(The title is a word pun in Italian, as the adverb “mathematically” is translated as “matematicamente,” which contains the word “mente,” which means “mind.”)
Introduction: Logic as the pillar of reason and mind
The great Gauss called it the queen of the sciences, and for Pythagoras, it was the keystone of the entire universe. We are talking about mathematics, of course, man’s faithful companion from the earliest times. But what characterizes mathematical thinking? Is it perhaps an abstraction of common concepts that, for unknown reasons, has become an autonomous and mature discipline? A discipline now so emancipated from any other form of thought that it claims to admit no opinions and boasts of such a long and refined elaboration that it transcends the very inherent limits of man to define with logical-rational rigor even the conception of infinity?
As much as I believe in the creative possibilities of humankind, it seems very doubtful that beings lacking a natural predisposition for calculation in a general sense could achieve such an accomplishment, especially in such a relatively short time. For this reason, I will try here to expound some considerations that arise both from many recent findings in the neuroscientific field but also from the needs of modern artificial intelligence, whose main task is no longer to come up with particularly efficient algorithms (as is the case with many sorting and searching techniques), but to attempt the most critical challenge of the new millennium: namely, the more or less faithful reproduction of the mechanisms underlying conscious behavior. For example, people like Rodney Brooks of MIT or Mark Tilden of Los Alamos National Laboratory have long ago decided to direct their robotic research toward an approach that we might call autopoietic, overly based on the adaptive capabilities that some particular structures (neural networks and related) possess, to let the machine learn as much about how to deal with various real-world situations as about the most appropriate representation of its surroundings.
It seems as if for some time now, intelligent systems engineering has been heading toward a kind of “de-mathematization” of the entire theoretical apparatus to give way to methodologies whose validity is not always guaranteed by more or less elementary theorems but rather by man’s confidence in the means that nature has employed to build, in the course of evolution, the current human intellectual level.
At this point, however, a question arises: is it correct to say that humans live (in a behavioral sense) without the aid of mathematics? In other words, does the development of mathematics owe itself to the creative work of humankind, or is it, like the study of natural laws, nothing more than a reasonable grasp of a somewhat autonomous reality? To try to answer this question, it is necessary to make a brief argument, which will be analyzed in its essential parts in the following chapters.
Many of the most complex mathematical results find no application in everyday life. Still, it is equally valid that evaluating the usefulness of such a significant achievement in today’s situation is extremely risky, just as it is absurd to attempt to understand modern art without carefully studying the path of development that has marked human history.
Moreover, reasoning in this way leads to the absurd point of considering that even the extension of elementary arithmetic concepts, such as addition or multiplication to numbers, is unlikely to come across as unnecessary. If 3 x 2 is an operation within everyone’s reach, this is not true if one replaces integer operands with slightly different values, such as 2.9999 x 2.0001. Yet the most immediate logical step, after the strict definition of the set of natural numbers, is precisely to consider non-integer quantities whose precision can be made large at will.
If one understands that half of 1 is a half and adopts the standard notation 0.5, no one prohibits further subdivision of this value. It is not possible in any way to find some logical mathematical result that prohibits endless iteration. Trying to prove what has been stated is entirely superfluous, but attempting to understand why reason leads us in a given direction is a far from trivial goal. It requires careful and accurate analysis of many aspects of mathematical thinking that go far beyond the boundaries of the discipline itself.
It is first of all essential to premise that the fundamental basis of this science is logic, and the first man to devote much research in this direction was Aristotle, whose goal in this particular field of knowledge was to define a kind of abstract language that was completely untethered from material concepts.
The domain then shifted from purely subjective experience to reason, which naturally had to be objective and universally valid. Based on this assumption, he analyzed the concept of proposition-which we here will treat exclusively as belonging to a particular set and arrived at the fundamental result that it is possible, in association with any proposition, to define a truth function whose result is always a value of a binary set. For example, if A is part of B, the statement is true; therefore, the function will yield a positive outcome. Otherwise, it will be false. It is not necessary to ascertain that the two conditions are mutually exclusive since it is always possible to imagine a division of the universe of discourse (i.e., the totality of compatible concepts under consideration) into two complementary parts and place any object in one of the two without indecision or ambiguity. Consequently, each proposition can admit (as a result of the decision function) only two values (true or false), and all other possibilities are excluded (Principle of the Excluded Third). If it is true that A is part of B, we can automatically say that the opposite is false; moreover, our reason leads us to eliminate all combinatorial possibilities: a proposition cannot be both true and false simultaneously.
