1
Reconside ing Mach in he ligh o he In e play o P ac ices
Ma ía de Paz ([email p o ec ed])
Depa men o Philosophy, Logic and Philosophy o Science
Uni e sidad de Se illa
Abs ac : The aim o his pape is o econside Mach’s posi ion in he ligh o he ecen
app oach on he in e play o ma hema ical p ac ices p o ided by José Fe ei ós’ book
Ma hema ical Knowledge and he In e play o P ac ices. In he i s sec ion he main
app oach o he book is p esen ed, wi h a cha ac e iza ion o he no ion o ‘scien i ic
p ac ice’ and he iple app oach used o unde s and ma hema ical knowledge:
cogni i e, p agma is and his o ical. In he sec ions 2, 3 and 4 Mach’s philosophical
iews a e p esen ed acco ding o his epis emological app oach. The i h sec ion is he
conclusion.
Keywo ds: Cogni i e App oach, His o y, Philosophy o ma hema ics, Philosophy o
science, P agma ism
In he pas ew yea s he e has been an in e es on highligh ing a ecen ‘ u n’ in
epis emology cha ac e ized as ‘ he p ac ice u n’. This shi conside s he p e ious one
ca ied ou by science s udies in he 1970s as a eac ion o adi ional philosophy o
science. This was ega ded as he a i ude acco ding o which “[ ]esea che s in science
s udies [...] we e p og essi ely mo e encou aged o pay a en ion o scien i ic p ac ices
in me iculous de ail and along mul iple dimensions, including he ma e ial, aci , and
psycho-social ones”
1
. Tha u n was pa ly a eac ion o a philosophy o science ha was
e y ocused on he analysis o scien i ic heo ies as comple ed p ocesses, which
dis ega ded he ac ual p ac ice o science, he de elopmen o expe imen al p ac ice, he
ac i i ies ha p omp ed he scien is o he o mula ion o new explana o y hypo heses,
o he p ac ical pa hs ha lead scien is s o he de elopmen o hose heo ies which
we e he explici objec o analysis.
1
Léna Sole e al., Science A e he P ac ice Tu n in Philosophy, His o y, and he Social S udies o
Science (New Yo k and London: Rou ledge, 2014), 1.
2
The p ac ice u n in epis emology ies o o e come some p oblems posed by he
science s udies u n, such as he excessi e ocus on expe imen which con ibu ed o
ein o cing he classical opposi ion be ween heo y and p ac ice. This was he case, o
some ex en , o he end known as ‘New Expe imen alism’
2
, bu especially o he
Empi ical P og am o Rela i ism (EPOR) in hei sociological s udies o expe imen al
wo k
3
. One possible way o o e coming his classical opposi ion is o pay a en ion o
he p ac ice o ma hema ics, since his is a discipline adi ionally linked o heo e ical
ac i i y.
The philosophy o ma hema ical p ac ice cons i u es he ocus o a en ion o
much ecen esea ch, pa icula ly since he c ea ion o he APMP (Associa ion o he
Philosophy o Ma hema ical P ac ice) in 2009. In his pape I would like o b ing he
a en ion o a speci ic con ibu ion o his ield in o de o unde s and, upda e and
econside Mach’s philosophical posi ion. I am alking he e abou he ecen book
published by José Fe ei ós Ma hema ical Knowledge and he In e play o P ac ices.
Al hough he co e o Mach’s philosophical posi ion is no conce ned wi h he
his o y and philosophy o ma hema ics, I will show ha Fe ei ós’ app oach can
enligh en se e al aspec s o Mach’s iews. In o de o a ain his goal, he i s sec ion
o he pape is de o ed o p esen and summa ize he gene al lines o Fe ei ós’ wo k. In
he second and in he hi d, a connec ion wi h Mach’s philosophical pe spec i e will be
p o ided and he o h is in ended o o e a conclusion based on he main lines o he
pape .
1. A iple app oach o he in e play o p ac ices.
Despi e being specialized on he his o y and philosophy o ma hema ics, mo e
speci ically on ma hema ical p ac ice, Ma hema ical Knowledge and he In e play o
P ac ices discusses se e al epis emological and philosophical issues ha a e impo an
and in e es ing o philosophe s wo king on how we acqui e knowledge, how i is
p oduced, how i becomes a cohe en body o sys ema ized and cons i u ed opics and
2
See Ian Hacking, Rep esen ing and In e ening, In oduc o y Topics in he Philosophy o Na u al
Science (Camb idge: Camb idge Uni e si y P ess, 1983) and Jed Z. Buchwald (ed.) Scien i ic P ac ice:
Theo ies and S o ies o Physics (Chicago: The Uni e si y o Chicago P ess, 1995).
3
See in pa icula Ha y M. Collins, “‘Son o se en sexes’, The Social Des uc ion o a Physical
Phenomenon”, Social S udies o Science, 11, no. 1 (1981): 33–62, doi: 10.1177/030631278101100103. A
good e e ence is also And ew Picke ing, “Agains Co espondence: A Cons uc i is View o
Expe imen and he Real”, in P oceedings o PSA 1986, eds. A. Fine and P. Machame (Pi sbu gh:
Philosophy o Science Associa ion), ol. 2 (1987): 196–206.
3
how a speci ic pa o knowledge is ela ed o o he human ac i i ies
4
. In his sense, i
could be asse ed ha he book does no conce n only ou way o knowing and
unde s anding ma hema ics, bu ou knowledge in gene al, and migh be ele an o he
knowledge o any o he scien i ic discipline. By asse ing his I would no like o deny
he speci ici y o ma hema ical knowledge –which Fe ei ós claims and shows in he
book: “ma hema ics is an essen ial pa and pa cel o he heo e ical wo kings o
science […], bu also dis inc i e and peculia , because he e he heo e ical wo k
becomes au onomous.”
5
I would only like o s ess ha he book de elops a heo y o
knowledge o a speci ic ield ha could be ex ended o o he domains o knowledge o ,
a leas , whose ex ension should be conside ed. In ac , his is pa ly he pu pose o his
pape , in which I ex end Fe ei ós’ iews in o de o unde s and hose o Mach, which
we e mainly ocused on he his o y and philosophy o physics, a he han on
ma hema ics. The heo y o knowledge p esen ed by Fe ei ós is made explici h ough
a iple app oach s a ed a he e y beginning o he book:
6
cogni i e, p agma is and
his o ical.
