# The “intellectual quotient” fallacy

Last update: 2022-03-02

## Abstract

We show that the concept of intelligence applied to real-life situations is inherently subjective. We conclude that there is no scientifically defined quantity that quantifies intelligence. We point that so-called “intellectual quotient” is an illusion caused by the conflation of scores in various puzzles with each other and the fallacious usage of descriptive statistics.

## 1 What we show

We will refute the follwoing claims:

- Intelligence per-se is within the scope of science.
- Intelligence exists as a numerical variable.
- Intelligence can be measured using so-called intellectual quotient tests.

For (1): A neccessary condition for an assertion to be scientific is that it makes a prediction about objectively observable properties of the real world [1].

**Natural science example 1:** Clasical electrodynamics makes predictions about electric fields, magnetic fields, currents and charges. Classical electrodyamics predicts the pattern of colors that will be projected in the sensor of a digital camera based on the physical properties of the lens which can be measured in a laboratory (geometry of each lens, geometric arrangement of lenses with respect to each other and the sensor, index of refraction as a function of frequency for frequencies in the visible spectrum) [2].

**Natural science example 2:** Thermodynamics make predictions about temperature and the flow of energy in very general terms. Suppose we have an ice bucket filled with 20 L of water and 10 kg of ice. We place the lightbulb in a black bag and then in the ice bucket. We turn the lightbulb on for 1 hour and measure the current and voltage, then we take out the ice and weight it. Thermodynamics and some background from basic electricity predict the mass of the ice left given the current, voltage and time.

**What is intelligence, in terms of measurements?** Intelligence refers to the behavior or thought of agents; for clarity of description we focus on humans. Given an objective description about the behavior of a person and the thoughts it ascertains to, what can be said about its intelligence? We will show it is entirely subjective.

## 2 Inherent subjectivity

### 2.1 Oil spilled in kitchen cloth

In 3 different identical kitchens, 3 different humans of the same age are cooking and accidentally spill oil in a kitchen cloth.

- Human A opens her computer-phone and searchers the web for “how to remove oil from fabric”. She finds an article that recommends washing with dishwasher soap. Following the article, she washes the cloth with water and dishwasher soap.
- Human B pauses while she thinks how to proceed. Human B has no experience removing oil. However human B knows that the dishwasher has a detergent and detergents create an emulsion between oil and water which allow the oil to become dispersed through the water with the help of mechanical scrubbing. She gets up and washes the cloth with dishwasher soap and water. The cloth becomes clean after some time washing.
- Human C immediately washes the cloth with water, same as human B. Human C does this because her family washed oily things with dishwasher soap and it worked, so she does it too. She knows nothing of emulsions and boundary effects.

Who showed the most intelligent behavior?

- One could argue that human A is the most intelligent because recognized that she needed more knowledge and she used the resources at hand to obtain the neccessary knowledge and solve the problem.
- One could argue that human B is the most intelligent because she made an inference from her existing knowledge to reach a solution to the problem.
- One could argue that human C is the most intelligent because she arrived at the solution immediately.

There is no objective standard by which one can pick one of the answers as the valid one. The matter is inherently subjective.

### 2.2 The mathematician and the “engineer”

Suppose we observe the following 2 persons:

- A mathematician. He has solved many long-standing mathematical problems in topology. As a recognition of his contributions he was offered a prestigious medal in mathematics in one occasion and 1 000 000 USD in another occasion. He turned both down. He lives as a recluse. He spends most of his time solving other conjectures that he does not publishes. He says: “Why publish? I do not want the people who also contributed to these problems to fight like dogs for the credit and exhibit me like one for having made the final step”. He states to be very unhappy, but that it does not matter. He says the main reason is that he is disappointed of mathematics as a social activity because of the unfairness in giving credit and the undue importance given to prestige.
- An electric engineer (at least in title). Both of his parents had been white-collar workers since before he was born and worked extra hours to pay their debs when he was a kid. Now he has a high-paying job as the CEO of a company he founded that produces electronic devices. By his own admission “I am hardly an engineer; I am good at putting the real engineers to work and make me money from that.”. He says that he is very happy.

Which one is the more intelligent? Let us suppose a conversation between 2 commentors.

M (pro-mathematician commentor): The mathematician is obviously more intelligent. He solved really hard problems. The engineer is an idiot and he knows it.

