Acknowledgements
We would like to warmly thank Colin Allen and two anonymous reviewers for addressing important comments, both on the experimental and on the philosophical side, which led us to significantly improve the first version of this article. Gabriele Ferretti has some specific acknowledgments to do. Since the finalization of the work for the publication of the present article has been done during the transition from one research position to another, Gabriele Ferretti would like to acknowledge support from both the corresponding fellowships. This work was supported by a NOMIS Fellowship, awarded by the Eikones-Center for the Theory and History of the Image at the University of Basel, Switzerland. This work was also supported by a Humboldt Fellowship, hosted by Professor Albert Newen at the Institute for Philosophy II, Ruhr-University Bochum, Germany.
1The following is arguably a fundamental question about human cognition: Does natural selection tend to favour individuals whose perceptual systems are epistemically reliable? Over the last few decades, different mathematical models have been used to address this question. It should be noted, however, that our question raises two different, albeit related issues, the first psychobiological and the second epistemological. The former is descriptive in nature and concerns the kind of perceptual strategies available to complex organisms under the pressure of natural selection. The latter is normative and concerns the cognitive evaluation of those same strategies. We will accordingly begin by addressing the epistemological issue raised by Donald Hoffman and his collaborators (henceforth, H & C, see, in particular [Hoffman, Singh et al. 2015], [Mark, Marion et al. 2010], but also [Hoffman, Singh et al. 2013] and [Hoffman & Singh 2012]) and then move on to engage closely with the current debate on perceptual strategies in evolutionary theory.
2We will briefly consider Godfrey-Smith’s approach [Godfrey-Smith 1991] in order to show that it presupposes perceptual accuracy rather than investigating it. We will then discuss a new proposal from H & C, which posits the idea that our perception is tuned to fitness and is therefore completely unreliable. Their approach to this issue will be the main topic of our paper. We intend to show that, contrary to their view, the platitude opining that fitness is one of the most important forces responsible for shaping our perceptual strategy does not imply that our perception is altogether unreliable. What led them astray, we suggest, is the fact that their model fails to take the relevance of environmental modifications into due account. A model that does take these into account, as we will try to show, integrates the idea that the acquisition of apparently useless information about the environment in fact increases fitness, at least up to a certain point. Moreover, according to the same model, even when two organisms make use of the same number of bits, modifications of the environment will in general favour the one that uses its information to target reality rather than utility. As a consequence, perceptual strategies aimed at acquiring reliable information about the environment will in the long run be favoured over those aimed at increasing utility.
3The plan is as follows. Section 2 briefly presents Godfrey-Smith and H & C’s approaches, section 3 sketches a theoretical framework within which our fundamental question can be stated in a philosophically exact manner. Section 4 tackles what is arguably the most powerful argument levelled by H & C against critical realism, namely one based on evolutionary game theory. In section 5 we put forward two alternative models, which assume that fitness covaries with the environment surrounding the cognitive agent. Then, we use our model to show that, in a biologically more realistic situation, knowledge of the objective structure of the world proves to be a rather valuable asset. We conclude by evaluating our results in section 6 and by listing the technical notions and equations we rely on in Appendices A and B.
4In 1991 Peter Godfrey-Smith developed an interesting framework based on signal theory to model the optimal relation between the inner states of an organism and the external world. His approach assumes the reliability of what he calls the “world-head” relation—i.e., the organism’s dispositions to act in response to a given stimulus—and it statically represents the relation between that stimulus and the inner state. The same framework was later generalized and couched in terms of Bayesian decision theory [see, e.g., Knill & Richards 1996]—i.e., by relying on Bayesian conditionalization to model the way in which the mind evaluates hypotheses about the distal stimulus based on its perception of the proximal one. This approach builds on a distribution of an a priori probability of the possible realities of a distal stimulus and an a priori distribution of likelihood of a certain perception with respect to those realities. Then, through Bayes’ conditionalization rule, the a posteriori probability distribution of the hypotheses corresponding to those diverse realities can be established. Within this broader framework, it is possible to study the role played by evolution in perception, as the framework allows for the idea that in the long run organisms will modify their a priori probability distributions in accordance with a certain utility function determined by biological fitness. One could hence try to establish whether this function will produce a priori probability distributions in the long run that would favour perceptual accuracy. Yet, to the best of our knowledge, this line of research has never been systematically pursued and philosophical reflections on the Bayesian approach [cf. e.g., Rescorla 2015] seem to presuppose the accuracy of perception rather than investigating it. For this reason, we will set these proposals aside in this paper and leave scrutiny thereof for a later occasion.
