Bailly F., Longo G. Mathematics and the Natural Sciences: The Physical Singularity of Life

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Incompleteness and Indetermination 83

ical, material hand is the very ﬁrst place for gesture; humanity, actually

the human brain, wouldn’t exist without the hand. This is not a matter of

metaphors but rather a concr e te reference to what we have been taught by

the evolution of spec ies: in the course of evolution, the human free hand

has preceded the human brain and has stimulated its development (Go uld,

1977). In fact, the ner vous sy stem is the result of the complexiﬁcation of

the sensor i-motor loop. The human brain is what it is because the human

hand is for this loop the richest possible tool, the richest possible animal

interaction in the world. Then, socialization and la nguage (becaus e of their

complexity a nd expressivity) and his tory as well, have done the remaining

work, up to mathematics.

What is lacking in formal mechanisms, or in other words their provable

incompleteness, is a consequence of this hand g e sture which structures space

and measures time by using well order. This gestur e originates and ﬁxes

in action the linguistic construction of mathematics, indeed deduction, and

completes its signiﬁcation.

2.1.11 Summary and conclusion of part 2.1

As we said at the very beg inning , mathematics has always been an essential

component of the theories of knowledge. So there lie, in our opinion, the

reasons of this focus, reiterated throughout the course of history: they are

due to the anchoring of mathematics in some of the fundamental processes

of our interaction with the world. The gesture which traces a trajectory, an

edge, the following of a prey by eye jerks (saccades), the memory of these

gestures, as well as counting to keep, divide, compare . . . are among some

of the most ancient acts carried out by the living beings that we are, and

they participate in mathematical construction. It even seems that writ-

ing began with the quantitative recording of debts, by the Sumerians (see

Herrenschmidt, 2007). In shor t, it may be that the ﬁrst great conceptual in-

variants have been proposed in their speciﬁcally mathematical maximality,

as developments, complex and constituted through human communication,

of the most fundamental originary organizing gestures for spac e and action

in space. The cognitive sciences and mathematics have all to gain from a

two-way interaction, through the examination o f the grea t problems of the

foundations of mathematics.

In particular, we proposed to analyze two features of mathematical rea-

soning, namely: the construction of mathematical concepts a nd the struc-

ture of mathematical proofs, in order to rediscover sense in mathematics.

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84 MATHEMATICS AND THE NATURAL SCIENCES

The idea is that signiﬁca tion is based on language ﬁrst, but also on ges-

tures, as forms of action and communication, in a broad sense, it actually

originates by interference with action; mo reover, we po int out that linguis-

tic symbols, which are essential to intersubjectivity as a locus for human

abstraction, ar e grounded as a last resort on gestures. Yet, mathematical

concepts and proofs are developed w ithin language, as a nee d for inter-

subjectivity in the context of communication; and the linguistic framework

brings further stability and invariance to mathematical concepts and proofs,

in particular since writing exists.

In our pe rspective, thus, the signiﬁcation of c oncepts and proofs relies

also, in contrast with a Platonist and a formalistic view, on some particular

features of human cognition. These features precede language, resort from

action and g round our meaningful g e stures. We gave two examples of them:

the complex constitution of the integer number line and of trajectories ,

from eye saccades to movement. The ﬁrst grounds the notion of the well-

ordered number line, a constructed mental image, on which the pr inciple

of induction used in mathematical proofs relies. Similarly, the ima ge of the

continuo us line/trajectory is the result of a variety of gestures, including

eye saccades. It founds and gives meaning to the s ubs e quent mathematical

conceptual (linguistic) construction.

The proof itself, as a particular case of deductive reasoning, relies on

gestures: this a consequence of the sketchy analy sis developed here of re-

cent “concrete” incompleteness theorems, which show exactly where formal

induction is insuﬃcient. Thus, gestures may be involved in mathematics

and proofs at diﬀerent levels. In sug gesting and grounding the construc-

tions of mathematical structures and in proofs, by completing the principle

of induction, as in the example. B ut also in the deductive structure of the

proof itself: the geometry of proof developed by Girard, as a new paradigm

of deduction (in co ntrast with “seq uence matching”, the tool of for malism)

could be used to uncover the organizing structur e s involved in proofs. In

this view, the explicit deductive process, as a result of an exigency of com-

munication, is constrained by language but implicitly involves structured

gestures, similarly as concept formation.

The cog nitive foundation of fragments of the mathematical practice

hinted at here is clearly an attempted epistemological analysis, as a ge-

nealogy of praxes and concepts, in languag e and before langu age. The

historical construction of mental images is a core component of it, as a key

link to our relatio n to space and time and a constitutive part of our ongoing

attempt to organize the world, as knowledge.

