aa r X i v : . [ phy s i c s . b i o - ph ] J a n The world is not a theorem
Stuart A. Kauffman and Andrea Roli Institute for Systems Biology, Seattle, USA Department of Computer Science and Engineering, Campus of Cesena, Alma Mater StudiorumUniversit`a di Bologna European Centre for Living Technology, Venezia, Italy
January 1, 2021
Abstract
The evolution of the biosphere unfolds as a luxuriant generative process of newliving forms and functions. Organisms adapt to their environment, and exploit novelopportunities that are created in this continuous blooming dynamics. Affordancesplay a fundamental role in the evolution of the biosphere, as they represent the op-portunities organisms may choose for achieving their goals, thus actualizing what is in potentia . In this paper we maintain that affordances elude a formalization in math-ematical terms: we argue that it is not possible to apply set theory to affordances,therefore we cannot devise a mathematical theory of affordances and the evolutionof the biosphere.
Prologue [L’universo] `e scritto in lingua matematica, e i caratteri son triangoli, cerchi, ed altre figuregeometriche, senza i quali mezi `e impossibile a intenderne umanamente parola; senza questi`e un aggirarsi vanamente per un oscuro laberinto. (Galileo Galilei, Il Saggiatore , Roma (Italy), 1623)
Egyptian papyri dating some 5000 years ago document the use of arithmetic and geometricnotions for solving practical problems, such as the need of measuring and subdividingthe soil [18]. After thousands of years, three centuries before Christ, we find the most “The universe is written in the language of mathematics, and its characters are triangles, circles, andother geometrical figures, without which it is humanly impossible to understand a single word of it; withoutthese, one is wandering around in a dark labyrinth.” Translation by S. Drake. Elements , by the Greek Euclid of Alexandria,a prominent and influential deductive theory—queen of the Sciences and true of the world:the language in which the universe is written is made of triangles and circles, as Galileostated in the XVII century. Nevertheless, an axiomatic and deductive system such asthe
Elements does not need to describe the world, as shown by non-Euclidian geometries.However, mathematical systems shine as pure crystals and one might expect them to beconsistent and complete, as indeed Hilbert did. These hopes have been destroyed byG¨odel, who has set the limits of deductive systems. Despite these limits, we currently relyon mathematical models for understanding systems of any sort, conscious of the possibleincompleteness of the deductions we make. But there is a kind of incompleteness we haveprobably overlooked: what can be entailed by a formal system is already contained in it,and we cannot expect to be able to deduce novelty, to entail the becoming of the biosphere.In this paper we maintain that the flourishing evolution of the universe cannot becaptured by a formal mathematical model: the world is not a theorem.We start by discussing the notion of affordances and the properties of the becomingof the biosphere, in Section 2. Section 3 illustrates the core of our thesis: affordancesdefeat any sound definition of set, so no set theory, hence no math, can be used to modelaffordances. Several objections might be raised to argue against our claim: in Section 4we analyze and answer to the ones we think are the most relevant. Finally, in Section 5,we observe that the essence of our statement can be already found in the late writings byWittgenstein and we summarize the main points of this paper, emphasizing that, ratherthan projecting a dark and negative perspective, our claim has a creative and positivepotential.
