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^`14819: Reference: [1] Fishler,MA; Firschein,A; Intelligence in eye, the brain and the computerua
6
ak24854: Reference: [2] Kampis,G (1991): SelfModifying Systems in Biology and Cognitive Science: A New Frame
9 @
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7 e34593: Reference: [5] Kampis,G (1991): SelfModifying Systems in Biology and Cognitive Science: A Newh
p32831: Reference: [8] Simon,HA; (1981), The Architecture of Complexity, in The Sciences of the Artificial, Eds.,
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kp15223: Reference: [7] Sarkar,S (1992): Models of Reduction and Categories of Reductionism. Synthese 91, 167194.
g29155: Reference: [6] Rosen,R (1991): Life Itself  A Comprehensive Enquiry into the Nature, Origin and
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" hei818209: Subsection: 2.3 Weaknesses in the holist position
% k24129169: Subsection: 2.4 Irrelevances to the debate
( ms /23818: Section: 3 Practical limits to modelling
+ f16136922: Section: 4 The number  complexity analogy!
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1 ren427991: Section: 6 Heuristics in the search for truthlo#
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: Th[12201: Reference: [2] Harnad,S (1990): The symbol grounding problem. Physica D 42, 335346.ren%
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F Refr27476: Reference: [4] Gabbay,DM (1994): Classical vs Nonclassical Logics (The Univiversality of Classical Logic).'
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M e: m34578: Reference: [2] Edmonds,B (forthcoming): What is Complexity?  the philosophy of complexity per se withr*
S ion626012: Subsection: 3.2 Limited computational resources+
W e ho43867: Reference: [1] Barwise,J; Perry,J (1983): Situations and Attitudes. MIT Press, Cambridge, MA. 352 pages. li,
[ to g71487: Reference: [17] Siegelmann,HT (1995): Computation Beyond the Turing Limit. Science 268, 545548.l m
^ ing075794: Subsubsection: 2.4.2 Analogue vs. digitalur.
b s id54155: Reference: [2] Bremmerman,HJ (1967): Quantal noise and information. 5th Berkeley Symposium on
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i 1] Z72235: Reference: [20] Turing,AM (1936): On Computable Numbers, with an application to the1
B Refn64325: Reference: [22] Yates,FE (1978): Complexity and the limits to knowledge. American Journal of Physiology2
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N5mpue34593: Reference: [5] Kampis,G (1991): SelfModifying Systems in Biology and Cognitive Science: A Newv
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P5nd f16034: Reference: [3] Pattee,HH (1995): Evolving selfreference: matter, symbols and semantic closure.
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S5tiop32831: Reference: [8] Simon,HA; (1981), The Architecture of Complexity, in The Sciences of the Artificial, Eds.,Ph
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f5ionr27476: Reference: [4] Gabbay,DM (1994): Classical vs Nonclassical Logics (The Univiversality of Classical Logic).
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j5Seco43867: Reference: [1] Barwise,J; Perry,J (1983): Situations and Attitudes. MIT Press, Cambridge, MA. 352 pages.yte
k5g71487: Reference: [17] Siegelmann,HT (1995): Computation Beyond the Turing Limit. Science 268, 545548.Ars
l5Inf16034: Reference: [3] Pattee,HH (1995): Evolving selfreference: matter, symbols and semantic closure.
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[8Bruce Edmonds
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[5Centre for Policy Modelling,
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fD (OIn the face of complexity, even an inprinciple reductionist may be at fDH(
&the same time a pragmatic holist.
UT`X Contents
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Abstract
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,^The reductionist/holist debate seems an impoverished one, with many participants appearing to ^UU
ladopt a position first and constructing rationalisations second. Here I propose an intermediate position of FolUU
Fipragmatic holism, that irrespective of whether all natural systems are theoretically reducible, for many bzUU
]nsystems it is completely impractical to attempt such a reduction, also that regardless if whether irreducible UU
bnwholes exist, it is vain to try and prove this in absolute terms. This position thus illuminates the debate UU
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Introduction
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`It appears to me that the reductionist/holist debate is a poor debate. Many of the participants f6
cappear not to be seeking truth or useful models about modelling or understanding phenomena but are manf6
eadsolely concerned with supporting previously decided positions in the matter, leaving any search for prf6
atbtruth solely within their chosen paradigm. The two camps have adopted distinct languages, styles, f6
albjournals, conferences and criteria for success and thus are largely selfreinforcing and mutually (f6@
haexclusive.
