=Paper= {{Paper |id=None |storemode=property |title=Lifeworld and Mathematics |pdfUrl=https://ceur-ws.org/Vol-672/paper3.pdf |volume=Vol-672 |dblpUrl=https://dblp.org/rec/conf/cla/Wille10 }} ==Lifeworld and Mathematics== https://ceur-ws.org/Vol-672/paper3.pdf

Lifeworld and Mathematics

Rudolf Wille

Fachbereich Mathematik,
Technische Universität Darmstadt,
Schloßgartenstr. 7, D–64289 Darmstadt
wille@mathematik.tu-darmstadt.de

Abstract. The article “Lifeworld and Mathematics” has been inspired
by well-known scientists, from whom are listed here: Edmund Husserl,
Jürgen Habermas, Reuben Hersh, Martin Heidegger, Hartmut von Hentig
and Knut Radbruch. Basic for this article is Husserl’s phenomenological
lifeworld analysis of the mathematized modern natural science. Haber-
mas, in whose social theory the concept of lifeworld has also a central
meaning, recommends a theoretic communicational approach for the life-
world analysis which answers the question about the intersubjective con-
stitution of the lifeworld in the sense of the american pragmatism. Hersh
has a pragmatic understanding of mathematics so that he views math-
ematics as a social, cultural, historical reality. Heidegger saw the basic
character of the modern knowledge attitude in the new knowledge claim
named the “mathematical” which is not deducable out of mathematics.
Hentig has offered the book “Magier oder Magister? Über die Einheit
der Wissenschaft im Verständigungsprozess”, in which he discusses gen-
erally the role of the science in our society, while doing so he considers as
necessary a democratically activated general understanding of scientific
actions. Already 1991 Radbruch has pointed out in a lecture at the Darm-
stadt Seminar about “general mathematics” the analysis of conditions
and realizations of the mathematical as task of general mathematics.

1 Lifeworld and Inworld

The concept “lifeworld” has been introduced into philosophy by Edmund Husserl
in his late work “The crisis of the european sciences and the transcendental
phenomenology” (see [Hu54]). With the term “lifeworld” Husserl names - in the
frame of his phenomenological philosophy - the concrete context of the world.
This context is intersubjectively experienced by humans in original evidence and
is therefore ordered before the objective-scientific cognition of the world. Husserl
founded on the “lifeworld” concept his fundamental critic about the development
of science. For him the scientists have chanced the ideal of objectivity, which has
to be understood only methodically, into an independently made “objectivism”
by which the relationship of the research to the responsibility of the acting human
being is lost. To overcome by it the occuring sense-crisis of science, Husserl
recommends to make conscious that and how the objectivistic imagined world
Lifeworld and Mathematics 23

arises out of human achievements which are founded on the lifeworld as the
extensive horizon of human cognition and action.
Husserl’s phenomenological lifeworld analysis of the mathematized modern
natural science has been extended by Alfred Schütz with a phenomenological
analysis of the social lifeworld [SL79]. Schütz understands the lifeworld as the
transcendental frame of possible everyday experiences of the recognizing and
acting subject. Jürgen Habermas, in whose social theory the concept of lifeworld
has also a central meaning, criticizes the approach of Schütz: On the one hand
Schütz tends to start out from an intersubjectively constituted lifeworld, without
to clarify how the lifeworld is intersubjectively produced (a problem of which
Schütz knew that also Husserl has not solved it); on the other hand he takes over
Husserl’s philosophical view of consciousness to comprehend the “experiencing
subject” as the last reference point of the lifeworld analysis [Ha81; vol.2, p.197f].
Habermas recommends a theoretic communicational approach for the lifeworld
analysis which answers the question about the intersubjective constitution of the
lifeworld in the sense of the american pragmatism. A convincing understanding of
mathematics has also to be pragmatically founded, as it is for instance realized
by Reuben Hersh in his 1997 appeared book “What is mathematics, really?”
[He97]. Hence, the treatment of the theme “lifeworld and mathematics” shall be
based in this paper on Habermas’ concept of lifeworld, which has therefore be
more explicated in detail.
Jürgen Habermas introduces in his main work “Theorie des kommunikativen
Handelns” [Ha81] “lifeworld” as a complementary concept to the “communica-
tive action”. Habermas understands action and therefore communicative action
as mastering of situations. “The concept of communicative action cuts out of
the mastering of situations first of all two aspects: the teleological aspect of the
realization of the mastering of purposes (or the execution of a plan of action)
and the communicative aspect of the interpretation of a situation and the achiev-
ing of an agreement. In the communicative action the participants pursue their
plans on the basis of common definitions of situations in mutual agreements.”
[Ha81; vol.2, p.193] For the unification of common definitions of situations the
lifeworld is fundamental because “communicatively acting subjects communi-
cate always in the horizon of a lifeworld. [...] This lifeworld background serves
as a source for defining situations which are presupposed by the participants as
unproblematic.” [Ha81, vo.1, p.107] For the situation-oriented communication
“the lifeworld is a reservoir of self-evident facts or unshakable convictions which
the communication participants utilize for interpretation processes. But single
elements and determined self-evident facts are only mobilized in the form of
consentaneous and at the same time of problematic knowledge if they become
relevant for some situation. [...Out of an understanding-oriented view] we are
able to think the lifeworld as a culturally transmitted and linguistically orga-
nized supply of explanation patterns.” [Ha81; vol.2, p.189]
To be able in answering the question about the intersubjective constitution
and reproduction of the lifeworld, Habermas goes back to the social psychol-
ogy of the american pragmatist Georg Herbert Mead. Mead, who is interested
24 Rudolf Wille