The notion of affordance has been originally introduced by Gibson [5] in psychology withthe aim of expressing the actions that an object enable to an animal observing it. Theconcept has been subsequently extended, and it is currently adopted in diverse fields such asbiosemiotics, cybernetics and robotics [2, 7]. In general and abstract terms, we can say thatan affordance is “the use of X to accomplish Y ”, where X may be an object, a living being,a situation, etc., and Y is in general an action or a behavior that typically leads to a goal.Heritable variations and selection make it possible for organisms to find new, advantageoususes afforded by objects or other organisms. For example, an empty snail shell affords thepagurus a house where it can hide and protect itself from predators, or a colony of bacteriathat is evolving in a new environment may discover a more efficient way to get to the foodthrough favorable mutations and selection. In the same way, new organs have emerged inorganisms as they afforded opportunities that enhanced their fitness in their environment.By organism we mean a Kantian whole : an organized being having the property that theparts exist for and by means of the whole [9, 13] that senses the world, chooses betweenwhat is good or bad for it, and acts. In general, affordances open possibilities either by someadaptive (possibly evolutionary) advantage, such as the case of Darwinian preadaptations2e.g. the evolution of the swim bladder [11]) or because of a choice of the organism—whichrecognizes an advantageous opportunity and acts consequently.An affordance is not a property of the object or the environment only, but it is anentangled property of both the object and the agent . In fact, affordances cannot bedefined in a non-circular way. Niches are prominent example of this, as they constitute thesubjective world of an organism [25].To discuss in more detail the properties of affordances relevant for our argument, wemake a brief excursus in a futuristic scenario. Let us imagine a robotic environment, maybeon a rocky extrasolar planet, where robots extract minerals, work in factories producingseveral materials and goods; moreover, by executing specific programs, they build andassemble the components to produce new robots. One day a robot, failing to decrease itsspeed fast enough after a declivity, accidentally stumbles onto a sharp and pointed rock andgets a dent on its aluminium side. The rock turns out to be made of obsidian, very usefulto the robots’ community. This accident has no consequences on robot’s functions, butopens up the possibility for the robot to detect rocks containing obsidian, which protrudesfrom the rock trunk, because the dent fits rather well with obsidian extrusions. The dentaffords the robot to identify obsidian rocks more efficiently and becomes then an “obsidiandetector” for the robot. Note that this sensor has a meaning for this robot and in thisparticular environmental niche.Let us imagine that a new robot is built with an error in the assemblage procedure andgets a bump on its aluminium side. We now have also a “dent sensor”. This bump enablesthe robot to detect dents in other robots and then exploit this information to identify therobot that is more efficient in finding obsidian and follow it. Should the robot with thebump be specialized to work to shape obsidian, then the dent in the first robot may servealso as a “bump detector”. Therefore, the two special robots can recognize each other andform a specialized team performing extraction and processing of obsidian more efficiently. Bumps and dents, and bumps and dents detectors, did not exist prior to the accidentsand they could not even be predicted. The accidental sliding of a wheel and the impurityin the aluminium opened up the possibility of bumps and dents. These accidents enabled new forms of behavior and cooperation in the niche where the robots operate: affordancescreate new affordances. It is also worth observing that bumps and dents, along with theirdetectors, have acquired a meaning because they are useful for a robot to do something, tobe more efficient in accomplishing a task. They have a meaning only in this specific nicheand for these robots. Now that they have a meaning, the robots can name and use theminside their control programs, which we assume to be adaptive to some extent. Observe By “agent” we mean the subject w.r.t. which the affordance gets a meaning. For example, an agentcan be an organism, an artificial system or even an inanimate object. The cause of this failure is that a wheel slid on the flat ground instead of rotating. The cause of the error is an impurity in the aluminium sheet that deceptively recursively attractedthe accumulation of more aluminum in a small area of the chassis. For the roboticists wandering how is it possible that the robots actually exploit bumps and dentswithout having specific pre-designed sensors, we observe that bumps and dents can lock the rotation ofthe robot and this event can be detected by a specific sensory-motor pattern [22]. meaning ; that is they refer to or are correlated accordingto some system with certain physical or conceptual entities. These semantic aspects ofcommunication are irrelevant to the engineering problem.” Let us now imagine that we want to equip the robots with a program that can predictaffordances and the emergence of new uses of objects. One might be tempted to thinkthat once a formal model of objects and relations is defined, e.g. by means of a suitablelogic language, it would in principle be possible to entail the evolution of this formalizedmulti-robot colony, at least in terms of the most likely scenarios. This is the same as tryingto predict the evolution of the biosphere, but there are no entailing laws for the becomingof the biosphere [11, 12, 16]. In the next section we detail this statement extending it toa stronger claim: not only there are no entailing laws for the evolution of the biosphere,but affordances and the evolution of the biosphere—and all the systems with analogousproperties, such as the technological market—are inherently not mathematizable.
