Abstract: Math contains cognitive biases. To support this astonishing observation, I begin by going back in the history of mathematics. By erasing any intention within them, we have at the same time lost track of complexity and quality. These intentions exist, but are now hidden in acronyms, in particular the ‘=’ with multiple meanings. The biases are those of cognitive stages arbitrarily making choices between these meanings, mistakenly believing to access the real per se.
The title is rowdy. It is easier to admit that equations include cognitive biases in their construction, which is already surprising for mathematical language. But the equation itself? When an equation is wrong, it is due to poor knowledge of mathematical language or its misapplication. Semantic error. The intrinsic coherence of language is not called into question. Let a calculation proceed without introducing an error, for example by entrusting it to numerical circuits: it always leads to the same result. How can we talk about cognitive bias inherent in calculation, in this case? Computers don’t make mental mistakes!
But by reasoning in this way, we presuppose that it is the real per se, through the machine, that speaks to us. The material world would express itself live! We just learned its language, we think. Wrong. Certainly reality has given birth to us. But we came out of a womb, not a mathematical matrix. We have both discovered and built this language, in relation to the world. We discovered the regularities. They still had to be interpreted. We are indeed the owners of the current forms of mathematics. The best proof is that they continue to evolve. While the mathematic of reality per se does not move, is independent of us.
Let us agree then to call the structure of reality per se the Mathematic, and the forms of our language about it mathematics. These languages lack a metamathematic to get seriously close to the Mathematic, and this article develops an important reason that prevents us from accessing them.
Observation and performance
A very important property of our common languages is absent from mathematics: the difference between observational and performative statements (John Austin 1955). Examples of observational: “the sky is blue”, “the water is cold”, “this equation has three terms”. Observational statements are manichean: true or false. Because they do not depend (or little) on the observer; the observation is that of reality per se.
The performatives transform human and physical reality: “I throw a pebble”, “I turn a tap”, “I declare this equation correct”. The success of a performative statement depends on the observer, on her ability to represent the world in the present and the future, to make it go towards the expected prediction. This separates performative statements into two subcategories: simple predictive and intentional. The simple predictive is content to accompany the world where it has spontaneously decided to go; the intentional subject it to a wish that is an offbeat representation of the world, an alternative to its usual course.
Remove any intention from the real threatens ours
Mathematical language lacks this distinction. Does this mean that reality per se does not contain the slightest intention? To conclude in this way creates some really critical pitfalls. On the one hand, our mind is excluded from the world, since its intention cannot be housed there. Embarrassing dualism for a Mathematic that wants to be monistic. On the other hand, if mathematics does not contain intentions in themselves, they nevertheless serve to describe them: they are the fundamental forces. There is no need to introduce movement or temporality into these intentions of reality per se. It is a road, a sequence of interactions and its inflections. Our mental intentions can likewise be assimilated to such inflections.
If mathematics is devoid of intention it is because historically we have chosen to strip it away. The human being has long considered herself unique within the living. A Chosen One, the sole depository of a divine will. The material framework could only obey this will. In case of failure, the explanation is that part of the divine intention escapes us. The rebellion of science against this religion was to grant its independence to reality. A somewhat radical reaction: science, by the way, has rid reality of anything that could recall divine and human intentions vicariously. The real does not really own its forces. It undergoes them, in the same way that the spirit undergoes the impulses of the body. Science has insidiously recreated an equivalent of a divine pantheon, moving it from the human mind to the real per se. The Gods of Olympus have become the Fundamental Forces. And the holy grail of physicists is to find the one God, the ultimate equation that would generate all reality.
Sterilization in the mathematical department
Did you startle? Probably, if you are at least keen on mathematics and physics. And for good reason! The scientific language with which your mind is imbued does not contain so much religious fervor! It is carefully wrung out. We think for others through the language of acronyms. This one filters any emotion. While the concepts themselves, as neural graphs, are perfectly apt to integrate intentions and sensations like any other. They are not waterproof. Mathematics can intrinsically create the most incredible enjoyments, which are associated with the feeling of mastery of the world, with the satisfaction of a calculating intention. But the language of acronyms performs a deep wash. The scrolling of the lines has the same effect as a 90° adjustment of the washing machine. The equation is sterilized from its impressions.
We are therefore faced with this delicate situation: reality per se could not be rid of its intentions, renamed ‘fundamental forces’, and it is our choice of language that has made them disappear. What is to be concluded? Should we not reintroduce intentions into mathematics, to return to a monistic reality, the one to which we can integrate our mind?
