Open Journal of Philosophy
2012. Vol.2, No.2, 165-170
Published Online May 2012 in SciRes (http://www.SciRP.org/journal/ojpp) http://dx.doi.org/10.4236/ojpp.2012.22025
Copyright © 2012 SciRes. 165
Some Remarks on the Physicalist Account of Mathematics
Department of Logic, Institute o f Philosophy, Eötvös Loránd University, Budapest, Hungary
Received February 28th, 2012; revi sed March 25th, 2012; accepted April 9th, 2012
The paper comments on a rather uncommon approach to mathematics called physicalist formalism. Ac-
cording to this view, the formal systems mathematicians concern with are nothing more and nothing less
than genuine physical systems. I give a brief review on the main theses, then I provide some arguments,
concerning mostly with the practice of mathematics and the uniqueness of formal systems, aiming to
show the implausibility of this radical view.
Keywords: Physicalism; Formalism; Philosophy of Mathematics
On the following pages I comment on a rather uncommon
philosophical approach to mathematics called physicalist formal-
ism.1 I give a brief review of the main theses, then I try to out-
line some of the main flaws of this approach. I am far from the
intent to be exhaustive in this respect, as I am far from a con-
sequent defence of an alternative approach. But I am convinced
that the few arguments given here, concerning mostly with the
practice of mathematics and the uniqueness of formal systems,
are enough to show the implaus ibility of this radica l view.
Mathematical Assertions Have No Meaning
The formalist-physicalist philosophy of mathematics lies on
the empiricist extreme of the “ideologies” concerning mathe-
matics. As a genuine empiricist view, it has its own questions
and problems to cope with by its own way. As Ayer put it in
this often recited paragraph:
“For whereas a scientific generalization is readily admitted to
be fallible, the truths of mathematics and logic appear to eve-
ryone to be necessary and certain. But if empiricism is correct
no proposition which has a factual content can be necessary or
certain. Accordingly the empiricist must deal with the truths of
logic and mathematics in one of the following ways: he must
say either that they are not necessary truths, in which case he
must account for the universal conviction that they are; or he
must say that they have no factual content, and then he must
explain how a proposition which is empty of all factual content
can be true and useful and surprising. [...] If neither of these
courses proves satisfactory, we shall be obliged to give way to
rationalism. We shall be obliged to admit that there are some
truths about the world which we can know independently of
Physicalism, as it is outlined in Szabó’s works, has to deal
with both kinds of challenge. On one hand, mathematics is said to
be an inductive science, so it cries for explanati on why its truths
(in a pre-theoretic sense) are usually thought to be necessary or
its “laws” to be more solid than that of physics.3 On the other
hand, physicalism asserts that mathematical statements do have
factual content, though far not in the usually preached sense. So
the urge for the explanation of useful and surprising results also
remains. Szabó is up to accomplishing both of these tasks, but,
I will argue, the results are unsatisfactory.
According to the physicalist-formalist a mathematical state-
ment (a theorem) is true if and only if it can be derived from the
given axioms. The truth-condition of a statement lies in the
existence if its derivability. It is meaningless to talk about
mathematical truth generally, only in accordance with a given
axiom system. True, there have not always been reliable axiom
systems, but that is only a contingent historical fact, and as such,
lies out of mathematics. Many proofs worked out by the
mathematicians of the past are of historical interest only, and
not regarded as strict, genuine proofs today.
Szabó (2003) distinguishes two different senses of truth. The
first kind is the truth of the theorems, i.e. the mathematical truth
in a given formal system. The second kind of truth is the em-
pirical truth, where a sentence refers to some empirical fact
through some semantics. Now the main formalist thesis of the
physicalist is this: mathematical objects and propositions have
no meaning, they are only meaningless marks on the paper
manipulated by some rules. Formulas do not carry Tarskian
truth, they can be true in the first sense of truth only. The ar-
gument says that if a mathematical proposition were a state-
ment of the factual world, the mathematician should carry out
experiments as the physicist does. That is clearly not the case
(especially if thought experiment is regarded as a special kind
of non-experiment).4 Moreover, even if one does associate
meaning to mathematical propositions, it is irrelevant to the
truth of the given statement. Even if one ascribes truth of the
second type to the axioms of some formal system, it is not
3Once I talked about Russell’s thought experiment (in Russell (1912)) to one
of my friend s , an i mmaculate in ph i l os op h y. “ Can you imagine that ther e ar e
immortal humans?”—I asked. “Yes, of course I can”—he said. “But can you
imagine that 2 × 2 is 5?” After a while he replied: “Yes, I can, but it makes
me rather angry”—indeed, in a more informal way cannot be literally cited
4There is a bit of muddle here, however. Following the physicalist, as we
will see, one would wonder what non-
hysicalist methods mathematicians
1See the works of Szabó L. E. (2003, 2009, 2010). Since the third one is a
lecture note , I will refer to the first two as long it is possible.
