A theory of intelligence: networked problem solving in animal societies

09/01/2009 ∙ by Robert Shour, et al. ∙ 0

A society's single emergent, increasing intelligence arises partly from the thermodynamic advantages of networking the innate intelligence of different individuals, and partly from the accumulation of solved problems. Economic growth is proportional to the square of the network entropy of a society's population times the network entropy of the number of the society's solved problems.

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1 Introduction: The General Collective Problem Solving Capacity Hypothesis

In this article, I consider the effects of networking on the emergence of intelligence in individuals and societies. The following hypothesis promotes and sustains this investigation:

The General Collective Problem Solving Capacity Hypothesis.

Society possesses a general, collective problem solving capacity.

The General Collective Problem Solving Capacity Hypothesis implies that the same general problem solving capacity that society uses, for example, to develop language, is used to solve problems in mathematics, science, business, musical composition and performance, sports contests, social interactions, politics and daily life. “All life is problem solving” [47]; all problem solving is a strictly analogous process.

Let’s adopt some notational conventions that will allow us to make the observations in the discussion that follows more precise. The formulas used in the definitions are sometimes modified by a subscript relevant to the context in which they are used.

Problem solving definitions.
  1. : a set of solved problems, as of time .

  2. : the set of solved problems that have been learned by an individual, or by a single problem solver in a network of problem solvers, as of time .

  3. or : the set of solved problems that have been learned by the average individual in a society, as of time .

  4. : a theoretical construct, representing the set of solved problems that an individual, who did not have the benefit of any social networking or language, would know, as of time . In past times, such an individual, with the benefit of living in a familial group but with only a rudimentary language, would be said to be living in a ‘state of nature, or that imaginary state, which preceded society’ [30, Book III, Part II, Section II, p. 501].

  5. : the set of all of a society’s—a social network’s—solved problems, as of time .

  6. : the set of language problems solved by a society, as of time .

  7. : the set of lexical problems solved by a society, as of time .

  8. Lex(t): the set

    of words in a lexicon, as of time

    .

  9. : the number of words in a lexicon, as of time .

  10. or : the number of solved problems in a set , as of time .

  11. : the total number of problems, solved and unsolved, as of time .

  12. : a percentage rate of increase per decade in the number of solved problems for some set of solved problems . For a single period of time, ,

    (1)

    or

    (2)
  13. : a percentage rate such that

    (3)
  14. from year to year is the number of decades, , the number of thousands of years, , or the number of years, as indicated by the context.

The hypothesis raises the

General Collective Problem Solving Capacity Problem.

Is there any way to prove The General Collective Problem Solving Capacity Hypothesis—that is, to show that the same problem solving capacity of a society is used to solve all types of problems?

We want to find a common attribute of different kinds of collectively solved problems that will demonstrate that society has a general, collective, problem solving capacity. Any such attribute should be objectively measurable; otherwise objective comparison is not possible. To narrow our scope of inquiry into what the common attribute might be, we consider how to characterize a problem solving capacity.

If Society A solves twice as many problems in a year as Society B, Society A has demonstrated, literally, twice the problem solving capacity—solved problem productivity—of Society B. Let a subscript indicate a set of solved problems pertaining to Society A and let a subscript indicate indicate a set of solved problems pertaining to Society B. Here, we consider the increase,

(4)

over a period of time, in the existing number of solved problems for a Society A, as indicative of Society A’s problem solving capacity. If the number of a society’s solved problems remained unchanging from one year to the next, then that society would not have a problem solving capacity, just an unchanging store of solved problems.

Suppose, in hypothetical circumstances, that the population of a human society is unvarying—constant—and that the proportion of the population that are problem solvers in the society is constant. Then, even if the number of problem solvers affects a society’s solved problem output, in these circumstance the number of problem solvers is not a factor. If Society A has times the problem solving capacity of Society B during the same period of time, then, adapting (4),

(5)

The ratio between the left and right sides of (5), of the number of problems solved in two different societies during the same period of time, is maintained for the rate at which problems are solved, that is,

(6)

If society has a general problem solving rate, then if its general problem solving rate results in increasing the number of solved problems of one kind, then the number of solved problems of all other kinds should also increase at the same rate. If we can identify different kinds of general collective problems for which the problem solving rates are the same, then we have evidence in favor of—and consistent with—the existence of society’s general collective problem solving capacity.

In light of the foregoing discussion, we have found one possible common attribute that we can use to test our hypothesis that society has a general collective problem solving capacity: the rate of collective problem solving.

2 The plausibility of a general collective rate of problem solving

2.1 General considerations

Suppose the capacity of any individual to solve any particular kind of problem is an instance of that individual’s general problem solving capacity. Then one can infer that the same applies to society, that is, that the capacity of a society to solve a particular kind of collective problem is an instance of society’s general collective problem solving capacity. For if individuals apply the same average amount of energy to solve an average problem, then society collectively will apply the same average cumulative amount of energy to solve an average collective problem.

There is accepted evidence that implies that individuals use the same average amount of energy to solve the average problem—that individuals have a general problem solving capacity—that is characterized by experts in the field of intelligence testing as a general intellectual proficiency. The

“tendency shared by most individuals to perform many different intellectual tasks at about the same level of proficiency …has been demonstrated repeatedly in statistical analyses of the interrelationships of performances on tests measuring different intellectual functions …” [35, p. 22].

