Here I quote several paragraphs from Ruzabin(#1) that
I mentioned in [1427]. Though I'm afraid my translation
contains errors, I hope these quotes could clarify his
thought on the matter.
"In order to study mathematical relations in pure form,
we have to exclude some kinds of characters and relations
of things from our thought. If we think simplistic, this process
of abstraction in mathematics seems throwing out non-
mathematical characters and picking up only mathematical
ones. Real objects, however, do not have characters
which mathematics may pick up directly. This can be easily
understood. So mathematical abstraction is not a simple
work excluding non-mathematical characters and
maintaining mathematical characters. Though empirical theory
on abstraction stands on the view of materialism, we are
not able to analyse processes of mathematical conceptualization
correctly. In order to study mathematical abstraction
scientifically, we have to consder dialectical character
of its conceptualisation process as well as approaching
the original nature of mathematical concepts materialistically."
(chaper 1)
"From empiricist theory of abstraction, abstraction of
mathematical characters like numers or figures is to
throw out non-mathematical characters one by one
and reach to the ultimate mathematical character.
But the thought of abstraction above simplifies
actual situation too extremely because characters of things are
uncountably many, and mathematical characters do not
exist in pure form in things unlike color or heat-conductivity.
Idealists utilized this weak point of empiricist concept of
abstraction in order to attack materialist view on
mathematics. But if we consider cautiously on means of
abstraction of conceptualization in mathematics,
dialectic-materialistic character of those means are to be
unveiled, and the fault of idealists' thought on specific
a priori recognition in mathematics becomes clear.
Let's look at the process of conceptualization of
natural number which is the starting point of the development
of mathematics historically and logically in order to
understand what is the abstraction by equalizaion.
For human beings today, the concept of natural nember
is too familiar though there are some points not yet
enough clearly understood. They believe that calculation
and comparison assume existence of natural number.
Human beings, however, did comparison and calculation
to various sets of things without clear concept of number
in a period of social development undoubtedly. This is
proved by historical facts. The concept of natural number
required rather progress of thinking power of abstraction
so that the concept appeared much later in fact."
(chaper 1)
after criticizing Bourbaki(#2),
"Mathematical structure reflects quantative relations and
forms of space in real world. These are not creation of
arbitrary thought of human beings but contain objective
characters and exist indepently from our consciousness."
(chapter 1)
"First, the structure and characters of a formal system
depend on characteristics of content theory which has
this system as formalization. Formal systems have
meanings because they have interpretations. Second,
we always need meta theory when we study fomal systems.
We describe characters of formal theory and means of
inference allowed there with help of meta theory. By
the way, the inference of this meta theory stands
on meanings and contents like inferences of content
theory. The conclusions are able to be accepted
intuitionally. Third, any formalization on enough rich
content theory cannot be perfect that study on
formal theory shows. In a certain meaning, inter-
relationship between a content theory and its formal
system is similar to the relationship between a thing
and its model. That is, a formal system cannot express
characteristics and characters of the content theory
perfect, like a model gives no perfect knowledge
about the original." (chapter 2)
"That mathematical study by formalists themselves, however,
cannot finish eternally proves that it assumes content
mathematics even as meta-mathematics. As the conclusion
by Goedel clarifies, the attempt to make mathematics based
on itsself, the effort to expell contents in mathematical
argument, the attempt to totalize mathematics into one
total axiom system, are to become unsuccessfull. This
invention thus assures These of materialism that concepts
and methodology of mathematics has thier origin only in
empirical facts and practices so that any attempt to
found mathematics not regarding its birth in empirical
facts must be unsuccessfull." (chapter 9)
#1 G.I.Ruzabin: O Prirode Matematicheskogo Znaniya, Moskow, 1968
japanese edition is tranlated by Saburo Yamazaki, Yasumitsu Shibaoka
Iwanami, 1977
#2 Nichola Bourbaki: Architecture of mathematic
Les grands courants de la pensee mathematique
presentes par F.Le Lionnais, 1962
japanese edition is edited Tamotsu Murata, Tokyo-tosho, 1974
in OPE solidarity,
Iwao
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Iwao Kitamura
E-mail: ikita@st.rim.or.jp