# Fact, Fiction, and Fitness

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## Abstract

**:**

## 1. Introduction

[Psychophysics] anticipates the discovery of general laws relating the sensations to physical attributes. That is, the measured sensations are expected to correspond systematically to the physical quantities that give rise to them.

Here “objective” means independent of any observer or observation: an “objective aspect of the visual world,” is a structure or state of the OIW. The philosopher Jerry Fodor was adamant that [20]:Definitely do compute explicit properties of the real visible surfaces out there, and one interesting aspect of the evolution of visual systems is the gradual movement toward the difficult task of representing progressively more objective aspects of the visual world.

Fodor is here using “true” in the sense of correspondence referred to above. The cognitive scientist Zygmunt Pizlo concurs that [21]:There is nothing in the ’evolutionary,’ or the ’biological’ or the ’scientific’ worldview that shows, or even suggests, that the proper function of cognition is other than the fixation of true beliefs.

Veridicality is an essential characteristic of perception and cognition. It is absolutely essential. Perception and cognition without veridicality would be like physics without the conservation laws.

OK, so (radical) embodied cognitive scientists can be realists. That is, they can believe that there is an animal-independent world, and that some of our perceptions and thoughts get it right.

## 2. Natural Selection

Similarly, the psychologist Roger Shepard proposes that evolution shaped our senses to internalize various regularities of the external world. In his article “Perceptual-cognitive universals as reflections of the world” he claims [29]:Our sensory systems are organized to give us a detailed and accurate view of reality, exactly as we would expect if truth about the outside world helps us to navigate it more effectively.

It is worth noting here that the assumption of an OIW underlies all of these statements.Natural selection has ensured that (under favorable viewing conditions) we generally perceive the transformation that an external object is actually undergoing in the external world, however simple or complex, rigid or nonrigid.

Looked at from an evolutionary point of view, the principal function of nervous systems is [...] to get the body parts where they should be in order that the organism may survive [...] Truth, whatever that is, definitely takes the hindmost.

Later he concedes, however, that “we do have some reliable notions about the distribution of middle-sized objects around us” [32]. It is now widely understood that the primary selective forces in human evolution, at any rate, are social [33]. The “world” to which human perceptions are adapted is, therefore, not just the presumed OIW, but is also a world of other experiencing organisms. While the social character of the human world is often explicitly acknowledged (e.g., by Trivers [28]), the OIW is still regarded as the “ground truth” by theorists of veridical perception.Our minds evolved by natural selection to solve problems that were life-and-death matters to our ancestors, not to commune with correctness.

## 3. Evolutionary Games

## 4. Four Theorems

**Total Orders Theorem**. The number of admissible payoff functions that are homomorphisms of total orders is $2\left(\genfrac{}{}{0pt}{}{n+m-2}{m-1}\right)$. Thus for any fixed m, the ratio between admissible homomorphisms of total orders and admissible payoff functions goes to zero as n goes to infinity. Additionally, even if we let m increase at the same rate as n, e.g., $m=n$, the ratio still goes to zero.

**Proof.**See Appendix A.2.

3D symmetrical shapes of objects allow us not only to perceive the shapes, themselves, veridically, but also to perceive the sizes, positions, orientations and distances among the objects veridically.

**Permutation Groups Theorem**. The number of payoff functions that are morphisms of the symmetric group, ${S}_{n}$, is $2n+n!$ Thus the ratio of these to all admissible payoff functions is $\frac{2n+n!}{{n}^{n}-{(n-1)}^{n}}$, which has limit 0 as $n\to \infty $.

**Proof**. See Appendix A.3.

**Cyclic Groups Theorem**. The number of payoff functions that are homomorphisms of the cyclic group is $(m,n)$, the greatest common divisor of m and n [45]. The ratio of the number of cyclically homomorphic functions to admissible functions goes to zero as n goes to infinity and $m\le n$.

**Proof.**See Appendix A.4.

**Measurable Structures Theorem**. Suppose the measurable structure on W has order k and is neither trivial nor discrete. Additionally, suppose that the measurable structure on V is not trivial. Then the number of measurable functions is bounded by ${m}^{k-1}+{\left(\frac{m}{m-1}\right)}^{k-1}{(m-1)}^{n}.$