As we shall see in the third part of this discussion, a few decades ago, a new conception of logic (called “fuzzy”) was born and developed by Professor Lotfi Zadeh of the University of Berkeley, which starts precisely from postulating the invalidity of the principle of the excluded third. The assumption in question is not eliminated altogether but only adapted to the speculative needs of man.
If the question is well posed, bivalence cannot be contemplated. Still, with appropriate strategies, it is possible to “break down” the binary subdivision into a variety of partial divisions and treat each of these with a decision function no longer of the true/false type but capable of providing a value representing the degree of membership in the particular set.
Surely, many people are familiar with this methodology, with the example of the half-full or half-empty glass. Indeed, without appropriate clarifications, one is inclined to think that in some situations, recourse to the principle of the excluded third is entirely misplaced: the statement “the glass belongs to the set of half-full glasses” does not imply that its opposite (” the glass does not belong to the set of half-full glasses”) is false because no one would dare to doubt that the glass is also half-empty.
In reality, this is a real mental illusion because, in everyday life, one hardly ever finds oneself in situations that require the definition of a problem in strictly mathematical terms: the preceding question is binary only if we make it “compatible” with logical thinking, that is, if we define sets in such a way that any doubt is replaced by certainty; this is always possible since membership in a given set, in this case, is determined by a point quantity, the level of the liquid contained in the glass; if we admit that there is a lower limit (0) and an upper limit (the height beyond which overflowing occurs), the range of variation can be divided into two equal parts and the proposition “the glass is half full/empty” corresponds precisely to the median value of the level.
At this point, it is crucial to realize that when we fill the glass up to the threshold of separation of the two parts, we arrive at an unambiguous condition to which we can (and must) correspond one and only one linguistic concept, defined short of synonyms: if, for example, we choose the phrase “half full” and assume that “half empty” is also acceptable we are also forced to say that the two expressions are logically equal, and therefore there is no point in asking whether the validity of one implies that of the other since this is guaranteed by definition itself. In mathematical terms, the question is much more trivial and can be solved by eliminating logically equivalent definitions: what matters is the intensive degree, that is, the level of the liquid that can take on all fundamental values between the minimum and the maximum, and whenever it is equal to L, the only true proposition is “the glass belongs to the sets of all identical glasses filled up to a level L,” all other possibilities being excluded.
Aristotelian logic is dichotomous and admits only reasoning that relies on an analysis of concepts whose outcome cannot be other than the rigorous true/false; however, in light of the diversity (even if only functional) implied by fuzzy logic, to say that the Greek philosopher “simplified” his investigations by using this stratagem is not only a gross error but a veritable inference denied by human experience itself. It could even be postulated that dialectical thinking is a reality linked to every man’s forma mentis, and this is corroborated both by the numerous results achieved by the experimental sciences (which are based on Aristotelian logic) but especially by the innate tendency to always think in a complementary way; it is immediate to realize that any idea (whether material or abstract) cannot be conceived without first tacitly accepting its opposite. This probably drove Aristotle toward the binary result- not a sterile worldview but a deep psychological analysis of the mind.
But what is the viewpoint of neuroscience in this regard?
Although this might seem rather strange, it has been shown (see ) that the mental representation of numbers follows a generally rectilinear pattern, called LNM – Numerical Mental Line – by the researcher Francis Galton, to whom this discovery is owed. As Umiltà and Zorzi have pointed out, it is wrong to think that numerical computation occurs through the same brain centers responsible for linguistic operations. In other words, man has matured in the course of evolution, an autonomous ability to handle mathematical matters, and only in the most trivial cases does he refer to memorized formulas with a low degree of criticality, for example, the results of the Pythagorean table.
Whenever one is faced with the need to calculate a result, there are two alternatives: the first is that of approximation, which, according to brain imaging results, is assigned to the right hemisphere, while the second is that of precision, understood as the logical-rational analysis of non-immediate problems, which is carried out primarily by the left hemisphere.
This separation of roles demonstrates how the brain’s information processing is preceded by a selective intervention that directs flows in the most appropriate directions. Thus, in our case, the particular nature of mathematical computation is “recognized” a priori and does not arise from some “linguistic abuse.” But the most interesting fact that confirms what was stated earlier is that LMN is inherently a bisectional representation.