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Incompleteness and Indetermination 85

2.2 Incompleteness, Uncertainty, and Inﬁnity: Diﬀerences

and Similarities Between Physics and Mathematics

With the view of developing this chapter’s theme ar ound the notion of in-

completeness, such as it appeared with force during the 19 30s, a per iod

which saw the ﬂo urishing of Gödel’s great logical theorems, and such as

it started to foster debates and arouse new perplexities in physics during

the same era, we will delve here into the conceptua lity of quantum physics

(to which results we will also be called to refer to on several occasions).

In the following chapter, from a broader point of view, we will attempt

to situate this ﬁeld of physics with regard to the other gr e at theories hav-

ing marked the XXth century: relativity and the theories of dynamical or

critical systems. These theories will prove to be less destabilizing on a con-

ceptual level, but will, nevertheless, contribute in clarifying cer tain aspects

addressed here , in a complementary and relatively simpler fashion.

Firstly, and somewhat trivially, we know that in physics the accumula-

tion of empirical proof does not suﬃce to account for the tota lity of the the-

oretical construction which represents phenomena. This is how, in physics

and a t a very ﬁrs t level, is manifested the incompleteness of proof principles

relative to construction principles (cf. Chapter 4 for the constructive role

of the g e odesic principle). However, this incompleteness has been thought

of as way beyond this ﬁrst level, both in cla ssical and in quantum frames.

2.2.1 Completeness/incompleteness in physical theories

2.2.1.1 From logic to physics

We ﬁrst return to a conceptual compa rison betwee n the pos itions of Laplace

and Hilbert on the one hand and of Poincaré and Gödel on the other. The

ﬁrst two refer to a sort of strict completeness of theories that are physico-

mathematical in one case, and purely mathematical in the other: they

would be complete in the sense tha t any statement concerning the future

would be decidable (this is the Laplacian predictability of systems deter-

mined by a ﬁnite set of equations) or, regarding logico-formal derivations,

the completeness or decidability of arithmetic (or of any suﬃciently ex-

pressive axiomatic, thus ﬁnitistic, theor y: this is Hilbert’s completeness

and decidability conjecture). The other two demonstrate incompleteness:

Poincaré, with his theorem on the three-body problem, showed unpre-

dictability of interesting non-linear systems, which we may understand a s

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86 MATHEMATICS AND THE NATURAL SCIENCES

undecidability of future states of the physical system (an explicitly given

statement about it would be undecidable or underivable from the e qua-

tions). Their dynamics will be said to b e sensitive to the initial conditions,

and in that, deterministic, yet unpredictable: minor varia tions, p ossibly

below the level of observability, could ca use major changes in the evolution

of the system. Gödel proved the unprovability of coherence and the intrin-

sic undecidability and incompleteness of arithmetic (and of all its formal

extensions), by constructing an undecidable statement, equivalent to the

formal assertion of consistency, which is thus also unprovable within the

system.

This analogy here is informal, as Hilbert asked for the decidability of

purely mathematical statements, while Poincaré’s unpredictability concerns

the relatio n between a system of equations and a physical process. Yet, on

one side Poincaré’s philosophical c ritique of Hilbert’s pro gram (the “sausage

machine of formal mathematics that would automatically prove all theo-

rems”) is based on the intuition of the “unsolvable” (“the provably unsolv-

able problems are the most interesting as they force us to ﬁnd new ways”

he writes in 1908). On the other side, recent results show an asymptotic

correspondence between dynamic unpredictability and undecidability. In

short, classical physical randomness as deterministic unpredictability may

be shown to be equivalent to algorithmic randomness, a stro n g form of

(logical) undecidability: they coincide for inﬁnite sequences or trajectories

in eﬀective dynamica l spaces (thus the corre spondence may be c onsidered

asymptotic, se e Longo (2009 ).

2.2.1.2 From classical to quantum physics

It can then be interesting to try to characterize the main types of physical

theories

themselves in terms of this relationship to completeness.

If relativistic theories may indeed proclaim theoretical completeness, in

the sense deﬁned above, it very well appears that critical theories on the one

hand and quantum theories on the other hand, may pre sent two distinct

manners by which they manifest incompleteness.

In the case of chaotic dynamic systems, for example, unpredictability,

as we observed, is associated with the sensitivity to the initial conditions

joint to the non- linearity, typically, o f the equational determination (which

we will call formal, see Chapter 5). It may, however, be o bserved that an

It is the distinction between relativistic, critical, and quantum theories which we will

detail in Chapter 3.

