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

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Randomness and Determination 273

present is, beyond any doubt, sta tistical mechanics and thermodynamics,

be it at equilibrium or in the study of irreversible pheno mena.

To acco unt for these phenomena and because ﬂuctuations (not neces-

sarily quantum) also ﬁll our universe at the same time a s there is a certain

degree of disorder, Boltzmann was lead to introduce the constant k

, which

bears his name. In a way, the latter measures an entropy, that is, a physical

quantity generally related to averages taken from collections. The elements

of these collections (a gas, for e xample, formed of atoms in movement sub-

mitted to random shocks) ar e animated by disordered movements of which

the average eﬀect of interaction is represented by the temper ature T (in

kelvins: K). In the simplest of cases, each of the system’s degre e s of freedom

is associated with an e nergy which is expressed under the form of: k

T/2.

It is to be noted that the other physical theories (gravitational, quantum,

electromagnetic) are likely to address isola ted elements (two masses, an

electron, a photon, in any case, elements of a limited number of degrees of

freedom). On the other hand thermody namics (the domain par excellence

of the relevance of the appearance and utilization o f k

) addr e sses situa-

tions where these e lements are very numerous (high number o f degrees of

freedom). This makes it dependent on a statistical mechanics approach (see

nevertheless point 2 above for a brief commentary regarding what Louis de

Broglie has called the “thermodynamics of the iso lated par ticle”).

7.2.3 Final remarks on quant um randomness

It is obviously not possible for us to provide a real conclusion: our approach

is partly conjectural and to suﬃciently support it there lacks gener al math-

ematical theorems (on the relationships between classical randomness and

chaos). Further developments in the physical theories themse lves are also

needed which could nurture a vision of contemporary physics thoroughly

uniﬁed or at least suﬃciently objective and discriminating regarding the

processes at play in q uantum interactions. Curre ntly, the paradigm, which

is dominant, which does not necessarily mean deﬁnitely established, rather

goes in the sense of the conceptions, which we have presented. But the

search for a causal interpretation of quantum physics referring for instance

to a (continuo us) sub-quantum environment is not closed, even if one could

consider tha t, since its inceptio n, its fecundity remains somewhat limited,

while progress in other directions is quite rich. We would above all like

to avoid ﬁnding ourselves in a way overdetermined by a priori views too

ideological in natur e , which would make us want and thus defend a total

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274 MATHEMATICS AND THE NATURAL SCIENCES

determinism or, conversely, an essential indeterminism of which in any case

the r e levance in terms of philos ophical implications remains to be demo n-

strated. Admitting that we, nevertheless, want to go beyond the operational

aspect of the scientiﬁc approa ch in order to question the set of signiﬁcations

that it mobilizes (which is also among our preoccupations), it appears to

us to be much more interesting to argue for the spatial/energetic/tempo ral

irreducibility due to the non nullity o f the h constant. In fact, this ir-

reducibility is the harbinge r of considerable developments yet to be estab-

lished, be c ause it seems in our view to be cor relative of these intrinsic prob-

abilities, which we have considered. Thes e are themselves the expression,

it appears, of the conditions imposed by indetermination, by the complex

character o f the wave function, by the non-c ommutativity, non-separability,

etc., in short, by quantum speciﬁcities. It see ms much more heuristic and

fecund to plea and argue for a speciﬁcity of living phenomena, which we

know to be physico-chemical in the analysis of all of its functionings, but

with its mode of existence no t reducing itself to the current theories of the

inert. A mode of ex istence which conduces, in order to be understood, to

the introduction of concepts as speciﬁc as those o f metaboli sm, of normal

or pathologica l, of living or dead, of phylogenesis or ontogenesis, hered-

ity or protension, even of “contingent ﬁnality,” joint to a proper notion of

(entangled?) probabilities.

Finally, it seems important to emphas ize the fact that the objective

character, which we are assess ing here, is a constructed, or constituted

objectivity. Its constitutive process obviously dep e nds on the standpoint

(preparation, measurements, choice of formulations , of convenient mathe-

matical structures and principles, etc.). But once constituted, it becomes

independent of the viewpoint in the s e nse that it resorts to abstract math-

ematical constants or structures, w hich, henceforth, prescr ibe it as much

as they describe it. As we have claimed several times during this book,

invariance and stability with respect to a change of coordinate sys tem and

measure (viewpoint) correspond to constructed objectivity in science.