There is thus neuroscientific confirmation regarding the hypothesis-more than reasonable-that the human mind cannot conceive of the indivisible, even if it takes refuge in the highest abstractions of mathematics; dichotomous logic, therefore, is not a forcing of thought but arises precisely from the structural-functional characteristics of it.
One particularly interesting fact, reported in , concerns a test during which subjects were asked to determine which distance between pairs of numbers was greater; it was seen that reaction times were much shorter when the distances were large, while they tended to increase if the two numbers were close together. The LMN bisection operation occurs correctly (in healthy subjects) in both cases, but when greater precision is required, it requires longer and more accurate brain processing; at this point, a dilemma arises: is it possible that the brain has a “blurry” view of the single point of the line-the number-and thus, in cases of more excellent proximity, is in difficulty in providing an immediate result?
Admitting this hypothesis is possible, it remains to be understood why a set (or succession) of numbers provides a mental image of a segment. At the same time, the minimal element is defocused and degenerates into a diffusion area. Of course, when talking about LMNs, it is implied that the interviewees are asked to think about numbers and not numbers. This seemingly subtle difference is actually of fundamental importance as it allows us to understand how the brain generates each sequence.
Should we be able to think of number as an autonomous, unrelated entity, we would be confronted with the paradox of the indivisible: for if we accept based on philosophical disquisitions, but rather on the experience of every man, any quantity must be able to be broken down into smaller quantities, we must also keep in mind that the LMN can never collapse into a single point; at most it can tend toward one atomic element without, however, being able to “disconnect” it from its neighbors. Consequently, when it comes to having to decide which of the two intervals is the shorter, it is reasonable to assume that if the extremes are relatively close, from a cerebral point of view, a partial overlap from the areolae occurs, which tends to confuse the two numbers making their discrimination more difficult.
This effect of “contamination” is, in a sense, analogous to that relating to the persistence of images on the retina: the ability to discriminate between two frames is less the closer they are temporal; from a certain point of view, the representation of numbers is related to the concept of a continuum which, in turn, is mathematically associated with the real number field. George Cantor has shown how this set is the “densest” so far known, and apparently, no one has yet been able to prove that there are other fields with more remarkable power of the continuum.
Moreover, all experimental sciences, primarily physics, have constructed models of natural phenomena that inevitably use real numbers. On the other hand, only by using this number field can it be shown that there is a univocal correspondence with the points on a line and that, therefore, the LMN is also, in a sense, a local representation of the whole set. From these considerations, it can be concluded that not only is logic the basis of any mental speculation, but that number, in its most metaphysical sense, is conceived by the human mind as an autonomous concept that enables it to represent a sequentially ordered manifold.
The linguistic necessity of mathematics
Natural language is one of mankind’s most significant achievements; it is based on experience and is strongly related to the kind of sensory perceptions characteristic of humankind; for example, we do not “see” a color, but if anything, we describe the effect that a particular wavelength of light produces in the brain through a linguistic term conventionally designated to fulfill that particular task.
A person blind from birth will not be able to understand what red, green, or yellow means; that is, in that individual’s mind, the process of linking signifier and signified cannot occur precisely because the missing key element is experience. In these terms, commonly used language is severely limited by its very nature: it is encompassed within the limits that the mental definition associated with it imposes, and it is subject to the hegemony of the type of empirical development of a given population; the French linguist Georges Mounin in  writes: ” …What language communicates is the totality of the experience we have of non-linguistic reality (at least potentially), insofar as it is familiar to all users of a language. …Languages do not analyze this reality identically, so they are not an invariable, unique and always the same cast of a given invariable reality, always seen in the same way in all languages; in short, languages are not universal nomenclatures. “.
On p. 63, the author shows how some African idioms have an enormous narrowness in the description of colors, mainly due to the association of colors with the characteristic pigmentation of vegetation; on the contrary, the long cultural-historical evolution of Europe has led to the definition of countless variants of a certain color hue and has found for each of them an adequate and unambiguous term.
It is clear, then, that what we persist in wanting to “call” is the memory, more or less vivid, of a given experience, which is the basic premise of any natural terminology. But can we talk about some experience in the case of numbers?
Accepting this hypothesis (which requires a certain tolerance !), the word “number” should be linguistically related to a kind of “perception in a broad sense,” and, given the universality of mathematical language, the particular kind of experience could in no way influence such perception. Still, it should possess a generality capable of transcending all social and cultural barriers.