I will focus your attention on an inescapable ‘word’ of mathematical language. It corresponds to the “is” of the common language. This is the “=” sign. Here are some phrases of the common language that all correspond in mathematics to the sign ‘=’ :
The first 3 locutions add the notion of reversibility to equality. When a physicist creates a mathematical model of reality she defaults to the reversibility of the sign ‘=’, its version ‘equivalent to’. It is the experiment that eventually modifies it to non-reversible, ‘leads to’ (=>) or ‘comes from’ (<=). We already detect here an intention of reality per se that is not specific to language.
The 4th locution, ‘correlated with’, conceals a more important notion: there is a change in quality, not explicit. The two terms of equality are not of the same nature, but the acronym does not take them into account. Serious oversight. The quality of a thing is not reducible to its quantitative representation. It is a whole dimension that is eliminated from the language of mathematics. At best, a trace is left with the help of units, on both sides of equality: a specific unit is granted to each side. But removing it doesn’t change the equation. In the flattened world of mathematical language, equations do very well without units. These units are ‘colors’ but the equation means the same thing in black and white. It is easy to get rid of this “colorful” dimension of mathematics that makes them unnecessarily more complex.
Complexity engendered or progenitor?
Complexity. That is the name of this obliterated dimension. Complexity is generally seen by mathematicians as generated by mathematics and not as imposing itself on it. Logical continuity of the idea that mathematics is a Platonic ideal space, popular among scholars. If we simply think that maths is a reflection of it, then we admit that we are speakers of this language and that there may be dimensions of reality “forgotten” by it. Whereas if we confuse real per se and mathematics, then it is math that generates complexity and there is no need to separate the two. But we have seen the crippling pitfalls of this attitude. I will avoid them by stating that the complex dimension is independent of mathematical language and is not spontaneously recognized by it. It only sees the effects. The reason lies in the very organization of our cognition, we will see that in a moment.
In order not to shock further, I did not add ‘approximant’ to the list of locutions covering the sign ‘=’. However, turning from mathematics to physics, this is what happens: ‘=’ becomes an approximation of reality. It is not the only acronym concerned. Infinity, for example, ‘∞’, has no proven equivalent in physics. And is not demonstrable. The limits of the measurements are those of our instruments. No certainty about the proper reality of the infinite(s). It can only be theoretical objects, exclusive property of language, allowing to describe certain appearances of reality without it being really infinite.
If the sign ‘=’ is so personal to each equation, let’s ask ourselves what about the other elementary words, especially the ‘+’. How are we sure that entities added together, all symbolized by a ‘1’, are strictly identical? They are never completely so, otherwise they would be a single entity. Our mental scene imposes at least different locations in space to individualize them. It is only as a word in language that the ‘1’ is identical.
The Mesopotamians distinguished two types of addition: addition and stacking. The stacking corresponds to our contemporary addition, two members of the addition identical in value. The addition consists in assigning a higher importance to one of the two, which the other complements. It reintroduces the notion of quality within elementary addition. André Deledicq, in ‘Dictionnaire amoureux des mathématiques’, praises the combination of the two additions into one, operated by modern maths. But if it is a step forward for reductionism, this loss of the qualitative is a regression for the complete representation of reality. Mathematics has moved away from it.
Our modern acronyms, progress or brakes?
Because it is not obvious that the two members of an addition are always qualitatively equivalent. Above all, it is not certain that the result, on the other side of the equal sign, is qualitatively equivalent to what is added together. The notion of quality is abandoned, as are all our impressions. Rough pruning.
The Mesopotamians made a foray into the world of qualitative mathematics, but it was not pursued. Euclid did not know the sign ‘=’. But, in light of what we have just seen, is this really a progress in mathematical language, or a brake? The creation of this unique symbol concealed the complexity in the language. The complex dimension was buried for 2,000 years.
I do not insist on it either. These examples are there to show that mathematical language cannot be analyzed independently of our cognitive mechanisms. These can produce math without being mathematical. They mix qualitative continuity and quantitative discontinuity. This is where the absence of a metamathematics of reality, from the physical to the mental, is sorely felt.
The cognitive bias incriminated here is simple and widespread: it is to imagine that our mental scene is reality per se. Certainly it wants to establish a correspondence. We must salute the efforts of theorists who refine our representations to make them an increasingly faithful mask. But they remain… images, neural configurations mimicking the information present in reality… the same theorists tell us. Let us keep a relative independence between reality and its descriptive language, even if it is as good intrinsically as math.
Neural configuration and concept are two sides of the same coin. It is both a philosophical consensus and the principle of a universal method, the details of which are here. This metaphor of the two-sided coin holds up remarkably well for the concept of relative independence that we need. Thanks to this tool, a fusion concept on one side is also the assembly of sub-concepts on the other side. Symbolic vertex of a neural graph. The subconcept system is a horizontal representation in the complex dimension, while the concept/subconcept hierarchy is the vertical axis of that dimension.