Be o e desc ibing his iple app oach and i s implica ions, I would like o ocus,
i s , on an analysis o he no ion o p ac ice p oposed by he au ho . Gi en ha my
cha ac e iza ion o Mach’s philosophy would be p ecisely in e ms o a p ac ical
app oach, i would be use ul o unde s and he no ion placed a he basis o his
concep ion.
Rega ding he no ion o ‘ma hema ical p ac ice’ p oposed by Fe ei ós, i s o
all I mus say ha he does no wan o in oduce a igid de ini ion,
7
bu one ha is
gene al enough o encompass he plu ali y o p ac ices ha he wan s o deal wi h. Tha
said le ’s see wha he au ho says: “By a ‘p ac ice’ in gene al I mean a ecognizable
ype o ac i i y ha is done –and can be augh and lea ned– by human agen s.”
8
This
appea s o be a e y gene al no ion, bu i shows ce ain cons ain s gi en he speci ici y
o Fe ei ós’ ocus on ma hema ics.
4
In his pa o he pape I use my p e ious book e iew o Ma hema ical Knowledge and he In e play o
P ac ices published on The Philosophical Qua e ly. See Ma ía de Paz, “Book e iew o Ma hema ical
Knowledge and he In e play o P ac ices, by José Fe ei ós.” The Philosophical Qua e ly, 68, no. 271
(2018): 406-408 h ps://doi.o g/10.1093/pq/pqx036 (online i s , 2017).
5
José Fe ei ós, Ma hema ical Knowledge and he In e play o P ac ices (P ince on: P ince on Uni e si y
P ess, 2016), 14.
6
Ibid., 3
7
Ibid., 17.
8
Ibid., 28
4
So, he says: “ma hema ical p ac ice is wha he communi y o ma hema icians
do when hey employ esou ces such as amewo ks (and o he ins umen s) on he
basis o hei cogni i e abili ies o sol e p oblems, p o e heo ems, shape heo ies and
(some imes) o elabo a e new amewo ks.”
9
When we apply he b oade no ion o
p ac ice o ma hema ical p ac ice, we can see ha i is necessa ily an ac i i y ca ied ou
by human beings –no o he communi y o beings p ac ices ma hema ics in a
ecognizable way–; his implies he analysis o a conc e e ype o knowledge, no an
abs ac ion in which he ea u es o he p ac i ione could be abs ac ed. Besides,
ma hema ical p ac ice is some hing done wi hin a communi y, so he app oach has o
conside he wo k o he single ma hema ician and he ela ion o o he ma hema icians.
I is also es ic ed conce ning he objec o applica ion. Al hough he o e s a e y
gene al no ion o p ac ice, when coming o ma hema ical p ac ice, his is de ined o
a ain conc e e goals (sol e p oblems, p o e heo ems, shape heo ies), bu he adds: “i
will always be possible o link such ma hema ical p ac ices wi h pa icula ma hema ical
heo ies.”
10
This goal es ic ion a ec s he app oach i sel , since he p o ides a no ion o
p ac ice use ul o de elop an epis emology o ma hema ics,
11
hus, able o say
some hing abou he kind o knowledge ma hema ics p o ides. The e o e, o de ine a
p ac ice he p esence o he p ac i ione , he communi y and he goal is clea ly needed.
In o de o ully unde s and he no ion o ma hema ical p ac ice we ha e o cla i y he
no ion o amewo k.
The e a e wo p ecise meanings o his no ion in he app oach we a e ying o
cha ac e ize. The i s is he symbolic amewo k which e e s o he pa icula kind o
cha ac e s, i.e. symbols used by ma hema icians in o de o ep esen p oblems and o
es ablish solu ions. The second is he heo e ical amewo k iden i ied by p oposi ions,
p oo s, p inciples o laws sha ed by a communi y, me hods o open ques ions, such as
conjec u es. As a esul , he amewo ks a e he ins umen s employed by a
ma hema ician and he communi y o a ain ce ain esul s when hey a e p ac ising
ma hema ics. Gi en his de ini ion, we ha e o ake in o accoun he link be ween he
amewo k and he ma hema ician, ha is, wha he au ho calls he ‘agen ’, and his pai
‘agen - amewo k’ is placed a he cen e o he p ac ice.
12
This is because he p ac ice
is de eloped wi hin a amewo k and hanks o a amewo k, since i cons i u es he
9
Ibid., 33
10
Ibid., 33
11
Ibid., 30
12
Ibid., 44
5
esou ces ha he ma hema ician has o sol e ce ain conc e e p oblems. And also
because he agen – he ma hema ician– is he one who p ac ices, he one who does
ma hema ics. F om my poin o iew, all hese ideas become clea e when we ake as a
whole he app oach ha Fe ei ós p o ides in he book, ha is, he iple pe spec i e
ha we e e ed o abo e: cogni i e, p agma is and his o ical.
Fi s , le us conside he cogni i e app oach. This e e s, as s a ed by Fe ei ós,
o an epis emological concep ion in which he subjec is se iously concei ed (‘no
epis emology wi hou a subjec ’).
13
The human subjec has de ini e cogni i e skills
biologically de e mined, bu his biological de e mina ion depends no only on he
b ain’s de elopmen bu also on he en i onmen , bo h na u al and cul u al, in which he
subjec g ows and li es. This en i onmen de e mines ce ain ea u es ha he agen has
and i is because o hese pa icula i ies ha Fe ei ós decides o call his subjec an
‘agen ’. He wan s o make explici he conc e e cha ac e ha he no ion o agen has,
opposed o he abs ac ness ha adi ional philosophical concep ions ha e a ibu ed o
he no ion o ‘subjec ’. The agen is endowed wi h a body and a social li e and can ha e
di e en es ic ions depending on he cul u al milieu she inhabi s.
14
I is no he same o
be a nine een h cen u y Ge man ma hema ician han a Babylonian magician ying o
es ablish as onomical measu emen s and hey do no sha e he same amewo ks.
Also, he emphasis on he agen ma ks ano he impo an ea u e: he ac ha
he subjec ac s and in e ac s, his is why Fe ei ós cha ac e izes his cogni i e app oach
as agen -based. He wan s o show ha ma hema ics is a human p oduc , and hus
p omo e an epis emology agains he obli ion o ac ion.