What has been demonstrated for individuals necessarily applies with greater force to society. In a society, differences in the problem solving capacities of individuals—for some the capacity is higher, and for others the capacity is lower, than average—offset each other. The larger the sample of problem solvers, the more likely it is that the average amount of energy spent by the average individual problem solver in solving an average problem is a good estimate of the society’s average amount of energy spent per individual solving the average problem. Then, for collective problems, the cumulative amount of energy required should be the same for different kinds of problems. This is a consequence of

The Law of Large Numbers

 when applied to the number of problem solvers. The larger the sample of problem solvers, the more likely it is that the estimated average individual problem solving capacity equals the actual average individual problem solving capacity for the entire society:

“…as the number of observations increases, so the probability increases of obtaining the true ratio between the number of cases in which some event can happen and not happen, such that this probability may eventually exceed any given degree of certainty”

[4, p. 328].

In principle, it should be possible to estimate the actual amount of energy used by the average problem solver to solve the average problem, from the point of view of the problem solvers.

The second reason why using average problems as data is likely to reveal society’s average collective problem solving rate involves the thermodynamics of problem solving.

If it requires a certain amount of energy to solve a problem, then to solve a set of problems requires a certain total amount of energy. If we assume that the number of problems solved by a society is proportional to the average amount of energy used to solve each problem, then—considering a solved problem as equivalent to information—

(7)

If we denote the quantity of information for Subject # 1 by , and the amount of energy used to create that information by , and similarly for Subject # 2, the proportionality of information and energy may be expressed as

(8)

We might put it this way: Sets of solved problems containing the same amount of information required the same amount of energy to solve them. If

(9)

for sets of information, then

(10)

for all and , where (9) holds.

The equivalence of ratios in (8) resembles the form of the relationship between heat and absolute temperature in thermodynamics,

(11)

where, in an ideal heat engine,

  • represents an amount of heat from a heat source added to the heat engine’s working substance,

  • represents the absolute (higher) temperature of that heat source,

  • represents the amount of heat removed from the heat engine’s working substance and added to the engine’s heat sink, and

  • represents the (lower) absolute temperature of the heat sink.

(A good account of the ideal heat engine is in [17, Ch. 44, Vol. 1].)

If the average problem solver has an average general problem solving capacity, then it must be that the same energy input can achieve, regardless of the nature of the problem, the same quantity of information output. That would imply that the analogy of the equality in (8) to the equality in (11) is exact. Whether society has an average general collective problem solving capacity is equivalent to asking whether (8) is true for different kinds of information obtained by general collective problem solving.

If a society as a whole confronts a particular problem, the society is more efficient if it diverts its problem solving energy resources to solving that problem only until the payoff—the benefit of obtaining a solved problem—for the energy required to solve the problem matches the payoff for alternative uses of society’s problem solving energy resources. If a society is adaptive, the competition among problems for the society’s finite problem solving energy resources should result in the same average level of solved problem productivity for different kinds of collective problems that have the same information output.

New solutions of problems, and improvements in existing solutions, are inventions—conceptual inventions—by problem solvers. Problems that confront society compete for society’s problem solving energy resources. Kenneth Arrow remarked that

“Invention is here interpreted broadly as the production of knowledge. From the viewpoint of welfare economics, the determination of optimal resource allocation for invention will depend on the technological characteristics of the invention process and the nature of the market for knowledge” [2].

Society collectively determines the ‘optimal resource allocation’ for the solving of its problems—how to allocate its resources to create knowledge.

The efficiency criterion for the allocation of problem solving resources implies that problem solving outputs for different kinds of problems should be, on average, proportional to their energy inputs, as in (8). In principle, there should exist an average amount of energy per average general collective problem, from the point of view of the collective problems.

Since the number of solved problems is finite and in principle, enumerable, we can number the solved problems sequentially, using 1, 2, 3, …, N; the subscript i pertains to a problem’s assigned number. Each solved problem, required a finite number of energy units to solve it; a standard energy unit, , is selected so that when we add up the total amount of energy required to solve all solved problems, the total amount of energy is

(12)

The average energy per solved problem is

(13)

Therefore, if there are a finite number of solved problems which each required a finite amount of energy to solve, then it must be possible in principle to calculate the average amount of energy that was required to solve the average problem. For there to be the possibility of calculating an average, it is necessary that the total number of solved problems, and the total amount of energy required to solve them, both be finite. A human society and its store of solved problems satisfy those requirements.

Human beings, for example, consume a finite amount of food, which supplies a finite amount of energy, during their lifetime. If (1) the average person consumes 2000 calories per day, (2) the proportion of daily calorie intake devoted to problem solving is known, and (3) if the number of problems solved over a period of time is known, then the average amount of energy used to solve the average problem could in principle be calculated. Similarly, the average amount of energy used to transmit the average amount of information from one individual directly to another could, in principle, be calculated.

All of these considerations imply that it is plausible that a society has a general collective problem solving rate.

2.2 Analogical generalizations

A model that successfully describes general collective problem solving for human societies should apply as well to general collective problem solving by other animal societies; general collective problem solving by human societies is a paradigm for general collective problem solving for all societies of animals. Studying collective problem solving in human societies has advantages over studying problem solving in other animal societies. Statistics about changes in problem solving capacities and economic productivity are more numerous and more available for societies of human beings than for societies of other animals.

We can study the emergence of general collective problem solving in human societies,

  • as an analogy for emergent networked processes generally,

  • as a paradigm for networked problem solving in animal societies, and

  • as a phenomenon consistent with the concepts and theories of thermodynamics.

