the invention of a fundamental social condition in
the new electronic environment. We let Hardy represent the extreme position
that mathematics can be considered as “gentle and clean” as it has no social
impact. We contest Hardy’s claim: “If the theory of numbers could be employed
for any practical and obviously honorable purpose, if it could be turned
directly to the furtherance of human happiness or the relief of human suffering,
as physiology or even chemistry can, then surely Gauss nor any other mathematician
would have been so foolish as to decry or regret such applications. But
science works for evil as well as for good (and particularly, of course,
in time of war); and both Gauss and lesser mathematicians may be justified
in rejoicing that there is one science at any rate and that their own,
whose very remoteness from ordinary human activities should keep it gentle
and clean.”
In
contrast to Hardy’s indications, we find that mathematics operates almost
everywhere, and that it operates as an integrate part of the socio-technological
structures of today’s society. The notion of technology we are going to
use is broad, including social, economic, cultural, military devises. When
we want to emphasise that by technology we not only thing of its mechanical
aspects but also of its manifestations in forms of operations and decision
making, we shall talk about technological actions. Sometimes also about
socio-technological actions in order to emphasise the broad implications
of such actions. In particular, we shall examine the construction of “trust” from
fundamental results in Number. We shall conduct our analysis by discussing
the following four questions.
First,
we want to give an example of “mathematics in a package” to which we relate
most of the rest of our considerations. The package we will open and look
into is a encryption packages used for the secure transfer of files and
emails over the Internet. In particular we will locate some of its mathematical
content. In this way we want to clarify the question “What is in the package”
Secondly, we want to emphasise
that this package operates in a way more “real” than clusters of mathematical
theorem and results normally do. Although the package represents knowledge,
and mathematical knowledge in particular, it does not operate simply as
a cluster of theoretical knowledge. It becomes relevant to ask: “Whose
package it it?” as it has become part of our social and economic reality.
Thirdly, we want to clarify
the question “What could be done by means of the package”. In the clarification
we will be more specific about the thesis of the formatting power of mathematics,
which condense the idea that mathematics operates almost everywhere, and
that this operation has a social significance. The aspects of the formatting
power which we will examine are: (1) By means of mathematics it is possible
to provide new space for socio-technological action. (2) By means of mathematics
it is possible to investigate details of a hypothetical situation. (3)
Mathematics becomes “locked in” in reality and becomes inseparable from
other aspects of society.
Fourthly, we want to discuss the question: “What
does these observations mean to social theorising?” In particular we will
consider what our observations related to the formatting power of mathematics
may have of implications for social theorising in general. An understanding
of how mathematics may be operating is not only of relevance for the philosophy
and the sociology of mathematics, but also for social theorising in general.
In particular, notions like “thrust”, “risk” and “reflexivity”, which play
a role in works of Giddens and Beck, represents aspects of the formatting
power of mathematics in. In developing a social theory of new social networks
on terra-silica, one must be aware that the nature of the social
interactions which can occur is founded as much on how mathematical capabilities
are appropriated as it is on the patterns of interactions which people
are used to on terra-firma. As a consequence, a part of the sociological
study of the dynamics of virtual networks and of the “informational society”,
as coined by Castells, must include the study of mathematical theories
and results as key “actors” in that socio-technical environment.
What is in the package?
Encryption, the process of encoding messages
to achieve confidentiality, is a technique which could most readily be
associated with warfare - coding of secret strategic information about
enemy movements, dissemination of false information to fool enemy intelligence,
leakage of intelligence information, and so on. Interest in cryptography,
the study of encryption systems and methods, has now expanded far beyond
the military sphere into the wider social spheres of commerce, and to personal
communications. Thus, a particular encryption software system as PGP (Pretty
Good Privacy), can offer new socio-technological possibilities. This in
turn can disturb alter forms of social relations in potentially profound
ways.
PGP
is a software package that was released in 1991 by a private individual
Phil Zimmerman (rather than a commercial software firm) in the USA to provide
electronic mail and file storage security. PGP is one of many security
packages that incorporate encryption systems. PGP supports the basic cryptographic
services expected of an electronic mail security system, namely, confidentiality
and authentication. Confidentiality means protection from unintended parties
reading the contents of the message or file being transmitted, and authentication
means the assurance of the correct identity of the message source and the
integrity of the message (that is, that the message has not been changed).
Before we start examining PGP in detail,
we will make some comments about cryptography in general. Suppose we call
the original message or the “plaintext” P. Then the secret message
or the “ciphertext” C is produced by a transformation of P by
an encryption function, say E. The relationship between P and C can
be represented by:
C = E(P)
The encryption function should have the
property that
P1 ¹ P2 Þ E(P1) ¹E(P2),
that is, different messages are encrypted
into different ciphertexts so that ambiguity does not occur. “Decryption” is
the process of recovering the plaintext from the ciphertext through some
function D:
D(C) = P
The functions E and D should
have the property that
D ° E(P) = P
Thus, if the plaintext P is encrypted
and the result decrypted, the result should be the plaintext P itself.
Stated in another way, the function D is the inverse function of E:
the decryption function reverses the action of the encryption function.
To
give an example, Ludwig Wittgenstein was known to use a simple cryptographic
tool when he wrote his private remarks: the first letter of the alphabet a was
substituted by the last letter z, the second b, by
the second to the last, y, and so on. In this way the message: P: “this
is a secret” will be encrypted as C = E(P): “eqpf
pf z ftvgte”. In this particle example we have the D is identical
to E.