in the complementary structure of the subjective and social world, desribes in
his book “Mind, Self, and Society” how speech becomes a medium of socializa-
tion and social integration and how personal identities and social institutions
develop through liguistic arrangements and normative steering interactions. So-
cialization and social integration, which produce symbolic structures of the self
and the society and with it competences and patterns of relationships, are exe-
cuted by acts of communication; however Mead does not analyses thematically
that these processes of communications are also be reflected in intersubjectively
constituted cultural knowledge. To make it more clear that the speech mediated
normative interaction - besides subjective competences and social patterns of
relations - also leads to a propositional differenciated linguistic communication,
Habermas distinguishes more clearly as Mead “between speech as a medium of
communication and speech as a medium of the coordination of actions and of the
incorporation of individuals.” [Ha81; vol.2, p.41]
This treefold function takes over the speech in the communicative action:
“Under the functional aspect of communication, the communicative action serves
the tradition and renewing of cultural knowledge; under the aspect of the coordi-
nation of action, it serves the social integration and the production of solidarity;
finally, under the aspect of socialization, the communicative action serves the
instruction of personal identities.” [Ha81; vol.2, p.208] In the communicative ac-
tion the constitution and reproduction of the lifeworld take place and to be more
precise by renewing and continuing of valid knowledge, through develpopment
and stabilisation of legitimate orders and of solidarity of groups as well as by
the training of sound of mind and competent acting persons. Habermas goes
with his lifeworld understanding beyond the area of knowledge because “the sol-
idarities of the about values and norms integrated groups and the competences
of incorperated individuals in simular manner as cultural tradition [...] flow in
communicative action.” [Ha81; vol.2, p.205]
As a reference system for describing and explaning, which concerns a life-
world as a whole and do not only regard occurrences, Habermas turned out
culture, society, and person as the structural components of the lifeworld which
corresponds the proceedings of the cultural reproduction, the social integration,
and the socialization. What Habermas wants to understand in this connection
by culture, society, and person, he lays down as follows: “I entitle “culture” as
the knowledge supply out of which the communication partners provide them-
selves with interpretations to come to an agreement about something in the
world. I entitle “society” as the legitimate order about which the communica-
tion participants organize their membership in social groups and secure with it
their solidarity. I understand under “Personality” the competences which makes
a subject to have speech and capacity to act, i.e. to renovate, to participate in
communications and at the same time to declare the own identity.” [Ha81; vol.2,
p.209]
To make clear how the lifeworld differentiates itself in its symbolic structures,
it recommends to describe in detail the three processes of reproduction and their
connections:
Lifeworld and Mathematics 25

• “The cultural reproduction of the lifeworld [...] secures the continuity of the
tradition and a sufficient coherence of knowledge for the everyday practice, re-
spectively. Continuity and coherence are measured by the rationality of the as
valid excepted knowledge.” Disturbances of the cultural reproduction cause that
the as valid excepted schemata refuse and “the resource > sense < becomes
scarce.” [Ha81; vol.2, p.212f.]
• “The social integration [...] provides for the coordination of action about le-
gitimately regulated interpersonal relations and makes constant the identity of
groups in a sufficient measure for the everyday practice. For this the coordi-
nation of action and the stabilization of group identities are measured by the
solidarity of the relatives.” Disturbances of the social integration cause that
the as legitimately regulated memberships are not sufficient and “the resource
> social solidarity < becomes scarce.” [Ha81; vol.2, p.213]
• “The socialization of the relatives of a lifeworld [...] secures for the again grow-
ing generations the acquisition of generalized abilities of action and provides for
the voting of individual life stories and collective life forms. Interactive abili-
ties and styles of life guidance are measured at the soundness of mind of the
relatives. Disturbances of the socialization course of events produce that per-
sons can protect their respective identity only with defensive strategies which
impair a fair realistic participation in interactions through which “the resource
> self − strength < becomes scarce.” [Ha81; vol.2, p.213]

After it has been described what the processes of reduction are capable to
achieve for receiving in each case the belonging component of the lifeworld, the
question remains what it can contribute respectively for receiving the two other
components. The cultural reproduction can stabilize legitimations for existent
institutions in the soceity as well as deliver for the person effectively forming
attitude patterns for the acquisition of generalized capacity to act. The sozial
integration can secure in the culture moral duties and commitments as well as
stabilize for the person legitimately regulated social memberships. The social-
ization can promote in the culture interpretation efforts as well as renew in the
society motivations for concurring norm actions. (s. [Ha81; vol.2, p.214ff.])