15
So, knowledge, speci ically
ma hema ical knowledge, is no some hing a ained by con empla ion, bu by
de eloping and pe o ming ce ain ac ions. Ano he ea u e o his cogni i e and agen -
based app oach is he ole ha he communi y plays in he de elopmen o
(ma hema ical) knowledge. The ma hema ician is no an isola ed indi idual; she migh
c ea e p oo s o heo ems, bu i is wi hin a communi y ha ma hema ics is ansmi ed,
and his is whe e new de elopmen s a e accep ed, modi ied, and ecei e he o m in
which hey will be augh . So, agen s by ac ing and in e ac ing p oduce and de elop
knowledge, by pe o ming ce ain p ac ices, which connec s wi h he second pa o he
app oach men ioned.
13
Ibid., 10.
14
Ibid., 60-61.
15
Ibid., 11.

6
This p agma is pe spec i e is indeed be e cha ac e ized, as he au ho does, as
p ac ice-o ien ed, meaning ha he oo s o knowledge should be ound in p ac ical
ac ions o human beings, hence he impo an no ion o p ac ice de ined and discussed
abo e, and also he emphasis on he subjec pe o ming his p ac ice who is called ‘ he
agen ’. In his sense, he e a e p o o-ma hema ical p ac ices which can be conside ed as
echnical ac i i ies in he physical wo ld such as coun ing and measu ing. These
echnical ac i i ies o p ac ices –which imply he manipula ion o conc e e objec s o
human-made ins umen s– could be si ua ed a he basis o mo e abs ac ma hema ical
p ac ices ha a e eached in u he s ages o knowledge. I is he cha ac e iza ion in o
di e en knowledge s a a ( om mo e echnical and basic o mo e abs ac and
complex) and he way he di e en s a a ela e o each o he , which esponds o he
in e play o p ac ices ha names he book. This ecou se o echnical p ac ices is
impo an o he heo y o knowledge p oposed, since h ough a de ailed s udy o some
his o ical cases, he au ho aims a suppo ing his philosophical s a emen ha he “ oo s
o knowledge and hough lie deepe han language”,
16
ha is, hese oo s may well lie
on he use o ins umen s and on he de elopmen o echnical p ac ices (coun ing and
measu ing being he bes examples o he case o ma hema ical p ac ice). This pe mi s
a link wi h he hi d ea u e, gi en ha he e is a clea ecou se o a his o ical su ey.
The his o ical app oach s esses ha in o de o unde s and ma hema ical
knowledge we ha e o s udy i s his o y. This implies a ision o ma hema ics as a
his o ical science, i.e., a science ha has a his o ical de elopmen , a diach onic
p oduc ion o concep s ha can be explained in a ime line; bu mo e impo an ly,
ma hema ics is a his o ically de eloped science in he sense o his o ically laden. This
means accep ing he po en ially con ingen aspec s o a science ha has oo equen ly
been unde s ood as a p io i and necessa y. Tha is why Fe ei ós cha ac e izes his
app oach also as hypo he ical. A his poin , he aces an impo an on ie di iding
basic om ad anced ma hema ics, as i is only in he la e whe e he hypo he ical
elemen s can be ound. The con ingency and hypo he icali y o ma hema ics do no
imply a po en ially ela i is ic cha ac e . Ad anced ma hema ics is dependen on ce ain
choices, bu hese choices a e no a bi a y since hey a e cons ained by he oo ing o
mo e abs ac p ac ices in mo e elemen a y ones – and hese las ones in echnical
p ac ices. This concep ion jus i ies he ele ance o s udying echnico-p ac ical
16
Ibid., 13.
7
de elopmen s in wha we ha e called abo e p o o-ma hema ics o he unde s anding o
abs ac ma hema ical concep s – jus like in Mach’s his o ical-c i ical me hod, as we
will y o show la e .
To sum up, he in e connec ion o his iple epis emological app oach migh be
unde s ood when hinking abou he meaning o e ms: he meaning is de e mined and
acqui ed by agen s h ough he p ac ical use o concep s and ools in he cou se o he
his o ical de elopmen o science (ma h in pa icula ). I is now ime o see his
app oach om a Machian pe spec i e.
2. The no ion o p ac ice in Mach.
The pu pose o his and he nex wo sec ions is o ame Mach’s philosophical
concep ion wi hin he iple app oach p esen ed by Fe ei ós. In his sense, pa o he
idea is o b ing Mach’s epis emology up o da e, o which in some espec s i will
imply o in oduce o eign elemen s o ein e p e his claims and no o ake s ic ly
concep s and elemen s p o ided by Mach himsel o e en o conside Mach’s
con ibu ions wi h ce ain lexibili y. We would s a by e hinking Fe ei ós’ no ion o
p ac ice in a Machian ligh and hen he iple app oach will ollow.
Fi s o all, in o de o jus i y ou applica ion o a p ac ical app oach o Mach, i
is impo an o conside ha , acco ding o Fe ei ós “a en ion o p ocesses, and no
me ely o mo e o less pe ec esul s, is cha ac e is ic o s udies o ma hema ical
p ac ice.”
17
I is p ecisely his app oach o science as a p ocess ha can be cha ac e ized
as p ope o Mach’s posi ion:
O dina y hough does no se e pu e knowledge, and he e o e su e s om a ious
de ec s ha a i s su i e in scien i ic hough , which is de i ed om i . Science only
e y g adually shakes i sel ee om hese laws. Any glance a he pas will show ha
p og ess in scien i ic hough consis s in cons an co ec ion o o dina y hough .
18
Indeed, om his quo a ion we can see ha science is a g adual p ocess and no a
comple e endea ou , i is seen as a p og ession in which e o s a e depu a ed, and i has
i s o igin in he imp o emen o common ideas (o dina y hough s). Mach concei es
17
Ibid., 29.
18
E ns Mach, Knowledge and E o . T ans. T. J. McCo mack and P. Foulkes (Do d ec h/Bos on:
Do d ech -Reidel publishing company, 1905 [1976]), 2.
8
science as an un inished ask, because each pa icula science has o ake in o accoun
he p og esses made in o he sciences “ o he sake o o ien a ion in i s own ield.”