Modern
cryptography refers to encryption and decryption functions that take the
form of mathematically based computer algorithms. In talking about modern
encryption, we are assuming a computerized process where the original message
in natural language has already been converted into a suitable machine-readable
representation. In fact, we can think of the machine-readable representation
as a number, i.e. we can think of P as a number.
Encryption
and decryption algorithms, except possibly those used in military and intelligence
work, are public algorithms. One can purchase (or in the case of PGP freely
download) these encryption packages. Although it may appear that making
algorithms public would make encrypted communication less secure, the consensus
has been on the contrary. Because the algorithms themselves are
public, confidentiality is established by the secrecy of a “key”. The key
is necessary for the execution of an algorithm.
Modern
cryptography can be classified into two distinct approaches: (1) conventional,
single-key or symmetric cryptography, and (2) public-key or asymmetric
cryptography. Symbolically, these processes of encryption and decryption
can be represented by the equations:
C = EK1(P)
P = DK2(C)
where EK1 is the encryption algorithm using the
key K1, and DK2 is the decryption
algorithm which uses key K2. For conventional encryption the two
keys K1 and K2 are the same, while for public-key encryption
they are different.
The
big step forward in cryptography was to avoid the need to use the same,
or closely related, keys for encryption and the decryption. Using the same
key for encryption and decryption requires a system of key distribution.
Somehow the key which has to be kept secret between the two communicating
bodies must be safely distributed from whoever generates the key to the
other. Any capture of this key by a third party compromises the confidentiality
of the communication. The public-key system, on the other hand, relies
on an “asymmetric”, or two key system. In this system, each party has a “public
key” which is known and shared by both the sender and the receiver (and
which could be known to the wider public without compromising the security
system), and a unique “private key”. Hence, a message is encrypted by a
sender A using the public key K1 of the receiver B;
this message, however, can only be decrypted by the intended receiver B using
their private key K2, which even the sender does not know. The strength
of the public-key schemes is based on the difficulty of determining the
private key K2 from the knowledge of the public key K1 and
the ciphertext C alone.
PGP
uses a combination of conventional encryption techniques and public key
encryption techniques to achieve confidentiality of e-mail transactions.
It is the public-key encryption schemes used in PGP that we will be examining
in some detail.
Opening the package
At the surface where PGP is implemented
and used, PGP’s mathematical artefacts are invisible. People implementing
PGP would typically download from the Internet or purchase from a vendor,
a whole package within which the various public-key and symmetric algorithms
reside. Whilst in some cases, a version of PGP may offer options in features
such as key sizes, the user would not look beyond the technical specifications of
the product to determine whether or not it is the appropriate, machine
compatible product to buy. Nor would the user need to be conversant with
the mathematical theories behind cryptographic algorithms. The package
would have a user-friendly interface which guides the users through the
process of getting encryption-enabled; encrypting a message; decrypting
a message; and digitally signing a message. For the users, the power of the product
is entrusted not in what they understand of the mathematics upon which
the algorithms are constructed, but in the vendors of the package with
whom they deal, and or the reputation that the whole package has established
among relevant professional communities.
The
PGP package is made up of a collection of modular entities; for example,
encryption algorithms such as IDEA (International Data Encryption Algorithm)
and RSA (Rivest-Shamir-Adleman) have been separately developed, tested
and distributed, and have been appropriated to be building blocks of PGP.
The design of PGP ensures that these algorithms are “seamlessly” linked
to serve the overall functions of PGP. When we ask “what mathematics makes
up the package?”, we are effectively asking what mathematics underpins
each of the different algorithms within the package. We will focus on the
mathematics which underpin the RSA public-key algorithm used in PGP, i.e.
the algorithm by means of which public and private keys are generated Effective
encryption systems requires two key features. It must be easy to implement,
but difficult to compromise. For public-key systems, implementation involves
the generation of the public and private keys. Compromising the system
involves among other things determining the private key from knowledge
of the public key and possibly some ciphertexts.
The
class of mathematical functions upon which the designers of public-key
systems have employed to achieve the requirements of the systems is what
is known as “trap-door one-way functions”. These are functions where the
function values are easy to compute, but where, given the function and
a function value, it is difficult to compute the value(s) which the function
acted upon, without additional information. An example of this is a function
that takes two prime numbers and computes their product; multiplying two
prime numbers is simple, but determining the prime factors of an arbitrary
number, especially a large number, is difficult. This prime-factorization
problem is in fact the mathematical basis of the RSA public-key algorithm.
The
main role in this public-key algorithm is played by two prime number p and q.
As any numbers, p and q can be multiplied, and we get n = pq.
However, if p and q are large enough primes, it will be very
difficult to determine p and q from n, even knowing that n is
a product of two primes. The degree of this difficulty represents the degree
of security in the PGP system. So, for the constructor of a system, it
is essential to get hold of two large prime numbers. For the hacker it
becomes essential to factorize.
Of particular relevance in the design of
a public-key encryption system is a result dating back to the days of Euclid.
There exist infinitely many prime numbers. A proof of this was constructed
by Euclid, and the statement of this result is known as Euclid’s Theorem.