Although Habermas takes over from Mead the processes of socializations
and social integration for the structure of the subjective and social world, he
criticizes that Mead establishes these processes alone ontogenetically and does
not clarify particularly the transition to norm conducted interaction as well as
the structure of group identities. To close this gap, Habermas refers to the 1912
written sociology of religions of Emil Durkheim [Du81] in which the roots of the
moral authority of social norms are uncovered. Durkheim analysed the religious
belief and the patriotism “as expression of a tribe development deeply rooted
in a collective consciousness which is constitutive for the identity of groups.”
[Ha81; vol.2, p.206] Habermas sees in this usage of speech in ritual secured
normative consent, which produce this collective consciousness, the phylogenetic
foundation for the linguisticly mediated norm-conducted interaction, which the
starting position presents for the social-cultural development.
26 Rudolf Wille

Where and how can one grasp empirically what for Habermas the lifeworld is?
Since Habermas speaks usually about the lifeworld of social groups, this suggests
to examine the lifeworld structures of those social groups. For this Habermas
holds the participant perspective for unsuited and recommends more likely “the
everyday concept of lifeworld with whose help communicatively acting persons
localize and date themselves and their statements in social rooms and historical
times. The persons meet themselves in the communicative everyday speech not
only in the attitude of participants but they also give narrative descriptions of
occurrences which happen in the context of their lifeworld.” [Ha81; vol.2, p.206]
This practice of narrations particularly has the function that the communica-
tion participants ascertain their personal and collective self-understanding. A
personal identity can only be trained if the succession of the own actions is un-
derstood as a narratively representable lifestory, and a social identity only if the
own story is recognized as interactively embedded into the narratively repre-
sentable story of collectives. “The collectives obtain their identity only to such
an extent as the expectations, which people make themselves about the lifeworld,
overlap sufficiently and concentrate to unproblematic background convictions.”
[Ha81; vol.2, p.206]
For this it recommends itself to examine the lifeworld of a social group on the
basis of the expectation which the group members make themselves about their
lifeworld, repectively. However, it has to be clarified that this approach turns
arround the line of sight: The individual is as a rule member of different social
groups of which the examination group is only one; hence, of the connecting
lifeworld expectations the single members of the examining group have not only
the overlaps to be identified but it has also to recognize the relation to the
examination group, respectively. To make these connections more capable of
thinking, it offers to introduce the net of all lifeworlds, which are represented
in the knowledge, behavior, and selfunderstanding of one person, as theoretical
concept of the lifeworld analysis; this concept shall be named the “inworld” of
this person. As an inversion of the inworld-definition one obtains the statement:
the lifeworld of a social group is the common of the inworlds of their members.
Obviously this statement reflects the propagated initial search stage.
To found this attempt further for the empirical lifeworld analysis, the in-
world concept shall be directly anchored into the social-psychology of Mead as
already Habermas has done it with his lifeworld concept. For this it has been re-
turned to Mead’s theory of identity whose structural characteristic is that Mead
subdivides the self-identity in a subjective ]I[ and a social ]Me[. The ]I[ is spon-
taneous, emotional, creative, and expressive, while the ]Me[ is reflected, rational,
conventional, and controlled. The ]I[ gives me the self-feeling, the ]Me[ represents
the we-experience. Mead writes himself: “The ]I[ is the reaction of the organism
on the attitudes of others [...]. The attitudes of the others form the organized
]Me[, and one reacts on it as an ]I[. [...] For the human it is important that
he receives the attitudes of others and adjusts his own identity or takes on the
battle. This recognizing of the identity of the single in the process of the identity
consciousness gives him the attitude of the self-assertion or subordination under
Lifeworld and Mathematics 27