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ineabstract arguments is unproductive and fairly irrelevant to practical enquiry. In this way I hope to yRf6@
leKplay a small part in refocussing the debate in more productive directions.
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[I will start by reviewing some of the features of the debate, the versions of reductionism e inf6=ano(
Section 2.1
), some weaknesses in the two sides which make it unlikely that there will be a resolution undf6=nato the abstract debate (
!Section 2.2
and
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3Section 6
2) which will hopefully open up more @H=whPimportant and productive questions asked in the conclusion (
6Section 7
5).
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The reductionist/holist debate
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Versions of reductionism
f6]The scientific method is not a well defined one, but one that has arisen historically in the
ucepursuit of scientific truth. From this practice some philosophers have abstracted or espoused a
cpurer form of ideal scientific practice, which is epitomized in the reductionist approach. It is din
(earound this that debate has largely centred. There are many formalisations of reductionism. Here are o@
onsome examples:
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iGEvery problem is effectively decomposable into subproblems
oni`H2.5The explanation of the whole in terms of its parts
%
ThbAll of these are subtly different. They all epitomise a single style of inquiry, that any 3
uiaphenomenon, however complex it appears, can be accurately modelled in terms of more basic formal A
p_laws. Thus they are rooted in an approach to discovering accurate models of the natural world, dinO
(anamely by searching for simple underlying laws. They range from the abstract question of whether a]
call real systems can be modelled in a purely formal way to more practical issues about the sort of ed k@
in2reduction preformed in actual scientific enquiry.
y N]In this paper I aim to show the irrelevance of the abstract question; that when faced with a Nrogchoice of action it is a very similar range of issues that face both the inprinciple reductionist and ablNshholist. So for the purposes of this paper I will take the abstract definition (*) as my target absolute Al@Ntl>definition of reductionism (and hence by implication holism).
h
,
Weaknesses in the reductionist position
pp
ataForemost in the weaknesses of the reductionist position is that the abstract reductionist thesis a
riiitself is neither scientifically testable nor easily reducible to other simpler problems. Thus, although
roimany scientists take it as given, the question of its truth falls squarely outside the domain of a
ti^traditional science and hence reductionism. Its strength comes from the observation that much
asuccessful science has come from scientists that hold this view  it is thus a sort of inductive h
N^confirmation. Such inductive support weakens as you move further from the domain in which the
\induction was drawn. This certainly seems true when applied to various soft sciences like n *
bsdeconomics, where it is spectacularly less successful. The current focusing on complex systems, is 8@
neBanother such possible step away from the thesis inductive roots.
F
f \A second, but unconnected support comes from the ChurchTuring thesis. Here the strength of lfT
ifethis thesis within mathematics is projected onto physical processes, since any mathematical model of ab
ihthat process we care to posit is amenable to that thesis. If you conflate reality with your model of it trp@
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;
co^Holist literature abounds with counterexamples to the reductionist thesis. Some of these are I
innseriously intended as absolute counterexamples. They tend to fall into two categories: the practical W
poc(and so unproved in an absolute sense) and the theoretical but flawed. An example of the former is 7the
ar`Rosens example of protein folding which he justifies as a counterexample to the ChurchTuring s@
"Thesis (CTT) on the grounds that
bI...thirty years of costly experience with this strategy has produced no
4Qevidence of this kind of stability... despite a great deal of work...the problem 9
Eis still pretty much where it was in 1960...this is worse than being eH
n Munsuccessful; it is simply contrary to experience.... [14]
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)bThis is a perfectly valid pragmatic observation, justifying the search for alternative approaches
ourto the subject. It does not, of course, disprove the CTT and in itself supplies only weak support for @
le?extrapolations to broader classes (e.g. all living organisms).