The implication of this theorem on the analysis of encryption schemes is
two fold: a hacker trying to determine the private key would have to go
through the process of searching for a pair of prime numbers over an infinitely
large set, while those generating the key pairs have the benefit of an
infinite set from which to choose a suitable pair of prime numbers. From
both perspectives, number theoretical issues becomes relevant. For instance,
how difficult is it to get hold of large prime numbers? The number of primes
between 1 and 20 is ###, while the number of primes between 101 an 120
is ### Does the density of primes decrease dramatically, when we search
among large numbers? The answer given by the Prime Number Theorem is: “yes,
but not too much”. For the hacker, this is a discouraging
result. Their search for possible prime pairs does not become easier as
the search set increases in size; for those generating the keys, they can
be assured that they have a fair chance of striking a suitable prime number
no matter how large a prime number they wish to have.
While
the search for prime factors of a given number x may be laborious,
the Fundamental Theorem of Arithmetic states that any natural number can
be factored into prime numbers in one and only one way (if we do not consider
the order of the factors). This effectively means that once a hacker finds
a set of prime factors whose product is the original compound number x,
they need not search further because these are the prime factors.
The Fundamental Theorem of Arithmetic also ensures that, when properly
chosen, the properties P1 ¹ P2 Þ E(P1) ¹E(P2) and D ° E(P) = P can be acknowledged.
In order to construct a system of encryption,
two large primes p and q has to be identified. The product n = pq is
calculated. Using classical results from Number Theory and given the number
n it is possible to determined two other numbers, e and d.
The public key K1 is then the pair (e, n) and the
private key K2 is (d, n). The process of both encryption
end decryption becomes simple mathematical procedures, just involving simple
arithmetic calculations. Thus, encryption is achieved by
:
C = Pe mod n
and decryption by:
P = Cd mod n
Thus encryption means raising the plaintext
(the number) P to the power of e, and dividing the result
by n. The
ciphertext C will then be the rest. Decryption will mean raising the the
ciphertext C to the power of d and dividing the result with n, the
rest will be the plaintext P.
The
product of the two primes is no secret, as n is part of the public
key (e, n). The calculation of the number d, however, will
break the code. The calculation of d is in fact also simple, in
case we know the two primes which multiplied gives the result n.
So, breaking the codes ends up in the task of making a prime number factorisation.
Constructing a code, on the other hand means identifying two fairly large
prime numbers.
Although
factoring is a simple process to describe mathematically, it is a time
consuming task when we are working with large numbers. “At present, a 200-digit
number that is the product of two 100-digit numbers cannot be factorised
in any reasonable time ... In fact, not so long ago, the most efficient
factoring algorithms on a very fast computer were estimated to take 40
trillion years, or 2000 times the present age of the universe.”
This result appears amazing. Thus, it takes
less that three lines (using the print on the present text) to write down
a 200-diget number. And this three-line number will provoke a task beyond
the shared effort of all computers of the world. Drawing as well upon all
present mathematical knowledge.
A new significance of mathematical formulae?
If we study classics in Number Theory,
like Elementary Number Theory by Hardy and Wright, then it is clear
that no kind of applications to anything external to pure mathematics are
considered. The study stays well within the walls
guarding the purity of mathematics from the contamination of the real world.
However, the significance of a mathematical theorem is relative to its
contexts. When a theorem is presented in relation to the derivation of
another theory or in a textbook, it might appear “clean and gentle” and
insignificant as far as its social impact is concerned; but when it appears
in an application package, such as PGP, its significance may be completely
different. In fact, this change of significance is an essential new departure
point for the philosophy of mathematics. In this section we will consider
how a number of what may be regarded as classical or fundamental results
in Number Theory, namely, Euclid’s Theorem, the Prime Number Theorem, Euler’s
Theorem, and Fermat’s Little Theorem have come to acquire significance
well beyond the walls of pure mathematics.
The
effectiveness of the RSA algorithm described earlier, and in particular,
its key generation algorithm, rests heavily on the ability to find suitable
pairs of primes p and q. To this end, one of what Hardy calls
a “real” mathematical theorem is central, Euclid’s Theorem. Hardy [OS1]states: “... Euclid’s theorem is vital
for the whole structure of arithmetic. The primes are the raw material out of
which we have to build arithmetic, and Euclid’s theorem assures us that
we have plenty of material for the task”. However, Hardy assumes that as important
as this theorem is in building the theories of arithmetic, it has little
practical relevance: “There is no doubt at all, then of the "seriousness" of
either [Euclid’s or Pythagoras’s] theorem. It is therefore the better worth
remarking that neither theorem has the slightest ‘practical’ importance.
In practical applications we are concerned only with comparatively small
numbers ... I do not know what is the highest degree of accuracy which
is ever useful to an engineer - we shall be very generous if we say ten
significant figures .... The number of primes less than 1,000,000,000,000
is 50,847,478: that is enough for an engineer, and he can be perfectly
happy without the rest.” Hardy not only predicts incorrectly the
uselessness of Euclid’s Theorem, but also the critical importance on the
size of numbers for some engineering applications. It is recommended that
the size of the prime numbers for RSA encryption is in the order of 75-150
digits.
However,
it is not enough that Euclid’s Theorem tells us that prime numbers are
abundant. In RSA (and other similar algorithms), it is critical, not only
that these numbers can actually be found in principle, but that they can
be found efficiently. Because there is no known way of analytically generating
prime numbers, a search algorithm, one being the Miller-Rabin algorithm
which involves testing the primarity (whether a number is prime or not)
of a randomly selected number, is used. The search is a probabilistic one, which
identifies whether or not a random number has a high probability of being
prime. The search for prime numbers can be compared with the search for
gold and diamonds.