To understand the reasonableness of what has been stated, it is helpful to consider some structural elements of natural language: the nouns, articles, conjunctions, prepositions, and some verbs; of course, my reasoning refers in a privileged way to the Italian language, but it is not difficult to find extreme functional similarities with Neo-Latin, Anglo-Saxon and Slavic idioms. In the case, on the other hand, of archaic languages or those related to poorly evolved populations, the situation is slightly more complex because of the use of particular paradigms based, for example, on polysynthesis of expressions. However, as the neuroscientist A. Oliverio stated in , ” …But why is it, we might ask, that all languages, despite having similar structures, do not resemble each other in grammatical hierarchy? According to Baker and numerous linguists of the Chomsky school, language would also have evolved as a strategy for communicating secretly, hiding information from competitors, and a difference between languages would, anciently, have served this “cryptographic” function. “. In light of this, worrying about apparent translation incompatibilities is unnecessary, bearing in mind that our goal is to show how meaning and signification, not syntactic and grammatical rules, have a profound relationship with mathematics.
It would be contradictory to assume that some people have developed a language phylogenetically based on the concept of numbers while others have done without it; it is our goal to show precisely how it is impossible to disregard it and, thus, that the above problem competes more with ontology. Then again, as Mounin points out, linguistic richness is determined by environmental and social conditioning, but this does not mean that an Aboriginal Australian cannot instantiate the objects of his or her daily life semantically in the same way as a European or an American. This is the strong point that fully supports the thesis of “perception in the broad sense of number,” which is also corroborated, in my opinion, by the physiology of the central nervous system and, in particular, by the positional staticity of Wernicke’s area, which is deputed to the understanding of meanings; in all human beings it is located at the same point and, in all human beings a lesion to the posterior temporal circumvolution of the cerebral cortex causes sensory aphasia.
This, though perhaps a bit too reductionist, leads me to the question: Why should the brain, presenting a structure independent of genetic heritage, give rise to a functionally different conscious self? I support the hypothesis that nature tends toward an ever-increasing economy of means to eliminate redundancies (e.g., the “pruning” of synapses during early childhood) and optimize the elements deputed to the organism’s functioning.
It makes no sense to assume that the product of brain activity depends only minimally on the architecture of neural circuits. It is much more logical and consistent to think that, at least macroscopically, the understood as an organ that implements its peculiar functions is an invariant of humankind, in the same way as every other apparatus and system that performs all vital functions. Those who despise the thesis of materialism and consider the heart or liver fundamentally different from the brain might reject my position. However, it should be pointed out that science must proceed based on hypotheses that are in agreement with experience, starting from the condition sine qua non that they can be refuted at any time by new developments in human knowledge; the purpose of reductionism is thus to maintain a continuum between acquired and established knowledge and the still mysterious areas of the brain and mind. Having clarified my point, let us move on to the analysis of the individual linguistic elements examined by referring to the strict definitions provided by the Grande Dizionario Garzanti della Lingua Italiana ® (the same can be achieved with any good English Dictionary):
- The Noun
It is the fundamental element of any natural language; it provides a phonic-graphic representation of both a material and an abstract object. For convenience, we divide nouns into two distinct categories: general and particular nouns; the former belongs to all terms that do not refer to any punctual object, while the latter contains all words related to very precise and uniquely determined realities.
The most common general noun is certainly “man,” it pierces any space-time barrier and defines most comprehensively the entire human race, past, present, and future; is it possible to exclude this noun from the entire linguistic heritage to analyze it in a universe of discourse stripped of all other references? First of all, it is evident that the subject of any examination is the person who conducts it; he or she is part of the category of men and himself or encompasses every peculiar characteristic of the genus, so it is obvious that one certainly cannot isolate the concept if one wishes to have cognizance of it.
But what happens in the mind of an individual who wonders what a man might be? The first effect is undoubtedly that of self-recognition: “I am a man”; the second, which is logically unified with the first, is the mental representation of the idea of “man” as opposed to all other possible ones: “I am a man because I am different from all the objects my mind can know outside the set of men.”
It is evident that contained in this statement is an implicit awareness of the existence of other entities that have all the qualifications to be classified as humans: the general name then is the synthetic representation of a set and, consequently, it must activate, at the mental level, all the associative mechanisms necessary for a complete definition not of the individual element, but instead of the entire class referenced by it.