**Proof**. See Appendix A.5.

## 5. Discussion: Does Natural Selection Favor Veridical Perceptions?

- We use the counting measure to prove that the probabilities of homomorphisms are zero. One might argue that this is the wrong measure. The main reason for using counting measure is that it is the canonical unbiased measure on finite sets of payoff functions. Proposing any specific biased measure would need careful explanation of why the logic of natural selection dictates this specific biased measure. We believe, however, that this burden cannot be met.
- The conclusions of our proofs are immune to the objection, “You cannot say whether something is veridical or not without first knowing what it is saying.” This objection assumes a representational account of perception, which is not required by our proof. Moreover, this objection is false on its face: an error-correcting code detects that a message received is not a veridical copy of the message sent, without knowing what the message is saying.
- One might wonder whether the theory of evolution can be an impartial arbiter in the debate over whether natural selection entails veridical perceptions. After all, does the theory itself not simply assume the veridicality of certain perceptions, such as organisms, species, physical resources, and (using some laboratory assay) DNA? How could the theory conclude against veridicality without refuting itself? This quandary has a simple solution, however. There is an algorithmic core to evolution by natural selection—variation, selection, and retention—which requires no commitment to DNA, organisms, and other such claims about the structure of the world. This algorithm, popularized as “Universal Darwinism,” applies to the evolution of organisms, but it has been speculated that it even applies to the evolution of art, music, memes, language, and social institutions [46,47].
- Our argument is based on evolution by natural selection. One can object that evolution is affected by many other factors—including genetic drift, pleiotropy, linkage, and constraints from physics and biochemistry—and that natural selection plays a relatively minor role.However, the standard evolutionary argument for veridical perceptions is that accurate perceptions are fitter, which is an argument from natural selection. To our knowledge, there are no arguments for veridical perception based on genetic drift, pleiotropy, linkage, or constraints from physics and biochemistry. Such arguments seem unlikely. It is hard to imagine how neutral drift, for instance, could favor veridical perceptions.
- Our argument focuses on just four structures: total orders, symmetric groups, cyclic groups, and measurable structures. There are, of course, many other structures relevant to perception, such as topologies, metrics, and partial orders. These structures also need to be studied, to see whether they are preserved by payoff functions. Ideally, one can hope for a general theorem, perhaps using category theory, that specifies all structures that are not preserved and thus not veridically perceived.
- One might object that many payoff functions are close to being homomorphisms of the structures of the world in, say, the sense of a ${L}^{2}$ norm, and thus that natural selection will shape perceptions to be close to veridical, if not precisely veridical. We reply that they will also be close to being homomorphisms of countless other structures that are not in the world, and thus that natural selection will equally shape perceptions to be close to countless non-veridical structures. There is no argument here for natural selection favoring perceptions that are close to veridical rather than close to countless non-veridical possibilities.

## 6. Conclusions

These brilliantly colored and kinetic visions…are immediate and vivid…I work using just one ’sense trigger,’ such as sound…listening to only one selection of music at a time, played over and over again until the painting or sculpture is finished. A work need not be completed in one day provided I listen exactly to the same music when I return to work.

Evolution is likely not done with the perceptual interface of Homo sapiens. It is still tinkering. Here we see the data structures of physical objects given novel use in hearing and taste. This application is clearly not veridical. Ball bearings are not a veridical presentation of Karo syrup; ivy is not a veridical presentation of angostura bitters. The physical objects that we normally see when we open our eyes are, no less than these synesthetic objects, non-veridical data structures. They are just satisficing solutions to the problem of compressing and presenting fitness information for action, planning, and reasoning.When I taste something with an intense flavor, the feeling sweeps down my arm into my fingertips. I feel it—its weight, its texture, whether it’s warm or cold, everything. I feel it like I’m actually grasping something. Of course, there’s nothing really there. But it’s not an illusion because I feel it.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

OIW | Observer-independent physical world |

ITP | Interface theory of perception |

## Appendix A. Proofs

#### Appendix A.1. Definitions

**Notation**

**A1.**

**Definition**

**A1.**

**Quasi-Definition**). A (“first-order ”) homomorphism of the same kind of structure in V and W is a function $f:W\to V$ that preserves this structure.

**Definition**

**A2.**

**Definition**

**A3.**

**Definition**

**A4.**

#### Appendix A.2. Total Orders Theorem: Counting Functions that are Monotonic, i.e., First-Order Homomorphisms Preserving (or Reversing) Order

**Lemma**

**A1.**

**Proof.**

**Theorem**

**A1.**

**Proof.**

#### Appendix A.3. Permutation Groups Theorem: Counting Functions Preserving Symmetry under the Symmetric Group S n

**Lemma**

**A2.**

**Proof.**

**Lemma**

**A3.**

**Proof.**

**Lemma**

**A4.**

**Proof.**

**Theorem**

**A2.**

**Proof.**

#### Appendix A.4. Cyclic Groups Theorem: Counting Functions Preserving Cyclicity on a Finite Group; or Periodic Functions on a Lattice