19
Taking ha in o accoun , i is clea ha , o Mach, science as a whole is pa o a
p ocess and hus, he esul s o science could ne e be conside ed as de ini i e o
pe ec , bu simply as pa o he no mal de elopmen . I is on his basis ha we can
s a in e p e ing Mach’s posi ion as a p ac ical app oach. Bu in o de o b ing i close
o he concep ion ha we a e dealing wi h he e, we ha e o be mo e speci ic and y o
see i i is possible, in some way, o concei e Mach’s no ion o p ac ice “as a
ecognizable ype o ac i i y ha is done –and can be augh and lea ned– by human
agen s.”
20
To a ain his goal, we should conside Mach’s s udies on mechanics in his
amous wo k Die Mechanik in ih e En wickelung his o isch-k i isch Da ges ell . As is
well known, his book s a s wi h he s udy o some machines, he le e and he pulley
in pa icula , in o de o explain he i s p o o-mechanical de elopmen s. Wha we wan
o emphasize he e is how he explains he de elopmen o he i s mechanical machines
when e e ing he wo k o A chy as o Ta en o: “he buil some mechanical de ices o
d aw ce ain cu es.”
21
And a ew lines la e he says: “I is p ecisely a dis inc i e ea u e
o he p ehis o y o mechanics ha , i has di ec ed i s a en ion owa ds he p ac ical
uses [...].”
22
These passages ein o ce he p ac ical pe spec i e we wan o emphasize.
Bu hey also poin in he di ec ion o unde s anding Mach’s no ion o p ac ice, which
has no been philosophically elabo a ed as Fe ei ós’. The cons uc ion o he de ice o
d aw cu es implies a ecognizable ype o ac i i y done by human agen s, since i is
peculia o human beings o cons uc ins umen s o de elop geome y. Bu his ac i i y
can also be augh and lea ned, since he e mus be a echnique behind he cons uc ion
o he ins umen ha can be ansmi ed o ep oduce he cons uc ion o simila
ins umen s.
As a esul , since he e y beginning in he science o mechanics we can ind he
p oduc ion and ep oduc ion o an ac i i y as he ounda ion o mechanical knowledge.
And, o cou se, he e is no possibili y o iden i ying an ac i i y, i.e. a p ac ice, wi hou
he p ac i ione –wha Fe ei ós calls ‘ he agen ’– A chy as in he case we analyze he e.
19
Ibid., 3.
20
Fe ei ós, Ma hema ical Knowledge and he In e play o P ac ices, 28.
21
E ns Mach, Die Mechanik in ih e En wickelung his o isch-k i isch Da ges ell (Leipzig: B ockhaus,
1883 [1901]), 9
22
Ibid., 9.
9
In he p ac ice e e ed, we can also see ha i aims a a conc e e goal – o d aw speci ic
ypes o cu es– which equi es he exis ence o an unambiguous objec o applica ion
( o sol e de e mined p oblems). The s udy o his p ac ice is also use ul o de elop an
epis emology o mechanics in he same sense as Fe ei ós de elops an epis emology o
ma hema ics. This is because Mach’s s udy is di ec ed o unde s anding he p ac ices a
he basis o he science o s a ics which is conside ed as he mos ancien o he
mechanical esea ches, whose o igin is p ecisely he de elopmen o hese machines –
he pulley, he le e – and also o p imi i e au oma ons. As we see, jus like in Fe ei ós’
epis emology, ce ain p ac ices a e able o be linked wi h ce ain heo ies.
The e a e s ill wo elemen s ha we need o discuss o b ing Mach’s iews unde
Fe ei ós’ pe spec i e. The i s one is he communi y and he second he amewo k.
Rega ding he communi y, i is p esen all o e Mach’s wo k. In his Mechanics, om
he e y beginning he conside s he exis ence o mo e han one man de eloping
p ac ices. Thus, he speaks abou “A chimedes and his successo s.”
23
Bu i becomes
e en clea e when he e e s o he p ocess o knowledge acquisi ion cha ac e ized in
Da winian e ms: “The acquisi ion o he mos elemen a y knowledge does no de ol e
upon he indi idual alone, i is p e-e ec ed in he de elopmen o he species.”
24
This
s a emen poin s a a global communi y o he de elopmen o he science: he
communi y o human beings. And he e i can be objec ed ha Fe ei os’ no ion o
communi y is much mo e es ic ed. And in ac i is, bu he es ic ion comes wi h he
no ion o amewo k. The gene al ea u e –and he mos ele an – o he no ion o
communi y is he ac ha science is no de eloped in isola ion and, gi en he e e ences
ha Mach made o he human species and se e al peoples ha ing mechanical p ac ices
–Egyp ians, Assy ians, e c.– he necessi y o he communi y o de elop knowledge and
p ac ices is beyond any doub .
We ha e asse ed abo e ha we equi ed a second elemen o unde s and Mach’s
pe spec i e om Fe ei ós’ and his was p ecisely he amewo k. The in oduc ion o
his no ion will allow us o ully explain Mach’s no ion o p ac ice acco ding o
Fe ei ós’ es ic ions and o es ic also he no ion o communi y ha we ha e de ined
in he p e ious pa ag aph. Fe ei ós’ conside s wo speci ic kinds o amewo ks: he
symbolic and he heo e ical. Rega ding his, Mach conside s he unde s anding o
phenomena as linked o he cons i u ion o specialized jobs and wo king associa ions.
23
Ibid., 16 and .
24
Ibid., 1.
16
pe cei e; an idealiza ion o he mo e ele an elemen s o sensa ions selec ed o sa is y
ou p ac ical ends o o o e come ce ain in ellec ual di icul ies. The p ocesses o
g ouping and es ablishing connec ions among elemen s a e a ied, bu hey do no
cons i u e a ‘ ep esen a ion’ o ou pe cep ion, as we said abo e. This is because
elemen s a e pa icula , ha is, un epea able and any idealiza ion o abs ac ion made by
us wi h he pu pose o o ganizing hem, o exp essing egula i ies, o , o cons i u ing
laws, is a dis ancing om o an aliena ion o pe cep ion.
49
As a esul , laws, heo ies and
all physical science a e no he esul o sensa ions hemsel es, bu o ou way o
g ouping hem ega ding ou cogni i e s uc u e which is subo dina ed o he expe ience
o he species. This cons i u es some kind o biological a p io i, ha we ha e al eady
men ioned, in he sense ha wha is a p io i o an indi idual is a pos e io i o he
ances o s, because hey ha e acqui ed i du ing he cou se o hei con ac wi h
phenomena.