7.3 Determination and Continuous Mathematics

It is to be observed that, in the preceding sections, we have identiﬁed “ran-

dom” with “unpredictable,” in both classical and quantum physics. In the

classical ca se, dynamic unpre dictability has provided us with the very def-

inition of randomness, as a consequence of the relations hip between math-

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Randomness and Determination 275

ematical deter mination on the one hand and physical measurement on the

other. In microphysics, we have highlighted that randomness is integrated

with the theory itself, that it is in a way the sta rting point, rooted in its

peculiar polarity b e tween knowing subject and object given by means of

mathematics, of the preparation of the experiments and of measurement,

all conducing to an “unpredictable” re sult, where only probability is at-

tained. By referring this time to the role of continuous mathematics, we

will return to the diﬀere nce between the two theoretical (and phenomenal)

ﬁelds, which, relative to rando mnes s, we have separated, res pectively, in

terms of epistemicity and objectivity (or intrinsic for quantum theory ).

When we use the classica l viewpoint or tools for the analysis or the

production of randomness (we observe a turbulence, throw a coin, a dice),

a preliminary analysis of the phenomenon or object is possible: we ana-

lyze the irregularities and the s tabilities of a ﬂuid, we look at the physical

structure, the symmetries of a coin, of a dice. This enables us to a scribe

probabilities to the process, which will follow, and to make physically well-

founded estimates regarding some elements of this process. In the case of a

coin or a dice, the object’s s ymmet ries and set of physical properties enable

us to ascr ibe probabilities to the occurrence of the var ious possible events

befo re they take place (1/2, 1/6, respectively). In short, it is possible to

separate the physical object and its properties from the process, to study

it b e fore the measurement relating to the dynamic of interest to us. This

is impossible in microphysics: prior to measurement, tha t is , before the

process w here randomness will be veriﬁed, it is impossible to determine

for the physical object its lis t of properties; pa rticularly, quantum non-

separability prevents us from isolating the “properties” of a quantum. In

microphysics, the only form of access to the world lies in the measurement

of pr ocesses. It is imp ossible to “look” at the photon, the electron, as we

can a dice, a coin, independently of its process of production, of evolution

and of meas urement, where proba bilities are intrinsic to observation.

Even when information on possible measurement at the beginning of

the process is given, for exa mple, when preparing an electron for the mea -

surement of its spin (by setting a direction, one knows a priori that it will

be “up” or “down”), there is no underlying classical theory enabling us to

conceive the exact theoretical determination of this spin, before and inde-

pendently of measurement, as the result of a determined state or even of

a trajectory. It is instea d possible to c onceive an analysis of the classical

dynamics of a coin which, as a solution to movement equations , would de-

scribe the c oin’s exact trajectory, a geodesic along the Euclidean lines of

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276 MATHEMATICS AND THE NATURAL SCIENCES

the barycenter and of a point of the edge, for example. The diﬀerent “hid-

den variables” approaches in quantum mechanics presumed these classical

theories of real underlying trajectories but, a s we recalled earlier, they do

not elude the non-local aspect of their speciﬁcations.

Moreover, the classical theory e nables us, conceptually, to go to the

limit of measurement. O nce again, we very well know that measurement is

always a n interval, in classical physics; however, the theory of dynamical

systems is given within a framework of continuous mathematics, with Eu-

clidean points and trajectories, which are considered as widthless lines. For

this reason, as Laplace rightly stated and we already recalled, an inﬁnite

and perfect intelligence, knowing the world point by point, could predict

everything, including throws of dice. In theory, this boundar y continuous

framework is, to this day, essential, because the imposing of an a priori

mathematically ﬁnite limit for measurement has no classical physical sense.

This then enables us to conceive of the complete determination, in partic-

ular a trajectory, which begins from a point; it is in this se nse that the

classical world is deter ministic. And randomness, as unpredictability, re-

mains epistemic: it is in the relationship between on the one hand the tool

of knowledge and of determination which is mathematics, and on the other

hand the object which we presume to be independent and measurable only

in a humanly approximated fashion. It is this (presumed) independence

which does not occur in quantum physics, where the object and objectivity

itself are constituted by the pr actice of k nowledge (pr e paration of the ex-

periment, measurement and their mathematization, or even their principal

mathematical consequence: the quantum object).