- 1 For instance, [Marr 1982, 340], [Trivers 2011, 2], and [Pizlo, Li et al. 2014, 227]. H & C’s views (...)
- 2 H & C focus on the case of visual perception, as they assume that the same will hold for other sen (...)
5In a series of articles (see, in particular, [Hoffman, Singh et al. 2015], [Mark, Marion et al. 2010], but also [Hoffman, Singh et al. 2013] and [Hoffman & Singh 2012]), Donald Hoffman and his collaborators (henceforth, H & C) have constructed a controversial, yet apparently powerful mathematical case, which gives a negative answer to our question. The currently mainstream view among scientists studying perception is that, in normal conditions, our perceptual representations are largely accurate.1 Had this not been the case—it is usually argued—our species would have long been driven to extinction [Geisler & Diehl 2003], [Yuille & Bülthoff 1996]. This is also true for every kind of biological system. In order to question this widely held assumption, H & C appeal to evolutionary games, which purportedly show that our visual2 systems are tuned to utility rather than objective reality and that we therefore have little or no reason to believe that our perceptual representations are ever, or even broadly accurate. The gist of their evolutionary argument is that individuals whose visual systems integrated this last property would badly lose an evolutionary competition to individuals whose visual representations were tuned to utility rather than objective reality. Their proposal has the undeniable merit of lending empirical meaning to various philosophical views concerning the adequacy and accuracy of our perceptual systems. The proposal is hence deeply thought-provoking both from scientific and philosophical standpoints.
- 3 Despite various attempts at defending a nonrepresentational view of perception [Noë 2004], [Chemer (...)
- 4 We are aware that there are many different philosophical theories of perception. However, as alrea (...)
6In this paper we will take issue with some of the arguments defended by H & C and put forward a new model, whose epistemological implications run counter to those purportedly supported by theirs. Our model suggests that—contrary to H & C’s view—an individual whose visual representations were at least partially accurate would be more successful from an evolutionary point of view. In line with contemporary philosophical theories of perception [Nanay 2013], [Siegel 2006], [Brogaard 2014], [Ferretti & Zipoli Caiani 2019] we assume that perception is a representational process,3 i.e., that it represents reality by ascribing various features to objects as well as to the visual scene as a whole.4 Our main goal is to provide a plausible and empirically testable model of visual perception, standardly conceived of as a process whereby a stimulus causally interacts with our visual system in turn giving rise to a more or less accurate representation of its source, i.e., a perception. It is generally acknowledged that perceptual representations have accuracy rather than truth conditions [Mausfeld 2002], [Siegel 2006, 2010] and [Koenderink 2014]. As a consequence, this paper will discuss the accuracy of our visual representations. It should be noted that the question we are facing is not “whether there is biologically useful information in the environment” as this is assumed to be the case. The important question is in fact “whether it is biologically useful to acquire this information, even if its acquisition is biologically costly”.
7Following H & C partially, let us consider an environment characterized by a set R of features, every subset of which can be a stimulus and can therefore cause a corresponding subset of the set P of possible contented mental states in a biological sensitive being. Then let us consider the best possible theory T of these two aspects of reality. The notion of a “best possible theory” is necessary to face our problem, since we are investigating the accuracy of the representation of the environment by a perceptive biological being. To evaluate this accuracy, we need an effective description of the two relata we are comparing, namely the environment and its representation. Moreover, the supposition of a best theory TR of the stimulus is not a petitio principii. Indeed, it is possible to object that we are investigating the accuracy of perception and assume that we know how the world is constituted but if perception is not at entirely accurate it is very improbable that we could possess this kind of knowledge. Nonetheless the epistemological part of our paper—and the same was true for H & C—is not devoted to establishing a proactive tool to measure the accuracy of perception. We instead prefer to outline a general theoretical framework which works to define exactly the problem we are investigating on. For this reason, presupposing a best possible theory of the stimulus is an idealization which is allowed.