50
In his way, he economy o hough is ela ed o Machian Da winism in wo
senses. Fi s , because i helps science o ge he adap a ion o hough o ac s and o
hough s be ween each o he wi h he leas possible numbe o symbols.
51
Second,
because economic media o adap a ion ha e been acqui ed by us no as isola ed
indi iduals bu as membe s o he species. The concep o economy is so c ucial
because i pe mi s he connec ion be ween he p agma is and he cogni i e app oaches.
Bu he a angemen o elemen s depends also on he in e es o he esea che ( he
scien is o he agen ), since depending on how sensa ions a e associa ed among each
o he , he esea ch belongs o one science o o ano he ( o physics, o ma hema ics, o
psychology, e c.). This is why Mach de ended ha physics is a science c ea ed wi h
limi ed and speci ic pu poses –in he same sense as ma h has conc e e aims o Fe ei ós.
In physics, he scien is has he powe o de e mining he own poin o iew in o de o
emphasize he mo e in e es ing ques ions o he . As we e e ed abo e, e e y science,
and physics in pa icula , is pa o a la ge body o knowledge. The e o e Mach
ecognizes he plu ali y and speci ici y o he sciences, being hus impossible o
49
C . Ma ía de Paz, Hen i Poinca é: Del Con encionalismo a la G a i ación (London: College
Publica ions, 2016), 151.
50
C . Rudolph Halle , “Con en ionalism and i s Impac on Logical Empi icism.” Philosophia Scien iae 3,
no. 2 (1998-1999): 100.
51
Gi en his es ic ion o media (symbols, p oposi ions, e c.), Mach’s economy has also been in e p e ed
as a a ian o Ockham’s azo (c . E win N. Hiebe , “Mach’s Philosophical Use o His o y.” In
His o ical & Philosophical Pe spec i es on Science, ed. Roge S uewe (Minneapolis: Uni e si y o
Minneso a P ess, 1970), 189 and Mu a Baç “S uc u e e sus p ocess: Mach, He z, and he no ma i e
aspec o science.” Jou nal o Gene al Philosophy o Science 31, no. 1 (2000): 40).

17
a ibu e o him any educ ionis in e p e a ion. This complemen a i y among he
sciences ecalls Fe ei ós wo ds, when he asse s ha ma hema ical p ac ices “do no
li e in a acuum o pu e hough ” and he goes on:
Ma hema ical ideas always depend on a echnical and p ac ical backg ound; hey a e
cons an ly linked (cogni i ely) wi h non-ma hema ical p ac ices. Among hese I coun no
only echnical p ac ices such as coun ing and measu ing, bu , especially, scien i ic
p ac ices o modelling ha a e a he co e o he sciences, in which ma hema ics plays a
cen al cons i u i e ole. In a nu shell, a ull explana ion o ma hema ical knowledge and
he web o ma hema ical p ac ices makes i necessa y o embed his ne wo k wi hin he
la ge web o scien i ic p ac ices.
52
In ac , he his o ical s udy de eloped on coun ing p ac ices such as he
Babilonian o he ones de eloped by ibal socie ies, whe he i is o making business
o dis ibu ing ood,
53
is eminiscen o Mach’s Mechanics and his s udy o mode n
abs ac mechanical concep s by oo ing hem in he s udy o machines cons uc ed o
sa is y echnical needs ( ecall ha Mach’s book s a s wi h he s udy o he le e ). In
bo h s udies we can see he link be ween speci ic scien i ic heo ies and echnical
p ac ices –basic a i hme ic and p imi i e s a ics, espec i ely. This in e ac ion –o
in e play be ween p ac ices is explained by Fe ei ós in e ms o hema izing,
54
meaning
ha he p og ess in science –in ma hema ics, conc e ely– is done by making explici by
means o heo y some aspec s ha ha e p e iously been implici in p ac ice. This
pe spec i e is as onishingly simila o Mach’s emphasis on ins inc i e knowledge,
55
which he also calls ‘in olun a y’ and which is p e ious o conscious –i.e. explici –
scien i ic knowledge. In Fe ei ós book, his is also p esen when asse ing he need o
pay a en ion o non-linguis ic o ms o knowledge, including aci o ms,
56
as is he
case o he use o diag ams in geome y –o diag amma ic hinking. In sho : o bo h
au ho s, he oo s o knowledge lie deepe han hough and language.
The ole played by his ins inc i e o aci knowledge implies he in e connec ion
be ween di e en le els o p ac ice. To Fe ei ós
57
we ha e ope a i e knowledge o
52
Fe ei ós, Ma hema ical Knowledge and he In e play o P ac ices, 163.
53
Ibid., 187.
54
Ibid. 84.
55
Mach, Die Mechanik in ih e En wickelung his o isch-k i isch Da ges ell , 1.
56
Fe ei ós, Ma hema ical Knowledge and he In e play o P ac ices, 46.
57
Ibid., 39.
18
di e en p ac ices and knowledge s a a and hey ha e sys ema ic connec ions. These
connec ions be ween le els a e no only a guide o de eloping u he knowledge – his
is also he eason why we can link speci ic heo ies o speci ic knowledge when making
epis emology–, bu hey also play a ole as cons ain s. The e a e se e al cons ain s
wi h a ole in he de elopmen o knowledge. Some o hem a e cogni i e
58
and o he s
a e in a-ma hema ical.
59
They es ic he o ma ion o new concep s. In some espec s
his is close o Mach’s concep ion o knowledge as a es ic ion o ou expec a ions:
An unp ejudiced conside a ion o he de elopmen o na u al science makes us see i s
o igin in he ac ha we begin by no icing in p ocesses hose aspec s ha a e o
immedia e biological impo ance o us [hence he cogni i e es ic ions], and only la e
p og essi ely ex end ou in e es o aspec s, ha a e mode a ely impo an . In he ligh o
his e lec ion, he ollowing ob ious o mula ion may pe haps be accep able: in o igin,
he ‘laws o na u e’ a e es ic ions ha unde he guidance o ou expe ience we
p esc ibe o ou expec a ions.