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Incompleteness and Indetermination 87

inﬁnite pr e c ision regarding the initial data is meant to generate a perfectly

deﬁned evolution (deterministic aspect of the system), or that a reinitial-

ization of the dynamic system with rigorously identical values leads to

reproducible results. This is in principle and from the mathematical view-

point, of c ourse, because physically spe aking, we are still within the context

of an approximation and the result of a measurement is always, in fact, an

interval and not a unique point. Hence, we may notice that an e ssential

conceptual discontinuity appears between that which pertains to a ﬁnite

level of approximation and that which constitutes a singula rity, a t the “ac-

tual” inﬁnite limit of pr e c ision (or, one could say, between the unpredictable

and the repr oducible).

In quantum theories, contras ting ly, as we shall see, whether it is an issue

of relationships of indetermination or of non-separability, unpredictability

is intrinsic to the approach, it is inherent to it. In this case, the degree of ap-

proximation matter s little: there is a sort of conceptual continuity between

ﬁnite and inﬁnite, the observed and measured behaviors do not present any

particular transition. Another way to observe this is by noticing that this

time, regardless of the rigor of the reinitialization of the system, the results

of the measurement will not necessarily be individua lly reproducible (prob-

abilistic character of quantum measurement), even if the law of probability

of these res ults can be perfectly well known.

It is randomness then which is at the center o f these theories, and this

by subtle diﬀere nce s from classical frameworks. This point is delicate and

we refer the reader to chapter 7 dedicated to this theme.

2.2.1.3 Incompleteness in quantum physics

In the debate on the c ompleteness of quantum physics, Einste in, Podol-

sky, and Rose n (EPR) highlighted three characteristics of a physical object

which they believed to be fundamental in order to be able to speak of a

complete theory (Einstein et al. 1935; Bohm, 1951):

• the reference to that which they called elements of reality (as some-

thing “be yond ourselves”);

• the capacity to identify a principle of causality (including in the

relativist s e nse);

• the property of locality (or of separability) of physical objects.

Bell inequalities (Bell, 1964) and their experimental veriﬁcations, no-

tably by Aspect a nd his team (Aspect et al., 1982), have shown that the

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88 MATHEMATICS AND THE NATURAL SCIENCES

third EPR postulate was no t corroborated: experience shows that two

quantum objects having interacted remain for certain mea surements a sin-

gle object, consequently non separable, rega rdless of their distance in space.

In short, two quanta having interacted, even if they are afterwards causa lly

separated in space, must be considered as a unique object, any measure-

ment of a value on the one would automatically determine the value of the

other.

Presented this way, the eventual incompleteness (or , conversely, com-

pleteness) of quantum physics se e ms to have nothing to do with what is

meant by completeness or incompleteness in logic

and in model theory,

which have been addressed in length (Löwenheim-Skolem, Gödel, Cohen

– the continuum hypothesis, up to the recent results by Friedman-Kruskal

and Girard, which we have also addressed). However, a deeper examination

reveals that what appears at ﬁrst glance as a lexical telesco ping may not be

completely fortuitous. To each of these characteristics “required” by EPR

in physics, one may, indeed, without distorting the signiﬁcations too much,

associate characteristics “req uired” by mathematics:

• elements of reality would be put into correspondence with proofs

that construct ex istence, that is, the eﬀectiveness of mathematical

constructions (which, axiomatically or not, and we have seen the

diﬀerence, cause mathematical structures to exis t);

• the principle of causality would be put into correspondence with

the e ﬀectiveness of the administration of proof (which presents and

works upon the ra tional sequences o f demonstrations, be they stem-

ming from formalism as such or not);

• the property of lo c ality (or of sepa rability) would be put into corre-

sp ondence with the autonomy o f ma thema tical theories and struc-

tures ina smuch as they would be “locally” dec idable (or that within

a formal theory, any statement or its nega tion would be demonstra-

ble).

Now it is precisely this local autonomy of theories, this “locality” in

terms of decidability, which seems to be contradicted by the theorems of

incompleteness in mathematics. T he latter indeed refer to a sort o f globality

of mathematical theories in that, for example, the adjunction of axioms to

a theory may render it if not “decidable”, at lea st “more express ive” (or

By accepting, for this discussion, to use the framework of arithmetic, of set theory

(ZFC type), and its models, a framework within which the questions relative to logical

completeness were ﬁrst raised.

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Incompleteness and Indetermination 89

capable of deciding previously undecida ble statements), but at the cost of

generating a new theory which require s the same treatment itself, because,

remaining formal, it would still be incomplete.

