Some Remarks on the Physicalist Account of Mathematics

doi:10.4236/ojpp.2012.22025

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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

Some Remarks on the Physicalist Account of Mathematics

Ferenc Csatári

Department of Logic, Institute o f Philosophy, Eötvös Loránd University, Budapest, Hungary

Email: [email protected]

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

Introduction

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

experience.”2

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

here.

4There is a bit of muddle here, however. Following the physicalist, as we

will see, one would wonder what non-

hysicalist methods mathematicians

could have.

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.

F. CSATÁRI

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-

straints.

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)

physical facts.

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

Mathematics

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.

166

F. CSATÁRI

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

all.

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

Mathematics

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-

ever.

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

Euclid.

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…

F. CSATÁRI

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,

for instance.14

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-

calist.

Gödel-Theorems and in General

Meta-Mathematical Activities Do Not Belong to

Mathematics

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

physics”…

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).

15Wiles (1995).

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.

168

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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