F. CSATÁRI
asserted by the upper Löwenheim-Skolem-theorem, say. And it
is far from clear how similarity is to be understood: my hand-
writing, a computer with an algorithm and some synaptic proc-
esses show very little similarity. As for those features which
happen to be similar in these systems, I would be puzzled to
show what Löweinheim-Skolem-theorem is to do with them.
What, Then, Does Belong to Mathematics?
A formal system is nothing more than axioms and rules,
given as particular symbols, after all particular arrays of ink on
a particular sheet. As it is a physical system, it must have some
kind of behavior as, for instance, the solar system has, in which
one can observe some regularities. What kind of “behavior” of
this “system” can be identified by an observer not contaminated
with platonic views from her schoolgirlhood—say, an alien?
There is no such behavior, unless one does not mean the prima
facie physical properties of the paper and the ink, which is
probably not what the phyisicalist is up to. Because it must be
seen, nothing mathematical follows from these physical proper-
ties.
For the strings of symbols to work as mathematics, the ink
appearances must be realized as tokens. Thus, they must be
realized by someone. But at this point we must postulate the
inter-subjective existence of symbol types. Without recognising
the types the given tokens belong to, one can identify exactly as
many “systems” on an ink-marked sheet of paper as in a hand-
ful of ashes poured on the table.
The First or the Last? (If at All…)
Taking a glance at the history of mathematics it can be
clearly seen, that philosophy and mathematics have always
been living close, if you like, in a kind of symbiotic relationship.
Also true, however, that the assessment of this relationship is
quite far from being unanimous. Some scientists assert the pri-
macy of philosophical considerations, claiming for clear phi-
losophical foundations before beginning with mathematics.
Errett Bishop, the father of constructive analysis was seemingly
unsatisfied with the practice of his contemporary mathematic-
cian fellows:
“There is a crisis in contemporary mathematics. And any-
body who has not noticed it is being wilfully blind. The crisis is
due to our neglect of philosophical issues.”20
But many maintain that philosophy has no legitimacy to lay
down different norms for mathematics. As David Lewis writes:
“How would you like to go and tell the mathematicians that
they must change their ways […]? Will you tell them, with a
straight face, to follow philosophical argument wherever it leads?
[...] [W]ill you boast of philosophy’s [...] great discoveries:
That motion is impossible [...], that it is unthinkable that any-
thing exists outside the mind, that time is unreal, that no theory
has ever been made at all probable by evidence [...]? Not me!”21
As often, the truth lies somewhere in the middle. Philosophy
should and would not in itself determine the right terms for
scientific practice and methodology. But it has the right to make
critical observations on the methodologies of sciences or even
normative recommendations. Let me give now an example out-
side of mathematics. Despite the fact that it is quite illegitimate to
assume causal connection between phenomena in statistical
correlation, many sociologists do assume causal connection in
those cases (Not to mention policy-makers). I think it is quite
right to tell them they are wrong. On the other hand, they have
the right not to change their well-tried ways with heavy traffic.
Similarly, it is equally right to draw the mathematician’s atten-
tion to the awkward consequences of the Axiom of Choice
every now and then…
The physicalist view, as Szabó (2010) makes it clear, is an
approach to mathematics following the philosophy-first princi-
ple (as opposed to the philosophy-last-if-at-all principle).22 As
such, similarly to the intuitionist program, it prescribes for
mathematics what it should be. Do not be misguided by the fact
that it does it in a disguise of description. A description so far
from the mathematics comprehended by common sense can be
suspected to be a normative manifestation.
But the position of physicalism is much worse than that of
intuitionism. The latter claims for serious methodological re-
strictions, and finds devoted followers among the practising
mathematicians—while, of course, the majority adheres to the
“Cantorian paradise”.23 At the same time the physicalist, in
aiming to show what mathematics really is, jettisons almost
everything—including the whole methodology. For, according
to this view, mathematical systems are nothing more than
physical systems; there is no a priory knowledge; deduction is a
special kind of induction; our only source of knowledge is em-
pirical observation. And, to be sure, physics has its own meth-
odology—why would another be needed?
Maybe the mathematician should be grateful to be so re-
lieved from the methodological burden, but I have doubts about
it. The situations is rather this: in trying to get rid of the all of
the “verbal decoration” surrounding mathematics, the physical-
ist, it seems to me, gets rid of mathematics itself. Following
him, the mathematician would be exiled not from the Cantorian
but from the mathematical paradise.
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Ayer, A. J. (1936). Language, truth, and log ic. London: Golla ncz.
Bishop, E. (1975). The crises in contemporary mathematics. Historia
Mathematica, 2, 505-517. doi:10.1016/0315-0860(75)90113-5
Gödel, K. (1944). Russell’s mathematical logic. In P. Benacerraf, & H.
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Szabó, L. E. (2003). Formal systems as physical objects: A physicalist
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Szabó, L. E. (2009). How can physics account for mathematical truth?
Preprint, URL (last checked 26 January 2012 ).
http://philsci-archive.pitt.edu/archive/00005338/
20Bishop (1975); recited in Shapiro (2000), p. 23.
21Lewis (1993); recited in Shapiro (2000), p. 30.
22A classifica t ion of Shapiro (2000).
23See: Hilbert (1926).
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