But we can probably push the analysis further than suggested by these

conceptual analog ies. If we refer to the interpretation of Gödelian incom-

pleteness in terms of discrepancy between constructio n principles (struc-

tural and signiﬁcant) and proof principles (formal), see part 2.1, that is, in

terms of incomplete covering, of semantics by syntax (all achievable prop o-

sitions are not formally der ivable, or, more traditionally: semantics exceeds

syntax), then a closer relationship may be established. This relations hip

concerns, among other things, the introduction and the plurivocity of the

term interpretation, according to whether it is use d in a context of model

theory or if it is taken in its common physical sense. In model theory, the

excess of semantics (cons truction principles) with regard to syntax (proo f

principles) is ﬁrst manifested in distinct interpretations. We mentioned the

existence of non-isomorphic models with the same syntax, that is, non-

categoricity (for example, the non-standard model of Peano’s arithmetic,

cf. above and Barreau and Harthong (1989)). Gödelian incompleteness

furthermore demonstrates that some of these models realize diﬀerent prop-

erties (elementary non- e quivalence).

In physics, if we accept to see the equivalent of sy ntax in the mathe-

matical structure of quantum mechanics a nd the equivalent of semantics in

their proposed conceptual-theore tical interpretations (hidden-variable the-

ories, for example, see below), then we would indeed ﬁnd ourselves in a

situation where the “semantics” also exceed “s yntax” and, consequently,

where a certa in fo rm of incompleteness (in this sense) would be manifest.

But more pr ofoundly, in the case of quantum physics, it is the excess of

the “possible” over the “a c tual” (that which is at play in the relationship

between the characterization of the quantum state and the oper ation of

measurement) w hich best illustrates this type of parallelism and of com-

parison. In other words, the result of a measurement corresponds to a

If we want to continue with the analogy and in parallel with classical logic, we will

also notice that this search for hidden variables – which prove to be non-local, however

– evokes i n a way the method of forcing which enables Cohen (1966) to demonstrate

the independence in ZFC of the hypothesis of the continuum (by constructing a model

that did not satisfy it, whereas Gödel had constructed a model that did). The hi dden

variables in question are indeed “forcing” within the physical theory they nevertheless

continue to respect, in a somewhat analogous manner – conceptually speaking – to the

one where forcing propositions are compatibly integrated with the original axiomatic

model, in logic.

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90 MATHEMATICS AND THE NATURAL SCIENCES

plurality of the potential states lea ding to it, each with its well deﬁned

probability. A sort of non-catego ricity of the states (non-isomorphic, even

if they belong to the same system) relative to the well deﬁned result of the

measurement, which operates here like a sort of “axiomatic” cons traint in

that it stems, as we have seen, from the universe of physical principles of

proof (empirical proofs are “constrained” by physical measurment).

The concept of incompleteness was understood by EPR in the sense that

quantum physics should be deterministic in its core and that its pr obabilis-

tic manifestations would only be due to the lack of knowledge of “hidden

variables” and of their behaviors, which actually amounts to saying “there

are hidden causal relations hips (not describ e d by the theory) between parti-

cles.” Now, the EPR argument is experimentally contradicted by the viola-

tion of the Bell inequalities (see above) and the property of non-separability

is indeed inherent to quantum physics. In this s e nse quantum theor y has

been shown to be indeed complete.

Can we nevertheless spea k o f incompleteness in a sense diﬀerent than

that of EPR without, however, it being totally extraneous? In other words,

would it be possible to fo rmulate a proposition that is undecidable in the

sense that it would be true according to one model and false according to

another? Let’s consider the crucial statement which can be attributed to

EPR: “there are hidden variables.” As we have just seen, this statement

is false according to the usual model (standard interpretation) of quantum

mechanics; yet a theo ry presenting the same properties as quantum me-

chanics can be constructed, in which this statement is true on condition

that an adjective is added: “there are non- local hidden variables”

“Lo cal variables” is an expression which is also equivalent to “variables attached to

particles” (they depend only on properties speciﬁc to particles that happen to be local).

To speak then of non-local variables is to express the fact that the variables which govern

the behavior of particles are not only attached to these particles, but may involve others,

which is another way to consider the non-separability we have just mentioned. As a

matter of fact, the distinction between two particles having interacted is a representation

that s tems directly from classical physics – be if it isrelativistic. For its part, quantum

physics proves to be fundamentally non-local. This is the reason why a type of true

proposition in a model (inexistence of hidden variables and non-separability) can be

false in another (existence of hi dden variables, but by specifying non-local variables).

In fact, more broadly, the controversy ﬁrst initiated by de Broglie continues among

some physicists (currently a smal l minority), with regards to the character of causal

determination (classical but hidden, undescribed) of quantum physics.