Those with an understanding of mathematics will undoubtedly have connected the above definition with that of numerical sets, and this is certainly not accidental since the two concepts are not only equivalent but, from a logical point of view, the general name can exist only and exclusively if one defines a priori a “container” concept capable of serving this purpose.
Such a concept is a number, that is, an abstract entity, but with a level of generality sufficient to justify any correspondence between material sets-constituted of elements that are not numbers, an appropriate subset of an appropriate numerical class. On the other hand, what makes general names interesting is the ability of the human mind to construct particular instances capable of eliciting proper awareness. Kant called this process figurative synthesis and, in my opinion, opened the door toward a more rational understanding of conscious thought.
If you ask an individual to think of a road, it is evident that you are providing all the necessary elements to activate brain processes. Still, you are not defining any particular case. Each person will think of different roads, more or less recurrent in experience, but there is no general rule to determine a priori which mental image will be evoked. Moreover, it is by no means sure that the memory is real: an individual can model a road based on his or her imagination or, rather, on the mind’s enormous capacity for generalization.
In logical terms, this process is analogous to what happens when asked to think of a number. Although the comparison seems to imply greater “sterility” in the latter case, it is good to remember that the amount of information associated with any memory is always finite and limited, whereas numerical fields-for the sake of convenience, let us think of the real one-have no limit, neither lower nor upper. In other words, referring back to Cantor’s significant result, synthesizing an actual number is undoubtedly the “freest” of all possible ones.
Of course, a person is much more likely to be convinced that he or she “possesses” (in the sense of synthetic capacity) many more memories of daily life than numbers, but this does not mean that humans have replaced perceptual objects with the quintessential mathematical abstraction; if anything, this observation should make us realize that the generalizing power of a number is infinite, to the point of “self-incorporating” into any mental representation.
For particular names, the matter is much simpler since they, from a semantic point of view, do not have to activate any synthesis: the content of the term bypasses all generalizing circuits to activate the mnemonic areas that contain the point information directly.
If this process cannot occur because the memory has been completely forgotten, neural circuits that give rise to the awareness of nonknowledge are activated. In mathematical terms, this amounts to looking for a particular element in a given set, and the outcome can only be binary. Again, it is easy for doubts to arise about such a strong statement. Still, because of what was said earlier, I remember that by choosing the set appropriately (expanding or narrowing it if necessary), any doubts about the possible bivalence of a concept are automatically dispelled.
In any case, for our purposes, it is not essential to accept the dichotomy; what matters is the logical progression from the general number to the particular, i.e., the process that allows us to distinguish, for example, Mario Bianchi from the element “man”; it is clear that Mario Bianchi is a man, i.e., he belongs to the set of men, but it is not true that a man is Mario Bianchi.
This means that “Mario Bianchi” is unable, as a special case, to represent the class it belongs to because it does not possess all the necessary characteristics. The particular name is a general name deprived of the attribute of generality and consequently always follows from the latter. Here, then, is the answer to the question about the genesis of linguistic concepts following the isolation of an element: this is not possible, and no matter how hard we try, we get a result that tends, as with LMN, to defocus the concept within the set to which it belongs without, however, succeeding in eliminating the binding constraints imposed by the general name.
In summary, we can say that a particular name is analogous to a definite number belonging to a set, and since human cognition of numbers always refers to the set (seen, for example, as a succession and represented with the LMN) and not to the point element, we can say that our mind must first “take possession” of a general name to then, be able to instantiate the particular case.
For obvious reasons, my discussion will focus on indeterminate articles (un, one, one, all), which, in the Italian language, have an immediate correspondence with the world of numbers. “One road” is a concept perfectly equivalent to the general name “Road” in the same way as “All roads,” however, the two statements have a very different logical placement: in the first case, indeterminacy leaves room for figurative synthesis, while in the second case, the whole is called into question, not as a group of features, but as an autonomous entity defining a class with particular peculiarities.
To understand what was said, compare the phrases “Think of one road” and “Think of all roads.” The former is tantamount to inviting the subject to associatively retrieve all the information inherent in the word “road” to be then able to synthesize a suitable particular case. At the same time, the latter, strictly construed, inevitably causes a paradox, a kind of “mental tilt” since the only way to arrive at a synthetic result is through deduction when, on the other hand, the article “all” necessarily requires induction. To solve the problem, the brain must call on all its resources, yet it can never arrive at an acceptable solution.