3 Estimating a problem solving rate

3.1 Rate estimation

In this section, we consider some aspects of finding a problem solving rate. We assume that in principle it is possible to enumerate all solved problems that an individual has learned, or that a society has accumulated in a store of solved problems, leaving aside the difficult problem of what counts as a single solved problem.

The output of a problem solving capacity consists of solved problems. When society successfully applies its problem solving capacity to solve a problem, society increases the number of already solved problems. If the number of solved problems increases linearly, then for any number of decades in the period of time, ,

(14)

If the number of problems increases exponentially, then for any period of time ,

(15)

or, having regard for the definition of in (2),

(16)

Like interest accruing on principal, the number of newly solved problems increases the total number of solved problems exponentially. If the accumulation of solved problems is analogous to the accumulation of interest, then it is likelier that (15) and (16) rather than (14) characterize the way that solved problems accumulate.

If we know the number of solved problems at an earlier year and also at a later year , it is possible to calculate the average rate per decade, of exponential growth in the number of society’s solved problems, by solving for in

(17)

Then, from (17), with and ,

(18)

(18) suggests three kinds of data can be used to determine society’s average rate per decade of collective problem solving :

  • Data that has measured itself, or a related rate equal to ;

  • Data that has measured and for some set of solved problems, which allows us to find the logarithm of their ratio as in (18), and to solve for ; or

  • Data that gives the ratio

    (19)

    but not the values of and separately, which allows us to solve for in (18).

With the rate per decade determined directly as in (18) when the values of and are known, or indirectly when only the ratio set out in (19) is known, we can calculate the average rate of increase in the number of solved problems per decade. If, using data for different kinds of solved problems, we calculate the same rate for them, we will have obtained evidence favoring the validity of the hypothesis that society has a general collective problem solving rate. The closer that the values of the rates for the different kinds of solved problems are to each other, the more persuasive such evidence is.

This is a refinement of the possible common attribute, the rate of general collective problem solving, for testing The General Collective Problem Solving Capacity Hypothesis. We can test not just whether the rates coincide, but we can impose a tougher test of whether exponential rates coincide over an extended period of time. If rates for two different kinds of problem solving are both exponential, but the functions describing each differ, the gap between the two rates would likely increase over time. If the functions coincide over a long period of time, the hypothesis has passed a sterner test.

As a consequence of The Law of Large Numbers, and from (19), it appears that when and are large, we will have more confidence that when their ratio in (19) is used to solve for in (18), we will obtain an estimate that is close to society’s actual average general problem solving rate, .

3.2 Error estimation

Suppose that we have estimates of both and , and that we use the equation in (18) to estimate . How accurate is as an estimate of the rate ?

By assuming the existence of an actual rate, , we can show that the longer the time period used for estimating that actual average rate, the closer is to , all other things being equal.

Suppose and are given, and that is the actual value of . Let

(20)

be the actual number of solved problems at year . Let be an estimate of society’s problem solving rate, and let be an estimate of the number of solved problems at year . If we use an estimated average rate, , to estimate based on , then, applying (17),

(21)

Let

(22)

Here is the error in the estimate of the size of resulting from using an inaccurate applied to to obtain a value for as an estimate of . Taking the ratio to reveals that a longer time period reduces the amount of error in . For, applying (21),

(23)

Taking, in (23), the logarithm of the right side of the first line and the logarithm of the last line, we find that

(24)

From (24), the error in the estimate of resulting from using to estimate is

(25)

The error in an estimate of becomes smaller as increases (and also as decreases). In percentage terms, the error is

(26)

As an example of the effect of the number of decades on an error in estimating , suppose the actual number of solved problems is known for a year which is 33.2 decades later than . If is 10% higher than as a result of imperfect data— is 10% of —an actual rate of problem solving of .00341 (3.41%), for example, per decade will be estimated to be higher by about

(27)

The error as a proportion of , which we here suppose to be 3.41% per decade, would therefore be

(28)

or about 8.4 percent. If =—if is 10% too high an estimate—over a period of 332 years, the estimated average rate per decade, , would be per decade, about 8.4% more than the actual value of 3.41% per decade used in this example. If the span of time was 839 years—83.9 decades—then a 10% error in would result in an error in the estimate of of about 3.33%. If the span of time was 3,742 years—374.2 decades—then a 10% error in would result in an error in the estimate of of .007469, which is less than three quarters of one tenth percent. The results of error calculations when is 10% too high are summarized in Table 1.

Number of years Size of error in %
100 27.9
332 8.418
839 3.33
3742 0.07469
Table 1: Percentage error in due to being 10% too high.

If —if is 10% too low an estimate—over a period of years, we can do a similar set of calculations leading to the results in Table 2.

Number of years Size of error in %
100 30.9
332 9.3
839 3.6
3742 0.0826
Table 2: Percentage error in due to being 10% too low.

If the error in the number of problems at year is less than 10%, whether high or low, then the size of the errors in Tables 1 and 2 would be even less. This discussion suggests that, all other factors being equal, a rate estimated by using a longer period of time is more reliable. Similar calculations apply to determining the effect of an error in estimating —the actual number of solved problems at year .

This error analysis suggests preferring data consisting of a large number of problems evaluated by a large number of people over a long period of time, to improve the accuracy of our estimate of society’s average general collective problem solving rate.

4 Testing the general collective problem solving capacity hypothesis: required attributes of data

4.1 Kinds of data

One possible way of estimating society’s average, general, collective problem solving capacity—society’s average, general, collective problem solving rate—is to identify a kind of solved problem that can be counted, and to then calculate the problem solving rate as described in subsection 3.1.