The complexity of an image is not the complexity of the medium
It is not necessary to believe this model of the complex dimension to understand what cognitive bias is. Officially some neuroscientists want to “flatten” neural configurations into a single large, complex system (Nick Chater 2018). The complex vertical/hierarchical axis is not completely eliminated. But the underlying echelon is pushed back to neural biology, instead of including the mental representations themselves. Reductionist vision that I would not discuss here. It suffices to admit that the image of reality itself, all scientific theories included, is the object of mimicry in this vast neural system. The complexity of the image is also a mimicry and not that of reality itself. It is the language self-organized by neurons that copies this complexity. It depicts it without itself having a complexity of the same nature.
Some might even argue that the mental picture has no complexity, as a simple puzzle of sub-concepts fitted to each other, but not self-organized together. They are only organized to achieve the result: a satisfactory mimicry of the complexity of reality. It is quite different. Our mind proposes different images of complexity and makes a choice among these alternatives, without the image itself possessing the complexity in question.
Where the cognitive bias appears
How do you know if the choice is the right one? This is impossible in the absence of an authentic theory of complexity, a theory of this dimension that imposes itself on mathematical language and not generated by it. Let us note this deficiency through the diversity of definitions of complexity, specific to each discipline, information, evolution, cognition: no metatheory unites them.
Mathematical complexity is that of one stage (or several) of cognition, not that of reality per se. This is where algorithms are intrinsically cognitive biases. They claim to describe the complexity of reality per se, when they are only reflections of it. Mediocre reflections in fact, since they hide it. Mathematical acronyms obscure this dimension. All equations should be revised, for a careful reassignment of the sign ‘=’, or its replacement by other acronyms when the terms, on both sides, differ qualitatively.
A tie with jump
The equations in this case signal a jump in the vertical axis of complexity. Our neural graphs simply add a new territory to their horizontal map. In order for them to effectively integrate this leap in complexity, this real overlay, I propose to replace in the unfolding of the equation the sign ‘=’ by a line break, and the appropriate acronym: ↩︎
This while waiting for a metatheory of reality that will make it possible to assemble these ‘line breaks’…
Mathematics is a rigorous language even in its treatment of uncertainties and fuzzy logic. Their intrinsic coherence is fascinating, to the point that some would gladly make them the very foundation of reality. However, we do not know why they have this structure and they constantly bring new surprises. This is the indication that they are undoubtedly the emergence of a more fundamental language of reality in themselves. The Mathematic exists, no doubt; but it is not ours. Ours are a humanized version, a well-ordered reflection of the structure of reality per se, but only a reflection that may miss some of its dimensions.
Our reality is a novelist who tells a story. We listen to the parts that speak to us and that our concepts can grasp. We take notes with mathematical shorthand patiently developed by our ancestors. But acronyms are sometimes too crude to describe the facts exactly. Between the lines lie other dimensions of history. We do not yet understand its complexity.
How to Do Things with Words, John Austin 1955
Dictionnaire amoureux des mathématiques, André Deledicq, Mickaël Launay 2021
The Mind is Flat: The Illusion of Mental Depth and The Improvised Mind, Nick Chater 2018
3 thoughts on “Equations, cognitive biases?”
Interesting conclusions, jeanpierre. So, let’s look at it a little differently. I think language also contains cognitive bias. However, to use a modern (?) characterization, language is a kind of algorithm: formerly called a tool. With mathematics, equations are also algorithms. By using symbols and manipulating quanta, this enables users of math to make sense of data and either confirm or refute postulates and/or relationships. Users of math recognize the value and necessity of this language, without which they would have a hard time making sense of what they do. Does a mathematician think cognitive bias a problem in accomplishing her work? I doubt that. Not everyone is adept, at, or even cognizant of math. Is that a problem?
I don’t think so. But I am not an authority because I am not a mathematician.
It’s interesting that you confuse equation and algorithm, Paul, because it’s a good example of the difference between observative and performative statements that also exists in math. The equation observes the relationships between variables. The algorithm performs a mathematical “recipe”, a series of steps to arrive at a result. The algorithm has an intention that the equation does not have.
I agree with you that a mathematician cares little about cognitive biases, just as any user of language cares little about its biases as long as it leads to the desired result. A mathematician immerses herself in a virtual mathematical universe that she thinks is universal. If she encounters an error, it means that she is not inside yet, that she took the wrong door. But I show in the article that mathematical language is a reduction of real math and can miss some of its dimensions. This is the case of the complex dimension, considered as a by-product of mathematical calculation.
I agree with your agreement with mine. We differ on the equation=algorithim piece. I follow your reasoning—I think—just do not agree with it. Differences are as they are.
Best, and thank you!