- 5 We assume, for the sake of simplicity, that it is possible to distinguish between the two theories (...)
8T will include two sub-theories—namely TR and TP—which deal with R and P respectively.5 If these two theories are developed enough, we will also have two corresponding state-spaces STR and STP. Roughly speaking, if a theory is based on n features, its state-space is n-dimensional and one of the features is represented on each dimension. The notion of “state-space of a theory” is useful, since the set of possible states of a system—according to the theory—is a sub-region of the state-space. Moreover, it is easy to build a correspondence between the two state-spaces of TR and TP and then evaluate its accuracy.
- 6 Thanks to an anonymous referee, who emphasized this problem.
9We will begin by briefly sketching the way in which H & C set out the problem of evaluating the accuracy of perception. We will then modify it in order to render it epistemologically more appropriate.6 However our modifications will not affect H & C’s main point which will be discussed in the subsequent sections.
- 7 Note that here the term “perceptual strategy” introduced by H & C does not involve any forms of be (...)
10Hence, leaving details aside, we can imagine a representation function F mapping STP regions onto STR ones—i.e., roughly, a model of our capacity to represent a stimulus given a perception—and an inverse causal function Q mapping STR regions onto STP ones—i.e., roughly, a formal representation of the causal link between stimuli and contented mental states. A perceptual strategy7 can now be defined as the composite function FQ mapping STR regions onto STR ones. A given strategy may be dubbed realist if FQrSR=rSR (where rSR is a region of STR)—i.e., if our perceptions perfectly mirror the stimuli that cause the corresponding contented states. As it does not seem plausible to suggest that biological perceptive systems generally implement this kind of strategy, it is useful to define a further strategy that we may call critical realist. A strategy is critical realist if there is at least a subspace of STR within which realism holds—i.e., within which, if rS'R is a region of S’TR and S’TR ⊂ STR’ then FQrS’R=rS’R. If, on the other hand, S’TR = ∅, then the resulting strategy will coincide with what H & C call interface theory.
11H & C distinguish an initial causal process which goes from the stimulus to the experience of the biological system and a further process which goes from the representation of the biological system to reality. In this way the evaluation of accuracy of perception becomes particularly simple, since the stimulus and its representation are in the same state-space. However, this model is not appropriate, because the causal and the representational processes are the same phenomenon seen from two different points of view. The former is seen from the perspective of the mind and the latter from the perspective of the environment. Indeed, there are not two different mental contents—what the biological system perceives and what the biological system considers real—but only one, namely the representation of the object caused by the impingement of the latter in the sense organs of the system. This means that accuracy must be evaluated comparing the TP and the TR state-spaces, that is STP and STR’, not comparing STR before perception with STR after perception, as proposed by H & C, which is a much more difficult task. In order to overcome this problem, we developed a mathematical framework, the details of which can be found in Appendix A. Here we shall briefly illustrate its result. According to the model, it is possible to measure the accuracy of perception of a biological system through the distance between the objective conjunct probability measure on the state-space of the stimuli µRR and the conjunct probability measure induced by the perception on the state-space of sensations µPP.
- 8 See Appendix A for further explanations. In any case, this equation means that representation and (...)
12In this context, it will not be necessary to go into the details of how to measure the distance between these two functions. It is indeed possible to define an inner product between probability measures and then an Euclidean distance. This distance (dRP) can be normalized so that when two measures are the same its value is “0” and when two given elements belonging to an algebra defined on STR xR and yR are such that “μPP(xP, yP ) = 1 − μRR(xR, yR)”8 its value is 1.
13At this point it is easy to establish that a critical realist strategy determines a value of dRP ≤ 0.5, whereas an interface strategy determine a value of dRP > 0.5.
14Going a bit deeper into the question of measuring the distance between µRR and µPP’ it is reasonable to suppose that there are sub-regions of STR such that dRP is almost 1 and others where it is <0.5. We may also suppose there to be cases in which the perception is quite accurate and others in which it is quite inaccurate. If this is true, one can define a “critical realist” perceptual strategy as involving a representative set of cases in which dRP is near 1. It is “representative” in the sense that for each different kind of sensitive nervous terminations there is at least a perceptive normal situation in which dRP is almost 1. Obviously, if this does not hold true then perception follows an “interface strategy”.