The blooming evolution of the biosphere is characterized by the emergence of new organ-isms, which in turn evolve new functions. New organisms and new functions open furtherpossibilities for adaptation and evolution: the biosphere evolves by expanding towards the adjacent possible [10]. Affordances play a fundamental role in this evolution, as they rep-resent opportunities for the organisms: they enable the development of new functions, andthe emergence of new niches and new organisms.The notion of affordance is not restricted to the biosphere, but we also find it in othercontexts, such as the technological market, where new technologies enable the emergence ofnew functions, which in turn make it possible to create further new technological devicesand methods. A typical example of affordances is that of the uses of an engine block:obviously it can be used as propulsive component in a car, it is used both as engine andchassis for tractors, we can crack open coconuts on one of its corners, its cylinder borescan host bottles of wine, the engine block can be used also as a paper weight, and so on. How long is this list? One might answer that the list is extremely long, but finite orsomeone else might argue that it is infinite. We actually argue that the affordances of anobject are indefinite . We first observe that the affordances of an object are not an intrinsic Quoted from [24], italics in the original text. See also a discussion on the possible uses of a screwdriver [11].
Library of Babel byBorges [1], where all the 410 pages long books composed of all the possible combinationsof characters are stored, what distinguishes a book from a printed random sequence ofletters is the meaning that the reader gives to the sequence of letters; the meaning dependsupon the experience, the history of the reader. In addition, the experience of the readerin turn depends upon the environment in which they live. In other terms, affordancescannot be defined non-circularly. For example, technological devices are sold by adaptingthem to the specific niche corresponding to preferences of the customers. And in biology,the mixed microbial communities living in the gut of animals are reciprocally adapted,and their composition and dynamics also depends on the characteristics of the host [14].Furthermore, the universe is non-ergodic [10, 16, 11], as its dynamics does not visit allthe possible states, as opposed to ergodic systems, which visit all their possible states ina time vastly longer than the present lifetime of the universe. In fact, the universe visitsjust a tiny fraction of its possible states; let’s think for example of the molecules thathave been formed vs. the ones that could have come to exist. Non-ergodicity prevents usfrom defining a phase space that can contain all the relevant variables—and thus defineequations and study the trajectories of the objects we are interested in—because we don’teven know what the relevant variables are. This holds also in the case of the biosphere,where the expansion of Hilbert space cannot be predicted within a universe lifetime timehorizon [3].Therefore, the uses of an object, i.e. its affordances, are not finite nor infinite butrather indefinite , and there are no deductive rules to list them starting from an initialone nor to deduce some uses of an object from the uses of another object. Since newuses arise all the time by Darwinian preadaptation—or exaptation—evolution cannot beentailed [11, 12, 16].A dramatic implication of this statement is that set theory cannot be applied in mod-eling affordances, because the fact that the affordances of an object are indefinite defeatsany definition of set. The main reason for this indefiniteness is that the collection of uses of X , alone or with other objects, is constantly changing, therefore there is no way to expressthe property of X belonging to a set once and for all. Let us take, for example, the axiomof extensionality , which is one of the axioms that define sets [8] and states that “If two5ets A and B have the same elements, then they are equal”. Let’s assume that we wantto denote by A and B the collections of affordances of objects X and Y , respectively, andsuppose that we can identify a specific use of X that we call u ; therefore, u ∈ A . In virtueof the indefiniteness of affordances, it is not possible to assert the truth of u / ∈ B , i.e. it isnot possible to determine that u is not and will never be a use of Y .Also the axiom of choice [20], which is introduced whenever a choice function cannotbe defined, cannot be applied. One might object that this could be a situation analogousto real numbers, for which the axiom of choice and set theory hold. But we observe thatreal numbers may be given as sequences of symbols from a finite alphabet, i.e. we knowthe symbols composing these infinite strings. In the case of new affordances coming toexist over time like the uses of the engine block alone or with other things, the uses havenot yet come to existence, therefore the syntactic names have not come to existence, so wedo not know the syntactic symbols we would use to name the elements of the set of uses ofan engine block. The evolution of the biosphere is indeed characterized by the appearanceof new functions and new organisms. The human heart is an affordance; it affords a meansto pump blood to the body. The loop of Henle affords a means to concentrate urine in thekidney [17]. And, neither of these affordances existed on the earth two billion years ago,so we cannot do set theory with respect to them.We are not claiming that there is no possibility whatsoever to model affordances, butthat it is possible only in the syntactic frame already defined for the model we have built.Coming to the robotics scenario, robots cannot predict a new use not already deductivelypresent, therefore the flourishing evolution of the biosphere is beyond math because thenotion of set, and consequently all the formalisms that derive from it, does not hold in thiscase. Of course, the possibility of making theories of the physical world still holds, but it isbounded by the objects and relations we decide to include; in other words, we see what isrelevant, we name it and we put it in the model, which is a syntactic formalization of thepiece of reality for which we want a theory [15]. Nevertheless, we are maintaining that anyattempt of modeling a system encompassing affordances or any theory trying to capturenovelty and open-ended evolution would be inherently flawed [4, 12, 13, 23]. We understand that our claim might sound either trivial or provoking, or both. In thefollowing we anticipate what we believe are the main objections that may possibly beraised against our argument.