Although this may seem absurd, initial knowledge needs classes to be called “accomplished.” Still, the mental constructions that result are so powerful that they can never be fully explored, which is why we are inclined to limit the universe of discourse and understand the word “all” not in an absolute sense would be correct but in a strictly relative sense. If, on the other hand, we improperly refer to the characteristics common to all elements of a set, the sentence should be understood as, “Think of all the peculiarities that the generic element belonging to the class of roads must possess,” which is perfectly equivalent to: “Think of a road.” Of course, equivalence exists by the informational material required to carry out the symbolic synthesis; from a purely semantic point of view, the two statements can safely be considered different even though to be able to effect any signification, one must still be able to have recourse to the synthesis and thus not to the particular element as such, but to the membership characteristics encoded in it.
The determinative article can be used as a simple “apposition” for particular nouns or to transform a general understood as a set of characteristics into the corresponding class, just as with the indeterminative article “all.” In general, the former case always occurs when a specification is added to the sentence in a broad sense: “The street where I live,” “Mary’s brother,” etc… At the same time, the latter can be implied in sentences such as “Man is a bipedal animal.”
It is evident that the addition of the characteristic “bipedal animal” is possible only if a definite category (“Man”) is defined as a priori that gathers to itself all elements having certain peculiarities.
To summarize: the indeterminate article (proceeding by analogy) is helpful but not necessary for pointing to the generic numerical element (a, one, one) or the whole class (all), while the determinative article can take on two valences, the first being fully equivalent to the last indeterminate case, the second, in conjunction with a specification, allows pointing to a particular element of a whole.
Conjunctions (we will only consider “and” and “or”) are par excellence, logical connectives that find a precise and essential place in Boolean algebra. Let us consider the propositions “A and B are C” and “A or B is C”: the former expresses the concept of A and B belonging to the set C, and thus, if C is defined through a collection of features, it states that both A and B possess the peculiarities necessary to be members of C.
The figurative synthesis of C (from knowledge of the above proposition) could then lean toward A or B (or a mixture of the two) without any well-determined rule; for example, asking a white man to think of “a man,” it is very likely that daily experience will lead him to synthetically imagine a white man, while the same request made to an Asian could culminate in the mental representation of a person with almond-shaped eyes and all the other physical characteristics of Asians.
However, this does not imply that the results can safely be reversed since what matters is the fact that A and B are both elements of C. The conjunction “and” thus makes it possible to group equivalent propositions by membership characteristics to simplify their form. Still, of course, one could do without it as long as one “unbundles” all the sub-phrases of a given statement: “A and B and C and … and Z are W” is equivalent to the union of: “A is W,” “B is W,” …, “Z is W.”
The conjunction “or,” on the other hand, is mutually exclusive and finds valid use in all propositions in which there is doubt about membership; it should be noted that while “A and B are C” increases the overall level of information to which each subphase contributes, “A or B is C” only assures us that one of the two elements is peculiarly eligible to be part of C, but gives us no helpful clue about the other. The only logical certainty we have is that if, for example, A belongs to C, B will not belong to it. However, the connective “or” cannot provide adequate data to know its location. Indeed, it must be kept in mind that to validate a proposition with the conjunction “or,” it is not necessary to be aware of the characteristics of both members: the sentence “Albert Einstein or “gklikj” was a man” is correct. Still, we arrive at this certainty despite having no idea what a “gklikj” might be.
Figurative synthesis from the connective “or” is based solely and exclusively on only one of the two elements and cannot, as a rule, arrive at a reasonable image of the other. The only case in which this is possible is when exclusion occurs between known and perfectly classified objects; in such a case, from a logical point of view, a kind of “coactive separation” of sub-phrases occurs, resulting in the elimination of redundancy due to the use of the connective.
For example, if I state that “My mother or my little dog is a man,” what I implicitly operate is equivalent to the disjunction of the two propositions, “My mother is a man”-Logically correct sentence-and “My little dog is not a man,” which is immediately changed to: “My little dog is an element of the set of dogs.”
Linguistically, then, conjunctions play a fundamental role in natural language. Still, their semantic “power” arises only from logic, which, through rationalizing thought and expressions, is the guarantor of their correct use and the mental results that flow from them. However, I suggest consulting a book on Logic highlighting all the more advanced speculative results for those who want to explore the subject further.
Referring to the Italian language, nine simple prepositions (di, a, da, in, con, su, per, tra, fra) have the linguistic role of allowing the formation of complements (the same happens in English with of, to, at, in, with, on, etc.