15Let us now consider the main argument that H & C rely on in order to show that Homo sapiens implements an interface strategy (henceforth, IF).
16Evolutionary game theory is arguably the best way to predict the evolution of a discrete phenotypic trait whose fitness depends on its frequency in the population [Rice 2004, 263]. Since we are discussing the evolution of different perceptual strategies which can coexist, the fitness of a strategy will depend on the frequency of the other.
17Mark, Marion et al. imagine a simple situation, in which an organism finds itself in an environment divided in three territories [Mark, Marion et al. 2010, henceforth, M & C]. Each territory has one resource, whose quantity takes discrete values from 1 to 100. Utility (which corresponds to fitness) is proportional to the quantity of resource present on the territory. They calculate the payoff for a realist strategy, i.e., a strategy that gathers all the available information, as well as of a critical realist strategy, i.e., a strategy based on only two possible perceptions, according to whether the quantity of the resource is above, equal or below a certain threshold β. In calculating the payoffs, the authors assume that the interface strategy will always provide a choice first, as the realist strategy will take longer to gather information (and the time spent will have a negative weight which is proportional to the quantity of acquired bits). Their calculations apparently show that the payoff for the realist strategy (assuming a cost of 1 per bit) will drive realism to extinction for any value of β. The same would happen if we increased the complexity of an environment. This result was indeed to be expected, given that the realist strategy was never very plausible to begin with as we observed above.
18M & C next consider a game in which 3 strategies compete, namely realism, critical realism and IF. At this point, though, instead of relying on the approach adopted in the preceding case—which could indeed be applied to a 3 × 3 matrix—the authors write:
- 9 This equation can be used to calculate the time derivative of the frequency of a certain trait, ba (...)
[...] unlike in two-player games, we cannot immediately calculate which strategy survives and which becomes extinct. Instead we compute the time derivatives of the frequencies of each strategy using the replicator equation. [Mark, Marion et al. 2010, 513]9
19The replicator equation is known to be of huge importance and influence for the study of the evolutionary system in biology [Cressman & Tao 2014] and [Nowak 2006].
- 10 Every point in the simplex represents a possible value of the three frequencies, whose value must (...)
20The article reports no calculations as such and just a triangular simplex,10 whose vertexes represent populations implementing only one of the three available perceptual strategies. Inner points correspond to mixed populations whose overall dynamics (represented by means of arrows) point toward the interface strategy from most points of the simplex, in case information has reasonable energetic costs.
21It is interesting to analyze the playing field of the three strategies in this case. As in the previous strategy, we have 3 territories and one resource whose values range from 1 to 100. Yet now the utility is a Gaussian with its peak at 50 and is not proportional to the quantity of the resource to be found on each territory anymore. As in the previous case, the realist strategy gathers all the information and the critical realist strategy relies on 3 perceptions (e.g., 3 colours). The same is true for the interface strategy. The difference between the two latter strategies lies in how colours are used (Figure 1).
Figure 1: Critical realist strategy (above); Interface strategy (below). On the x-axis the quantity of resource; on the y-axis the utility. The difference between critical realist and interface strategies is expressed through diverse distributions of colours. The different colours are represented through the diverse types of filling: points, horizontal lines and diagonal lines. It is evident that the use of colours in the interface strategy is more useful—in terms of fitness—than its counterpart in the critical realist strategy.
22The quantity and utility of a resource are represented on the x and y-axes respectively. While the critical realist strategy simply associates the three colours with the increasing quantity of the resource, disregarding the utility curve, the interface strategy only keeps track of utilities. As a consequence, the latter graph is divided in three non-adjacent zones, which track utilities rather than quantity. When quantity and utility are non-monotonically related, each strategy must be charged with the costs associated with gathering information and calculating its corresponding utility. The authors modulate the cost of knowledge to make it comparable to the cost associated to knowledge of utility.
- 11 The authors propose situations in which the stage is different, but they never consider the case i (...)
23This stage has been clearly and intentionally set to put critical realism at a disadvantage, since, by keeping perceptual complexity fixed,11 the interface strategy has a running start. For instance, a situation where IF is disadvantaged was pointed out by Anderson [Anderson 2015]: organisms often tend to homeostasis and hence they have an interest in knowing whether the quantity of a given resource is above or below a certain threshold. In this kind of case, the critical realist strategy would have an advantage over its interface counterpart. However, H & C’s model is biased for a more profound reason.
