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.

REFERENCES

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colorable. Illinois Journal of Mathematics, 21, 439-567.

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.

Putnam (Eds.), Philosophy of mathematics (2nd ed., pp. 447-469).

Cambridge: Cambridge Uni versity Press.

Hilbert, D. (1926). Über das unendliche. Mathematische Annalen, 95,

161-190. doi:10.1007/BF01206605

Leacock, S. (1911). Gertrude the governess or simple seventeen. In: J.

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Book.

Lewis, D. (1993). Mathematics is megethology. Philosophia Mathe-

matica, 3, 3-23. doi:10.1093/philmat/1.1.3

Shapiro, S. (2000). Philosophy of mathematics. Oxford: Oxford Uni-

versit Press. doi:10.1093/0195139305.001.0001

Russell, B. (1912). The problems of philosophy. London: Williams and

Norgate.

Szabó, L. E. (2003). Formal systems as physical objects: A physicalist

account of mathematical truth. International Studies in the Philoso-

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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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