In order to better highlight the correlated role of determination and of

continuo us mathematics within the classical frameworks, in their autono my

relative to the physical object (a pre c onsitituted), let’s return to a previ-

ous remark. Mathematics performs this passing to the continuous limit in

many abstract constructions, which are at the external boundary of the

“rational” (ratios between integer magnitudes, for Greek thought). We are

thinking, for instance, of the sequences of rational numbers converging to-

wards an irrational, let’s say

√

2: this theoretical limit produces

√

2, which

is not a rational number. It therefore exists within a conceptual universe of

actual limits (the geometric construction of

√

2, which so deeply troubled

our colleagues in Greece, to the po int of leading some of them to the brink

of suicide, is the true beginning of mathema tics). The understanding stem-

ming from real numbers as Ca ntor–Dedekind conferred to the mathematics

of continua a very robust foundation, as it provided this continuum made

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Randomness and Determination 277

of points b e yond this world with an immense mathematical eﬀectiveness,

stability, and conceptual invariance. It has also e nabled the development of

modern physics, w here the inﬁnite and the passages to the limit play a cru-

cial role well beyond diﬀerential calculus. However, it is clear that it does

not have any “physical sense” if we are referring to physical measurement,

although diﬀerential and alge braic calculus, in the continuous framework,

are at the center o f classical physico-mathematical determination. In short,

classical determination is a “limit notion” and has sat at the core of the

mathematics of co ntinua a nd mathema tical physics for a long time.

The dimensionless points, Euclid told us, are the exact departing points

for trajectories, for widthless lines, he continued, determined by equations,

as Newton, Laplace, and Einstein explained. It is ther e fore continuous

mathematics which enables us to conceive, on the one hand, of theoretically

perfect classical determination, and on the other, of the unpredictability of

physical evolution, somewhat sensitive to the boundary conditions, follow-

ing the unavoidable imprecision of physical meas urement. Because it is the

very idea o f a conceptually possible continuous substrate, which highlights

the approximation of measurement: a universe where the space-time is dis-

crete, digital for instance, would be exact, becaus e it would allow fo r exact

measurements, digit by digit, as separable points, and a n exact a c c e ss to

information, just as the digital machine a c c e sses its databases.

Let’s note that even in turbulence theory, the framework provided by the

Navier-Stokes equations is continuous, therefore deter minis tic, in this limit

sense, although speciﬁc to highly unpredictable phenomena. And this yields

further important diﬃculties for pr e dictio n, even of the theore tical type, due

to the absence, even in our day, of proof for the unicity of solutions and

therefore for the unicity of the possible trajectory, once given the boundar y

conditions.

Yet none of this remains for quantum physics, as we argued: it is the-

oretically impo ssible to pa ss to the threshold of poss ible measurement, to

refer to a continuous substr ate, b e it purely conceptual. The theory begins

with Planck’s h constant, an inferior boundary of measurement, an inde-

termination intrinsic to the ma thema tics of quantum mechanics. There are

not, in quantum space, any dimensionless points, p ossible departures for

widthless trajector ies: they are proscribed by the theory. In fact, there are

no more trajectories whatsoever within spa c e -time, in the cla ssical sense:

there lies the radical watershed constituted by quantum physics, after two

thousand years of physics of trajectories , from Aristotle to Ga lileo and

Newton to Einstein. In this sense, randomness is intrinsic to the theory,

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278 MATHEMATICS AND THE NATURAL SCIENCES

participates in the construction of objectivity, itself becoming “objective.”

7.4 Conclusion: Towards Computability

In the previous se c tions we tried to understand the notion o f randomness as

unpredictability, in two diﬀerent theoretical frames (classical dynamics and

quantum mechanics). The epistemic nature of classical chaotic dynamics

was stressed as based on

• possible alternative understa ndings (deterministic vs purely statistical),

• an underlying continuum structure that allows us to conceive per fect

determination, in the sense of complete inﬁnitary predictability (God,

says the theory, who would know the world by C antorian points, one

by one, would b e able to predict all future events).

In c ontrast to this, there are no alternatives to measurement as proba-

bility value nor hidden continuous variables in the prevailing interpretation

of quantum mechanics, thus the “theoretically objective” nature of quan-

tum randomness (there is no such conceivable God in quantum theories).

Moreover, the entanglement e ﬀects (non-separability and non-locality) ar e

at the core of the pr e vailing interpretation of quantum physics and, as

stressed above, they contribute to the peculiar nature of quantum random-

ness. This interpretation is also the basis of modern approaches to quantum

computing and crypto graphy, since deterministic hidden variables are in-

compatible with curr e nt security quantum protocols.

Many computer scientists are familiar with the mathematical deﬁnition

of randomness proposed by Martin-Löf, related to the Kolmogorof approach

and widely developed by Chaitin and others. In short, Martin-Lö f used

the classica l computability theory to give the notion of “passing any eﬀec-

tive statistical test” and deﬁned by this the so called “inﬁnite ML-random

sequences.” We do not present them here formally (see Calude, 2 002), but

we just quote one of its major conse quences, which highlights the sense of

the approach:

an inﬁnite ML-random sequence has no inﬁnite recursively enumerable

subsequ ence.