60
Despi e he simila i ies, i seems o be some kind o subjec i e elemen in Mach’s
o mula ion ha is no p esen in Fe ei ós, since he la e in oduces he cons ain s o
gua an ee he objec i i y o ma hema ical concep s. And we canno a ibu e a
concep ion o objec i i y o he Machian o ma ion o concep s, since physics is an
economic desc ip ion wi h he pu pose o adap a ion, no o explain na u al p ocesses.
Consequen ly, science is a desc ip ion o he elemen s g ouped by he scien is s o
sa is y ou needs, and as such, any s a emen o exis ence based on physical laws is as
supe luous as naï . Howe e , as we will y o show la e , his does no imply a
comple ely ela i is ic concep ion o knowledge and we will equi e he his o ical
app oach o p o e his.
4. How o ame Mach in he his o ical app oach.
In he p ac ical-p agma is app oach ha we ha e jus shown, we can see a good
oppo uni y o emphasize he in e play o p ac ices. Bu i is e en mo e pa en when
ecou se is made o he his o ical app oach. To apply his o Mach’s pe spec i e, i
would be be e cha ac e ized as his o ical-c i ical, which is p ecisely he me hod he
58
Ibid., 62.
59
Ibid., 257.
60
Mach, Knowledge and E o , 351.
19
used o s udy he de elopmen o mechanics. This means ha his o y p o ides he
p i ileged milieu o he unde s anding and cla i ica ion o concep s, guided by a
c i ical –philosophical– pe spec i e. Mach conside s ha he p ope analysis o science
o sol e p oblems is no he logical, bu he his o ical one, because he de elopmen o
scien i ic concep s depends on his o ical con ingencies. As Cohen asse s, o Mach, “a
heo y o his o y mus supplemen epis emology, and pe haps e en comple e i .”
61
This
ele ance played by his o y equi es a diach onic p esen a ion o science in o de o
show he links be ween Eh ah ungen o echnical p ac ices and heo ies o
Wissenscha en, c ea ing a ne o links. O in Fe ei ós wo ds:
The whole his o y o ma hema ics can be p esen ed as he g adual de elopmen o a
ne wo k o links connec ing di e en co e no ions ha ini ially lie sepa a e, i.e., as he
c ea ion o a delica e apes y o spide web es ablishing b idges be ween he disc e e and
he con inuum, numbe and geome y, algeb a and analysis –and, la e , opology–
e ce e a.
62
This implies a link be ween he plu ali y o he p ac ices ha can go beyond he speci ic
domain o a discipline –such as ma hema ics– anscending hus i s limi s and
connec ing wi h o he disciplines. A clea example is he use o ma hema ical me hods
in physics. As a esul , he in e play o p ac ices is no only om echnical o p o o-
ma hema ical p ac ices o mo e abs ac ones (bo om-up), bu also be ween di e en
sciences, in such a way ha “concep ual unde s anding depends on he web o p ac ices
and knowledge s a a ha an agen commands.”
63
This en ails ha he acquisi ion o
meaning is no dependen only on use o on his o y, because new connec ions be ween
p ac ices no p e iously ega ded a e able o modi y he concep ual unde s anding o he
ma hema ician. O he wise concep s will always ha e he same meaning o will only
a y depending on di e en uses. In he p ocess o concep o ma ion di e en
cogni i e esou ces migh apply, such as imagina ion and abs ac ion. A good example
is he ole o idealiza ions in oduced in Euclid’s Elemen s.
64
This migh a i s appea
as a ansi om echnical o ma hema ical p ac ices, bu i is a di e en p ocess han he
one we e e ed abo e, his is a op-down p ocess in which he cons i u ion o ideal
61
Cohen, “E ns Mach: Physics, Pe cep ion and he Philosophy o Science.”, 142.
62
Fe ei ós, Ma hema ical Knowledge and he In e play o P ac ices, 40.
63
Ibid., 101.
64
Ibid., 113 and .
20
igu es is p e ious o he cons i u ion o hei scien i ic ela ions as exp essed in he
scien i ic amewo k ha o ms he hi een books o he Elemen s.
I is a ac ha Mach conside s he a iabili y and con ingency o scien i ic
concep s as well. Concep s a e unc ional labels which acili a e he ansmission o
knowledge. In his s udy o mechanics he shows how he basic p inciples o his
discipline a e ixed by means o simple and con enien conjec u es based on he laws o
mo ion, which a e g oups o elemen s economically s uc u ed as a es ic ion o ou
expec a ions. Acco ding o his, laws and concep s a e pos ula ed o ou con enience,
o p o ide an explana ion o a conc e e phenomenon, i.e., wi h he pu pose o es ic ing
ou ield o possibili ies o ac ion and o ma k he p ope cou se ha ou hinking
should ollow o ge a be e adap a ion. The con enience o he p inciples is based on
an economic image o explain phenomena. I is based on p agma ic c i e ia in o de o
ob ain a solu ion o o ien a e ou sel es in he wo ld, bu only one among he mul iplici y
o possible enligh ening solu ions. As a esul , scien i ic heo ies a e concei ed as
o de ed a angemen s o elemen s wi h unc ional alue di ec ed owa ds ac ion and
adap a ion. Howe e , as we men ioned be o e, scien i ic concep s a e no me e names o
ela i e concep s, “ hey a e s able and ich in con en because hey a e cha ged o
his o y and expe ience.”
65
The unc ional ole o concep s is no an a bi a y one, hey
a e no only names, hey ha e empi ical bu con ingen o igins and his o y makes hem
s able, as happens wi h some g oupings o elemen s such as he ‘I’.
66
I is ue ha
acco ding o Fe ei ós, ce ain s a emen s o ma hema ics a e simply ue o alse, such
as hose o elemen a y numbe heo y. Wha makes hem ue, acco ding o his
a gumen , is he meaning and p ope ies o numbe s as hey a e concei ed in he con ex
o basic coun ing p ac ices. This makes his posi ion di e en o Mach, since u h is no
a alue ha he wan s o include in his heo y o knowledge,
67
and we canno see i he
aces a on ie di iding basic om ad anced science as Fe ei ós does wi h a i hme ic
–meaning ha u h is applied o basic a i hme ic and, as we will show, hypo he ical is
used o ad anced ma h.