E (pro-engineer commentor): Is that so? The mathematician may have talent but it only serves to make him unhappy whereas the engineer has talent too and is successful.

M: The mathematician cares about mathematics and that is what he is good at. I call that success.

E: Why is he so unhappy then?

M: He is rightfully disappointed of the situation.

E: If he does not care about his own happiness, I call that being dumb, no matter how good he is at mathematics.

M: He does not have to be a recluse. He chose to be. He could be rich if he wanted. He could have taken the 1 000 000 USD prize and have a bigger bussiness than the engineer. The engineer is rich because he was at the right place at the right time.

E: But the mathematician did not do it. He had his opportunity and did nothing with it. The engineer created his own opportunity to succeed.

M: The mathematician was not handed the proofs magically. He created his own opportunity too.

In the dialogue we see several points that bear on the meaning of “intelligence”:

- Whether “success” should be judged by the person’s own standard or by the standard of society.
- If one says to judge on the person’s own standard, is any standard acceptable?
- If one says to judge based on the standard of society, at which point in time? Suppose somebody attained an outcome which was considered success and is now considered failure. Is this evidence of being intelligent for having succeeded or being unintelligent for having failed?

- Whether it is the capacity to do something or is actually doing (possibly aided by the circumstances) what matters.
- Whether doing something one wants that brings the opposite of hedoism is intellgent behavior or unintelligent.

Again, several plausible answers can be given and there is no way to settle the matter because it is inherently subjective.

### 2.3 Trivial identities

Humanity has been doing proto-mathematics [3] since before formulas were introduced to express propositions. Algebraic identities of the real numbers that are trivial to write and are self-evident now were not so back then. Thus we can immediately tell that (`a` + `b`)(`a` − `b`) = `a`^{2} − `b`^{2} whereas a proto-mathematician 5 000 years ago would have needed some time to verify an inexact translation [4] of this identity in natural language: “The substraction of 2 numbers times the addition of 2 numbers is the same as the sum of the squares of each number”. What does this say of our intelligence?

- We are more intelligent. We count our increased resources as part of our intelligence.
- We are not more intelligent. We count our increased resources as external to our intelligence.

## 3 What is the meaning of IQ tests?

### 3.1 Blatant lies

**If there is no numerical variable called intelligence, then what do intellectual quotient tests measure?**

They do not *measure* anything. A measurement is an objective quantification of a physical quantity.

So-called IQ tests are games. The better the player does according to the rules chosen by the game designer, the higher the score. Usually they are puzzles where the player has to chose the correct answer among several possibilities given. For example, in Raven progressive matrices the player is shown a cuadricule of 2×2 or 3×3 cells. Each cell has geometrically simple shapes or colors. The shapes or colors vary in a regular pattern among rows, columns or diagonals of the cuadricule. One cell of the cuadricule is missing and the player has to chose the one that matches the pattern.

**Lying by massaging the data.** To add to the deception that these puzzles have any meaning beyond being simple games, the true score is massaged by a transfer function chosen so as to force the massaged scores into a Gaussian distribution. The means is almost always 100 and the standard deviation is often 15. Note that the means, standard deviation and the fact that the massaged scores have a Gaussian distribution are not a property of the data. They are chosen in advance and the data is coerced into it. The choice of 100 and 15 are likewise meaningless and used because of preference and habit.

**Lying by mislabeling the data.** No matter what the game, the massaged score is presented under the label of “IQ”. In this way the different games are conflated as if they were the same.

### 3.2 Seeing past the lies

What can be said about these scores? The same as for any other game scores: They are *by definition* the performance of a particular player in a particular session of play in a particular game. By taking the average of scores among different game sessions one gets an estimate of how good the player is *in that game*. By taking an average of scores among different people, one can get an estimate of how good that sample is at that game.