24To demonstrate the limits of this example, let us now consider the following case. Suppose that the organisms in question are sparrows and the resource little worms. Worms evolve, so we can imagine a random mutation that significantly decreases their size. Since sparrows tend to catch the bigger worms, the new trait spreads rapidly across the population. This considerably alters the ratio between the utility of the resource and its quantity. The sparrow implementing an interface strategy will accordingly still “think” that the same number of worms is needed in order to maximize utility and hence will lag behind in terms of fitness. Its critical realist competitor will “know better” and will accordingly move to an area where either more worms or bigger worms are to be found. We take this case to show that slight modifications in the environment can bring about a serious disadvantage for the interface strategy. Indeed if a perceptual strategy completely tuned with utility in a certain environmental scenario does not also gather some apparently useless information, it is in danger of not “understanding” the importance of possible environment modifications in terms of utility.
25The present section is divided into two parts. In its first part, we consider a case in which two different perceptual strategies—critical realist and interface—have the same number of bits available. This is a situation similar to the one considered in the preceding section. In its second part, we consider a case in which a critical realist strategy collects more information than an interface one.
- 12 In a first version of the paper we neglected this very important case. We would like to thank an a (...)
26In the first situation, the difference between the two strategies is that they use bits in different ways—i.e., whereas the critical realist strategy attempts to at least partially represent the environment, the interface strategy uses information in order to target utility only.12
27Our main point is that, given a static environment, a strategy targeting utility will clearly outcompete one aimed at representing reality. In a case where the environment changes, the opposite will be true however. We will now try to show this by means of a very simple model according to which an interface strategy will initially prevail over a critical realist one and yet this trend will reverse in due time and because of modifications in the environment.
28Let us then consider an organism of genus ω, itself divided into two species ωCR and ωIF which differ in perceptual strategy only. Consider next two environmental features—x and y—interacting in a way such that, as time goes by, their global utility for ω fluctuates. The quantities of x and y will fluctuate as well—i.e., the phases of their oscillations will be such that their overall utility will fluctuate as well.
29Let us imagine a situation similar to that of beats, where the sum of two sinusoids which are partially out of phase (measured by ε) will form a sinusoid with a smaller frequency. The quantity of x and y could, e.g., be given by:
x = a cos t, y = b cos εt.
30We assume that, as time passes without genetic mutations but with the natural cycle of the environment, the objective utility function for ω will be:
31This means that U will indeed oscillate as time goes by. However, it will not be the mere sum of x and y, but rather a smaller quantity which decreases inversely with respect to constant I. While our utility function appears quite specific, it is in fact more general than may seem the case at first sight. In this regard, it should be noted that that U covaries with x and y. It indeed seems reasonable to assume that there will be a function connecting utility to environmental features. U’s third term, in particular, establishes that the connection between x and y is non-linear. This term becomes more relevant as I increases—i.e., when I is high, the difference at time 0 between interface and critical strategy becomes smaller. This means that, in a static environment, an interface strategy will prove more useful. All in all, it seems quite natural to suppose that environmental features will interact in a non-linear way. Imagine, for instance, that an organism’s calorie intake comes either from lipids or from carbs. It seems reasonable to expect that its utility will not be proportional to the mere sum of these two resources, as it will also presumably depend on their relative quantity. Moreover, it is also reasonable to expect that an organism tracking utility has higher fitness in a static environment. On the contrary, in a changing environment, a critical strategy will prove more useful.
32Note also that for t = 0, the following will hold:
33We define UIF (UCR) as the received utility function by ωIF(ωCR), given its peculiar way of encoding information. Let us now suppose that UIF does not change with time and is given by the following formula:
34On the contrary, ωCR knows the value of x and y with the same accuracy as ωIF evaluates a and b. This means that ωCR and ωIF will have the same quantity of bits available.
35Since ωCR knows x and y, it can evaluate a utility function of this kind:
UCR = x + y = a cos t + b cos εt.
36That is the mere sum of the two variables.
37At this point the distances between the real utility and the utility evaluated by the two strategies can be defined in the following straightforward way:
dIF = |U − UIF| and dCR = |U − UCR|.