(
unjAn example of the latter is Fishler and Firschein[5], where they give the spaggetti, the string,
pl\the bubble and the rubberband computer as examples of machines that go beyond the Turing !
CT]machine. These examples follow a section on the BusyBeaver problem, which is interpreted as t/
d abeing a function that grows too fast for any mechanistic computation. The examples themselves do p=
Znot compute anything a Turing Machine can not, but merely exploit some parallelism in the K
sfemechanism to do it faster. The implication is that since these compute these specific problems fY
tiafaster than a single Turing machine, this is sufficient to break the bounders of the BusyBeaver tg
cocproblem. Of course, the speed up in these example (which are of a finite polynomial nature) is not ?exu
oadsufficient to overcome the busybeaver limitation, which would require a qualitatively bigger speedan@
]up.
e(pa\Another example is that used by Kampis[8], that humans can transcend the Goedelian of balimitations on suitably expressive formal systems. He argues that because any such formal system es dwill include statements that are unprovable by that system but which an exterior system can see are tihehtrue, and humans can transcend this system and see this, that they thus escape this limitation. He then armsites Churchs example of the conjuction of all unprovable statements as one we can see is true but r@#that is beyond any formal system.
ma ficThe trouble with this is the assumption that humans can transcend any formal system to see p ihidthat the respectively constructed Goedelian statement is true. Although us humans are quite good at behiothis, the assumption that we can amounts to a denial of the CTT already, so this can not possibly used isis`as a convincing counterexample! If you state that the truth of the above is evident to us from bll hviewing the general outline of Goedels proof, i.e. from a metalogical perspective, then there will be a+haaother unprovable statements from within this metaperspective. Here we are in no better position s9ndlthan the appropriate metalogic for deciding this (without again assuming we are better and begging heG@ uthe question again).
U s`Churchs conjunction of unprovable statements gets us no further. We can only be certain of its Thcs etruth as an reified entity in a very abstract logic (which itself would then have further unincluded hqy lunprovable statements at this level)  otherwise we are merely inducing that it would be true based e SFI
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M+yo "rucThere is an essential difference between analogue and digital in the abstract. You can not pr"et^encode all analogue values as digital, only some of them. This can be proved with a classical $"aediagonal argument. This means that literally we can not talk about most analogue values, except as a o2@"itlcollective abstraction (let x be a real number...), as there are no finite descriptions of them.
@ $ caThe digital and analogue can arbitrarily approximate each other, thus the colour of a pixel on a N$ a[VDU is composed of different wavelengths (analogue), which is encoded by the computer as a q\$unbbinary number (digital), which is encoded as voltages in circuits (analogue), which correspond to j@$Fenergy levels (digital).
x(BaThe natural world may, at root, be analogue or digital, we do not know
9. Even with matter and B"benergy one could argue that the quanta are a result of observing a continuous wave function. Thus B`arguments which rely on a fundamental difference between the analogue nature of reality and the si@B"Rdigital nature of formalisms and the modern computer, must be somewhat arbitrary.
Sas`This is not to say that either simulating the analogue by the digital (or visca versa) does not re@Stipresent considerable practical difficulties.
a`
aAbility to modify hardware
he o\The ability of an organism to modify its own hardware, for example when a protein acts on ch cjits own DNA (e.g. to repair it), or at least acts to effect the interpretation of that DNA into proteins, esbis sometimes compared to a Turing machine which cannot directly effect its hardware (as usually @gu
defined).
9^This separation of hardware and software is arbitrary unless physicality can be shown to be %onbimportant attribute, effecting what can be computed. Otherwise, there is nothing to stop a Turing 3e `machine simulating such a change in hardware (including its own). For example, imagine a Turing utAatbmachine which could execute an instructiontype that could change one of its own instructions. It Oa)fwould seem at first glance that this new machine goes beyond the usual version, but this is not so. ]ar\A normal Turing machine can compute exactly the same functions as the new enhanced machine, mpkacebecause although it cannot change its own instructions, those simple instructions can be combined in ay@ntIa sophisticated way to simulate the computation of the enhanced machine.