In
order to have any confidence that a computer search algorithm has any practical
potential, however, we need also to have some assurance that prime numbers
occur “often enough” in the random search of the infinite set of integers.
As mentioned, the Prime Number Theorem provides us with the assurance about
the frequency of occurrence of primes that is needed. This theorem is another
of what Hardy calls a “deeper” theorem of mathematics: “When we ask these
questions, we find ourselves in quite a different position [to the shallower
inquiry which can be handled with Euclid’s Theorem alone]. We can answer
them, with rather surprising accuracy, but only by boring much deeper,
leaving the integers above us for a while, and using the most powerful
weapons of the modern theory of functions. Thus the theorem which answers
our questions (the so-called Prime Number Theorem) is a much deeper theorem
than Euclid’s or even Pythagoras’s.” But the theorem does not simply provide
us with a pure knowledge of Natural Numbers. It gets significance not observed
by Hardy.
Another
theorem which takes on a new significance in the light of the implementation
of the algorithm for searching for prime numbers is Fermat’s Little Theorem. Propositions derived from Fermat’s Little
Theorem give an upper bound on the probability of a number being composite;
at this point in time, this is the best sort of test that is practicable. Another Number Theoretic result known
as Euler’s Theorem forms the basis for producing the desired encryption-decryption
relationship to make the algorithm achieve its cryptographic purpose.
As
resources for implementing a software package, “old” pieces of mathematical
results are brought into play, not only to provide the basis of a central
algorithm as Euclid’s Theorem does for RSA. They also provide a resource
for deriving efficient ways of implementing the algorithm and as an insurance
that certain checks are adequate. Thus, the mathematical theorems such as
those mentioned above become the basis of trust that people implicitly
invest in a package such as PGP to provide secure communication.
In particular the thrust is
based on the difficulty of making prime factorisation of a big number.
This is a mathematical observation, as mathematical research not yet has
been able to provide an efficient algorithm. But it is not proved impossible
to identify much more efficient hacker-friendly algorithms. In this sense
the degree of security depends on the present state of mathematical research,
and furthermore on how this research is conducted in terms of publicity
of relevant results.
Although
we have indicated the number of classical mathematical results that have
acquired new significance in cryptography, we also need to note that mathematical
results in new areas of mathematics have also been incorporated into cryptography.
To give one example, bounds on how hard the problem of a brute force attack
by prime factorisation is in terms of computational effort, are derived
from results in a relatively new field of mathematics, complexity theory.
This theory provides us with the rate of growth in computational effort
when key sizes are increase, i.e. the number of decimals of the prime numbers
in question. Thus not only are classical mathematical results being applied
in new areas of practical significance, they are also being challenged
for their “truth” value, in a practical sense..
Whose package is it?
We have now got an impression of what is
in the package. Let us now look at the package as a whole entity. This
makes sense, even though the package is made up by mathematics. By discussing
the two questions “Whose package is it?” we will try to illustrate in what
sense mathematics “materialise”. Questions addressing the package is quite
different from questions we can address bit and pieces of mathematical
knowledge. The classical perspective of the philosophy of science and of
the philosophy of mathematics becomes inadequate. The package is a new
entity.
If
those pure mathematicians who generate the resources for cryptography stay
at an arm’s length, or further, from the applications of their results,
who are the social actors who package these results into packages such
as PGP? How do these people interact with each other? Do their interactions
represent a new pattern of knowledge production, or are they similar to
those in the traditional paradigm of mathematical research?
As
already mentioned, PGP was written by Phil Zimmerman, who was fascinated
by cryptography, and found a career niche in the area in college when he
started to seriously research the literature and publish in the area. As will become clearer, the birth of PGP
is attributable to a combination of his fascination with cryptography and
his ideological conflict with the US government. Specifically, PGP is claimed
to have been “proposed as counter-terrorism legislation which would have
placed limits on the use of encryption technology by U.S. citizens”. The
intention was that PGP would become available before this law came into
effect, even though the Bill eventually failed to be passed into law.
Being “needs
driven”, especially where the need is related to some practical political
and legal developments, the creation of PGP can hardly be compared to the
derivation of the sorts of “useless” results and theorems in Number Theory.
PGP is an applications package, developed with the full intention of use.
Zimmerman released the package on the Internet, making it freely available
to the public. This immediately constituted a breach of the US Arms Export
Control Act, and constituted illegal export. In particular, PGP, being
an encryption software, was one of the items listed in what is called the
US State Department’s Munitions list, a list of items that cannot be exported
with a special license, and which includes other items such as machine
guns, missiles, and bombs.
Zimmerman
was then further challenged by RSA Data Security, the company holding the
patent for the RSA algorithm, for theft and infringement of their patent,
because according to them, PGP resembled a proprietary software written
by Rivest of RSA Data Security, and marketed in the mid 1980s.
By
releasing the technology on the Internet, Zimmerman gave the “average” US
citizen a powerful tool for protecting their privacy from government authorities.
According to Zimmerman: “I wrote PGP from information in the open literature,
putting it into a convenient package that everyone can use in a desktop
or palmtop computer. Then I gave it away for free, for the good of democracy. … This
technology belongs to everybody. …Today, human rights organizations are
using PGP to protect their people overseas. Amnesty International uses
it. The human rights group in the American Association for the Advancement
of Science uses it. It is used to protect witnesses who report human rights
abuses in the Balkans, in Burma, in Guatemala, in Tibet.” Certainly, mathematics is in operation.