38Note that at t = 0 UCR = a + b. This means that at t = 0, dCR > dIF t = 0, that is UIF is better than UCR. The crucial questions thus become: What will happen as time goes by? Which strategy will perform better? It is easy to calculate that dIF > dCR for t > 0.
39According to our model, then, when the environment is held fixed and each organism has the same number of bits at its disposal, a strategy aimed at increasing utility will outcompete one aimed at representing reality. As the environment changes, however, the opposite will be the case.
40While our model is admittedly limited in scope, the assumptions upon which it is based seem quite reasonable. We take those assumptions are the following:
- Environment changes.
- Many environmental features display an oscillating pattern.
- Utility is not in general the mere sum of two such features.
- A constant utility function is not appropriate to represent utility in a changing environment.
- Knowledge of the environmental features’ behaviour, while itself insufficient to locate the real utility function, nonetheless seems a reasonable starting point to assess utility in a changing environment.
41In light of the above, we maintain that generally in a changing environment a critical realist perceptual strategy will outperform an interface strategy, even in a case in which two organisms have the same number of bits available.
42Let us now move on to the second situation—the one in which the critical realist strategy collects more bits than the interface one.
- 13 The relation between an organism and its environment is an extremely complex subject matter. On th (...)
43Most mathematical literature on evolution focuses on the way in which a phenotypic or genotypic trait evolves given a certain fitness [Rice 2004]. This can change on some models over time due to modifications in the environment [Gillespie 1991, chap. 4], whereas our goal is to investigate the impact of environmental changes13 on fitness variations of a biological system. To our knowledge, no mathematical studies in this direction have been carried out to date. Our very simple model focuses on the variation of fitness due to changes in the environment, without modifying genotype and phenotype of the population. To simplify our calculation, we presuppose asexual reproduction and no mutations. Moreover, the environment can change discretely generation by generation.
- 14 Number of offspring.
- 15 We are aware that this model seems to reveal nothing about how knowledge of fitness-relevant featu (...)
44Let us consider a set {ωK}K=0,...,N of populations differing only in their capacity to acquire knowledge from the environment. Imagine that a population ωK of a certain species at generation i has absolute fitness14 wi. This population lives in a complex environment that can be modelled by means of N possible, biologically relevant dichotomic changes concerning a certain set of given environmental features. Obviously, the number of features an organism can collect is smaller than N and the relation between N and this number is given by the discrete capacity of environmental discrimination of ωK. No mutation occurs in the sequence of generations. Suppose that ωK has a quantity K(0 ≤ K < N) of information about the features—equivalent to K bits—and that acquiring K information about each one of them has an increasing cost Kc/(N − K) in terms of fitness, where c is an average unit of cost. We use this kind of cost because when ωK acquires little information the cost is low but when the information obtained grows the cost increases sharply due to the excessive time employed.15 In this case, the cost associated with the acquisition of information gets bigger and bigger. When K goes to N, the harm suffered by ωK tends to 0 but increases probabilistically as K gets smaller. On the other hand, if K increases, the cost for gathering information becomes bigger and bigger; if K = N, the cost would be infinite.
- 16 Each change could be represented by 1 bit.
- 17 Here we follow [Eldredge 1998], according to whom important changes in the environment are a funda (...)
45After a generation, the environment will have undergone a certain number of changes,16 only an amount f of which are biologically relevant. Let us assume—plausibly enough—that most of these changes are either irrelevant or harmful, hence, in particular, most changes that have an impact on fitness will be slightly harmful.17 The probability distribution that ωK lacks information about each and every change will be uniform. Since “N − K” is the number of bits unknown by ωK and the whole number of relevant information is N, then the uniform distribution gives (N − K)/N = 1 − K/N for each information. Moreover we assume that the newborns have to gather all knowledge useful for their fitness obtained and also that is there is no costless transmission of knowledge from one generation to the following. Let’s say that the average negative impact of these changes on fitness is w−. Therefore, the fitness wi+1 of generation i+1, given the fitness wi of generation i, is given by the following difference equation (1):
46wi+1 = wi − f(1−K/N)w− − Kc/(N − K). (1)
47Kc/(N−K), that is the third term of the right part of (1), is the cost for population ωK of gathering K bits of information about its environment, whereas the second term is the biological damage for ωK due to its ignorance. This equation should tell us which value of K in ωK will prove advantageous over generations, in a changing environment. By solving the equation, (the details of which can be found in Appendix B), we define Kmax as
48
(2)
- 18 An anonymous referee criticized our choice only to consider extinction. Yet, as is well known, 99% (...)