`tlKSeveral such essential characteristics of such physicality are possible.
harVThe presence of noise in analogue systems (for this see Section 2.4.4 below).
(imZA fundamental difference between matter and symbols (as in
[10]). This is closely @,connected with the problem of measurement.
wa owYThat arbitrarily small changes in the initial conditions have significant effects on the s cioutcomes. The significance of this is either that noise (for this see
Section 2.4.4
) can then be mdYsignificant or that due to its analogue nature you can never know the initial conditions oHsa9sufficiently (for this see
Section 2.4.2
above).
AouTWe are thus left with the argument as to a fundamental dichotomy between matter and aAntgsymbols. Whether or not this turns out to be a fundamental distinction, it is not clear why this would su@Arabeffect of the ability to modify ones own hardware (or simulate such a modification in software).
h
hiNoise and randomness
; ]Noise is a random input into a systems processes. Such randomness can be defined in several sIeways. It can be any sufficiently variable data which originates from outside the scope of a systems sWndjmodel of its world, and is thus unpredictable. It can be data which passes a series of statistical tests. e@s AIt can be a pattern which is incompressible by a Turing machine.
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cfrom the purpose of this paper. Suffice to say that all of these (that I have seen) seem flawed if pr@@
omMintended as an absolute counterexample to the ChurchTuring Thesis.
cN
taaThe basic trouble that the holist faces in arguing against reductionism, is that any argument is n\
tahnecessarily an abstraction. This abstraction is to different degrees formal or otherwise. To the degree whj
blethat it is informal it allows equivocation and will not convince a skeptic. To the degree that it is x
aabsolute/formal it comes into the domain of mathematics and logic where the ChurchTuring thesis e
th^is very strong (by being almost tautologous). While informal arguments can be used with other H
d aholists, in order to argue with a reductionist a more formal argument seems to be required.
c
l jIt appears that it is a necessity limitation regarding the nature of expression itself that makes @
al5any such complete demonstration impossible.
h
nd
%Irrelevances to the debate
ex
h^Associated with the debate on the absolute question, but not central to it, are a host of old
thechestnuts that have not been shown to be relevant, but are often assumed to be crucial. There is not d
thhspace to deal with them all or thoroughly enough to convince a believer of their inadequacy, but I list in
helsome of the more frequent of them below. Despite their weakness to determine the absolute question, er
@
g 6they each have strong practical consequences.
h
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IDeterminism
m.
itWWhether natural systems are deterministic or not, in an absolute sense, seems to be an l a<
e cuntestable question. Both the deterministic and indeterministic viewpoints adequately describe the re J
it]observed world. Thus the abstract question of whether a real system is deterministic must be XH
ndUirrelevant to the abstract question of the validity of the reductionist thesis
8.
f io^Artificial situations can be categorised as deterministic or otherwise. For example in a game tt,]your next move may be determined by the rules or you may have a choice. Within the framework ey gyou are considering, there is either a mechanism for determining your move (i.e. the players strategy equ Djfor the game) or not. If there is, then the move is determined by that, if not, it is undetermined  i.e. cthere is simply not enough information to determine this by any process (mechanical or otherwise). ar ngNote that whether the move is determined does depend on the framework you are considering, but stiticwithin any particular framework (however general), whether something is determined is not relevant ion@ sOto the absolute question of whether the situation is reducible or not.
que
it\However, this does highlight the practical importance of choosing the appropriate framework is
c bfor a problem. The framework greatly effects the practicality of modelling a system. In the above
e cexample in a framework which includes the players strategies, it may be possible to model the game m f@
r Pbut impractical if you choose merely the rules and possible sequences of moves.
th
mo
^Analogue vs. digital
f! rmZThere is a basic difference between what can be theoretically modelled using analogue and /hatformalisms. Sometimes this is on the grounds of the importance of noise (for this see Section 2.4.4 ew=@er below).