Objection
If we could find a suitable level of description of the universe, then affordancescould be modeled as trajectories in this huge space. The relevant trajectories would beextremely rare in this space, as some specific given real numbers are among reals; thisset might even have measure zero, but still exist. And if the set of uses of an object isuncountable, then we can anyway apply set theory and, in particular, the axiom of choice6or picking elements from the sets, as we do for real numbers.
Rebuttal: the assumption that it is possible to find a suitable level of description ofthe universe means that we are able to define a suitable phase space, and so that wehave identified the relevant observables, symmetries and laws for completely describing theuniverse. But this is the consequence of a deliberate and subjective act of the modeler,who defines what are the most pertinent and interesting parameters and symmetries tobe used for defining the phase space. It is not an objective procedure [15]. This choice ishistorically dependent, as it is based on the universe observed by the modeler at a giventime, and since the universe is non-ergodic and new things and functions come to existcontinuously [3], it is not possible to define a complete phase space for the evolution of theuniverse once and for all. It is of course possible to provide an explanation a posteriori ofan affordance, as we can explain heart once we observe and recognize it in an organism,but not the coming to existence of heart.
Objection
Physicists always use math to devise models of the physical world and arewell aware of their limitations: some features of the evolution of the biosphere can stillbe entailed, knowing that some approximations will be introduced and there will be somediscrepancies between expected and actual results.
Rebuttal: in the case of affordances, we don’t know the sample space for the reasonsstated above, and therefore we cannot define a probability measure, nor can we define whatis random, so we cannot estimate our error [16].
Objection
We can always add to the model new objects and new relations.
Rebuttal: but still what can be deduced by the model cannot be novel, as it is alreadyformalized. Semantics comes first, then comes syntax.
We are often pervaded by a profound sense of wonder when we observe the flourishingand creative power of the biosphere, and not seldom we are also amazed by the way ani-mals and humans find creative solutions to problems or invent and build new tools. Thesephenomena are enabled by affordances, which also constantly change and appear. Sincecenturies we use mathematics to model phenomena of interest, and so provide explanationsto understand what we observe or to control systems and processes. Some intrinsic limitsof mathematical systems have been already raised in the Twentieth century, but mathe-matical models are still successfully applied in a plethora of cases. We would expect to be7ossible to formalize the evolution of the biosphere, to formally model affordances. Nev-ertheless, we face here an inherent limit: the properties of circularity and non-ergodicitycharacterizing affordances defeat any sound applicability of set theory, and consequentlyall math depending on it. In synthesis, we are claiming that the evolution of the bio-sphere, besides the impossibility of being entailed, is inherently not mathematizable: theworld is not an algorithm. (Apologies to Pythagoras, Plato, Neoplatonists, Newton, Bohr,and . . . ) Starting from a discussion on the meaning of words in a language and moving tothe foundation of mathematics, Wittgenstein had to some extent already contended thatmathematical inferences do not bring new knowledge. In
Remarks on the Foundation ofMathematics [26] he indeed maintains that surprise cannot come from an inference—and,if any surprise should come, it is simply because something was not yet understood. Weare aware that our claim might be considered as a negative result that brings shadow andfurther limits the range of mathematical understanding of the reality. On the contrary, webelieve that this incompleteness is a way to be creative [11] and achieve a deeper awarenessof the world: “We are of Nature, not above Nature” [13].
Epilogue
But how many kinds of sentence are there? Say assertion, question and command? Thereare countless kinds; countless different kinds of use of all the things we call “signs”,“words”, “sentences”. And this diversity is not something fixed, given once for all; butnew types of language, new language-games, as we may say, come into existence, and oth-ers become obsolete and get forgotten. (We can get a rough picture of this from the changesin mathematics.) (Wittgenstein, Philosophical investigations, first. publ. in 1953, 4th ed. 2009, Wiley-Blackwell, UK)
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