The
free distribution of the package poses a direct threat to commercial and
more highly guarded systems, while providing ready access to anyone wanting
to acquire and implement a secure e-mail system. The free access has in
fact posed direct threats to RSA Data Security, Inc who holds the patent
for RSA algorithm in the USA (and which expires in the year 2000), and
Zimmerman was accused by Data Security Inc of breaching the law by exporting
PG. Free export of PGP also posed a direct challenge to the USA Government,
and the NSA (National Security Agency), who have closely guarded the design
of the DES algorithm used in PGP by legally limiting the key size to 56
bits, while the exported PGP enabled 128 bit keys to be used; this, in
terms of security for the user is more advantageous, but for Governments
who may want to intercept and access transmissions between ‘suspect’ entities
is entirely inconvenient.
Mathematical
knowledge, as expressed in theorems, conjectures and the like are made
public through scholarly publications, and then diffused and distilled
in textbooks, university lectures and academic meetings. So how can we
understand the legal disputes about license and export regulations which
surrounded the initial release of PGP? In PGP, as in other encryption and
mathematically based software packages, we see a fusion of mathematical
results, which together achieve certain functionalities as a package. The
Fundamental Theorem of Arithmetic alone does not enable powerful encryption;
however, its application to cryptography, together with a number of other
mathematical results and reasoning, produces a powerful algorithm such
as RSA. Being a complex algorithm, with specific functionalities, it acquires
commercial, as well as political value.
It
has been the assumption in every (classical) philosophy of science that
knowledge is public. For instance, the whole notion of the “Third World”,
as discussed by Karl Popper, expresses the idea that knowledge constitutes
an entity which cannot be privatised. The entities of Popper’s Third World
are not put into any package. They are claimed to be free flowing entities.
But knowledge-in-a-package is different from knowledge-in-a-free-flow.
It can be patented, it can be exported, it can also be made freely available.
Knowledge-in-a-package operates completely differently from knowledge-in-a-free-flow.
The
question “Whose mathematics is it?” does not really make sense. The answer
appear rather simple. Mathematics is public. However, the question “Whose
package is it?” is a tricky question. A “wrong” answer to this question
could bring a person to court. Ownership can be maintained, and Robin Hood-like
actions can be carried out. This shows that we have to with a particular
entity which has left the realm of Popper’s Third World and entered the
real world included all it business issues.
What is done by means of the package?
Mathematics can “materialise” out of its
abstract entity of symbols and theorems and emerge as part of a functional
entity that drives a computer package. This package provides a specific
meaning to the more general claim, that knowledge and information become
intellectual technology. The
package can be installed and implemented, and its implementation “makes
a difference” on a number of fronts - social, political and technological.
A package underpinned by mathematics, functions in sharp contrast to any “scientific
theory”. It materialises into both the physical world and people’s economic
and social reality.
What
the examples of mathematisation cited in the Introduction and other numerous
examples do not fully reveal are the mechanisms by which mathematics penetrate,
pervade, and constitute reasoning in our socio-technological sphere. What interests us here is not only that
mathematics is being used to model social systems, but that mathematics
is being used to create new social realities. We could talk about a packet-built
reality. Thus, mathematics is no longer only attempting to provide a more
or less adequate picture of reality from which certain conclusions can
be drawn (for example, a map of a city). The utility of mathematics in
providing a picture model of physical phenomena was what gave mathematics
the “marvel” which Eugene Wigner wrote about in his famous essay “The Unreasonable
Effectiveness of Mathematics in the Natural Sciences”. But mathematics can do more than describe
what is already there. Mathematics can also create and become part of reality
itself, such as the socio-economic reality created and built further upon
by mathematical models of an economic system. In this sense, we contest
the limited notion that “mathematics and science help us to analyze existing
ideas and their embodiment in ‘things’, but [that] these analytical tools
do not in themselves give us those ideas.” We argue here that mathematics operates
in social and technological contexts and in doing so mathematics not only
represents but also constitutes reality. Mathematics is not only operating
in Popper’s Third World, say, as a servant for other of the inhabitants
of this world as Natural Science for instance.
In
this section we will concentrate more generally on how mathematics generates
and constrains socio-technological actions. We
will see how mathematics, in particular, influences the way we construct
our social realities and operates within the space of possible actions.
It would be absurd to say that the space for possible socio-technological
actions is constituted by mathematics alone, but we argue that mathematics
plays a role in the construction of such spaces and also in the way we
investigate and choose between alternatives. We will call the alternatives
within a space of technological actions, hypothetical situations.
We
shall try to clarify the question “What is done by means of the package” by
consideing the following three ideas representing the thesis of the formatting
power of mathematics: (1) By means of mathematics new technological alternatives
are presented - alternatives that are not possible to grasp and identify
without mathematics as a tool for analysis and construction. However, mathematics
also limits the set of hypothetical situations that are presented, as mathematical
construction is only one way of expressing a sociological imagination.
(2) By means of mathematics we can investigate particular details of situations
not yet realised. A particular strength of mathematics is to enable hypothetical
reasoning, which refers to reasoning about technological details of a not
yet realized technological construction. However, mathematics may also
produce blind spots concerning the effects of such a not yet realized construction.
Those effects cannot be foreseen and they emerge only when the technology
has been implemented. (3) When choices are made and the technological ideas
are translated into new technological and, hence, social realities, mathematics
simultaneously “enters” this reality in a concrete form. Mathematics assumes
an essential functional role within technological packages, and once that
happens, the mathematical influences on this reality become inseparable
from the other social realities in which it is acting. Mathematics will
have become “socialised”.