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Incompleteness and Indetermination 91

2.2.1.4 Incompleteness v s indetermination.

Sometimes, confusion is set in between the concept of incompleteness (as

presented by EPR) and that of indetermination (as highlighted by Heisen-

berg). So let’s try to further explain the r e lationships betwee n incom-

pleteness and indetermination, which do not cover the same conceptual

constructions.

The issue of incompleteness such as raised by EPR leads, we have see n,

to the search for “hidden variables” which would “explain” the counter-

intuitive behavior of quantum phenomena. It is indeed possible to elabo-

rate hidden-variable theories, but these variables are themselves non-local,

therefore simply postponing the intuitive diﬃculties. In this regard, it is

better to preserve the canonical version o f quantum physics, the Bell in-

equalities and the experiments by Aspe c t which hig hlight the property of

quantum non-separability (that for two se parated quanta, which have pre-

viously interacted, any measurement on one would determine the value of

the other). By highlig hting the fact that one of the origins of this situation

stems from the complex number (state vectors or wave function) character

of additive quantities (principle of superposition), whereas that w hich is

measured is a real number and refers to the squares of the moduli of these

quantities. We will return to this in a moment.

Quantum indetermination (“the uncertainty principle”), for its part, mo-

bilizes somewhat diﬀerent concepts: it consists in the treatment of explicit

variables (non-hidden), such as positions and momenta, of which the asso-

ciated operators will prese nt a character of non- commutativity. In short,

measuring ﬁrst one observable, then the other doe s not commute: a fun-

damental prope rty which will lead, moreover, to the development of non-

commutative geometry, the current insight into the space of microphysics.

It is in fact an issue of the constraints, which weigh the Plank constant

(small but non- null) up on physical measurement, precluding simultaneous

measurement with an “inﬁnite” prec ision of two conjugated magnitudes such

as positions and the corresponding momenta, which we have just mentioned.

If we wanted to roughly distinguish the two types o f conceptual am-

biguity introduced by these quantum properties with regard to habitual

intuition, we would say that incompleteness refers rather to an ambiguity

of object (Is the object local or global? How is it that according to the

nature of the experiment it appears to manifest either a s a particle or a

wave?

), whereas indetermination leads to a n ambiguity regarding what is

Situations with regard to which Bohr was led very early on to introduce the concept of

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92 MATHEMATICS AND THE NATURAL SCIENCES

meant by the “state” of the system. A q uantum object of which we k now

the precise position would be aﬀected by an imprecise velocity; c onversely,

the precise knowledge of a velocity would entail an “indeterminacy” regard-

ing the position occupied – in a certain way, a quantum object which we

could manage to “stop” would occupy all spac e .

In fact, both versions of these quantum spe c iﬁcities refer to a diﬃculty

of describing quantum phenomena “classically” within time and space such

as we experience them usually, while their description within their own

spaces (a Hilbert space, for instance) is perfectly clear.

Regarding space, we will no te several traits, which make o f our usual

intuition of space (and even of the Riemannian manifold of general rela-

tivity) an instrument which is unadapted to properly represent quantum

phenomenality:

i) Firs tly, a s we have indicated above, quantum quantities are deﬁned at

the onset on the ﬁeld of complex numbers C, as opp osed to classical

and relativistic quantities which are deﬁned at the onset of the ﬁeld

of real numbers R. It stems from this that in quantum physics, what

is added (principle of superpositio n) is not what is measured (complex

amplitudes ar e added, their square norms are measure d). At the same

time and for the same reason, quantum objects so deﬁned (a wave

function or state vector, for example) are no longer endowed with the

“natural” order structure associated with real numbers.

ii) Then, as we have seen, the deﬁnition of the observables makes it so

that some among them (corresponding to the conjugated magnitudes)

are no t commutable, as opposed to the classical c ase. In the context

of the geometrization of physics, this nec e ssitates the introduction of a

geometry that is itself non- c ommutative, thus breaking with all previ-

ous traditions: as access and measure of space is non-co mmutative, in

Connes (1990) geometry has be e n rec onstructed acc ordingly, by using

Heisenberg’s non-commutative algebras as a starting po int (since Klein

and Gelfand, we k now how to reconstruct geometry from – commuta-

tive – algebraic structures).

iii) Finally, the inquiry may lead even further, with regard to the rela-

complementarity (in the sense of a complementarity speciﬁc to the quantum object, s us-

ceptible to manifest, according to the type of measurement performed, either as particle

or as a wave), which was the object of many controversies.

For example, with inseparability, everything seems to occur as if an event locally well

deﬁned in a state space, that of the deﬁnition of quantum magnitudes, was to potentially

project itself upon two distinct points in our usual state space.