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maN Efu^Again whether we use traditional styled or (grounded) selfreferential systems for modelling, \@Eb6relates not to abstract but pragmatic considerations.
o`
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Simultaneity
} coWIn most existing computation devices, computation proceeds sequentially. Even parallel fdevices are usually arranged so that their computations are equivalent to such a sequential approach. bycLikewise in almost all formal systems, facts are derived via an essentially sequential proof. Even ome@ayBwhen the proof is not sequential in nature, its verification is.
6r [Natural systems, however, seem to work in parallel. Von Foerster gives an example of a box 56veewith many block magnets in it
, the box is shaken and when opened they are arranged in a very non6madrandom way, resulting in an attractive sculpture to an exterior observer. The two views of the box, E6r cinternal and external are simultaneous and different. It is claimed that such simultaneous and (in 6re6ac_some cases) irreconcilable viewpoints, mean that a single consistent formalism of a metamodel ex@6 dincorporating both viewpoints is impossible.
9YIf you have a parallel system there will be either some conflict avoidance or a conflict h9oacresolution mechanism (where by conflicting I mean exclusive). Of course, it is quite possible to se%9en\have cases where (as in the above box of magnets example) there are views that appear to be 39Nafconflicting, but you wont have conflicts within the same context, this is impossible if a consistent A9y ilanguage is used. Here it is not the simultaneity that is the problem but the reconciliation (or lack of 6OH9reXit) of the same thing from within different frameworks (see
LSection 3.4
K below).
] ?in^On the other hand, the difficulties of reconciling different simulataneous streams means that k@?6Toften the only practical option is to accept such different views as complementary.
f UT UTh
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os_Although it may be hard to prove practical limits to modelling any specific problem, there are oid@
h$many general practical limitations.
ni`
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ex
e,bIt seems we (us and our tools) are part of a finite universe, and are thus also finite. Any model
vidwe make, use or understand will also be finite. Quite apart from this our formal communications e
o(written articles) are definitely finite. Thus any practically useful model that we want to share will em @
tialso be finite.
iteIn these circumstances the fact that an Turing machine (which is essentially infinite) could ?
hbcompute something, may not be relevant if the mapping from this abstraction to an actual computer
rabmay mean that the computation is impractical. Thus the abstract question of the CTT is superseded .@
ll4by the question of the practicalities of modelling.
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wXcalculate that quantum mechanics imposes a limit of
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wiIcomputational limit, even if they are theoretically computable.
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woComplexity
y
fiZComputational Complexity is concerned with the computational resources required, once the
fprogram is provided. It does not take into account the difficulty of writing the program in the first
biplace. Experience leads me to believe that frequently it is the writing of this program that is the more t@
nidifficult step.
a :beWMore fundamental is what I would call analytic complexity. This is the difficulty of upp:
fanalysing (producing a topdown model) of something, given a synthetic (bottomup) model
R[4]
Q. :t cWhether or not this difficulty is sometimes ultimate, few people would deny that such difficulties g j:gexist and that the exist arbitrary levels. Given that our analytic capabilities will always be limited wouH:wif(see above in
USection 3.2
T), such complexity will always be a practical barrier to us
e.
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JContext
ut i\Not all truth can be expressed in a form irrespective of context. The very identity of some is$ nithings (e.g. society) are inextricably linked to context. Thus we will have to be satisfied that, for at e2lihleast some truths, it will not be practical to try and express them in a very general context and hence ep@:hacquire the hardness of more analytic truths (like all bachelors are men). It is true that we can Ng elaboriously express larger and larger metacontexts encompassing subcontexts, but this will involve \or_the construction of more and more expressive languages
V and require disproportionately more jt icomputational power  this will make this sort of endeavour impractical, beyond a certain level
A. x acChoosing an appropriately restricted context is one of the most powerful means at our disposal for @Co.coping with otherwise intractable situations.