The meaning of this strong incomputability property should be clear:

there is no way to predict or compute inﬁnitely ma ny values of the in-

tended se quence. As a matter of fact, if you had a total rec ursive function

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Randomness and Determination 279

that could output the date and the results of inﬁnitely many lotto or bingo

games, you would be very happy (and the sequence would not be random).

Inﬁnitely many, of cours e , otherwise your sequence would be “just” eventu-

ally random, which is mathematically the same, for inﬁnite sequences.

How does this mathematical deﬁnition relate to our analysis within

physics, in terms of dynamics or processes? In the mathematical approach,

based on computability, there is no reference whatsoever to an underlying

physical process generating the sequence, whether it is the lotto, dies or

coin tossing. By this, it is truly general and it applies as well to a quantum

sequence of, say, an electro n’s spin-up/s pin- down, interpreted as 0s and

1s (of course, this latter se quence is r andom or unpredictable for rather

diﬀerent reasons, as we said extensively: in particular, the measurement

itself contributes to produce the states).

In other wor ds, given a chaotic deterministic dynamics or a quantum

phenomenon to which we can associa te an inﬁnite sequence of integers, this

is ML-random. Clearly, one has to specify how to obtain a seq u e nce from

the intended proc e ss (e.g. by writing numbers or signs o n dies, coins . . . by

quantum measurement). In the case of a deterministic dynamics in a chaotic

regime, once associated a measure and values to the process, the system

becomes unpredictable a fter “enough” time

. A fully general and precise

statement of this implication is still to be given. Tha t is, how to relate,

in full generality, physical randomness to ML or algorithmic randomness.

Yet, some recent results in two PhD theses, by Hoyrup and Royas, co-

supe rvised by the second author characterize the dynamics of the process e s

whose chaotic behavior engenders exactly ML-random sequences. It turns

out that in classical dynamics, Birkhoﬀ r andomness is equivalent to a (weak

form of) ML-randomness in suitably eﬀectivized dynamical systems, see

Gacs et al. (2009).

The use of recursion theory allowed Martin-Löf to set on robus t grounds

previous work on randomness, which was done before the invention or a suf-

ﬁciently widespr e ad use of this theory. The notion of “passing any eﬀective

statistical test,” but also its consequence in terms of inex istence o f recur-

sively enumerable subsequences, cla riﬁes two major aspects of randomness

Some techniques, such as the analysis of Lyapounov exponents, can give an estimate

of this time: for an ago-antagonistic process modeled by the logistic functions w ith a

minimal level of observability of, say, 10

−15

, one has to wait about 50 iterations, then

the kneading sequence becomes random. As a consequence of the work by Laskar on

the evolution of the solar system (see references), after 1 million years (not much in

astronomical tim es) the i terated analysis of the position of Pluto wheteher it is still in

or out of the system is a random or unpredictable sequence.

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280 MATHEMATICS AND THE NATURAL SCIENCES

as unpredictability.

First, it provides a frame where one c an show that unpredictability is

stronger than undecidability. As we already mentioned, both classica l and

quantum unpredictability, which we identiﬁed with physical randomness in

their contexts, yield ML- randomness as a mathematical notion. In turn,

the latter implies a strong form of undecidability, for a sequence: it cannot

contain any inﬁnite recursively enumerable subsequence. It is then fair to

say that unpredictability is str onger than undecidability, as non recursive

enumerability, in a context where these two notions can be compared.

Second, Martin-Löf’ appr oach allows us to better understand the in-

terplay betwe e n subject/object in (scientiﬁc) knowledge. There is neither

randomness in nature, nor unpredictability. The world is no t random nor

unpredictable, per se: this simply makes no sense. Both randomness and

unpredictability pop out in the rela tion between the world and a knowing

subject: in order to predict one needs someone to pre-dicere (Latin for to

say in advance). If nobody says, there is no unpredictability nor physical

randomness. In quantum physics, there is even no “physical object” with-

out measurement and mathematics. Now, recursion theory is an eminently

linguistic theory: it was born as and it is a matter of algorithms, given in

words within formal systems, over sequence s of letters or of 0s a nd 1s. By

this, it gives an important contribution to the analysis of physical random-

ness, w hen this is deﬁned, as we did, in terms of unpredictability. As a

matter o f fac t, ML-randomness is based on the no tion of eﬀectiveness for a

test, that is, for an a c tivity of someone who wants to test or try to predict

the evolution of a sequence. In no way does the relevance of computabil-

ity in the analysis of this interplay betwe e n a knowing subject and the

world prove that there is anything intrinsically computational in the world,

as many claim. On the contrary, it works on the neg ative side: it serves

only to set a limit to our linguistic (algorithmic) eﬀort to s ay something

about the world (to pre dict) a nd this by a “negative” notion of algorithmic

unpredictability o r ML-randomnes s.