We
suggest that the three aspects are generally applicable to the role of
mathematics as resources for socio-technological actions. However, we will
discuss the aspects, first of all, with reference to the example of cryptography
and PGP.
By means of mathematics it
is possible to provide
new (maybe limited) space
for technological actions
In the past a combination of logical operations
and substitutions seemed adequate tools for encryption. However, large
scale industrialised cryptography is not easy to handle with any classical
approach. It would not be possible to imagine the development of public-key
cryptography without the development in its mathematical basis. Informed
by Number Theory, a completely new approach to encryption became identified.
Naturally, it is not that Euclid’s Theorem, the Prime Number Theorem, Euler’s
Theorem and Fermat’s Little Theorem to get with observations about trap-door
one-way functions provide a new possibility. But scial and economic interest
and mathematical resources creates new spaces for technological actions. In
this way we can consider mathematics as an essential resource for a technological
imagination, by means of which we grasp not yet realised technological
alternatives
The
technology of public-key encryption is by no means a product of “natural
evolution” or accident. It is a product of a deliberate effort to improve
on traditional cryptography that became increasingly inadequate as the
sole approach to providing confidentiality and authentication for new cryptographic
needs in the so-called information society. These two trends: the mathematical
developments relevant to cryptography and the emergence of social needs,
were catalysts in the conceptualisation of the possibilities for new alternatives
in encryption.
That
mathematics can open new spaces for technological actions, we see as a
general observation related to mathematics. This is not limited to cryptography.
Mathematics opens possibilities for new approaches in large scale economic
management, by providing possibilities for experimenting with different
economic politics by means of simulations models. Simulations models can
in fact be used in any form of technological constructions, like design
of bridges, airplanes, cars, etc. All such simulation models help to locate
a space for technological actions within which to operate.
However,
as mathematics opens a new space of technological actions, mathematics
can also create blind spots for other openings and for different approaches
to resolving threats to privacy. Mathematically based encryption only allows
certain questions to be asked about the problem perceived in the present
situation. If the solution space is defined within the bounds of mathematics,
then fundamental changes in social relationships, for example, which are
equally valid ways of addressing the perceived threat to privacy, may not
be considered. The choices that are considered become limited to choices
in encryption schemes and particular realizations of the schemes.
In
viewing a threat to privacy as a problem of current methods of data communication,
rather than a fundamental social problem, a “fix” of the technological
process of communication by technological means becomes a “natural” alternative.
Once the root of the problem is identified as a technological one, then
this precludes a wide range of questions and possibilities by means of
which the perceived problem is investigated. Thus, the notion of thrust
and privacy may include many other aspects that those that can be tackled
by proper cryptographic procedures. But such aspects may be downplayed
when the space of possible solutions are created by mathematical tools
As
mathematics opens new spaces for technological action, mathematics may
also create limitations, as the identified technological possibilities
may come to represent the only space for technological action. What mathematics
does provide is, thus, both a new area for technological development as
well as a trap, which encapsulates technological imagination, and separate
it from other non-mathematically based forms of sociological imagination.
This is a general observations related to all situations where mathematics
establish a resources for opening spaces for technological actions.
By means of mathematics it
is possible to investigate
(maybe rather particular)
details of a hypothetical situation
By a combination of mathematical tools
various investigations of particular details of encryption can be carried
out. Complexity theory allows some approximate bounds to be made about
how easily a hacker can succeed in determining the private keys of a public-key
encryption scheme. Number theoretic tools such as “number sieves” provide
cryptanalytic tools to assess the security of schemes such as RSA, and
facilitate investigations of new sources of methods.
The
point is that all such investigations can be carried out on the basis of
hypothetical situations. We need not construct any actual system in order
to investigate details of hypothetical situations. In fact, it is a strength
of mathematically based technological imagination that it can be carried
out on the basis of hypothetical situations. Mathematics helps designers
of encryption algorithms to envisage hypothetical scenarios of hacking,
or computer memory requirements, and so on, which are essential to meeting
the functional requirements of the package. Mathematics helps us to predict
the effectiveness of the algorithm design: for example, how many bits in
the ciphertext would be affected with a change in one bit of the bit sequence
of the key; or how many iterations are needed to realise a certain level
of “confusion” and “diffusion”.
However,
mathematics only allows us to investigate particular details. So, when
an encryption system is implemented, there might occur malfunctions, which
simply has not been considered. Mathematically based hypothetical reasoning
has limitations. It contains blind spots, and from these blind spot risk
structures may emerge, such as the spread of a bug in the system in a potentially
far-reaching way, because of the Internet environment in which systems
such as PGP reside. Another risk is the effect of the technologically established "web
of trust" being compromised in some way.
Although
there are mathematical techniques of risk assessment, analysis and management,
these tools are very limited in their use in the sorts of risk scenarios
described above. Mathematical concepts of risks are expressed in terms
of probability of a risk event, and the “cost” determined in some way of
such an event. In situations where the probability is very very low (a
very rare event), there is not the basis upon which to calculate the corresponding
risks in any meaningful way. Hence in terms of encryption systems such
as PGP, much “trust” is invested in the mathematical rigour of the system,
but mathematics in unable to resolve the risks associated with the technology
going wrong.