For a collection of solved problems to qualify as general collectively solved problems they must be intrinsically an accomplishment of society acting as a single problem solving entity. The total number, per year, of books published, words printed, things invented or manufactured, might be representative of the problem solving output capacity of a society. A language is a good example of a set of solved problems that involves all of society. The development of language is intrinsically an accomplishment of a society. A language is a collective, emergent phenomenon. It is, in its essence, a network phenomenon—a medium for transmitting information within a network. To use language as a way of testing society’s general collective problem solving productivity requires us to identify a feature of language that consists of a countable subset of solved language problems, with data available for different points of time in order to estimate the problem solving rate.

By comparing the cumulative number of solved problems for one kind of solved problem as of two different years, we convert the problem of measuring the implicit energy content of a set of solved problems to one of finding a countable feature of a particular set of solved problems. What counts as a separately solved problem? We need to find a set of solved problems where the criteria for deciding what counts as a solved problem are relatively consistent and easy to determine. It would be most helpful if we could identify a set of already compiled and counted solved problems.

Suppose it is possible to enumerate a set of solved problems. Suppose that, at an earlier year , a society has an accumulated store of solved problems, where

(29)

To test the statement implied by (8)—the amount of information is proportional to the amount of energy used to create that information—we need to numerically compare the implicit energy contents of different bodies of information. To compare their implicit energy contents, we need to assign a number to their respective amounts of information. Since it is difficult to compare different kinds of problem solving, such as lexical problem solving and mathematical problem solving, by means of a common criterion, we seek statistics that correspond to the amount of information for the same kind of problem solving at different points in time.

Another possible way of estimating society’s average, general, collective problem solving rate might be to use a statistic that is, indirectly, equivalent to it. We will explore that possibility in this section as well. Examples of statistics about averages that might be useful are the average test scores of individuals in IQ testing, and the average economic productivity of individuals in a society.

We will gain confidence in the correctness of our hypothesis that society has a general collective problem solving capacity if we find similar general collective problem solving rates for different kinds of general collective problems. The closer the rates are to each other, and the longer the period of time during which they are close to each other, the more confidence we will have. We therefore seek representative statistics to measure society’s average general collective problem solving rate. In this section, we will consider the criteria for choosing such data. First we consider the evaluative nature of society’s general, collective—and enumerable—problem solving rate. Then we consider how it is possible to use statistics about an average individual problem solving rate for comparison to society’s general collective problem solving rate.

4.2 Economic growth data

Economic statistics are a possible source of evidence about society’s general collective problem solving rate. If society’s store of solved problems increases, then economic growth should also increase. If a society’s general collective problem solving capacity results in an increase of the number of its solved problems, that should improve the economic well-being of its citizens. We might put it this way:

and

.

This chain of inference suggests that individual intelligence and economic growth are related, since society’s general collective problem solving capacity is the networked general problem solving capacity of its individual citizens.

Individual intelligence—both learned and innovative problem solving—is used to develop and to evaluate a society’s improved and new technologies. Economists infer that improvements in technology enable economic growth [48]. Since improvements in technology derive from the application of the problem solving capacities of individuals and of society to increase the productivity of some process, then so do improvements in the economy. Increasing society’s intelligence—its general collective problem solving capacity—should increase economic growth. If we can improve our understanding of intelligence, then we may improve our understanding of how society’s general collective problem solving capacity leads to economic growth.

Intelligence and economic growth are also related in another way: both are the product of emergent processes. A brain’s networked neurons manifest themselves as

one single mind. Similarly, society’s—a social network’s—

“dispersed bits of incomplete and frequently contradictory knowledge which all separate individuals possess” [27, p. 77]

leads, in an economy, to a market solution that

“might have been arrived at by one single mind possessing all the information” [27, p. 86].

Individual intelligence and the development of a market economy are both emergent processes. In emergence, disparate but networked parts arrange themselves to have a capacity that none of the parts individually has. If we can improve our understanding of how networking plays a role in the emergence of individual intelligence, we may improve our understanding, by analogy, of how markets and collective intelligence emerge. So, for economists, there are at least two reasons to study the nature of intelligence: first, to relate intelligence to economic growth, and second, because how individual and collective intelligence emerge may provide an analog for how markets emerge.

4.3 The evaluative nature of collective problem solving

If we decide to use enumeration as the way to estimate , then the type of solved problem that is the object of our investigation must have a definite solution that is discrete and countable. Problems concerned with political and social interactions often do not qualify as discrete and countable. In contrast, problems in a mathematics test, or inventions invented during a period of time, qualify as being discrete, countable, and capable of being identified as being solved.

Society, mostly, does not collectively invent new technologies like new light bulb designs. There are too few technological problems confronted by too few individuals who, as inventors, have problem solving capacities that are too unrepresentative of the average person, for us to consider most technological innovations and inventions as indicative of society’s average general collective problem solving rate. It must be the case that society’s average general collective problem solving rate measures the rate at which members of society solve evaluative problems. Even for language, which is a communication technology for a social network, created through the solved problem contributions of many individuals, evaluation plays a prominent role.