The example of Raven progressive matrices. As a game, the 60-answer form of the Raven progressive matrices is enjoyable the first time. As an intellectual challenge it is not a challenge at all. If it is asked “How useful is Raven progressive matrices to evaluate intelligence” first we have to fix the question and expectations. The inqury can be rephrased as: “Given somebody’s score in Raven progressive matrices, what can be said about this person?”. Very little. The patterns are trivial given knowledge of modular arithmetic and boolean operations. Therefore a correct answer is evidence that the players knows modular arithmetic and boolean operations; an incorrect answer is an indication of not knowing them, being sleepy, tired or distracted. It is possible that a player does not know modular arithmetic or boolean operations and formulates these concepts when trying to arrive at a simple description of the pattern in order to extend it to the missing cell. In my opinion a player like that, if also a very young person, has good prospects for computer programming or mathematics because both require the practitioner to notice patterns that can be expressed in mathematical logic and reason about them. The score of the game does not tell us what the person thinks while playing it so it is a moot point. To form an opinion about a person it is much more useful to make the person play the game and describe the reasoning in arriving to answers.

The considerations that apply to other games passed as “IQ tests” are analogous. For none of them can one conclude that the game measures intelligence because there is no physical quantity called “intelligence” and one can not conclude that it evalutes the general concept of intelligence because it is inherently vague. At most one can conclude that the performance in a specific game says something about a very small aspect of one’s particular opinion of what is intelligence. IQ tests are not special for this. This also true for many other human activities: Other games like Go and Chess, proving mathematical theorems, managing a bussiness, formalizing mathematical theorems in a proof assistant, writing software, eloquent conversation to obtain a result favorable to one’s goals (i.e.: manipulation). For example: Knowing that a person has written thousands of lines of computer-verified proofs tells one that such a person is competent in formalizing theorems, which says something about the intelligence of that person in most people’s opinion of what is intelligence.

## 4 Fallacious arguments from correlations

Among practitioners of the pseudosciences of sociology and psychology there is a widespread lack of ability to tell apart rigorously sound arguments from unsound arguments which are superficially similar to sound arguments. In the case of the concept of IQ the pseudoscientists who advocate it take the methods of statistics. They point that “IQ” (without making a distinction of which specific puzzle their data is a massaged score of) correlate with some parameter which they take as a proxy of success like income or self-rated happiness and argue that this is evidence that the scores in the puzzle “measure” intelligence, which we already established is wrong in the general case. There are additional mistakes:

- Observational statistics can only prove a correlation.
- That their purported parameters are “success”. Why is it success to self-describe oneself as happy? There are many negative things in the world that an individual can not significantly change for better like wild animal suffering, global warming, exhaustion of non-energetic minerals, the destruction of Earth by the Sun when it becomes a red giant. The pseudoscientists have taken for granted that none of these warrant unhappiness. The pseudoscientists posture entails that Jeremy Bentham’s posture “I would rather be an unsatisifed human than a satisfied pig” is a priori considered unintelligent.
- The assumption that “If X correlates with our parameteres of success then X is intelligence” is unjustified. Counterexample: Height and sex correlate with income.

## 5 Mathematical developments inspired by the concept of intelligence

The concept of intelligence serves as an inspiration and motivation for some mathematical developments like game theory introduced in von Neumann, Morgenstern (2007) and universal intelligence by Legg and Hutter. By hypothesis, the desirability of each possible outcome for each agent is a number called the payoff. It is outside the scope of mathematics to assign a payoff to real-world scenarios. In this way the subjectivity in defining intelligence is sidestepped.

## 6 Notes

- See von Neumann, Morgenstern (2007) § 3.8 “Principles of measurement: preliminaries” for commentary about the nature of measurements. .
- See Hecht (2002) for details. .
- In proto-mathematics the knowledge of numbers was by anaology with physical proccesses like counting and marking a string with the diameter of a circle, then putting it around the circle and seeing how many times the diameter fits around. In mathematics proper, numbers are either
*postulated*into existence in the axioms,*constructed*as sets in set theory or*defined*as a type in type theory. since before formulas were introduced to express propositions. Algebraic identities of the real numbers that are trivial to write and are self-evident now were not so back then. Thus we can immediately tell that ( `a`and`b`are 2 variables. Saying they are 2 numbers adds the constraint that`a`≠`b`. of this identity in natural language: “The substraction of 2 numbers times the addition of 2 numbers is the same as the sum of the squares of each number”. What does this say of our intelligence?

## 7 References

- E. Hecht (2002) “Optics”, 2nd edition. OCLC: 1169827289.
- J. von Neumann, O. Morgenstern “Theory of games and economic behavior” (2007). OCLC: 487207512.