2Ayer (193 6), pp. 72-73.
granted that it will be come down to the theorems derived from
the axioms. In other words, whether a theorem derived from
semantically true axioms is semantically true or not is a con-
tingent fact of the world (would there be semantically true
axioms at all). Maybe there are mathematical intuitions of
which we have perceptions the same way we have perceptions
of the physical world,5 but the truth mathematical statements
concern with is independent of them. Note that this account is
not simply a rewording of the old thesis: all mathematical
statements are tautologies, i.e. analytical truths. As we will see
in a moment, we cannot have analytical knowledge—simply
because we cannot have a priori knowledge whatsoever.
The success of mathematics in sciences (and above all, in
physics) is often considered as a strong argument for that it
does embrace some crucial structures in the “real world”.
Moreover (as the so called Quine-Putnam-argument suggests),
mathematics is indispensable in physics, and we have to have
ontological commitments toward the indispensable. Szabó
claims that this argumentation can be easily undermined. First,
mathematics has much more than what is needed in physics.
Second, reliable predictions are not made by mathematics itself,
but by a complex physical theory with its semantics (of course,
mathematical statements do not carry Tarskian truth). Third,
there are more than one mathematical theories applicable in,
say, physical predictions, and, forth, the appropriate mathe-
matical theory is chosen by the physicists, not by external con-
Since mathematical theories (in themselves) are nothing
more than formal systems, it is meaningless to talk about in-
tended interpretations, standard models, intuitive arithmetic or
set theory and so on. True, however, says the physicalist, that
the axiom systems may not be entirely arbitrary: usefulness can
be a criterion for choosing them. Anyhow, systems taken as
“formal”6, and among them the ones with interest for mathema-
ticians are chosen along convention or usefulness—factors
lying outside of mathematics. I am not sure how conventional
and useful Peano-arithmetic is, but seemingly the physicalist
does not take great pains to explain why some formal systems
are interesting for mathematicians, why not others.
There Are No Structures to Be Represented
After the semantic apologetics for formalism, Szabó turns to
the main genuine physicalist theses as the results of some onto-
logical considerations. Though mathematical propositions have
no meaning, they express objective facts about the formal sys-
tems they stem from. To be more precise: they express facts
about the particular signs constituting the given system and the
rules by which they are combined. Rules are also laid down by
signs. The ontology of formal systems is thus so simple: there
are only physical objects (signs) as participants in (contingent)
The “physicality” of formal systems has two sides. First, the only
way for one to be informed of the truth of some mathematical
fact is through a physical process. Of course, physical processes
are required for one to be “informed” of anything—let them be
historical speculations, artistic impressions or astrological con-
stellations. Doing mathematics in the head or divine revelations
are not exceptions—in a physicalist account there is always an
underlying neurophysical process. Now, the knowledge of the
truth of some mathematical proposition is a truth-condition of
the given proposition. But how about other propositions? I
mean, according to the physicalist, is the knowledge of the truth
of the sentence: “It was a wild and stormy night on the West
Coast of Scotland”7 a truth-conditions for it? If not, what the
special status of mathematical statements lies in?
Second, mathematical propositions (the true ones, i.e. which
can be derived) express objective facts of the physical world of
the system. Thus, says Szabó with irony, we do have the onto-
logical commitment that the Quine-Putnam indispensability ar-
gument claims for: mathematical statements “talk” about formal
systems, i.e. strings of symbols, and we do believe in the exis-
tence of these entities.
It must be clear that the formal systems so understood are
particular, concrete systems. There are no abstract things, ab-
stract structures represented by them. It would be a categorical
mistake to assume various “isomorphisms”. Mathematical know-
ledge is not conventional (though choosing the particular topic
(i.e. system) is), not perfect, not a priori, not certain. Mathe-
matical truths reveal “contingent facts about a particular part of
the physical world”.8
Probably the most striking assertion of the physicalist is that
there is no certain knowledge gained through deduction, since
the latter is only a special case of induction—namely the one
dealing with the facts of formal-physical systems. This claim is
uncommon though, it follows from what is said so far. If the
truth-condition for a mathematical proposition is nothing more
than the knowledge of the given propositions, then there is no
difference between knowledge and truth in this case. Formal
systems are particular physical systems of which we can gain
information by observations—thus induction.