The his o ical con ingency is exp essed by bo h au ho s in he a iabili y o
concep s. Fe ei ós conside s he a ia ion o concep s as dependen on he web o
65
Michel Pa y, “Mach e Duhem, l’épis émologie des sa an s-philosophes.” In Épis émologie e
ma é ialisme ed. Oli ie Bloch (Pa is: Klinsiek, 1986), 200.
66
Mach, The Analysis o Sensa ions, 23.
67
Pie o Go i, “Nie zsche as Phenomenalis ?” In Nie zsches Wissenscha sphilosophie: Hin e g ünde,
Wi kungen und Ak uali ä ed, Helmu Hei , Gün e Abel and Ma co B uso i (Be lin: De G uy e , 2012),
347.
21
p ac ices domina ed by an agen and, as a esul , a di e en web o p ac ices implies a
di e en concep ualiza ion.
68
To Mach, concep s ha e an in insic his o ical and
e olu iona y a iabili y, explaining hus he di e en concep ual possibili ies depending
on he his o ical backg ound and he en i onmen . Fe ei ós names his his o ical
app oach also as hypo he ical, because o he dependence ha ad anced ma hema ics
has on ce ain choices. These hypo heses a e cons i u i e o some ma hema ical
domains,
69
ounding hypo he ical sys ems ha “a ise ou o a ichly embedded his o ical
de elopmen ”.
70
Simila ly, Mach s a es ha “a p oposi ion in na u al science always has
a me ely hypo he ical sense”,
71
meaning ha p oposi ions in physics a e a om being
no ma i e o de ini i ely es ablished. In his sense a new pa allelism migh be d awn
be ween bo h au ho s, pa icula ly i we ake in o accoun ha Fe ei ós s a es ha “ he
p esence o hypo heses a he e y hea o ma hema ics es ablishes an impo an
simili ude wi h physical heo y”
72
and he de elops a hypo he ical concep ion o
ma hema ics o ad anced –no o elemen a y– ma h.
To Mach, p oposi ions a e hypo he ical because hey canno be ue o alse, bu
hey a e also cons i u i e because hey p o ide he mos bene icial concep ion o
mankind –human agen s– bo h in biological and cul u al espec s.
73
They a e a
condi ion o possibili y –hence hei cons i ui eness– o ou o ien a ion in he wo ld.
Howe e , he emphasis ha Mach places on a oiding any u h concep ion has oo
equen ly p o oked he associa ion o his hough wi h me e ins umen alism o an i-
ealism. This is denied by Cohen by emphasizing ha o an ins umen alis e idence and
his o y a e usually no ele an .
74
This is ne e heless p oblema ic since one can hink
abou e idence as empi ical adequacy –a concep e y dea o ins umen alis s such as
an F aassen; and one can conside he ole his o y plays in ano he well-known
ins umen alis like Duhem –Duhem’s case is mo e con o e sial gi en his no ion o
‘na u al classi ica ion’, bu o en e in o such a dispu e would equi e a di e en pape .
75
A mo e in e es ing pe spec i e has been ecen ly p o ided by Go i in which he
68
Fe ei ós, Ma hema ical Knowledge and he In e play o P ac ices, 103.
69
Ibid., 156.
70
Ibid., 157.
71
Mach, Knowledge and E o , 356.
72
Fe ei ós, Ma hema ical Knowledge and he In e play o P ac ices, 154.
73
C . Mach, Knowledge and E o , 361.
74
Cohen, “E ns Mach: Physics, Pe cep ion and he Philosophy o Science.”, 141.
75
The e a e se e al discussions o Duhem’s na u al classi ica ion such as Sonia M. Dion, “Pie e Duhem
and he inconsis ency be ween ins umen alism and na u al classi ica ion.” S udies in His o y and
Philosophy o Science 44, no. 1 (2013): 12 and Al edo Ma cos, Pie e Duhem. La iloso ía de la ciencia
en sus o ígenes (Ba celona: PPU, 1988), 153 and ).

22
cha ac e izes Mach’s posi ion as a hypo he ical ealism, in e p e ed as a hi d way
be ween ealism and an i- ealism.
76
Go i uses Mach’s Da winism o mi iga e his explici
an i- ealis pe spec i es –i.e. he idea ha scien i ic laws do no co espond o s a es o
ac s. Adap a ion o hough s o ac s equi es some kind o en i onmen o be adap ed
o. I do no wan o in oduce he e he no ion o ‘ex e nal wo ld’ as Go i does, because
his would mean he e u n o he dis inc ion be ween mind and wo ld ha Mach wan s
o dissol e. Bu in some sense, Go i is igh because adap a ion implies he exis ence o
ce ain cons ain s ha migh be impo an in he de elopmen o he indi idual o o he
species, and as such, also o science.
This idea migh b ing e en close Mach’s and Fe ei ós’ iews, and hus we will
be e y glad in he a ainmen o ou goals o his pape . Howe e , he e is no way o
a oiding Mach’s asse ions ega ding he impossibili y o dis inguishing eali y om
d eam:
The ques ion which is o en asked, whe he he wo ld is eal o whe he we me ely d eam
i , is de oid o all scien i ic meaning. E en he wildes d eam is a ac as much as any
o he . I ou d eams we e mo e egula , mo e connec ed, mo e s able, hey would also
ha e mo e p ac ical impo ance o us.
77
Consequen ly, despi e Da winism, i seems di icul o combine such ideas wi h a
ealis ic concep ion, e en wi h a mi iga ed one.
Wi h he his o ical-c i ical and hypo he ical concep ion, we come o he end o
ou analysis. The ele ance o his o y o Mach is emphasized by asse ions such as: “le
us no le go he guiding hand o his o y. His o y has made all; his o y can al e all. Le
us expec om his o y all”.
78
This places also he emphasis on his o ical con ingency,
and we hink ha he ollowing wo ds could be a ibu ed o Mach, since we ind he e
no only he con ingency and he his o y, bu also he in e connec ion o p ac ices, he
ecu ence o aci o ins inc i e knowledge and he necessa y in e ac ion be ween he
plu ali y o he sciences. Bu mo e impo an ly, i jus i ies he ex ension ha we ha e
made o Fe ei ós’ epis emology o adap i o Mach’s iews:
76
In Go i, “Wha Does I Mean o O ien Onesel in Science? On E ns Mach’s P agma ic
Epis emology.”
77
Mach, The Analysis o Sensa ions, 11.
78
Mach, His o y and Roo o he P inciple o he Conse a ion o Ene gy, 18.