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de
VRosen introduces an analogy between what he calls complexity (i.e. things that arent
iebmechanisms) and infinity; the reductionist/syntactic approaches to modelling correspond to finite
nebsteps. He claims that many systems (including all living organisms) are unameanable to such steps
e band qualitatively different  they correspond to infinity. Thus he postulates that to model these
s^complex systems require some transcendental device, like taking limits or some form of self@
Vreference.
dis
re`I wish to alter this analogy and hopefully deepen it. I wish to take an analogy between numbers b
vecand complexity. This size corresponds to the difficulty of modelling a system in a descriptive toperf$
spadown fashion given a language of representation and almost complete information (model) from the o2
cabottomup perspective of it components
=. Thus infinite size would correspond to infinite such i@
atXdifficulty  i.e. impossibility of such modelling (which roughly corresponds to Rosens he8bS_
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rsecomplexity). The abstract debate would then correspond to the question Are there systems with t@
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g $
re_Here we need to examine what we mean by the existence of such systems. The problems of 92
`showing that such systems exist are remarkably close to those involved in showing that infinity @
Mgexists. You can not exhibit any real manifestation of infinity, since the process of exhibiting is m, N
oriessentially finite. Even if we lived in a universe that was infinite in some respect, you could not show \
ma complete aspect that was infinite, only either that an aspect appeared unbounded or that a rj@
me?reasonable projected abstraction of some aspect was infinite.
x
kNote that I am not saying that infinity is meaningless, merely that it is always an abstraction of ,
rsjreality and not a direct exhibitable property of any thing. That infinity is a very useful abstraction is
incundeniable  it may be possible to formulate much of usable mathematics without it, but this would ste
st^surely make such symbolic systems much more cumbersome. So when we say something is infinite,
edkwe are talking about an abstract projected property of our model of the item, even if the thing is, in nfi@
ocXfact, infinite. It is just that exhibiting is essentially a finite process.
iv
ni_I suspect that the same is true of the irreducibly complex. A language of irreducible wholes inf
twis useful in the same sense that infinity is useful, but only as an abstraction of our model, irrespective wa
lof whether these wholes exist. If they do not exist, the language of the holist is still useful as an ab
_abstract shorthand for systems whose complexity is potentially unbounded. If they do exist the ver
onrlanguage of wholes would still be necessarily abstract, i.e. not referring to direct properties of real
rthings, even if the systems referred to were irreducible. It is just that exhibiting such systems @
re:(especially formally) is essentially a reductive process.
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teXThe computability of a welldefined function is a purely formal question. A function is
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atlcomputable if an index exists such that a universal Turing machine (UTM) with that index calculates sej
ns_that function. For example, consider an enumeration of halting computable programs in order of x
ofbsize. Every function represented by programs in this class is computable without us being able to
ncompute (or write) the programs for these functions, otherwise we could use this enumeration to compute H
la0the halting problem
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ed]we have the means to find the program to compute it. In other words, the characterisation of o
altreducible as computability is too strong for actual use in reducing a problem. To be able to really il
nepcompute something we have to be able to follow the instructions (or have a machine do it) and write
sthe program to do this (or really compute that). This can form a very long chain (the program that mer
omdcomputes the program that computes the program that...) , but eventually it has to be grounded in a th
abqprogram we are able to write ourselves, if the final program is to carry out our intentions. I call this i
islintensional modelling (or intensional computing, depending on the context). This is what we can
hajcompute (using computers as a tool), as opposed to what can theoretically be computed by a Turing @
d machine.
.
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ingto the surprising conclusion that there may be some computable functions that we cant compute o bJ
hej(intentionally), even by using a computer. We may, of course, come across a few of these programs X
haccidentally, for example by genetic programming, but we can never be certain of this and verifying pf
esethat these programs meet a specification is itself uncomputable in general (although you may be able et@
se9to analyse a few such particular examples sufficiently).
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theaccident (say as the unintentional result of a genetic programming program) this may be as difficult $
toZto model and understand as its more tangible cousins. The code may be so complex and self2
tlreferential that it was as difficult to decode as the mechanisms in normal life (as we know it). So
@
leven if (a suitable version of) the reductionist thesis were true in the abstract, we might still be forced
aN@
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fmUT UTh
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UU
rbSo whichever is our belief about the abstract reductionist/holist question, we are left with very
easimilar pragmatic choices of action when faced with an overly complex problem. Here reductionist
Thktechniques will be of little practical value for us as limited beings and we have to look to other
y \alternatives if we want to make progress on them. Whether you choose another (possibly less
od]successful) approach, depends upon the tradeoff between the difficulty of reduction and the
ia^importance of progress (of what ever kind) being made onthat problem. In the end, the biggest
hpractical difference between a reductionist and a holist is often only that a reductionist then chooses f
canother problem where the reductionist technique has more chance of success and the holist chooses @
is5alternative avenues of attack upon the same problem.