the community. Through it he reaches to a definitive identity.” [Me73; p.218,
237] The ]I[ and the ]Me[ are therefore constitutive for the identity formation of
humans.
The representations of the own inworld have obviously their place in the
]Me[; there they will be constituted and traditionalized by the processes of the
cultural reproduction, the social integration, and the socialization taking place
in the social groups of the lifeworlds which belong to the inworld. That and
how these processes can be effective, Mead clarifies with the spiritualization of
the gesture arranged interaction: “The transmisson of gestures is a part of the
proceeding social processes. [...] The development of the language, in particular of
the significant symbols, made it possible that just this external social situation
is included in the attitude of the individual.” [Me73; S.230] According to the
theory of Mead, the cultural reproduction for the knowledge and thinking of the
individuel is essential because “one must insert the external social world, which
one has entered into one-self to be able to think.” [Me73; S.243] With this the
]I[ and the ]Me[ are absolutely necessary. For the social integration “one must
take over the attitudes of the other to belong to a community; [...] on the other
hand the individual reacts constantly on the social attitudes and changes in this
process just this community.” [Me73; p.243] Finally Mead makes available also
the special meaning of the socialization: “We can only realize ourselve in so far
as we recognize the other in his connection to us. While the individual takes
over the attitude of the other, he is capable to realize himself as identity. [Me73;
p.273]
In the sense of Mead one can say: the inworld of a person comes into being
in that way, that the lifeworld attitudes of the others, which belong to the
same social groups as the person, are getting in the ]Me[ while doing so the
respective reference is taken to the social group it belongs to. With the takeover
of the lifeworld attitudes the ]Me[ gives a form to the ]I[ and with it to the
self-identity which is determined conventionally [Me73; p.253]; i.e. the inworld
delivers for the person’s thinking and acting a conventional background which is
in particular important for the identity formation. Since the lifeworld attitudes
of the others are an expression of the collective consciousness of a social group,
the inworld concept is understood both ontogenetic and also phylogenetic. It
shall be again clarified: the concept of inworld builds on Habermas’ concept of
lifeworld for whose arguments Habermas goes back to Mead’s theory of identity
and Dürkheim’s theory of collective consciousness; constitution and reproduction
takes place in the inworlds - as in the lifeworlds - in the communication of action.

2 Mathematics and the Mathematical

What is now the theme of mathematics in this paper? About which understand-
ing of mathematics can we use for our theme? This is not easy to answer because
it has always been difficult to say what mathematics is all about. Even the com-
monly given definitions of mathematics, as they are presented in the current
books of reference, are scarcely sufficient. Deputizing the 1998 edition of the
28 Rudolf Wille

Brockhaus-Enzyklopädie [Br98] we optain:

“Mathematics [is] one of the oldest sciences, emerged out of tasks such as count-
ing, calculating and measuring, which were based on practical (first of all sci-
entific and technical) formulations of questions for whose treatments originally
numbers and geometric figures as well as their mutual connections were taken.
Thereby the concept of number and the elementary geometric concepts have
been developed. Until today mathematics receives strong impulses from the at-
tempt to contribute to the description of scientific, economic etc. occurrences.
The field of activity of mathematics was essentially extended by abstracting the
original meaning of the studied objects and led to a “science of formal systems”
(D.Hilbert). According to that one understands under modern mathematics the
science of the abstract structures and logical conclusions, which are determined
by commitments of a few basic acceptances about the relations and connections
beween elements of arbitrary size. It belongs to its essential tasks to set up the
most general connections without contradictions between these quantities, out of
which it results conclusions in form of statements (theorems) on a purely logical
path. Mathematics is characterized by a high precision of its concept system,
strictness of its methods of proving and a strongly deductive character of its
presentation.”
The cited text describes obviously the common understanding of mathemat-
ics which dominates under mathematicians and does this definitely by a concise
and convincing manner. So what is unsatisfactory about this concrete defini-
tion of mathematics? To say it shortly: it lies on that what is left - in general
the relation of mathematics to the community of communication of thinking
and acting humans. Fortunately, in 1997 there has been appeared a book which
presents and substantiates a human related pragmatic view of mathematics: it
is the already mentioned book “What is mathematics, really?” [He97] of the
mathematician and philosopher Reuben Hersh. The approach of Hersh is based
on that what mathematicians really need and do for their work, and is not based
on a somehow ideal of mathematical objects and activities. Typical for Hersh is
the following course: The mathematical work comes thereby into action that one
discovers a problem which is connected with the existing mathematical culture;
then one works on a problem and needs help and encouragement in the case of
present difficulties; one suggets finally a solution for which one needs agreement
and critics. How isolated and self-absorbtionally a mathematician may work,
he finds nevertheless origan and confirmation of his work in the community of
mathematicians. [He97; p.5]
His pragmatic understanding of mathematics delivers Hersh also the argue-
ments to keep one’s distance to the formalism and the platonism, the two main
opinions of the nature of mathematics. [He97; p.7ff] Concerning the formalism
he critisizes that mathematics may be understood in parts as a rule guided game,
but with that it remains unclear how rules can be made, developed and evalu-
ated with regard to their applications. He reproaches the platonism that it does
not give answers how the immaterial mathematical objects may come in contact
Lifeworld and Mathematics 29