23
Con ingency and he hypo he ical a e ine i ably ins alled a he hea o mode n
ma hema ics, al hough he highly in e connec ed web o ma hema ical and scien i ic
ac i i y (pas and p esen ) ac s in such a way ha i becomes disguised and can e en seem
in isible. Needless o say, such a iewpoin , i co ec , mus a ec also ou
epis emological unde s anding o o he sciences, beginning wi h physics.
79
5. Conclusion.
We s a ed his pape by ecognizing he ecen u n in epis emology in o de o
p esen a con empo a y pe spec i e ha akes se iously he ma e ial, aci , and psycho-
social aspec s o knowledge. I is a ac ha Fe ei ós’ app oach is one which
co esponds o his pe spec i e. Bu mo e impo an ly, we ha e used his con empo a y
app oach as a ui ul guide o enligh en Mach’s iews. This has enabled us o ecognize
also hese ea u es in he hough o an almos o go en hinke in con empo a y
philosophy. Mach is usually mo e emembe ed as an inspi ing igu e o he Vienna
Ci cle, a he han by his own wo k. His books, indeed, a e a ely s udied and a e no
p esen in many philosophy syllabi. Fo una ely in 2016, wi h he occasion o
commemo a ing a cen u y o his passing, new s udies ha e come ou –such as Go i’s.
In his wo k, I ha e ied o show ha Mach’s iews migh ha e a ole in
con empo a y epis emology by b ie ly upda ing and econside ing hem in he ligh o
ecen de elopmen s in he ield. In some espec s, he idea o making so much
emphasis on ‘ ecen u ns’ –as Sole e al. do– shows mo e a p e en ion o being
o iginal and hus emembe ed in he u u e, han a eal inno a ion o a g ound-b eaking
con ibu ion. I am no denying he e he o iginali y o some o he s udies p esen in
Sole ’s wo k o in any o he s, bu only claiming o he necessi y o emembe ing he
pionee s o he s udy o scien i ic p ac ice. This is some hing e y p esen in Fe ei ós’
wo k, when e e ing o he wo ks o Dedekind, Riemann o Hilbe . And in his sense,
his book could ma k a new u n in philosophy, one ha conside s his o y se iously and
is no ashamed o ecognizing he ele ance o p edecesso s o o combining old and
new s udies.
Mach is no quo ed all along Fe ei ós’ book, and he e is no need o i ,
because, as we said, i is a book abou he epis emology o ma hema ics and Mach was
no a ele an igu e in his a ea. In ac , an impo an con ibu ion o Fe ei ós’
app oach is his abili y o show how a science ha is e y commonly linked o he idea
79
Fe ei ós, Ma hema ical Knowledge and he In e play o P ac ices, 162.
24
o ‘ heo y’ is also p ac ical. In his philosophy o ma hema ics, heo ies and heo ising
hemsel es become widely conside ed and analysed as p ac ice, ecognizing ha
scien i ic heo y i sel is be e unde s ood in e ms o heo e ical p ac ices a he han in
he classical sense o heo y.
80
This is a e y impo an aspec o scien i ic ac i i y ha
Mach was no able o encompass wi hin his iews. Bu despi e all his p oblems and his
old- ashioned pe spec i es, Mach’s –as Fe ei ós’– is a philosophy based in scien i ic
p ac ice, i.e. a philosophy ha akes in o accoun he ac ual p ac ice o science, which is
p obably he mos se ious way o making philosophy o science.
We ha e shown ha i is illumina ing o analyze Mach’s philosophy wi h a basis
on he iple app oach b ough up by Fe ei ós, al hough adap ed and modi ied. As we
said, Mach’s concep ion would be be e unde s ood as cogni i e and e olu iona y,
p agma is and p ac ice-o ien ed (he e we choose he same ca ego ies), and his o ical-
c i ical. The e a e o cou se impo an di e ences be ween bo h au ho s, bu he e a e
also one hund ed yea s o di e ence be ween hei wo ks. And hese wo agen s ha e
de eloped hei wo ks wi hin di e en amewo ks: Mach was an expe imen al
physicis and Fe ei ós has always been a p o essional philosophe ; Mach was a
his o ian o physics and Fe ei ós has been –and is– a his o ian o ma hema ics; e c. The
p inciples, conjec u es and p oblems ha hey a e con on ed wi h a e di e en beyond
any doub , and so is he con ex . Thei philosophical p ac ices a e, consequen ly,
di e se. Bu hey ha e impo an elemen s in common, as we ha e ied o show in his
pape .
I would like o inish wi h a inal app ochemen be ween bo h: he dissolu ion
o he dicho omy be ween na u e and cul u e. Fe ei ós says:
In ac , he na u e/cul u e dicho omy is only alid on a i s app oxima ion; when alking
o cogni i e ac o s unde lying ma hema ics, cul u al elemen s mus be aken in o
accoun . Thus, my s andpoin is be e desc ibed as a o m o p agma ism han o
na u alism.
81
All his wo k is pe aded by he a emp o a oid simple and classical dicho omies ha
es ablish igid on ie s ins ead o being help ul o de eloping a cohe en epis emology
abou a conc e e domain o knowledge. The same happens wi h Mach’s pe spec i e
80
See also he wo k o Joseph Rouse, How Scien i ic P ac ices Ma e (Chicago: The Uni e si y o
Chicago P ess, 2002) o a p ac ical conside a ion o heo ies.
81
Fe ei ós, Ma hema ical Knowldege and he In e play o P ac ices, 185-186.
25
when he s udies machines, o conside s he es ablishmen o a ame o e e ence o
de elop he ine ia p inciple: he does no conside hese ideas as me ely cul u al, as
opposed o na u al. The de elopmen o science as a cul u al ac i i y is a na u al a i ude
o human beings, i is a way o adap ing hei hough o ac s. A philosophical posi ion
which a oids simplis ic dicho omies u ns ou o be ui ul because i is able o
app oach he eal p ac ice o science in all i s plu ali y. This p e en s he applica ion o
oo igid ca ego ies ha a e unable o explain he ichness o p ac ices p esen in
di e en con ex s and cul u es, and which make science a mi aculous ou come o one
pa icula ci iliza ion. A philosophy ha wan s o app oach he plu ali y o p ac ices
mus be one which dis inguishes di e en shades o g ey.
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