*4.3. Unreasonable Misunderstandings of Mathematics*

There is one more type of fallacy, which at first sight, may look as clearly belonging to the Idols of the Number as they involve mathematics, but actually could be placed in the next category of the Idols of the Common Sense, in spite of the fact that they misguide not lay people, but highly respectable mathematicians or physicists. The example can be the naive reflection on *The Unreasonable E*ff*ectiveness of Mathematics in the Natural Sciences* by famous physicist Eugene Wigner [38]. Wigner mused on what he considered a mystery: "The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it" [38]. He concluded his paper with: "The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our ba fflement, to wide branches of learning" [38].

The leitmotif of the paper is like a bewilderment of someone who, referring to his shooting skills, after watching the arrows in the center of the target's bull eye, forgot that the arrows were shot first and

only after this, were the concentric circles drawn. Wigner's surprise is one more piece of evidence for the fragmentation of science which started to be considered as the normal state. In the past, there was no separation of mathematics and physics, therefore, the work on physical theories was not di fferent from the work on mathematical problems. There is no surprise (although apparently for Wigner there is) that informal, intuitive associations between di fferent domains of scientists' activities acted as cross-pollination between mathematics and physics, even if, very often, the formal association might have been never considered or achieved. Mathematical theories frequently went much beyond the interests of physical theories and the connection was lost.

Wigner's article could have been just an amusing anecdote about an absent-minded famous physics professor who suddenly realizes that instead of doing his job in physics, he is doing mathematics. However, the sensational title of the paper and the Matthew e ffect caused a lot of damage by creating a frequently invoked false mystery. It is hard to believe that *Unreasonable E*ff*ectiveness of Mathematics* was written by the founder of the studies of symmetry and group theory in quantum mechanics, who received the 1963 Nobel Prize in Physics for his contributions to the theory of the atomic nuclei and elementary particles through the discovery and application of fundamental symmetry principles.

An explanation of "[t]he miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics [...]"can be found in the works of other giants such as Hermann Weyl [39] or the 1977 Nobel Prize in Physics laureate Philip W. Anderson [40], who gave the answers to the role of mathematics in physics and other disciplines exactly in terms of symmetry and group theory. The apparent miracle turns out to be just an expression of the identity of mathematical and physical theoretical inquiries as summarized in a sentence from Anderson's famous article "More is Di fferent": "It is only slightly overstating the case to say that physics is the study of symmetry" [40]. Yet the title of Wigner's article became a meme which persists in confusing lay people. Whatever damage has been done, the case is an excellent example of the dangers of the Idols of the Number which include the belief in the reality of a strict demarcation line between mathematical and physical inquiries.

#### **5. Idols of the Common Sense**

It is again necessary to start this section from a disambiguation. Here, too, we have to be aware of the possibility of the confusion caused by equivocation. Common sense has two separate, although convoluted, traditions of study and associated with them are multiple ways of understanding this expression. Common sense in the understanding presented by Aristotle in *De Anima* is a capacity to identify shared aspects of things. Various expressions involved in the analysis of this capacity in humans and animals were later subsumed in the later translation into the "*common sensibles*" (and in modern psychological terminology called "binding"). Aristotle excluded existence of the sixth sense (although sometimes he addressed this capacity as the first sense), but rather considered the common sense a faculty by which *common sensibles* are perceived together as a single object.

Further evolution of this synthesizing faculty was long and too complex to be presented here, as this way of thinking definitely does not belong to the Idols of the Common Sense. Actually, it should be the subject of intensive studies within Natural Philosophy as a main tool for its integrative functions. An extensive study of common sense as the capacity to integrate information was published by the present author elsewhere [41].

There is another use of the expression "common sense" as a skill of using everyday experience common to all people from a more or less culturally homogeneous community in making decisions or normative judgments including judgments of the truth or falsity of statements. These type of skills are usually associated with "streetwise wisdom". Very often these skills are transmitted by language or learned by observation in the social environment. They may be of grea<sup>t</sup> practical value and they may be, in some situations, the only means to reduce complexity of the environment, i.e., they are necessary for everyday intelligent behavior. One of the main objectives of robotics and AI study is to develop in artificial systems the capacity of such common sense. Thus far, this objective has never been achieved.