o
bedThe point is that there is no necessity to prejudge this decision for every case, neither to always la
s dsay that alternative types of knowledge are worthless, despite the importance of the problem nor to es
pfsay that it is never worth abandoning a problem because of the type of knowledge that is likely to be $@
anzgained about it. I call these the extreme reductionist and extreme holist positions respectively.
2
baTo hold to the extreme reductionist position in a practical sense, one must surely claim that no v@
e `problem is so much more important than other more susceptible problems to be worth swapping the tiN
icsort of analytic knowledge that results from reductionist approaches for other types of knowledge. tio\@
m;This can be a result of one of several subsidiary claims:
@
j av[that all problems are practically susceptible  this would amount to denying any practical ecey@th#limitations upon ourselves at all;
o a [that there are always a near indefinite supply of equally important problems  denying any e p@ Lreal difference in the importance of problems, regardless of circumstance;
he
tUor that alternative forms of knowledge are always effectively worthless  presumably @
nd+including the reductionist thesis itself!
pec
aTo hold to the extreme holist position in a practical sense, one would have to claim that either t
bthere was no advantage to reductionist knowledge as compared to another type in any circumstances
nor that a problem was so important that it was not appropriate for anyone to research any other, more @
mamenable, problems.
o ubXThese would both be extreme positions indeed! I know of no one that holds them in these th
dcforms. The rest of us fall somewhere in between in practice: we accept that there are some lwbworthwhile problems where the reductionist technique works well and we also accept that there are &he\problem domains where the chances of a reductionist technique working are so remote and the iv4@geLproblem so important that we would value other forms of knowledge about it.
g B hebThis does not mean that we will all have the same priorities in particular cases, just that these Plandecisions are essentially a pragmatic ones differing in degree only. Once attention switches from the ^anosterile abstract question of whether in principle all problems are amenable to a reductionist approach seale c(and thus implicitly excluding the extreme positions outlined above), we can start to consider the ns znokrich set of possible strategies for making such choices in different cases
. This is has been up to now pra@ptAa largely uncharted area, but one that might pay rich dividends.
h8bֲT
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cakreductionist technique before using it (just like any other technique) regardless of the answer to the 8b4@
,abstract reductionist/holist question.
AnB te[Such an awareness opens up the possibility of a more systematic study of ways of combining Pthatechniques for the successful elicitation of knowledge. Ways which would include developing more ^ceffective ways of using different types of knowledge to further guide and refine that search. Such be lRecpractical heuristics are maybe all that we finally have  rejecting both blinkered singlestrategy Iz@@approaches and the extreme relativism of anything goes.
UR UT`
Acknowledgements
h(
)\I would like to thank the members of the Principia Cybernetica mailing list with whom I
s ahave had many useful discussions on this subject, especially Don Mikulecky, Jeff Prideaux, Cliff e@
ueJosyln and Francis Heylighen.
UP UT`
References
f thQb^
WBarwise,J; Perry,J (1983): Situations and Attitudes. MIT Press, Cambridge, MA. 352 pages.
(UupS
bBremmerman,HJ (1967): Quantal noise and information. 5th Berkeley Symposium on th@Uhe3Mathematical Statistics and Probability, 4: 1520.
d ihMmoMCutland,NJ (1980): Computability. Cambridge University Press, Cambridge.