In short, physical (deterministic) unpredictability (at least under the

form of Birkhoﬀ randomness, mentioned above) corresponds to algorithimic

randomness, a strong form of undecidability or incomputability. Thus,

computable processes correspond to predictable physical evolution. This

shows that the world is far from being a big digital computer (for a reﬂection

on the “interfaces of incompleteness”, see Longo, 2011).

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Chapter 8

Conclusion: Uniﬁcation and Separation

of Theories, or the Importance of

Negative Results

8.1 Foundational Analysis and Knowledge Construction

This book started oﬀ with a foundationa l analysis of mathematics a nd

physics, which it developed at length. The separation of construction prin-

ciples and proof principles was the starting point: with this distinction,

we highlighted analogies and diﬀerences between mathematics and physics,

within the framework of a great conceptual unity. If the theoretical a naly-

sis of physics was always conducted with a search for solid conceptual and

mathematical foundations, it is mainly the analysis of the foundations of

mathematics that was motivated by the quest for more or less “absolute”

certitudes, from the end of the XIXth century onwards. These motivations,

expressed energetically by the founding fathers, were large ly due to the

crisis stemming from the loss of Euclidean referents in the relationship to

space. After 2,500 years, Euclidean unity b e tween the space of the sens e s

and the space of physics crumbled jointly with the end of the absolute coor-

dinates of Newtonian time-space. Likewise, during the XIXth ce ntury, an

extraordinary boom in mathematics was accompanied by fa lse theorems or

false proofs for good theorems, by paradoxes and by contradictions. The

quest for “unshakable certainties,” to use the words of Hilbert in a letter

addressed to Brouwer, was presumably justiﬁed. It provided a new math-

ematical boost in logic, the development of mathematical logic, which was

largely centered on ar ithmetic, and, by these means, provided us with the

extraordinary arithmetic machines, which are changing the world.

Today, however, and also thanks to the stubborn eﬀorts of the logicists

and formalists of the beginning of the XXth century in providing us with

rigor, with eﬀective and unambiguous formalisms, there is no more need, in

mathematics particularly, for this certa inty to guide foundational analysis.

281

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282 MATHEMATICS AND THE NATURAL SCIENCES

And our motivation in conducting such reﬂections is almost opposite to that

which still haunts those who seek certainty in the predicative hierarchies of

arithmetic induction. In fact, we have tried to go some distance with regard

to the constitutive principles of our sciences, w ithin the framework of an

analysis we have deter mined as be ing o f cognitive type. Here, the logical

presuppositions were subordinated to the great options of our human (or

even animal) practices, but this was mostly done in the attempt to address

another: biology. The invariants of action in our space of life are in our view

constitutive o f these conceptualities, which are expressed by symmetries,

order and geodesics, at the center of the conceptual relationships, which we

have tried to highlight between mathematics and physics. It is thus that we

even consider logic as being constituted by a practice, the dialogical praxis,

such as it was initiated in the Greek ag ora: lo gical “laws” a re the invariants

of discourse , of the argumentative dialog. And as with any practice, its

role in the constitution of meaning can and must be readdressed by the

dynamics of knowledge, particularly when, from a given discipline, one is

confronted by another. Our critical analysis of the physico-mathematical

foundations is instrumental to the need for new concepts and principles, in

biology in par ticula r.

So a foundational analysis, our s, which aims at a “culture of knowledge,”

which tries to reinitiate discussions concerning not only the constitutive

principles of a given body of knowledge, but also that which is implied, the

so-called obviousness of common sense. Too often in sc ience, the researcher

is immersed in a level of technicality which removes him/her from the sense

of his/her origins and from the blinding force of common concepts. Such

a researcher is then at risk of remaining imbued with “common sense,”

deprived of any critical view of his or her own knowledge, and of making

rational-universal propositions, which are at risk of being relativized into

forms of abso lutes.

Our critical methodologica l attitude does not exclude the commitment

to a strong theo retical choice that is motivated, comparative, and supported

by proof. But even in s uch case, critical reﬂection regarding foundations

should be at the center of any scientiﬁc activity and accompany the posi-

tive scientiﬁc proposition, which must no t be confounded with the quest for

“absolute foundations .” By opposing this quest for absolutes, the non logi-

cist ma thema tician, the one who does not look for foundations in deﬁnite

and non- human laws, may indeed appreciate the so-called “foundationless”

approaches to knowledge. We could even s ay the same for the physicist

who appreciates the relativizing ana ly ses of Einsteinian time and space, as