This
also bring us to a general observation. A technological imagination, supported
by mathematics, can address hypothetical situation, and can in fact investigate
particular details of such an hypothetical situation. This is the general
idea of providing any simulation model. Then it is not necessary to realise,
say, the bridge construction for carrying out a detailed investigation
of how it might operates in stormy weather with a maximum of car passing.
And certainly such hypothetical reasoning may included blind sport. This
is exactly behind the blind spots of mathematical based hypothetical reasoning
addressing events related to not yet realised technological constructions,
that the risk of the risk society emerge. The carefully estimated almost
not existent risk related to the operations of atomic power plans illustrates
both that decisions making is related to mathematical calculations, and
also that mathematical based hypothetical reasoning contains blind sport.
Mathematics becomes “locked-in” in “reality”
and becomes inseparable from
other aspects of society
When installed, the PGP package becomes
part of a larger technological construction, and it operates within this
construction. This mechanical expression of the package is of interest
to us. What remains to be seen of the mathematics when the cryptographic
elements are built into a package? The interfaces with the user certainly
reveal no traces of, for instance, Euler’s Theorem, Euclid’s algorithm,
or any of the fundamental results of mathematics which underpin the logic
of the encryption algorithm that is used. Many users would see PGP as a
package within a bigger bundle, say an e-mail service, such as the commonly
used Eudora software. In keeping with the nature of these higher level
applications, the mode of interactions would be “user friendly”, with a
pulldown menu which guides the user through the services which we might
require, rather than through the underground corridors of the algorithms
from where the essential services are being delivered.
Without
the mathematics, there is no package. Without the package there is no technological
construction. As new rooms can be added to a house by using bricks and
mortar, extensions can be added to the network society by using packages
as bricks. The technological construction, however, is much more that the “machinery”.
As mentioned in the Introduction, we see technology as inccluding also
organisationan matters and proceedures for decision making. It also includes
the construction of security and of thrust. .
Trust
is established and secured by a mathematically based technology. For those
arenas where encryption has become the norm, trust that has been a fundamentally
social attribute has been transformed into a technologically defined attribute.
Within these arenas, the social realities, therefore, are intrinsically
determined by the technological package that characterises the nature of
the relationships within them. Trust, security, privacy and several other
social phenomena become reshaped in terms of spaces for technological actions
made available by means of mathematics.
The
inventor of the Diffie-Hellman key exchange (an optional key exchange algorithm
within PGP) claims: “As human society changes from one dominated by physical
contact to one dominated by digital communication, we will have many opportunities
to choose between preserving the older forms of social interaction and
asking ourselves what those forms were intended to achieve. ... The area
in which technology can most clearly make a positive contribution to privacy
is encryption. If we assert the individual’s right to private conversation
and take measures in the construction of our communications systems to
protect that right, we may remove the danger that surveillance will grow
to unprecedented proportions and become an oppressive mechanism of social
control. Fortunately, the fight for cryptographic freedom, unlike the fight
against credit databases, is a fight in which privacy and commerce are
on the same side.”
That
mathematics materialise and “socialise” is not just a phenomenon related
to packages of cryptography. Mathematics has materialised in many systems
for decision making. As indicated in the Introduction such decision can
concern medical actions. It can become part of the design for working conditions.
It is an integrated bard of the identification and measurement of risks.
The estimations of “acceptable” levels of unemployment have reaches a new
sophistication be means of simulation models. Thus, “acceptable” no longer
simply refers to moral standards addressing how big a part of the adult
population which should be left at the margin. Now acceptable also includes
estimations related to the national product, to the level of salary, which
is determined also by the degree of employment. Thus the notion of “acceptability” become
re-interpreted in terms of detailed cost-benefit considerations by the
help of mathematics.
What does these observations mean to social theorising?
Mathematics represents a powerful resource
for socio-technological action: The space of technological actions is involved
in rapid and unpredictable developments. New technological options are
generated, but the implications of realising these options are only accessible
by hypothetical reasoning. Technological developments are supported by
mathematics, because mathematics often helps to establish hypothetical
situations and analyse particular aspects of (some of) these situations.
Eventually, mathematics becomes part of the social reality in which the
technological actions are finally carried out.
We have illustrated this idea
with reference to the PGP package. But we find that these three aspects
of the formatting power of mathematics characterises general aspects of
mathematics in action. We will now try to point out a few implications
of this attempt to clarify the notion of the formatting power of mathematics.
In particular, we will emphasise implications for social theorising. We
find that it is essential to social theorising to pay attention of to how
mathematics is operating in social affairs. Any social theorising which
includes notions like “thrust” and “risk” are in need of a clarification
of how such phenomenon is constituted also by means of mathematics. More
generally, we find that overall sociological notions like “reflexivity”,
which refers to the feed-back process which, according to Beck, provokes
the risk of the risk society, needs an understanding of the formatting
power of mathematics in order to obtain an adequate interpretations. An
sociological interpretation of the “informational society”, to use a term
suggested by Castells are in need of an understanding of how mathematics
operates in packages, as mathematics stuffed packages are important bricks
in the network of the network society. Naturally, mathematics does not
in itself provide resources to establish new social forms and relations,
but it represents society’s overall dispositions for socio-technological
action. It can produce new spaces for socio-technological actions and help
to specify aspects of this space. As a consequence we find that any theorising
which address general features of social development, and which includes
consideration expressed in terms of “thrust”, “risk”, “reflexivity” and “informational
society”, must pay attention to the propensities for social development
established by mathematics.