with mathematicians out of flesh and blood and how the strange parallel exis-
tence of the physical and the mathematical reality can be explained. For Hersh
these questions can be only answered when one understands mathematics as a
social, cultural, historical reality. In this sense he describes the mathematics as
follows:
“A world of ideas exists, created by human beings, existing in their shared
consciousness. These ideas have objective properties, in the same sense that
material objects have objective properties. The construction of proof and coun-
terexample is the method of discovering the properties of these ideas. This branch
of knowledge is called mathematics.” [He97; p.19]
In the second part of his book, Hersh refers views about mathematics out
of the philosophical history from Pythagoras until the present time; for this
he describes two parallel running developments: that of the “main stream” at
which mathematics is respected as superhuman, abstract, ideal, unfailing, and
eternal, and that of the “humanists and mavericks” at which mathematics is
understood as a human activity and as human creation [He97; p.91f.]. Hersh
counts to the main stream first of all Pythagoras, Plato, Descartes, Spinoza,
Leibniz, Kant, Husserl, Frege, Russell, Carnap, to the humanists and mavericks
Aristotle, Euclide, Locke, Hume, Mill, Poincaré, Sellars, Wittgenstein, Popper,
Lakatos, Wang, Tymoczko, and Kitcher. It is surprising that Hersh reckoned
Husserl to the main stream and nevertheless quoted comprehensively out of
Husserl’s late work as for instance: “without the ’what’ and the ’how’ of its pre-
scientific materials, geometry would be a tradition empty of meaning.” [He97;
p.166] The lifeworld conception, which is here mentioned by ’prescientific ma-
terials’ and honours Husserl more as a “humanist”, does not come up without
a reason in connection with geometry because the geometry is the science from
which the concept “lifeworld” is introduced into Husserl’s late work [Hu54].
This first explicit connection of mathematics and lifeworld shall be shortly
presented in its basic thoughts. First Husserl explains the change of the antique
conception of science to the modern science conception. For him the unheard
new finding was the “idea of a rational infinite universe with a systematic ra-
tionally controlled science”. He clarified this idea by rearranging the euclidean
geometry to the modern geometry: The finally closed apriori of the greek geome-
try changes to a universal apriori to which the infinite ideal space belongs and an
“infinitely - despite the infinity - in itself closed homogeneous systematic theory,
which allows to construct - from axiomatic concepts and theory ascending - each
thinkable drawn shape in the space of deductive uniqueness. [...] Interested for
these ideal shapes and consequently concerned with them, to determine them
and to contruct out of already determined shapes new ones, we are ‘Geometers.
And just as equally, also for the further sphere, which is also concerned with
the dimension of the number, we are mathematicians of the ‘pure shapes, the
universal form of which is the self idealized space-time-form. [...] How all through
human contributions arising achievements they [the pure limes shapes] remain
objectively recognizable and available, also without that their sense formings
must be explicitly renewed.
30 Rudolf Wille

But now it happened in the course of history that the theory of the objec-
tively intended ideal forms developed itself into a sytematic, thoroughly useful
science - the modern mathematics - and made itself independent in the view of
autonomous objectivity, without that a real understanding of the actual sense
and the internal necessity of the constituted achievement of abstraction was
present. According to Husserl “the real evidence is missing and missing further,
in which the perceiving-achieving human can give himself an account not only
about that, what he is doing new and with which he is opperating, but also about
all through sediment resp. tradition closed sense-implications, hence about the
permanent assumptions of his creations, concepts, statements, and theories. Is
the science and its method not equal to an obviously very useful and reliable
maschine which everybody can learn to handle correctly, without to understand
at least the internal possibility and necessity of such services?” [Hu54; p.52] To
be able to counter the bad state of affair, Husserl retains it for urgent to reflect
the relation to the lifeworld - “the in our concrete worldlife us permanently as
real given world” - and to the human as its subject.
From the philosophical view this means according to Husserl : “Clearness
about the origin of the modern intellect and with it - capable of the not highly
enough evaluated importance of mathematics and the exact natural sciences
- about the origin of these sciences [... since the exact natural science] from
the beginning and furthermore in all its changing of senses and irrelevant sense
interprerations of decided meaning for growing and being of the modern positive
sciences, in the same manner of the modern philosophy - really of the spirit of
the modern european humanity in the first place .” [Hu54; p.58f]
More detailed as Husserl, his follower Martin Heidegger has treated the ques-
tion about the basic understanding of the modern natural science and - to be
more precise - in his 1935/36 presented lecture “Grundfragen der neuzeitlichen
Metaphysik”, of which the text has been published 1962 as a book with the title
“Die Frage nach dem Ding” [Hd62]. Heidegger characterizes the basic feature of
the modern natural science as follows: “The basic feature has to consist of that,
what the basic movement of the science as such standardly dominates as equally
original: it is the work intercourse with the things and the metaphysical model
of the thingness of the things.” [Hd62; p.52] Heidegger sees this basic character
of the modern knowledge attitude in the new knowledge claim which he names
the “mathematical”.
The “mathematical” is according to Heidegger not deducible out of mathe-
matics, but mathematics is itself only a determined form of the mathematical (cf.
[Ra91], [Wi98]). For being able to determine what is meant with “the mathemat-
ical”, Heidegger goes back to the origin of the word in old greek: τ α µατ µατ α
means the learnable. For Heidegger the learning is at this a taking and an acqui-
sition, by which the taking occurs through knowledge-taking and the acquisition
through application. Since Heidegger views human knowledge in the basic po-
sition of thinking to the things, respectively, and to the being anyhow, it is for
him “the basic learning of such taking where we this, what at all a thing is, take
into the knowledge. [...] This original learning is [...] some learning [...], at which
Lifeworld and Mathematics 31