>
t,(d \
MEdmonds,B (forthcoming): What is Complexity?  the philosophy of complexity per se with in:in[application to some examples in evolution. In Heylighen,F; Aerts,D (eds), The Evolution of sm HesKComplexity, Kluwer, Dordrecht. Also available electronically at http:// liV@mb'www.fmb.mmu.ac.uk/~bruce/evolcomp.
lisd(I\Fishler,MA; Firschein,O (1987): Intelligence in eye, the brain and the computer. Adisonkyr@IliWesley, Reading, MA.
e(Pisa
FGabbay,DM (1994): Classical vs Nonclassical Logics (The Univiversality of Classical Logic). :PtiWIn: Handbook of Logic, in Artificial Intelligence and Logic Programming. Vol. 2. (Eds: ): @Pin:Gabbay,DM; Hogger,CJ; Robinson,JA), OUP, Oxford, 359500.
h4tiJ
:Harnad,S (1990): The symbol grounding problem. Physica D 42, 335346.
(DmbT
7Kampis,G (1991): SelfModifying Systems in Biology and Cognitive Science: A New ng@DitWFramework for Dynamics, Information and Complexity. Pergamon Press, Oxford. 565 pages.
ex(n.Z
Kampis,G (1995): Computability, SelfReference, and SelfAmendment. Commuincation and @av0Cognition  Artificial Intelligence 12, 91109.
( mmZLofgren,L (1968): An Axiomatic Explanation of complete SelfReproduction. Bull. Math. @ tBiophys. 30, 415425.
3QPattee,HH (1995): Evolving selfreference: matter, symbols and semantic closure. s@3icACommunication and Cognition  Artificial Intelligence, 12, 927.
P(( oZ
Marquis,JP (1995): Category Theory and the Foundations of Mathematics: Philosophical 6@og$Excavations. Synthese 103, 421447.
5D(.4YRosen,R (1985): Organisms as Causal Systems Which Are Not Mechanisms: An Essay into R.isVthe Nature of Complexity. In: Theoretical Biology and Complexity: Three Essays on the `@.k VNatural Philosophy of Complex Systems. (Ed: Rosen,R) Academic Press, London, 165203.
8bR'@
b
p
Pi8bR'@
Q
euiUUhVkFor a classic account of the inevitable emergence of complexity in biological systems see
h[22]
gn A8b8b
c
`d8b8bhy`>8bʋS
d
>
q
7l8bʋ
8
8urUU`@3RUnless this question is also as untestable and irrelevant as that of determinism.8bc'@
e
T
r
LT8bc'@
M
osUUhC5Such a mechanism as Harnad suggests in
<[7]
;.8b'@
f
i
P 8b'@
Q
=n UUhF:For more on the definition of complexity see
@[4]
?.SFI
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a'@(V
Rosen,R (1991): Life Itself  A Comprehensive Enquiry into the Nature, Origin and @ i:Fabrication of Life. Columbia University Press, New York.
$
XRosen,R (1993): Bionics Revisited. In: The Machine as Metaphor and Tool. (Eds: Haken,H; hy2@8Karlquist,A; Svedin,U) SpringerVerlag, Berlin, 87100.
@h!
8_
Sarkar,S (1992): Models of Reduction and Categories of Reductionism. Synthese 91, 167194.
s tNh*.U
[Siegelmann,HT (1995): Computation Beyond the Turing Limit. Science 268, 545548.
\('C_
Simon,HA; (1981), The Architecture of Complexity, in The Sciences of the Artificial, Eds., j@''@4MIT Press, Cambridge, Massachusetts, pages 192229.
Fox(
niH
iTuring,AM (1936): On Computable Numbers, with an application to the @
?Entscheidungsproblem. Proc. London math. Soc., 2 42, 230265.
YVon Neumann,J (1966): Theory of SelfReproducing Automata. University of Illinois Press, u@e,Urbana, Illinois.
h8icG
pWeinberg,S (1977): The First Three Minutes. Basic Books, New York.
XRo(Jic^
fWimsatt,W (1972): Complexity and Organisation. In: Studies in the Philosophy of Sciences. @Jpr?Vol. 20. (Eds: Schaffner,K; Cohen,R) Reidel, Dordretch, 6786.
S ((ed\
BYates,FE (1978): Complexity and the limits to knowledge. American Journal of Physiology el@mp235, R201204.e T8bc'@
i
p
f
P8bc'@
Q
AecUUh
, =For an overview on this problem in biology see
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