True, mathematical truths appear necessary and certain truths
for many. The reason is that formal systems usually have stable
behavior among the physical systems and do not require “ex-
ternal” observations. But since “observations” and “(thought)
experiments” provide the only way to get in possession of
mathematical truths the certainty so gained is exactly the cer-
tainty of physical truths. According to the physicalist’s conso-
lation, the “certainty available in inductive generalization is the
best of all certainties”.9
Scientific Practice Does Not Belong to
One of the main morals of the physicalist approach is, I think,
that mathematics is not what we usually mean when talking
about “mathematics”, be the concept however vague. Of course,
rational reconstructions often require the revision of commonly
used concepts for the sake of exactness. But I regard such a radi-
cal change in the concept of mathematics unjustified. Moreover,
5As Gödel puts it in his defence of platonism: “[T]here is nothing in the least
absurd in the existence of totalities containing members, which can be de-
scribed (i.e., uniquely characterized) only by reference to this totality. [...]
Classes and concepts may [...] be conceived as real objects [...] existing
independently of us and our definitions and constructions. It seems to me
that the assumption of such objects is quite as legitimate as the assumption
of physi cal bodies an d there is quite as much reason t o believe in their ex is-
tence.” Gödel (1944), p. 456.
6According to Szabó it is mere convention that the rules and “symbols” o
chess are not regarded as an axiomatic system.
7“[…] This, however, is immaterial to the present story, as the scene is not
laid in the West of Scotland. For the matter of that the weather was just as
bad on the Ea st coast of Irelan d.” Leacock (1911), p. 45.
8Szabó (20 09), p. 9.
9Szabó (20 03), p. 10.
Copyright © 2012 SciRes.
my taste draws me to approaches with more bias toward scientific
practice: as philosophical concerns can be of crucial importance
for the science, philosophy likewise should not be blind for the
actual scientific practice.
True, Szabó does mention the practice of mathematics:
“Of course, very much depends on how we understand the
practice of the mathematician. I think, that in the same way like
ordinary people who dream about movie stars but live with
their partner, mathematicians rave about various platonic ob-
jects, but if they are seriously asked what they are confident
about, they reduce their claims to mere if-thenisms. All the rest
is just “folklore”. And this holds not only for the more complex
branches of mathematics but also for arithmetic and set theory.
When the number theorist says “there are infinitely many prime
numbers”, or simply “7 is a prime number lager than 5” then
(s)he means that all these concepts as “larger” and “prime”, etc.,
are defined with formal rigor, and that the statement in question
is a theorem within the corresponding formal framework. This
is true even if (s)he proves it by simple calculations, since it
was previously proved that the employed algorithm is correct.
[...] Thus, concerning the rigorous, scientifically justified, non-
folkloristic part of the claims of mathematicians, it is far from
“unquestionable” that they are committed to something more
than if-thenism .”10
Maybe I understand scientific practice in a different way.
When a mathematician is working on a given problem always
uses a language somewhat similar to the ordinary one, and, of
course uses the tool of formal derivation in the given formal
system. But the latter alone, isolated from the sloppy words and
thoughts would hardly “work” as mathematics, we would have
only ink-marks on a piece of paper, exactly as the formalist
insists (I will address this point below). Maybe mathematicians
are “Sunday-formalists”,11 but they do their work on weekdays.
Undeniably, formal methods are of extreme importance for
them, but other things have role in their scientific practice as
well. Even when working with their pencil and paper, mathe-
matician quite rarely do something like deriving proposition
from a bulk of axioms.
I do not intend to inquire into what is in the mathematician’s
head. Assessing what she believes in, what she is committed to
is not an easy mission. A much more modest project is just to
regard what she does when doing mathematics. I am convinced
that a philosopher dealing with mathematics should always
keep one eye on this.
Historical, Psychological Curiosities Do Not
Belong to Mathematics
It is easy to agree with this statement. However, glancing at
mathematics from historical or psychological perspective could
have some morals for the philosopher.