49Kmax (2) establishes the best value of K in order to postpone extinction.18 To understand whether our model is a good argument favouring a critical realist perceptual strategy, we must compare (2) with our epistemological definition in the preceding section. The distance dRP between the conjunct probability measures on STR and STP depends on the number of faithful causal connections imposed by Q. We assume that the faithfulness of perception of one feature is independent from that of another. This is reasonable, since there are no clear reasons for connecting the faithfulness of knowledge of different features. Let us assume that the probability distribution of ignorance on information is uniform. Then the probability of not knowing a piece of information about a feature is given by (1 − K/N). Therefore the rate of success of conjunct probability measures in STP is proportional to K2/N2. In this case dRP is equal to zero because the two measures, µPP and µRR’, are the same; otherwise dRP is proportional to 1 – K2/N2. Now, when returning to (2), we note that if f, the number of harmful environmental changes, is high, then Kmax tends to N, that is K2/N2 tends to 1 and the distance dRP tends to 0, that is a critical realistic perceptual strategy is favoured from an evolutionary point of view. To sum up, among the {ωK}K=0,...,N populations we investigated, the ω~N (the population with K ≈ N) is favoured. And the ω~N population has a critical realist perceptual strategy.
50Clearly this result is only partial. In a biologically more realistic situation, one would also have to consider the non-deleterious aspects of the environment. Moreover, objective laws of the environment are not constituted only by association of features, but rather as more complex relations. Finally, different organisms could deploy the usefulness of the acquired information in various degrees. A much more complete study would be needed to obtain more conclusive proof that accuracy is a good goal to pursue from a selective point of view.
51If perception is the only way to acquire information about our environment and perception turns out not to be even partially accurate, then investigating Homo sapiens and its environment would amount to inspecting our subjectivity. Yet modern science’s moral and cognitive mission also consists of pursuing fallible and revisable attempts at formulating justified hypotheses about Homo sapiens, its origins and the world it inhabits. Many cultural milieus encourage the idea that empirical science cannot make any justified claims about the external world, independently of the way in which that world is perceived or thought of. If perception were completely inaccurate, this idea would be reinforced. Whether and the extent to which human perception is an accurate representation of the world is an epistemological matter which can be empirically investigated at least indirectly by using evolutionary mathematical models. We showed the limits of H & C’s attempts at establishing the negative impact on fitness of an accurate representation of the world. We also used a simple model to show that, in a more plausible biological situation, the at least partial accuracy of an organism’s perceptual representations proves evolutionarily successful.
52Our model is clearly only in sketch form at this stage and certainly requires further development. Indeed, we are confident that, given reasonable assumptions concerning what should count as an accurate perception, it should be possible to empirically investigate the comparative fitness of different perceptual strategies along the lines suggested by H & C. However we also believe that such investigation should carefully take into account modifications in the environment.
53We can suppose that TR is a physical theory and TP a theory from the psychology of perception. In establishing this comparison between STR and STP we will have two problems: The first is that the features of TR—wave-lengths, masses, objective distances, etc.—are completely different from the features of TP—colours, weights sensations, perceived distances, etc.; the second is that we must find a criterion to connect causally a point of STP to a point of STR. In other terms, how can we know that a certain change in colour seen by a biological system is the effect of a certain change in the world?
54Before confronting these difficulties, we establish that in our setting, a perceptual strategy is simply the causal relation between STR and STP which is the mapping Q from sub-regions of STR onto sub-region of STP. R is the part of reality which is perceptually relevant for a given biological system. Therefore the dominion of Q is all STR. Clearly Q is not one-to-one, since two different stimuli can be perceived as equal. Moreover Q is not onto either, since the biological system can perceive something that is not real (hallucination).
55As is the case in H & C’s framework, from our perspective it is easy to define what is direct or naïve realism. Q is a direct realistic strategy if it is a mapping from TSR to TSP only for a very small sub-region of STR and for the most part of STR it is an identity in STR. Like H & C, we assume that this is not a biologically reasonable perceptual strategy.