As we mentioned earlier, in the case of indeterminate articles, very often particular nouns are obtained from a general noun followed by a specification, e.g., the phrase “The men of Rome” is composed of a reference to the class “man” and an appropriate addition, “of Rome,” which limits the set of possible values.
Thus, the role of complements is to allow the definition of subclasses generated by the set of features related to a “parent” class and the conditions imposed by the analysis; again, logic is the only guarantee of success since, to construct correct propositions, the conditions specified by the complement mustn’t disagree with the peculiar features of the set. Therefore, to be able to validate such a proposition, it is impossible not to have recourse to a conditional figurative synthesis, which, however, is much weaker than a normal synthesis because its success depends not only on reason but also on experience. As usual, let us consider a virtual experiment and evaluate the reaction of a subject who is asked to think of “a man from Mars”; it is evident that the conditional figurative synthesis is subject to the knowledge of the all but deterministic rule that there are no men on the planet Mars; even if we admit that a living race on Mars has peculiar characteristics, this does not mean that it is not permissible for the brain to picture a Martian with the features of an earthling.
Indeed, the request is misplaced in that there is an assumption in it that if Mars is populated, there will be elements classifiable as humans; quite different is the issue if one is asked to think of “an inhabitant of the planet Mars.” In this case, the synthesis can take place only if an appropriate set of characteristics that the population of the red planet must possess has been predefined; if this has been done, even by imagination, it is easy for images of little green men with antennae to come to mind, but if, on the contrary, there is no awareness of the existence of such a class then logic forbids the formulation of propositions such as the last one we have taken as an example since otherwise one runs the risk of plunging into a vortex of ambiguity and moving from the domain of reason to that of subjective opinions.
So, complements are permissible only if, through them, it is possible to complete a figurative synthesis based on reason and experience. Still, their indiscriminate use is the object of logic’s warning because the conditions they impose on the proposition must never be at odds with the generating characteristics of the class under consideration.
The fundamental structure of a linguistic proposition is Subject-predicate-complements. For our purposes, we will assume that the verb to be and its functional synonyms represent the key elements of class membership, so there is no point in discussing them further.
A special case is that relating to the comparison of particular names, “A is B”; from a logical point of view, there are two alternatives: the first assumes that A is identically equal to B, so the proposition is tautological, while the second is the classic case where B is a general name. Hence, we return to the main case. As for the other verbs, they generate propositions that can always be traced back to the stereotype “A belongs to the set B”; the figurative synthesis of mental images arising from sentences composed of subject and predicate is very often related to personal experience, and it is not always easy to be able to make appropriate generalizations. However, it is interesting that the mind rejects “gaps” and takes refuge in pure imagination whenever experience is absent or too limited. The result is the formation of erroneous ideas and beliefs that sometimes, unfortunately, condition one’s entire existence; logic, which stems from reason, rejects any element that does not find a proper place within rational reasoning and thus preserves the user from self-made failures.
Maintaining control of one’s life based solely on logic can appear disappointing (although it remains the basis of any mental speculation). Still, I am convinced that adherence to the simple rules of inference can ensure a better approach to various situations in which we are driven to the synthesis of realities whose truth we cannot in any way assert.
Fuzzy thinking and the concept of probability
When Professor Lotfi Zadeh formulated his theory on fuzzy sets, he started from the famous consideration that humans can make decisions based on partial and imprecise input information, which is why programming automatic systems to handle particularly complex situations is much more challenging than manual performance of them by a human operator. In , Bart Kosko, the leading “follower” of the fuzzy philosophy, cites the example of parking a motor vehicle in reverse; this task is performed naturally by the vast majority of motorists, but it requires mathematical modeling so complex that automation of the process is beyond the reach of commonly used computers. This gap can be filled by trying to program systems not by following Aristotelian logic but rather by referring to fuzzy logic.
We will assume here that the reader has minimal knowledge of the fundamentals behind this way of approaching the solving of a variety of problems. However, I would like to remind the reader that the basis of everything lies in the type of set membership function used. In the dichotomous case, it necessarily had to be binary; in the fuzzy case, however, it provides a continuous degree of membership that varies between 0% and 100%. According to this principle, an element A can belong to different classes with the only condition that totally the degree does not exceed unity (100%); in a way, this way of reasoning is very similar to the concept of probability, and here I will try to explain why.