It is obvious that strict, formal, axiomatic systems, the only
instances of mathematics according to the physicalist, have not
always existed. Indeed, they had only been brought about in the
wake of the foundation projects in the late nineteenth and early
twentieth century.12 Even not drawing far-reaching conclusions
from this fact, the physicalist must realize that the (only) objects
there are for mathematics (in his view) cannot be ind ep end en t o f
human activity. Of course, he can say, human activity was in-
dispensable for exhibiting some other systems in physics: (at
least) instruments were needed to detect them. Now it would be
of interest to show that the activity resulting in the exhibition of
formal systems is by no means different from those giving us
the possibility to talk about protons, neutrons and electrons, for
instance. Or if it is different, showing its specific nature may
have some morals.13 But the task does not seem to me trivial at
Moreover, historical consideration can shed some light on
how does mathematics (in the usual sense of the word) really
work. One can consult, for instance, the physicalist’s only ex-
planation for the existence of the actual mathematical objects:
the alleged utilitarian factors behind the fact that some privi-
leged systems are in the spotlight and not the infinitely many
possible others—those only useful in physical theories. In the
light of history, this assertion does not seem to be wholly con-
vincing. True, there are theories whose births were inevitably
urged by practical problems, e.g. analysis. But most of the
theories are rather like non-Euclidean geometries: there were no
practical constraints whatsoever as midwives at their births.
And there is a bulk of mathematical theories whose practical
applications are far from clear.
As for psychology, it exhi bits time a nd again vari ous patt erns
of pre-theoretic mathematical ability in adults, children or ani-
mals by strict, behavioristic means. Again, without going too
far with the consequences, this could at least suggest that the
actual systems dealt with in mathematical practice are not en-
tirely arbitr ary.
I will not go into details on these issues here. My intent was
only to indicate that the position of the physicalist can be
weakened by historical or psychological considerations (too).
True, one can disregard these perspectives and put normative
claims on mathematics. I will address this point soon. But let us
stay with mathematical practice for a little more while now.
Intuitions, Heuristic Do Not Belong to
It is a well-known fact that mathematical results reach far
beyond human intuitions. Some say, there was a point some-
where around the birth of non-Euclidean geometries, when
intuitions and modern mathematics divorced inevitably and for
good and all. The situation is far not so black and white, how-
First, intuitive insight and “intended interpretations” can be
blamed for those structures (axiomatic systems, if you like) that
we regard as mathematics today, even if they had grown over
human intuitions. It is not unthinkable that the physicalist
would be ready to admit this point, only he would add as usual:
this is a contingent “historical” fact.
Second, much depends on what we mean by intuition. We
can insist that it is something like an eternal, unchangeable,
mental vista which covers some areas of mathematics but not
others. But we can be more permissive. If we let the concept to
encompass any kind of, say, intuition-like heuristics, then we
10Szabó (2010), p. 21.
11A label coined by Reuben Hersh: (contemporary) mathematicians, when
asked of the nature of mathematics usually incline toward a formalist expla-
nation while in their practice they work with their objects as “real” objects.
12Of course, t h ei r o ri gins can be t r aced down in the work s of Leibniz or even
13Since the physicalist regards formal systems as flesh and blood object, it
seems to me he must accept that they are simply manmade—at least all o
the symbols used for mathematics are such. Hence, one could think, maybe
some “epistemology of artefacts” should be applied on them and methods
akin to car testin g or literary criticism…
Copyright © 2012 SciRes. 167
can confidently assert that intuition can be trained. In this case
practicing mathematicians have much more intuition concern-
ing complex structures than other earthborn beings.
Szabó (2010) declares that his physicalist account of mathe-
matics concerns strictly the context of verification, not heuris-
tics. In his opinion, creativity, intuition, and, what is more,
human thought as such plays only a marginal role in the verify-
cation of mathematical propositions, for, essentially, proofs are
nothing more than serials of formulas derived mechanically
from the axioms. And these derivations often run far beyond
the intuitive insight of a human. Nowadays, says Szabó, one
cannot claim a proof to be surveyable, as it is clearly seen in the
rather complex computer-based proof of the four-color theorem,
First, it is worth to note in connection with the four-colour
theorem, that the computer-based part of its proof is far not so
complex. The thing is, that there are simply too many cases of a
given (graph-theoretical) constellation-type for a human mathe-
matician to check even along a lifetime. Yes, this part of the
proof is mechanical, but not in the sense the physicalist would
like to see. The (computerized, mathematical) system does not
check what follows from the axioms of first-order logic along
with the axioms of set theory and the definitions of graph the-
ory, but it simply executes the algorithmic rules specified by
the programmer (mathematician). The “rest” of the proof does
require creativity and human thinking (including programming).
And not necessarily in a weak sense: without them there would
be no theorem whatsoever, (so there would be nothing to de-
clare as a mere physical fact, as a mere consequence of some
physical syste m).