For a word, society’s evaluative consensus is required before it become part of the lexicon. A new word presents word users with these problems: is this new word useful, convenient, efficient, and an improvement on the available alternatives, so that it is worth the effort required to learn it? For a consumer product, the consumer’s problem is not, how can I invent this? The consumer’s problems are instead: is this more useful, convenient and efficient than available alternatives? should I buy this? For a new scientific theory, a scientist’s problems include: should I accept this new theory and adjust my research, texts, and teaching accordingly [33]. Legislators must solve evaluative problems such as, does this proposed legislation remediate a social, economic, or political issue without itself causing harm? if passed, will this legislation be viewed favorably by my constituency? Jurists must solve problems that include: which solution is consistent with existing legal principles? which possible solution best resolves the legal issue involving the parties? which resolution is consistent with an appropriate remedy? Society must also evaluate how to best receive, store, retrieve, process, transmit and record solved problems. We infer that society’s general collective problem solving is evaluative. If a varied or new idea does not pass society’s evaluative muster, for most practical purposes, it has no effect. If no one pays attention to an idea, the idea might as well not exist.

Consider encoding a computer software program into computer code as analogous to encoding abstractions into words. Computer software continually improves as the result of the efforts of thousands of computer software engineers. Language, analogously, also continually improves, but emergently, and over much longer periods of time involving all of a society; language is a large-scale, open source,111‘Open source’ is an observation of Michael Shour, Toronto. software engineering project.

Just as energy is spent to devise computer software, so also is energy spent to improve language.

For an ensemble of independent systems which are uniformly random, James Clerk Maxwell, Ludwig Boltzmann and J. Willard Gibbs showed in the late 1800s that the statistically most likely state is one in which the same average amount of energy occupies equal volumes. By analogy, the same amount of energy being required to solve the average problem is a statistically more likely state. Considering the distribution of energy among solved problems, which each required the same average amount of energy to solve, is analogous to considering the distribution of energy among equally sized ‘volume elements’ [6, p. 50] or ‘region elements’ [45, p. 124] in statistical mechanics.

To demonstrate that the same amount of energy is required to solve any kind of average problem, as distinguished from specialized non-collective problems, we need to use, as data, evaluative problems that any person in society can solve, and which are evaluated by many people. It should not matter what the nature of the problem is, or who the particular problem solver is. Specialized problem solving will not reveal society’s average general collective problem solving rate. If we successfully choose as data average collectively solved problems, there is an increased likelihood that society’s average general collective problem solving rate will be revealed. This increased likelihood arises for the reasons set out in Section 2.

Society evaluating the merits of a proposed solution to a given collective problem is equivalent to networking the individual evaluations of all members of society. If a large number of people contemporaneously evaluate the merit of a proposed solution to a given problem, and if each person solves several sub-problems in such an evaluation, then , the total number of such solved problems, is large. Just as, in recording the results of a large number of flips of a coin, we expect the ratio of heads to tails to better reflect the true proportion of the chance of heads compared to the chance of tails, so we expect that a large number of solved evaluative problems produced by a large number of people will better enable us to estimate society’s actual average general collective problem solving rate.

Therefore, measurement of society’s average general rate to solve evaluative problems can help test the hypothesis that a society has an average general, collective, problem solving capacity.

4.4 When an average of individual rates coincides with a collective rate

4.4.1 Problem solving in IQ tests

Statistics about average IQ test scores are available. IQ tests are normalized by the producers of IQ tests—adjusted so that the average IQ of a contemporaneous reference group is at all times 100. The rate of change in the normalization has been measured. Therefore, it is possible to measure the rate at which average IQs increase by comparing normalizations of average IQ test results for IQ tests administered at earlier times to average IQ test results for IQ tests administered at later times. We look at IQ tests because they are a particular instance of indirectly measuring an average individual general problem solving rate. Since some statistics about average IQs are available, the rate at which average IQs increase can be compared to society’s average general, collective problem solving rate in other areas, such as lexical growth and lighting efficiency.

An IQ test has a standardized set of test questions, designed to measure an individual’s problem solving capacity for different kinds of problems, to be completed over a prescribed amount of time. The IQ test has two fixed variables—the test questions and the amount of time to complete them—out of three. The third variable, the number of correctly solved problems out of the total number of problems , helps determine the IQ test scores of each person because both sets, and the set of problems included in , are enumerable—can be counted. The proportion,

(30)

can be measured and compared to a contemporaneous standardized average proportion of problems correctly solved by a reference group of people who have taken the test. The test, having sampled an individual’s problem solving capacity by questions designed to test different kinds of problem solving skills, indirectly and in part, estimates (we infer), for the person tested, the proportion of society’s knowledge—the proportion of solved problems—that the person has learned.

An individual’s problem solving capacity is applied to, among other kinds of problems, the problem of how to learn society’s solved problems—how to acquire information from other individuals by socially networking with them, and from society’s store of solved problems. If, consistent with The General Collective Problem Solving Capacity Hypothesis, an individual’s problem solving capacity is a general capacity, it follows that the rate at which an individual learns society’s solved problems is approximately equal to the person’s general problem solving rate. Therefore, measuring what a person has learned is an indirect way of measuring an individual’s general problem solving rate. If a person’s problem solving rate is positive, then, if it is consistently applied, the person’s own store of solved problems, acquired to a large extent by learning society’s solved problems, should increase.

If the intelligence of a person is equivalent to the person’s general problem solving capacity, then if an IQ test estimates the person’s general intelligence, it at the same time measures the person’s general problem solving capacity. If an IQ test is an accurate estimate of a person’s intelligence, then informally we might speak of ‘a person’s IQ’ as if we are speaking of that person’s intelligence. But an IQ test, even in this informal sense, is only an estimate of the person’s ‘actual IQ.’