the receiver takes only that, what he already has for some reason.” [Hd62; p.56]
From that it results for Heidegger: “The µατ µατ α, the mathematical, that is
such at the things what we actually already know, what we therefore do not take
out of the things, but already in a certain way bringing them with us.” [Hd62;
p.57] Differently said: “The mathematical is such basic position to the things, in
which we make us clear what they are already presented us. The mathematical
is therefore the basic assumption of the knowledge of the things.” [Hd62; p.58]
Herewith Heidegger sees the central significance of the mathematical for modern
thinking, since in the nature of the mathematical there lies a “will for the new
design and self-foundation of the knowledge form as such.” [Hd62; p.75]
As what can the mathematical grasp thinking and acting? In thinking the
mathematical is the actual learnable about the things and delivers with it the
knowledge form, i.e. it is a formal thinking or more pointed: a form thinking. For
Heidegger the nature of the mathematical form thinking lies in its marking as
sketch by which it is settled “what we actually hold about the things, as what
they and how they are recognized in advance.” [Hd62; p.71] Such an axiomatic
sketch delimits the area of the things which are thought by the sketch in the
given knowledge form. In the mathematical as such sketch thinking, Heideg-
ger sees the basic feature of modern thinking and knowledge, with which it is
meant in particular a liberation from the middle-age thinking to the reformation
and self-foundation of the knowledge form. Since in his book [Hd62] about the
confrontation with Kant’s “critics of pure reason” Heidegger focusses the de-
termination of the mathematical on the taking in learning and moves back the
acquiring with the argument that the taking (as knowlegde) precedes necessarily
the using (as appropriation). [Hd62; p.55]
If one however changes the relation of Kant’s transcendental logic into Peirce’s
pragmatical semiotics, as Karl-Otto Apel has presented it in his article “Von
Kant zu Peirce” [Ap76], then the objective validity of knowledge for the single
consciousness stands not more in the foreground, but the intersubjective com-
munication about truth, rightness, and truthfulness of statements; i.e. the claim
of objective rationality will be replaced by the communicative rationality. The
pragmatic reference allows to see the taking of knowledge and the using of ap-
propriation in their interrelation, because each using produces also something
more of taking. Pragmatically the mathematical has to be understood as form
thinking which causes a mutual process of conception and application. In this
way the mathematical thinking is always constituted intersubjectively, because
the figures and operations of thinking have already been formed by processes
in a community of communication. The mathematical has therefore its place in
the lifeworld of the scientifically oriented communities of communication where
“lifeworld” is understood here in the sense of Habermas.
But in which relationship does mathematics stand to this lifeworld? For clar-
ifying this, it must be made understandable how it comes to the specific forms
of the mathematical which we call “mathematics”. According to Hersh’s under-
standing of mathematics, there are recognizing, thinking, and communicating
humans which cause these formings. Out of figures and operations of thinking
32 Rudolf Wille

concerning their mathematical forms, which are always activated again and again
in communications, there are formed in a process of progressive conventional-
izations determined formal systems of knowledge, which constitute a culture of
formal thinking: the mathematics. It is frequently difficult to say why certain for-
mal systems of thinking belong to the repective mathematics-culture and others
not; for instance, the system of integral proportions belongs to todays mathemat-
ics, but not the tone system of music. What finally a mathematic-culture takes
up from the conventionalized form-thinking (or again separates out; see [La72]),
that is decided in a complex process in the respective community of communi-
cation of mathematicians. There are the integrated conventions of thinking and
the connected explanation patterns and convictions which make up the lifeworld
of the respective mathematics-culture. With the progressive specialization, the
community of communication of mathematicians becomes subdivided into fur-
ther subgroups which respectively form their own lifeworlds what the general
communication about mathematics makes it increasingly more difficult. Thus,
it is today not easy to find out what still belongs actually to the lifeworld of
the whole community of mathematicians. The real question about the relation
of mathematics to the lifeworld of our present society escapes therefore a simple
answer.