On the other hand, some contemporary proofs like Wiles’ on
Fermat-conjecture15 are surveyable. True, maybe it is not sur-
veyable for me, but it is for the professionals of the topic—or
topics: the proof exploits numerous highly esoteric mathematic-
cal results. Anyway, I think that surveyability is a rather vague
and, in addition, unimportant concept. The crucial thing lies
elsewhere. It is hard to defend the claim, that a proof like the
one by Wiles is nothing more than a mere mechanical deriva-
tion from the axioms. Indeed, it would be hard to model such a
proof as a mechanical process. For one to build a computer
program deriving the proof of Fermat-conjecture (say, in the
style of Wiles), it seems to me, he must already know and un-
derstand the ideas applied in the proof given by Wiles!
Note that according to Szabó (2009) the “mechanism” of a
formal system can be observed as a neurophysical process of
the brain (Well, in a more correct wording: a serial of neuro-
physical processes in a brain accomplishing derivations is to be
regarded as a genuine formal system). Note further that we
cannot, certainly, talk about “representation” of a system, of a
proof or something like that, because there is nothing to rep-
resent. There are only particular physical objects (and “physic-
cal” rules) remained for us as mathematics, ink-colored paper
pulp, conducting wires, firing neurons.16 But as far as I know
we are quite far from the understanding of brain processes,
quite far from more or less justly regarding human brain as a
Turing-machine (even though it is often regarded so despite of this
fact), and very far from identifying the neurophysical process of
adding five to seven. Though I understand that the formal sys-
tems thus revealed are of crucial importance for the physicalist,
this area is so slippery that I think it is better to keep or atten-
tion on ink, paper and machines.
As far as I can see, the only plausible understanding of for-
mal systems as physical phenomena is the following. Assume a
Turing-machine with the axioms of logic, arithmetic, deriva-
tions rules etc. programmed in it.17 We can venture, that the
machine will derive the proof of the Fermat-conjecture at some
point we only have to wait long enough, since it will derive all
of the theorems of the system if we wait infinitely long. Give a
monkey a typewriter and infinite time, it will sooner or later put
down the complete works of Shakespeare.18 So far so good, but
what can we do with that fact? Can mathematics build on this
result, and the infinite many other results so derived? Can they
be grabbed, picked up or isolated?19 (And by whom?).
Sure, one can construct an arbitrary axiom system, since the
adherence of mathematicians to the ones in use is another “con-
tingent and unimportant” fact. Sure, she can translate it into
algorithms to check the mechanical consequences by a Tur-
ing-machine. Sure, she will get arbitrarily long proofs/theorems
(during an arbitrarily long time), which happen to be quite un-
surveyable in every respect. But it seems also sure for me:
maybe they can be seen as a genuine part of mathematics from
some philosophical reconstruction, but they will be completely
unimportant for and will have nothing to do with mathematics
as a science. Mathematics as practice has also nothing to do
with the bigger part of mathematics as so-labeled by the physic-
Gödel-Theorems and in General
Meta-Mathematical Activities Do Not Belong to
Following Hilbert, the physicalist draws clear distinction
between mathematics consisting of systems (somehow) built up
by meaningless signs and activities dealing with these systems,
namely me ta-mathemat ics. The latte r can make real me aningful
statements about the formal systems (i.e. mathematics), which
can be true (or false) in the semantic sense. In fac t, meta-mathe-
matical results are physical theories on physical formal systems
expressed in some genuine language—even though this lan-
guage (or better: a part of it) highly resembles to the (meaning-
less) sign serials of the formal systems. It follows that the theo-
rems of mathematical logic are to be confirmed by empirical
means: by observations.
But what is the empirical content of Gödel-theorems, Löwen-
heim-Skolem-theorems, the theorem on the independence of
continuum hypothesis etc.? Do they really assert a feature of a
particular physical system? (Which one?) Maybe even the
physicalist does not intend to claim that there is only one par-
ticular system the above theorems go for. On the other hand, it
is quite implausible to say, that the confirmation goes like this:
we pull out similar systems from a hat, one after other and
check whether the given theorem holds in it or not. It is far
from clear what feature of this meaningless serials of signs is
17But notice that going only that far is not at all unproblematic: at this point
we have already much more than the mere axioms at play.
18Will it really, is another question. Could every finite sequence of formulas
e enumerated by this “method”, the monkey will die, the typewriter will go
wrong in a (rather short) finite time. True, these are only “contingent facts o
19Of course, it is a contingent social fact for the physicalist whether a given
theorem is of importance or not.
14See Appel, Haken, Koch (1977) and Tym oczko (1979).
16Since these are the only things out there, they cannot be regarded as tokens
not having type s. I w ill address this rather problematical point in a moment.
Copyright © 2012 SciRes.
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-
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-
“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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