Whenever our knowledge of reality cannot reach a satisfactory level, we are often led — implicitly following the teachings of the physicist-mathematician Laplace — to turn toward an approach (probabilism) that, while allowing us to know only certain aspects of the problem, does not leave our minds in the shadows of ignorance. As Kosko says in , ” …I believe, then, that probability or “randomness” is a psychic instinct, a Jungian archetype or mental propensity that helps us organize our perceptions, memories or most of our expectations. Probability gives an orderly structure to mutually conflicting causal predictions about how the future will evolve in the next instant, season, or millennium.”
What we call “probable” is indeed somewhere between “true” and “false.” Still, unlike inferences without any foundation, probability is based on logical-rational reasoning and arrives at unequivocal certainties. When we say that the average age of a population is 35, we do not mean to imply that all group members are 35 years old, but instead that it is logical to consider a numerical value (the mean) that informs us, along with the variance, about the frequency of favorable encounters with individuals whose age is relatively close to a desired value.
The mean and variance are deterministic data; the approach, however, is not. But what is the relationship between probabilism and fuzzification of logic? When we talked about the LMN, we saw that the concept of number appears inherently fuzzy; it generates not a point, as analytic geometry would have it, but rather an areola that overlaps with adjacent line sections.
This process stems from the brain’s need for continuity and, in a sense, has “afflicted” man since his appearance on Earth. Yet, it reveals quite clearly what Bart Kosko wrote in his masterful essay on fuzzy logic: man sees probability not as a recourse “of convenience” but rather as a necessity that transcends even the boundaries of physics.
At this point, it seems evident that what has been stated so far is refuted by psychology. Still, the problem is quite different and requires careful analysis that does not limit itself to the effects but instead seeks to reach the causes. First, it should be said that fuzzy logic is not an alternative to Aristotelian logic. Still, if anything, it sets before us speculative questions from a different point of view: to say that an element belongs to several sets with a degree for each is equivalent to saying that it can belong to the different sets but that, indeed, it is always possible and reasonable to suppose the bisection of the universe of discourse and the consequent placement of the element in one of its complementary classes. However, this operation is generally quite difficult to perform since it presupposes a comprehensive knowledge of the problem and does not admit of any approximation; in this sense, Lotfi Zadeh’s statement is analogous to Laplace’s invitation to make use of the calculus of probabilities whenever the portions of nature under consideration present such complexity that any other approach is impossible.
However, it is also evident that this choice should not be prioritized over determinism. Otherwise, there would be a risk of increasingly reducing the degree of human knowledge, both in the case of the natural sciences and also in the area studied by psychology. Everyday life is not easily handled by resorting to the powerful mathematical techniques used in physics or engineering, but neither can it be said that our way of reasoning disregards the need for precision; if anything, it is the desire to “own” concepts in the most complete way that pushes us toward a seemingly less valid approach.
Just above, I said that the symbolic synthesis of propositions about which one does not have much experience is approached by the brain operating a kind of “improvisation” that uses the data possessed and tries to predict what the value of a specific statement might be based on more or less consistent constructions, this can be read in a “fuzzy” key: unless it possesses no useful information (which makes any approach impossible) the human mind unconsciously turns to probabilism and “frames” the data in a way that is fuzzy enough but precise enough to give rise to satisfactory mental images.
For example, an infant has no cognizance of the difference between being hungry and being thirsty; whenever his body signals a glucose or water shortage, the crying signaling mechanism is activated; following this, he is fed with milk, and, in this way, both needs are met.
A child’s mind cannot correctly make a dichotomy until he reaches weaning and a sufficient level of consciousness. Yet, he will always be able to briefly figure out the meaning of “I am hungry” and “I am thirsty” even without precise information. Somehow, he assesses the probability that a given sensation has characteristics in common with another and makes a classification such as, “I experience a sense that resembles a generic element of the set consisting of from the sensations of hunger, but at the same time it may be a member of the set consisting of the sensations of thirst.”
Fuzzy logic, therefore, allows for more rigorous handling of all those situations that, while obeying Aristotelian logic, are difficult to understand and treat.
- C. Umiltà, M. Zorzi, I numeri in testa, Mente&Cervello n.2 March-April 2003
- G. Mounin, Guida alla Linguistica, Feltrinelli
- A. Oliverio, Prima Lezione di Neuroscienze, Editori Laterza
- B. Kosko, Il Fuzzy Pensiero, Baldini&Castoldi
- I. Kant, Critica della Ragion Pura, Editori Laterza