An IQ test result, as expressed in (30), is analogous to society’ collective production of solved problems. Society faces, in a given period of time, some total number of problems, , some portion of which—the set of solved problems in —are solved during that period of time. An individual’s general problem solving capacity is therefore analogous to society’s general collective problem solving capacity. If society’s store of solved problems—its knowledge—increases, individual general problem solving capacity should increase.

IQ researchers reason that it is not a change in the biology of human beings that leads to the increase in average IQs that has been measured by them in the past few decades. The human brain could not evolve that quickly. Moreover, if intelligence increased that rapidly, then, extrapolating back in time, that would imply that the geniuses in the past, say 2000 years ago, achieved the results they did with meager intellectual resources, insufficient to accomplish what they did. It is therefore unlikely that the average innate, or basal, problem solving capacity of individuals has changed much over the past two or three thousand years. This has left researchers suggesting TV, diet, education, and so on, as possible factors that might explain why average IQs have been increasing.

Suppose that every person has an innate, or basal, general problem solving capacity. The innate, or basal, general problem solving capacity must be positive, otherwise it would not be possible for a person to figure out how to learn and remember the solution to a problem solved by other people—to learn information.

The few hundred problems that comprise an IQ test approximately measure the general problem solving skill of an individual. An individual’s problem solving skill includes two components. One component depends on how well the person has learned the society’s solved problems: how much knowledge the person has acquired from the social network and from society’s store of solved problems. Another component depends on how much innate, or basal, problem solving capacity a person has—a capacity they were born with—to retrieve learned knowledge from that person’s own memory and to process that learned knowledge—to vary, adapt and invent solutions to problems novel to the person using that knowledge. Comparing the average IQ for a group of people over a period of time measures a society’s general collective problem solving rate by indirectly measuring the average rate at which the effectiveness of society’s store of solved problems has increased, for all those individuals who have learned those solved problems. If the quality of society’s store of solved problems increases, the store of information of the socially networked individual should improve in proportion.

4.4.2 The average IQ and society’s IQ

In this part we try to demonstrate that the rate of increase in the average person’s general problem solving rate equals the rate of increase in society’s average general collective problem solving rate. It would follow that the rate of increase of average IQs and the rate of increase of society’s collective IQ should be equal. If that is so, we can validly use the rate of change in a society’s average IQ as a means of estimating society’s general, collective problem solving rate.

First, we wish to demonstrate, with reference to IQs, that an average of a society’s individual rates of problem solving is equivalent to the society’s general collective problem solving rate, in principle.

We will use the following additional definitions, some of which are analogous to the problem solving definitions which begin on page 1.

IQ Definitions.
  1. : the IQ—intelligence—of an individual, as of time .

  2. : the average IQ—intelligence—of a set of individuals, as of time .

  3. : a theoretical construct, representing what an individual’s IQ would be without any social networking or language, as of time .

  4. : a theoretical construct, representing what the average individual’s IQ would be without any social networking or language, as of time .

  5. : the set of solved problems of an individual that relate to social networking, as of time .

  6. : a function of the population of a society, as of time .

  7. : a function of the number of a society’s solved problems, as of time .

  8. or : a proportion, between 0 and 1, of , as of time .

  9. : a proportion, between 0 and 1, of , as of time .

  10. For both and , subscripts and, for the average’ value for a group of individuals, , are also used.

  11. : the base of the logarithmic function that measures part of the capacity of an individual to socially network.

  12. : the base of the logarithmic function that measures part of the capacity of an individual to network with solved problems.

The individual IQ.

In this first step, we relate an individual’s IQ to its component factors.

Some proportion of is based on the person’s innate problem solving capacity—the problem solving capacity of a person determined by the person’s genetically inherited, physiological capacity, . That is,

(31)

is also proportional to a function of , the amount of society’s information—the number of solved problems that the individual has learned. The more an individual knows, the higher their working, individual IQ. The relationship between an individual’s IQ and a function of the number of solved problems learned is linear. This must be the case for the following thermodynamic reasons.

An individual, during their life, expends energy in an approximately linear way, proportional to time.

(32)

An individual, has the capacity to add to their existing personal store, or set, of solved problems, , proportioned to the energy used by them to acquire—learn—those solved problems, as follows:

(33)

From (33), we infer

(34)

The more that an individual knows—the more solved problems that the individual has learned—the more resources the individual has to use to solve newly encountered problems. So

(35)

Since an individual’s use of energy is proportional to the advance of time, and the number of society’s solved problems acquired by an individual is proportional to the energy used to learn them, it follows that the increase in an individual’s store of solved problems is proportional to time. That implies that an individual’s IQ increases linearly along with the linear increase in a function of the number of solved problems that the individual has learned. Hence, similarly to (35),

(36)

The portion of society’s solved problems acquired—learned—by an individual is only a proportion of a function of all of society’s solved problems. That is,

(37)

or

(38)

where —the ‘’ factor—represents the proportion of a function of all of society’s solved problems. So, combining (36) and (38),

(39)

A solution to a problem that was solved by society can contain the solution to a problem that confronts an individual, or can help solve the individual’s problem by analogy or by other methods.

From (31) and (39), it follows that, without regard to the effect of social networking,

(40)
Learned solved problems as a factor in IQ.

(40) implies that an individual’s capacity to solve problems, which is represented by on the left side of (40), arises from the individual applying their innate IQ to learn a portion—a subset—of the problems that society has solved.