3 Inworld and Mathematics

In view of the considered difficulties, not the relation of mathematics to the life-
world of our society shall be directly looked for, but the mathematical and the
mathematics in the educational process of the single, i.e. in the constitution and
reproduction of the inworld of social individuals. The extension to the mathemat-
ical recommends itself because also in the educational process of an individual
the conventional mathematics as a forming of the mathematical is acquired. Ac-
cording to Jean Piaget, the evolutional psychological roots lie for the education
of the mathematical in the existing coordinates of actions which were already
present before the development of speech; then “these coordinates form a logic
of actions which, as far as she is concerned, presents the starting point for logic-
mathematical structures.” [Pi73; p.51] In Mead’s description of the development
of speech (from the gesture helps over the symbol helps to the norm regulated
interaction) it becomes clear how further form thinking develops. In general the
cultural communication, the social integration and the sozialization contribute
so early for it that a fullness of basic patterns of the mathematical could be
acquired. How far these basic patterns are activated during the learning of the
mathematical, this has been studied impressively by the interpretative teaching
research with the key concept of the so-called “framing”. (s. [Go80], [MV91])
Nevertheless it is up to now not much known about what the mathematical
and the mathematics mean for the constitution and reproduction of the inworld of
single humans. How difficult answers are about this question, it refers already on
that ambivalent views of many humans about mathematics. The global opinion
that arithmetical and logical thinking is learned through teaching mathemat-
Lifeworld and Mathematics 33

ics is in any case no help as long as it cannot be satisfactorily answered what
is meant and wanted, respectively, with arithmetical and logical thinking. As
a special obstacle there is the dominance of the mechanistical understanding of
mathematics as it is for instance expressed in the narrowing of the understanding
of logic as the doctrine of the forms of thinking onto calculations of logical con-
clusions. An even more dangerous development is the progressive specialization
and instrumentalization in mathematical research which reduces the coherence
and continuity of the common knowledge in the society of mathematicians and
produces in this way an always larger distance between mathematics and the
world.
For mathematics Husserl’s demand is urgent as before to remember the rela-
tion to “the in our concrete world-live us as constandly real given world” and to
the human as its subject. In the sense of Husserl’s demand we tried hard in our
Darmstadt research group since more than twenty years to answer the question:
What can and shall mathematics be for the general public? [Wi96b] For answer-
ing this question we had to clarify the conception of the self-understanding of
mathematics, of its relationship to the world as well as the questions about sense,
meaning, and connection of mathematical action. We received in particular great
help by the book “Magier oder Magister? Über die Einheit der Wissenschaft im
Verständigungsprozess” [Hn74], in which Hartmut von Hentig discusses generally
the role of the science in our society, while doing so he considers as necessary
a democratically activated general understanding of scientific actions. The sci-
ences must, as Hentig demands, “examine their disciplinarity and this means
• to uncover their unconscious purposes,
• to declare their conscious purposes,
• to select and to straighten their means,
• to present publicly and understandably their possible consequences and
• to make accessible their way of discouvery and results about the common
language.” [Hn74; p.136f.] “The always more necessary growing restructuring
of sciences in themselves - to make them better learnable, mutually available,
and more generally criticizable (i.e. also beyond specialized competences) - can
and must be performed with patterns which are taken from the general forms of
perceiving-, thinking-, and acting-forms of our civilization.”
We have tried in making serious with Hentig’s demands and to come to a gen-
eral understanding of mathematics. For this we have approached the required
restructuring of sciences for different subdomains of mathematics, which has
been shown eminently fruitful as well in teaching as also in research. However
critical reactions and discussions have shown that the reconstruction should be
seen in the frame of the more extensive concept of “general science”. What is to
understand under “general science”, I have tried to carry out in a contribution for
the THD-lecture sequence “Responsibility in the scienses” (1987). [Wi88] There-
after all efforts belong to the general science which try to disclose sciences and
to make them accessable, so that the general researchers can critically explain
the possible consequences and outcomes of scientific doings. General science is
therefore not an autonomous part of science, but as part of any scientific disci-
34 Rudolf Wille