**Definition**

**A5.**

**Theorem**

**A3.**

**Proof.**

#### Appendix A.5. Measurable Structure Theorem: Counting Measurable Functions, that is, (Backward) Homomorphisms Preserving Algebra or Partition Structure

**Definition**

**A6.**

**Definition**

**A7.**

**Lemma**

**A5.**

**Proof.**

**Proposition**

**A1.**

**Proof.**

**Definition**

**A8.**

- 1.
- The collection of subsets in the partition corresponding to the algebra $\mathcal{W}$ on W will be termed the base of the algebra and will be written as $\{{W}_{1},\dots ,{W}_{k}\}$.
- 2.
- The order of an algebra is the number of sets constituting its base. For example, the order of the trivial algebra is 1, and the order of the discrete algebra is the size of the underlying set.
- 3.
- The characteristic of an algebra is the multiset giving the sizes of each of the elements ${W}_{i}$ of the base: we say that the characteristic is ${\{{m}_{i};{l}_{i}\}}_{i}$ if there are ${m}_{i}$ subsets of size ${l}_{i}$. Thus ${\sum}_{i}{m}_{i}{l}_{i}=n$, where n is the size of W. (In other words, the characteristic of an algebra is a partition, in the usual sense, of the number n. Saying that two algebras have the same characteristic is an equivalence relation on the collection of algebras: either algebra can be obtained from the other by a simple renumbering, or permutation, of the set W.)

**Lemma**

**A6.**

**Proof.**

**Claim 1**: Any algebra of order k can be obtained from any given one by means of a finite sequence of steps of the above kind.

**Proof.**

- If at least one of a and b belong to multi-element sets of ${\mathcal{A}}_{1}$, we can perform a basic move, as above, to bring them into the same basic set of the new algebra ${\mathcal{A}}_{1}^{\prime}$.
- If both of a and b belong to singleton sets of ${\mathcal{A}}_{1}$, then because of the non-discreteness, there is another base set C with 2 or more elements. Pick an element $c\in C$. Make these elements companions by performing three basic moves in sequence, as follows:$\left\{a\right\},\left\{b\right\},C\dots \mapsto \left\{a\right\},\{b,c\},C\setminus \left\{c\right\}\dots \mapsto \{a,b\},\left\{c\right\},C\setminus \left\{c\right\}\dots $Of course, $C\setminus \left\{c\right\}$ is not empty, so these moves preserve the order of the new algebra.

**Claim 2**: If $\mathcal{W}$ and ${\mathcal{W}}^{\prime}$ have the same order, then the number of $\mathcal{W}/\mathcal{V}$ measurable functions is the same as the number of ${\mathcal{W}}^{\prime}/\mathcal{V}$ measurable functions.

**Proof.**

**Lemma**

**A7.**

**Proof.**

**Theorem**

**A4.**

**Proof.**

- (i)
- $1\in A$. Then ${f}^{-1}\left({V}_{1}\right)={W}_{1}\cup C$ for $C\subset \{n-k+2,\dots ,n\}$ and ${f}^{-1}\left({V}_{2}\right)={C}^{\prime}$, where ${C}^{\prime}$ consists of the remaining elements of W: i.e., ${C}^{\prime}:=\{n-k+2,\dots ,n\}\setminus C$.
- (ii)
- $1\notin A$. Then ${f}^{-1}\left({V}_{1}\right)={C}^{\prime}$ for $C\subset \{n-k+2,\dots ,n\}$ and ${f}^{-1}\left({V}_{2}\right)={W}_{1}\cup C$.

**Corollary**

**A1.**

**Proof.**

**Remark**

**A1.**

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**Figure 1.**Assignments of fitness payoffs: (

**a**) Fitness payoff is a linear function of the amount of stuph. (

**b**) “Veridical” sensory map that is homomorphic to this function. (

**c**) “Non-veridical” sensory map that is not homorphic to this function. It is less fit than the sensory map shown in (

**b**).

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Prakash, C.; Fields, C.; Hoffman, D.D.; Prentner, R.; Singh, M.
Fact, Fiction, and Fitness. *Entropy* **2020**, *22*, 514.
https://doi.org/10.3390/e22050514

**AMA Style**

Prakash C, Fields C, Hoffman DD, Prentner R, Singh M.
Fact, Fiction, and Fitness. *Entropy*. 2020; 22(5):514.
https://doi.org/10.3390/e22050514

**Chicago/Turabian Style**

Prakash, Chetan, Chris Fields, Donald D. Hoffman, Robert Prentner, and Manish Singh.
2020. "Fact, Fiction, and Fitness" *Entropy* 22, no. 5: 514.
https://doi.org/10.3390/e22050514