The
unpredictability of the development of the space for technological actions
is one of the forces in the development of “thrust” and “risk”.. In the
encryption example the unpredictability of the degree of “thrust” and “risk” is
closely related to the unpredictability of the results of mathematical
research, for instance related to algorithm theory. The unpredictability
is linked as well to the medium in which the package operates and is exchanged.
The medium, that is the Internet, has a global reach, and so the Internet
effectively plays a role in “selling” a new ideology. The risks that are
perpetrated by the same medium are also cause for concern, and constitute
another example of the sorts of incalculable risks that pervade our risk
society. A fundamental problem faced by any encryption package is that
when a hacker breaks the code, they will typically not let anyone else
know. People using a broken encryption scheme may continue to communicate
with each other, believing that the scheme is secure. Another risk, is
the continued use of compromised encryption keys in public-key systems.
The risk perceived by governments is that “unlawful” or politically subversive
actions may be taking place without their knowledge. Any sociological understanding
of the nature of the distribution of thrust and risk can draw upon an awareness
of their mathematical constituents.
The
formatting power of mathematics is elusive, unlike economic interests and
political agendas. The formatting power is invisible. But being invisible
does not mean being “not-real”. The powers of mathematics are able to interact
with other “powers”. We find that the formatting power of mathematics is
real, both in a physical sense (as for example, models of magnetic and
gravitational forces), and in a sociological sense (as in models of economic
and political forces), precisely because mathematics can interact with
both kinds of power. And one of the implications of this interaction is
the emergence of new risk structures. This provides an opening not only
for an understanding of the nature of the risk society, but also for a
further interpretation of the remark of Beck’s that “reflexivity” takes
place outside the control of the democratic institutions of society (as
well as outside the attention of sociology). Refexivity refers to fundamental but unnoticed
feed-back processes where the results and consequences of certain technological
actions, returns to out reliaty, however in an unexpected form. Certainly
not in a form which was conceptualised as part of the planning process.
We can think of forms of pollution as an example of processes of reflexivity.
The creation of new risk- structures is a more general expression of this
observation.
If we want to obtain an understanding of
social actions including the possible implications, it becomes important
to consider how the space of technological actions functions in the process
of social development. Social and political decision making is related
to the creation and explorations of such spaces. In particular, it becomes
important to consider the role mathematics is playing as a resource for
both the construction and the exploration of particular details of hypothetical
situations. Sociology cannot pursue its goal of providing a basic understanding
of social agency without paying a special attention to the formatting power
of mathematics. How can sociology interpret a society which resides and
unfolds in terms of the “informational society”, without a sound appreciation
of how the space of technological possibilities has come to unfold?
These
observation can also be expressed in terms of suggestions for the philosophy
of mathematics. This philosophy has concentrated on understanding the nature
of mathematical knowledge, which has led to analyses of abstraction and
formalisation. Thus,
analyses, as for instance carried out by Imre Lakatos, has concentrated
on investigating particular mathematical ideas as part of a process of
theoretical development. Lakatos fully accepted the idea that mathematical
knowledge is part of Popper’s Third World, and he tried to describe the
rational development of this free-flowing knowledge in terms of proofs
and refutations. However, observations related to the formatting power
of mathematics provokes a different emphasis in the philosophy of mathematics:
How can it be that we are staying in a world in which so much mathematics
in fact is operating? How do we cope with the living conditions of staying
in this technological constructed environment? The new significance of
mathematical formulae, which we have mentioned, becomes an essential departure
point for any philosophy of mathematics which does not want to consider
the constructions of mathematics form within only.
The
significance of mathematics changes dramatically when the mathematical
ideas are investigated as part of a package. This represents a unit of
particular importance for understanding the social role of mathematics.
The questions: “What is in the package?”, “Whose package is it?” “What
technical effects does the package have?” represents new questions to be
discussed in a philosophy of mathematics. It is a new task for the philosophy
of mathematics to engage with mathematics-in-a-package in order to identify
how spaces for technological action are constructed, and how a formatting
power of mathematics might be exercised.
We
have emphasised that by means of mathematics it is possible to provide
new spaces for technological actions. Let us finally pay attention to the
notion of “action”. Mathematics is a resources for actions of a grand variety.
It is not the case that mathematics provides these actions with any particular
quality. These actions must therefore be addressed by a critique and by
ethical considerations as any other kind of action. In this way the thesis
of the formatting power of mathematics shows the necessity of opening the
considerations about mathematics in two ways. Thus, the philosophy of mathematics
must open its perspective and consider what actions mathematics makes part
of. And in particular, social theorising must open its perspective in order
to include observations of mathematics based technological action in order
to provide adequate interpretation of the propensities of the informational
society.
Acknowledgments
We wish to thank Marten Blomhøj, Gunnar Bomann,
H. C. Hansen, Mike Newman, Miriam Godoy Penteado, John Reizes, Carl Winsløw,
and Warren Yates for their critical comments and suggestions for the improvement
of the earlier versions of this paper.
Ole Skovsmose
Centre for Educational Development in University
Science
Aalborg University
Fredrik Bajers
Vej 7B
DK-9220 Aalborg
Oest
Denmark
Tel: (+45) 96
35 97 80
E-mail: osk@dcn.auc.dk
Keiko Yasukawa
Faculty of Engineering/Faculty
of Education
University of
Technology, Sydney
PO Box 123 Broadway
2007
Australia
Tel +61 02 9514
2437
FAX +61 02 9514
2435
E-mail: keiko@eng.uts.edu.au
