An Initial Taxonomy of the Fundamental Cognitive Mechanisms used in Mathematical Creation/Invention

We describe an explicit initial list of the most fundamental cognitive mechanism used by our minds when we are doing formal research in mathematics. Some of these abilities are basic mental mechanisms used everywhere in human reasoning, other ones are more domain-specific abilities, which play their most outstanding roles during mathematical creation/invention.

1. Conceptual Blending

This is the mental ability of combine genuinely two ideas or concepts for creating a new one which contains essential features of initial conceptual spaces and, at the same time, being semantically more complex than the any of them. In the literature there are several compuationally-feasible formalizations of conceptual blending, which have been able to co-discover several mathematical concepts (see for instance [31], [7], [33], [34]. For more general information about the central role that conceptual blending plays in conceptual creation, the reader can see the website http://markturner.org/ blending.html

2. Analogical Reasoning

The ability to find commonalities between two objects living in multiple conceptual environments, seems to be a very omnipresent cognitive process within all the scientific disciplines. In particular, the invention of well-known results in physics as the Rutherford atom model and the discovery of the complex numbers in pure mathematics can be seen as emerging from an analogical process [55], [39]. In general, outstanding mathematicians of the twentieth century like AndreWeil pointed out the prominent role that analogical reasoning played in the development of modern mathematics [59].

3. Metaphorical Thinking

Pure mathematics has seen a tremendous explosion of metaphorically-inspired creative results during the last seventy years. For instance, new whole areas have appeared based on the effort of trying to understand one specific mathematical sub-domain in terms of others. As illustrative examples we can

mention the development of modern algebraic geometry mainly within the Grothendieck’s school by means of introducing categorical and homological methods to the classic framework described in terms of classical varieties and their rings of coordinates [35]; the quite outstanding solution of a classic number theoretical problem as Fermat’s Last Theorem by means of the metaphorical usage of new conceptual frameworks coming from Iwasawa Theory, the theory of modular forms and the theory of elliptic curves [60]; and integration of seminal methods coming from algebraic topology to fundamental notions of modern algebraic geometry as the development of a homotopy theory for schemes [45], among others.

4. Conceptual Substratum

This is the cognitive ability of choosing generic syntactic representations for every possible instance of an specific (mathematical) concept. For example, suppose that one should solve the following elementary question: Why when we add two even (integer) numbers the result is again an even number? This seems to be true for small pairs of numbers 2 + 6 = 8, 12 + 18 = 30 and 214 + 674 = 888. Now, for getting a general proof of this fact, we should consider syntactic expressions which can allow us to ‘represent’ the even numbers in a compact way. Therefore, we typically come up with a (mental) representation of the form 2*n. This means that essentially we are able to represent the collection of even numbers simultaneously with the single expression 2*n, where we assume implicitly that n is an integer. On the other hand, if we know that a number c can be written as 2*d, where d is an integer, then by definition c should be an even number. In conclusion, we have found a compact

(morphologica-lsyntactic) expression for representing every even number in a unified way. Now, let us consider again the former question with the former representation in mind: First, we need to consider two (potentially different) even numbers, so we consider (or ‘imagine’) a first even number 2*a and a second one 2*b, where a and b are integers. Second, we sum these numbers generically, namely, we obtain the expression 2*a+2*b. In addition, we check if the final syntactic expression corresponds to an even number. Thus, we try to give it the desired form 2*#, where # is a natural number. So, we factorize the former algebraic expression and get an expression of the form 2*(a+b). Lastly, we realize that this number has the desired form 2*x, where x = a+b is an integer. In conclusion, we ’proved’ an affirmative answer for the former question by performing symbolic operations on morphological generic representations for even numbers.

5. Conceptual Lining

This is the dual mental mechanism of conceptual substratum, and basically conceptual lining allow us for converting fixed syntactic representations into global (mathematical) notions. For instance, taking the former example, we consider the symbolic expressions 2*m and m 2ϵN, and then conceptual lining is the mechanism responsible for generating the well-known mathematical concept of the even (positive) numbers.

6. Syntactic Simplification

By doing a meta-analysis of the origin of the proof of a normality criterion for forcing algebras [10], we identify a kind of ‘omnipresent’ process applied widely and implicitly through the mathematical

literature, i.e., the process of starting to solve a conjecture and to get more insights about its solution by making reasoning in small and concrete examples. So, as shown in [10], this criterion was formulated from the very beginning by studying the easiest examples of forcing algebras in the ring of polynomial in 1 and 2 variables, and defined the simplest possible forcing data, i.e., monomials like x; y and xy. Thus, in this fashion, one starts to increase gradually the generality of the monomials, e.g. x^a; y^b and x^cy^d, (where a; b; c and d are natural numbers), until it is possible to obtain a compact and clear condition that was proved subsequently. Now, one specific sub-goal towards AMI is to obtain a meta-formalization of the former heuristic processes, which can be seen as a kind of main particularization, particularization on (exponents, superor sub-indices, etc.), and syntactic simplification. Moreover, they operate in the former example over the symbolic representations of polynomials.

7. Syntactic Generalization

This mechanism encompasses all the dual former sub-mechanisms, i.e., sub-abilities allowing for gradually getting more complex conceptual substrata fulfilling an specific property. For example, when a mathematician proves a conjecture for all natural numbers, by proving it first for all prime numbers; then for all perfect powers of a prime number; for all numbers with at most two prime divisor; and finally for all numbers with (at most) n prime divisors.

8. Conceptual Meta-Conjuntion

This is the ability responsible for combining conceptual subtrata an combining it in a conjunctive and disjunctive way. This cognitive process can be seen a very primitive and simplified form of doing conceptual blending. However, it should deserve an additional category within our taxonomy, because it is use more frequently than conceptual blending and, strictly speaking, this mechanism does not exceed the semantic complexity of the input concepts.

9. Conceptual Complement

This mechanism is typically used when there is a kind of fixed universal conceptual space U behind the formal setting in which we are solving a conjecture. Specifically, we are using a particular (mathematical) concepts C and, at some point, we need to be able to characterize the formal complement of C with respect of U, in a explicit syntactic way. Let us consider the former example of the even numbers, in some cases we need to be able of describe in a generic way (for example as a conceptual substratum) the elements (within the universe of the Natural numbers) that are not even numbers, i.e., the odd numbers. So, typically we would consider expressions of the form 2n + 1, (where n is a natural number) for solving specific conjectures.

Additional Cognitive Abilities

One of the main stages for the fulfillment of AMI is to obtain a complete list of these kind of cognitive abilities. This is one of the current sub-projects of the AMI-team. So, the former mechanisms were

explicitly described in order to give an initial and brighter idea of how such a global AMI-cognitive taxonomy would look like. Additional cognitive mechanisms in current research are related with inductive and abductive reasoning, syntactic generation and elimination of hypothesis and thesis and the proof’s method of Reductium ad absurdum (proof by contradiction).

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