pline and also subdiscipline. “General mathematics” names the part of general
science which is relevant for mathematics. Their efforts aim also in particular
to answers concerning the question what mathematics can and should mean
for the general public. In the sense of general science, general mathematics is
characterized by * the attitude to open mathematics for the generality and to
make it principally learnable and criticizable, * the presentation of mathematical
developments in its sense giving, meanings and conditions, * the imparting of
mathematics in its lifeworld connection even across the boarders, * the explana-
tion about destinations, procedures, conceptual values, and legitimate validities
of mathematics.
Since almost twenty years we discussed the theme “general mathematics” in
seminars, colloquia and on meetings. Already 1991 Knut Radbruch has pointed
out in a lecture [Ra91] at the Darmstadt seminar “general mathematics” as
task of the general mathematics the analysis of conditions and realizations of
the mathematical; for this he formulated the following: “Since the mathematical
is the actual object of research of general mathematics, such a mathematical
must also try hard about explicit and implicit traces of that mathematical in
all regions of rational reason and cultural forms. An analysis of applications and
situations of applications of concrete mathematics can - but must not - show the
way to such traces of the mathematical.”
The Darmstadt Seminar has received interestedly Radbruch’s proposal and
discussed comprehensively Heidegger’s understanding of the mathematical and
tried first of all to trace out the everyday thinking ([Le97],[LP97]). An important
method for it was to examine general actions of thinking on their formal part
and its possible abstractions in conventionalized mathematics as for instance:
abstracting concritizing mathematizing simplifyimg

allocating counting modelizing synthesizing

analysing differentiating ordering systematizing

arranging formalizing presenting tabulating

categorizing generalizing reasoning visualizing

classifying idealizing schematizing tabulating

combinating illustrating specializing transforming

comparing interpretating structurizing typifying

For this, a reader [SAM96] was put together with the presented list of think-
ing actions which may take to the listed words respectively more descriptions of
meanings from different restructurings, to get in this way a possibly broad foun-
dation for tracing out the mathematical: Helpful for the examination of thinking
actions were the long-standig experiences in “restructuring” of mathematical
theories in the sense of general mathematics, which has been performed most
Lifeworld and Mathematics 35

intensively in the fields of linear algebra, mathematical logic and mathematical
order and lattice theory. ([Wi81],[Wi82],[PW86],[Wi87],[Wi96a],[Wi97],[Pr98])
The efforts concerning general algebra, in particular about the restructuring
of parts of mathematics, have led out on manifold modes to connections be-yound
mathematics. In this way we have executed in the frame of formal concept ana-
lysis and conceptual knowledge processing already more than 200 projects of
applications in cooperation with experts out of varying research fields. This has
us not only brought “application projects of concrete mathematics” for analysing
the mathematical mentioned by Knut Radbruch, but - which was even more re-
vealing - the possibility to recognize the mathematical and the mathematics in
the thinking of non-mathematicians during the concrete work on tasks and prob-
lems which are important for them. We have generally made the experiment that
serious difficulties with mathematics have its ground less in lacking knowledge
about mathematical concepts, results and procedures, but in the missing con-
fidence with the self-evident truth, conventionalisms and convictions which are
used today by mathematicians in their inner-subjective communication. Since
this intimacy cannot directly be imparted, it arises for us sustainedly the basic
question: with which educational processes can we reach an appropriate relation
of inworld and mathematics?
For a self-understanding of todays mathematicians it shall be extensively
named a deficit which the non-mathematicians usually have, namely about the
self-understanding that sets in the modern mathematics is generally seen and
used as something unquestional existent, that even the question concerning the
existence of assumed objects is positively decided in the rule by the return to
suitable sets. How little this self-understanding exceeds over the society of math-
ematicians, we experience every year with the freshmen that the written set
thinking is for them not at all well-acquainted. The schools arrange still the
picture of mathematics of the 19th century, whose central objects of thinking
are still numbers and geometric figures. In this way one has in our community
no difficulties with a judgment such as “three and four is seven” and is also not
asked: “do you mean with it apples or houses?” A number such as 3 or 347 is
accepted as a meanigful object of thinking, and in a relation to objects whose
quantity is counted with it, this must be not named or only as well assumed.
But if one says: “Think a set inside of you and than the set of all its subsets!”,
then one usually releases a lack of understanding, at the best it may be inquired
what do you conclude about the total set and their subsets. It is interesting that
one hardly gains difficulties with the request: “Think about a tree with all its
branches!” or even: “Draw this tree!”.
For the freshmen of mathematics it approximately lasts one or two years until
they have taken over the self-understanding and conventionalisms as the foun-
dation of mathematics. How this happens in detail and how it can be well-aimed
supported, this is considerably a riddle. Perhaps the acclimatization succeeds in
set thinking more easily for younger students, when there are given set theore-
cical insides already on the level of schools. Sets are nothing else than concept
extents of which the concept intents are forgotten (as numbers are quantities
36 Rudolf Wille

which do not pay attention to qualities of numbers). In any case it would be
a great win for our society, when with the lifeworld thinking of concepts as its
mathematical part, the thinking of sets would be more self-evident.

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footnote 1

1
This article is an English version of the German article “Lebenswelt und Mathe-
matik” published in: Ch. Hauskeller, W. Liebert, H. Ludwig (eds.): Wissenschaft ver-
antworten: soziale und ethische Orientierung in der technischen Zivilitation. agenda
Verlag, Münster 2001, p.51-68.