Yet, we have to be careful about engaging both types of intuitive capacities in situations when the environment is very di fferent from the environment in which intuitive skills have been acquired. Even more dangerous is mixing the intuitive and rational methods of inquiry involving higher levels of abstraction.

### *5.1. Beware of What Escapes Awareness*

Idols of the Common Sense represent fallacies resulting from making conclusions based on individual, everyday experience, unaided by any systematic methods of critical thinking about the matters far removed from this experience. However, the origin of these type of fallacies is the result of the negligence of the recognition for both rational and irrational forms of inquiry and resulting confusions. When we ignore the role of the intuitive capacities as primitive and not deserving attention, they take over the functions of rational capacities and confuse them. In the presentation of the Idols of the Number, the central fallacious forms of inquiry were generated by the illusionary distinction between quantitative and qualitative methodologies of inquiry and the neglect of structural analysis accompanied by misunderstanding of mathematics and its role as a tool of inquiry. In the Idols of the Common Sense, the central confusion regarding the complex relationship between the rational and intuitive forms of inquiry, in mixing their analyzing and synthesizing roles is accompanied by the neglect of logic.

The distinction and relation between the rational and intuitive forms of inquiry was studied in my earlier publications [33,41,42]. For the purpose of this paper, it will be su fficient to consider the distinction between the inquiries involving the language-based reasoning organized and controlled by logic and the engagemen<sup>t</sup> of the human capacities to organize perceptions which escape linguistic and logical control. The most important capacity of the second type is our ability to integrate information into indivisible units which, in the rational form of inquiry, is called an "object". The examples of the interaction between the two forms can be found in the presence of the word "thing" in Aristotelian writings, which he never tried to explain or to define, or in the struggle to conceptualize the notion of a set (Cantor, Husserl and many others) which ultimately was abandoned by giving the notion of a set the status of a primitive concept.

The further consequences of the Idols of the Common Sense are especially detrimental for the study of the complementary objective and subjective forms of inquiry leading to the belief in their opposition and in the dominant and exclusive role for the former. The distinction here was explained very briefly in Section 2.2 in the terms of invariance, but was extensively discussed in my earlier publications [23].

If we want to study Contemporary Idola Mentis for the purpose of preventing errors and fallacies in the Contemporary Natural Philosophy, we have to avoid unjustifiably rigid rules and exclusive restrictions to the existing methodology of science. There is nothing wrong in the study and development of methodologies engaging human intuition and its capacities. There were many highly recognized mathematicians and physicists (e.g., Henri Poincare) who openly declared the primary role of their intuition in their achievements.

Yet, the collective experience of mathematicians and physicists provides examples of the abuse of what was considered a systematic use of intuitionistic methodology, in particular, the refusal to accept the excluded middle rule of logic. The most notorious was the abuse by Leopold Kronecker, of his power of being the editor, to veto the publication of works submitted by the founder of set theory, Georg Cantor, on the grounds that this was not mathematics. We are not concerned here about the development of clearly formulated programs to modify logic or other tools of inquiry, as long as they follow the rules of evaluation of intellectual activity and are not just expressions of individual belief or personal preferences. So, the excesses of intuitionism, even if being harmful, do not belong to the Idols of Common Sense. For this qualification, it is necessary that the common sense deviation from logical or systematic, methodological rules is without any awareness of it. The actual Idols of the Common Sense are the cases when the intuitive forms of inquiry developed in the familiar environment from everyday experience encroach on the functions of rational capacities.

The classical example of the fallacy belonging to the Idols of the Common Sense has its own name, "Linda the Bank Teller". Amos Tversky and Daniel Kahneman, studying extensional and intuitive reasoning, created a story about a fictitious character called Linda for the participants in their research [43]: "Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations. Which is more probable? 1. Linda is a bank teller. 2. Linda is a bank teller and is active in the feminist movement." More than 80% of participants in the research chose answer 2.

We should not be surprised at the above. Probability theory and logic are notoriously counterintuitive. This is a natural consequence of the di fferences between competences of the rational and intuitive capacities. It is significant that the most confusing for untrained people are problems related to conditional probability (in particular Bayes Theorem) and to inferences involving implication. However, even the use of simple connectives such as "and", "or" turns out to be problematic for students, if they cannot use Venn diagrams (i.e., set theoretical representation).