As to the innate IQ factor on the right side of (40), the physiology and capacity of the average human brain has likely not changed much if at all over the past few thousand years. If a person’s problem solving capacity is proportional to the physiological capacity of their brain, then it follows that the innate, or basal, problem solving capacity of the human brain—and the average individual innate IQ—has likewise remained unchanged over the past few thousand years. Therefore, we assume that, at least in the recent past of human beings,

(41)

Based on all of these considerations, it must be that any rate of change in the number of solved problems that an individual learns is a result of applying to increase their store of solved problems.

Since

(42)

we interpret (40) as demonstrating that an individual IQ test score is one possible metric for measuring the problem solving capacity, or IQ, of an individual that arises when an individual’s innate, or basal, problem solving capacity, or innate IQ, is applied to a subset of society’s solved problems. An individual’s innate, or basal, problem solving capacity, or innate IQ, is also involved in solving the problem of how to learn that subset of society’s solved problems. An individual’s store of solved problems cannot increase without the individual applying their innate, or basal, problem solving capacity; without an individual invoking their innate, or basal, problem solving capacity, the individual’s store of solved problems remains static. (40) suggests that the innate IQ on the right side of (40), as an invented, notional, metric, measures, at least partly, an individual’s problem solving capacity, which is equivalent to the processing part of the left side of (40).

This relationship is analogous to a relationship in thermodynamics, between energy and absolute temperature. Absolute temperature is an invented metric—and is not a function of time—such that the absolute temperature is, for a given closed system, proportional to the system’s heat energy. If the heat energy in the system doubles, the absolute temperature doubles. By measuring the absolute temperature, one can indirectly measure any increase in the given system’s heat energy content. The ratio of heat energy to absolute temperature for a given closed system is its entropy, which can be described as the ratio

(43)

where is the heat energy of the system, , the small Greek letter eta, represents the entropy of the system, is the base of the logarithmic function, is the absolute temperature of the system, and can be considered to be the state of the system, or the number of individual energy components in the system [45, p. 118, for example]. (There is a constant that multiplies , but we will suppress that for now, as if .)

Absolute temperature, as a source for our analogy, and an individual IQ test score, as a target of an analogy, does not work exactly. As was mentioned above, the standard IQ test score, based on a metric of 100 as the average IQ, is adjusted over time; the standard average 100 IQ is not absolute. In the target of our analogy, for a given individual and also for the average individual, the number of solved problems per IQ point increases over time, while for a given thermodynamic system, the amount of energy per degree of temperature is unchanging.

The problem solving capacity of an individual as it is measured by an IQ test score is affected by education—how much the individual has learned, as implied by (40). If, having regard for (40), we treat problem solving work as being the result of an individual applying using what the individual has learned, then if we want to focus on a problem solving capacity of an individual without the enhancement of education, we should use the theoretical, or notional, unchanging of an individual as a metric of that individual’s problem solving capacity, analogous to absolute temperature as a metric of heat content for a given system.

For an individual, is unchanging during their life, and so, for that individual, is analogous to absolute temperature, which is an unchanging metric for a thermodynamic system. Similarly, for a set of people—the average innate IQ for that set of people—is similarly an unchanging metric for their average problem solving capacity, or rate. If the average innate IQ has remained relatively constant over the past few thousand years, then it may be considered to be more analogous to the idea of absolute temperature than is individual (composite) IQ which changes, and as well, is partly a function of what a person learns. The idea of an innate IQ is entirely theoretical, because it would be difficult in practice to isolate a person’s innate problem solving capacity from what the person has learned.

Let’s consider how an innate problem solving capacity of an individual interacts with a network of solved problems. To simplify our analysis, and consistent with the foregoing, we first assume that this occurs in the absence of a social network. By way of an approximate analogy to (43),

(44)

(the change in information is analogous to energy, because new information is created by the input of energy, and because as the number of solved problems increases, the problem solving rate increases),

(45)

(the invented IQ metric, but at the level of the notional , is analogous to absolute temperature), and

(46)

(as in (43), forming an analog to a definition of entropy used in thermodynamics), where the base of the logarithm, , is applicable, for a particular individual, to a network of solved problems—what might called a network of abstractions or ideas.

From (46), and multiplying the left and right sides by , we have

(47)

The rate of change, on the right side of (47), in the number of solved problems learned by a problem solving individual, arising from the application of an individual’s innate IQ, is equal to the individual’s composite intelligence, , since, for the left side of (47),

(48)

It follows from (40) and (48), now having regard for the proportion of that an individual knows, that

(49)

In the second and third lines in (49), and likely change over the course of a person’s lifetime. changes as the result of collective problem solving; can change as a result of additional energy spent by an individual in learning solved problems—increasing the proportion of society’s solved problems that the individual has learned.

The factor

(50)

in (49) exactly corresponds to the general form that entropy takes in thermodynamics, .

The situation represented by (49) is a theoretical, or notional, one in which individuals have direct access to a store of society’s solved problems, but without any social networking.

The implication of (49) is that the factor set out in (50) multiplies innate IQ, and as well, that the more ‘solved problem’ targets an individual’s innate IQ can network with, the more ‘intelligent’ that individual will be.

Social networking as a factor in IQ.

We now consider the effect of social networking on , an individual’s IQ. For the purposes of this article, I accept as true the following observations:

“As the child negotiates the early developmental levels through countless emotional exchanges with her caregiver, she develops an implicit understanding of her society’s attitudes towards beliefs and social practices, norms and values, power hierarchies and the kinship system, and so on [24, p. 325].
…it is the pattern of reciprocal, co-regulated affective interactions in the early stages of development that helps the child differentiate her own individual personality and helps the group determine its collective personality” [24, p. 332].