56To investigate more realistic perceptual strategies, we assume that on STR it is possible to establish a σ-algebra AR, that STR has a good structure to be a sample space of a probability measure. Let us call the latter . Now, take two generic events belonging to AR and let us call them xR and yR. μR(xR) and μR(yR) are both probability measures, therefore also:
µRR = µR(xR ∧ yR) = µR(xR) µR(yR/xR) = µR(yR) µR(xR/yR)
is a probability measure.
57Let us assume that Q applied to AR imposes a σ-algebra AP on STP. Now, let us consider the probability distribution μP on this imposed algebra. If xR and yR belong to AR, let us define xP = Q(xR) and yP = Q(yR) respectively. If μR(xP) and μR(yP) are probability measures, then also:
μPP = μP(xP ∧ yP) = μP(xP) μP(yP/xP) = μP(yP) μP(xP/yP)
is a probability measure.
58Now, a brief explanation is in order. The probability distribution μRR is a good representation of how couples of real features behave insofar as μRR at least describes partially what happens objectively. It is clear that to have a more complete objective description of stimuli we should consider more complex aggregates of features and not just couples of these. However in this context, to consider only couples is enough to clarify the epistemological problem we are working with.
59“Q” answers to our second problem, i.e., Q establishes the causal connections between events in STR and events in STP.
60On the contrary, to solve our first problem, namely the inhomogeneity between STR features and STP features, we compare the two conjunct probabilities distributions μPP on STP and μRR on STR rather than xP with xR and yP with yR . It should be noted that µPP is induced on STP by Q applied to events in STR.
61To solve the equation (1), we get (3):
wi = w0 + im (3)
62where m = −fw−(1 − K/N) − Kc/(N − K) and w0 is the initial fitness of ωK.
Figure 2: On the x-axes the number of generations; on the y the fitness. Representation of the family of intersecting straight lines of equation (1). More iest are drawn to show how their value can vary when K increase or decrease.
63Representing the equation’s solutions as m’s slope changes (Figure 2), we get a family of intersecting straight lines that pass through the point P(0, w0). Step i corresponds to ωK’s extinction (call it iest). This result, which is obtained when wi = 0, has the general form:
64
(4)
65The bigger m is, the greater the extinction step. The extinction values (4) covary with K. Remember that we are considering a set of populations with different rates of knowledge represented by the number K. Notice that the value of m will certainly be non-positive insofar as the non-negative values of f, w−, c, N and K vary. This happens to be why we are concerned only with harmful environmental modifications for the sake of simplicity. Moreover, m cannot obtain the full range of negative values, but only those obtained by letting K vary from 0 to N, i.e., we consider only those populations having from 0 knowledge to all relevant knowledge. Reordering the expression for m, we get (5):
66
67We can now see that m increases as K gets bigger if:
68
assuming c ≤ fw− to have a value of K ≥ 0.
Figure 3: The values of iest which interest us are those between K=0 and K corresponds to the maximum value of m(Kmax) (the circled segment).
- 19 Which is the value which allows interface to win in M & C.
69If we set19 c = w−, we see that m’s values get bigger for K if:
70
71it should be noted that 1 ≤ f ≤ N, i.e., we assume that there is at least 1 deleterious variation of the environment. Hence, if f increases, Kmax (2) tends to omniscience (K → N), i.e., the values of K for which m increases are all except N. Recall that as m increases, iest gets bigger. On the other side, if f is small, K → 0, i.e., it is not relevant for the fitness of ωK to gather information. This means that, if the environment changes significantly, it pays to have information. For instance, if at least 1 out of 10 changes in the environment is harmful for a given organism, information, even if partial, will be convenient. Indeed, in this case we have:
72
73One can see that if N—i.e., the number of biologically relevant information of the environment—is big—as is plausible—the values of K for which iest increases are very high with respect to N.
74To sum up, the number of harmful environmental changes will be very high for fairly complex organisms and it will thus be reasonable to suppose that ωK with values of K≈N is favoured, i.e., populations with a good knowledge of their environment. What our simple model shows, then, is that a population with a high K will win out in a changing environment.
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