How to socially network is a problem solving exercise, beginning with an individual’s earliest childhood moments. Each ‘emotional exchange’ requires each party to the exchange to solve the

problem of how to participate in that exchange. A child has to decipher her caregiver’s signals, and has to solve the problem of how to signal back to the caregiver. ‘All life is problem solving.’

Analogous to the way that we considered the thermodynamics of how an individual networks with a society’s solved problems and abstractions, let’s consider how an individual networks with other individuals in their society—how the individual solves the problem of how to socially network—but ignoring, for now, society’s network of solved problems. We can infer that the capacity of an individual to socially network is analogous to their capacity to solve social networking problems. We can consider the set of individuals with whom a given individual networks, or equivalently, the set of solved problems, corresponding to ‘solved’ problems that establish social relationships, , to consist of a population connected, directly and indirectly, to an individual. That is

(51)

represents the person’s social network. Since an individual must solve problems of how to network with other individuals, by analogy to (46),

(52)

By analogy to (49)

(53)

where is the base of the logarithmic function applicable for the whole society.

In light of (40), and since an individual’s social network arises from the application of , any rate of change that a person experiences in their social network ultimately is a result of applying to the problems of how to socially network with other individuals. It must be therefore, that, similarly to (48),

(54)

The situation described by (53) is a theoretical, or notional, one in which individuals socially network, but do not network with solved problems.

The factor

(55)

in (53) also corresponds to , the general form that entropy takes in thermodynamics.

Now we suppose that an individual can contemporaneously network with other individuals, a situation described by (53), and with a set of solved problems, a situation described by (49). If we blend together the two situations described by (49) and (53), we obtain a combined formula that describes individual intelligence, as follows:

(56)

The three components, or factors, of an individual’s IQ in (56), appear to be:

  • the individual’s innate IQ;

  • a function of the individual’s knowledge;

  • a function of the individual’s social network.

There is likely a fourth factor that affects individual intelligence, represented by the combined effect of the environment, culture and infrastructure, to the extent not already subsumed within the three components just mentioned.

With respect to (56) the population of a society can be estimated. It is possible, if The General Collective Problem Solving Capacity Hypothesis is valid, to estimate the number of solved problems by finding an enumerated set of collectively solved problems proportional to that number. That leaves us with the problems of measuring, for an individual, the other four parameters in (56), namely and for each of society’s population and its solved problems. represents that proportion of society’s solved problems that an individual has ‘networked’ with. , the base of the logarithm of the number of society’s solved problems, must have some thermodynamic relationship to society’s store of solved problems due to its relationship to entropy. may represent how well an individual is socially networked, or how adept at social networking an individual is. , the base of the logarithm of society’s population, must also have some thermodynamic relationship to society’s population. These four parameters, the two ’s and the two ’s in (56), may be difficult to measure for an individual, but they have been measured as averages for nodes in some networks.

Later in this article we discuss the problem of how to characterize and measure the following four parameters in (56):

  • .

Average IQs.

We can evaluate the rate of change in the average individual IQ in terms of its component factors, by using (56).

Let the subscript in the following be used to identify the different individuals comprising the society under consideration. For a society consisting of individuals, the average of the society’s individual IQs is:

(57)

To simplify notation for the next equation, let the following represent the indicated values, all at a common time .

  • : the proportion of the number society’s store of solved problems learned by individual ;

  • : the number of society’s store of solved problems;

  • : the base of the logarithm for the logarithmic function of the number of society’s solved problems, for individual ;

  • : the proportion of the society’s network connected to individual ;

  • Pop: the population of the society;

  • : the base of the logarithm for the logarithmic function of the society’s population, for individual .

Using this notation, and combining the results of (56) and (57), we obtain, letting the subscript represent the average for the society’s individuals, at a time ,

(58)

We infer that the averaged values of the two s and s that help determine the IQ of the average individual in a society are related to the s and s with the subscript that apply to individuals in (58). Consistent with our hypothesis about the way a general problem solving capacity works, the same general problem solving capacity applied to learning society’s problems should apply to solving problems of how to socially network. Consistent with The General Collective Problem Solving Capacity Hypothesis, each component factor in (58) achieves its average value in a similar way. In other words, the average of the product on the right side of (58) should equal the product of the averages. So we infer, based on (58), that, at a time

(59)

(59) should apply for any rate of population growth, including a rate of zero population growth. For the past few thousand years, a human being’s genetically endowed problem solving capacities have been unchanged, which implies that the average innate, or basal, IQ has been unchanged during that period of time, consistent with (41). If we take the derivative with respect to time of the left side of (59)—average IQ—for a period in which there is no population growth, and of the product on the right side of (59), we have

(60)

because , as in (41), does not change.

In other words, (60) predicts that the average rate at which average IQs increase, for a period during which the population size is relatively unchanging, should equal the average innate IQ times the average rate at which the logarithmic function—the entropy—of an enumerated set of collectively solved problems increases.

Society’s IQ

Analogously to the situation of an individual, a society’s IQ is proportional to its store of solved problems, considering society as if it were one (networked) individual having a ‘single mind’ in itself:

(61)

Not all solved problems in a society are perfectly networked, but only a proportion, . That is,

(62)

If we take the derivative with respect to time of the left and right sides of (62), then we have