# 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).

**Bibliography**

[1] Alexander, J. (2011). Blending in mathematics. Semiotica, 187: 1–48. [2] Apostol, T. (1976). Introduction to Analytic Number Theory. Springer. [3] Atiyah, M. F. and Macdonald, I. (1969). Introduction to Commutative Algebra. Addison-Wesley, Reading, MA. [4] Beeson, M. (1989). Some applications of gentzen’s proof theory in automated deduction. In International Workshop on Extensions of Logic Programming, pages 101–156. Springer. [5] Besold T. R., Bou F., Cambouropoulos E., Codescu M., Confalonieri R., Corneli J., Hedblom M., Kaliakatsos-Papakostas M., Kühnberger K.-U., Kutz O., Gómez-Ramírez D., Mossakowski T., Neuhaus F., Pease A., Plaza E., Schorlemmer M., Smaill A., and Zacharakis A. (2017). Conceptual blending in dol, evaluating consistency, conflict resolution. In Confalonieri R., Pease A. and Schorlemmer M. et al. editors, Concept Invention: Foundations, Implementations, Social Aspects and Applications, Cognitive Technologies. Springer. [6] Bidoit, M. and Mosses, P. D. (2004). CASL User Manual. Lecture Note in Computer Science 2900, Springer. [7] Bou F., Corneli J., Gómez-Ramírez D., Maclean E., Peace A., Schorlemmer M., and Smaill A. (2015). The role of blending in mathematical invention. Proceedings of the Sixth International Conference on Computational Creativity (ICCC). S. Colton et. al., eds. Park City, Utah, June 29- July 2, 2015. Publisher: Brigham Young University, Provo, Utah., pages 55–62. [8] Boy de la Tour, T. and Peltier, N. (2014). Computational Approaches to Analogical Reasoning: Current Trends, chapter Analogy in Automated Deduction: A Survey, pages 103–130. Springer, Berlin, Heidelberg. [9] Brenner, H. and Gómez-Ramírez, D. (2014). On the connectedness of the spectrum of forcing algebras. Revista Colombiana de Matemáticas, 48(1): 1–19. [10] Brenner, H. and Gómez-Ramírez, D. (2016). Normality and related properties of forcing algebras. Communications in Algebra, 44(11): 4769–4793. [11] Bridge, J. P. (2010). Machine learning and automated theorem proving. Technical report, University of Cambridge, Computer Laboratory. [12] Bundy, A. (2011). Automated theorem provers: a practical tool for the working mathematician? Ann. Math. Artif. Intell., 61:3–14. [13] Bundy A., Atiyah M., Macintyre A., and Mackenzie D. editors (2005). The nature of mathematical proof. Philosophical Transactions of the Royal Society A, 363 (1835). [14] Bundy A., Jamnik M., and Fugard A. (2005). What is a proof? Phil. Trans. R. Soc A, 363(1835): 2377–2392. [15] Codescu M., Neuhaus F., Mossakowski T., Kutz O., and Gómez-Ramírez D. (2017). Conceptual blending in dol, evaluating consistency, conflict resolution. In Confalonieri R., Pease A. and Schorlemmer M. et al. editors, Concept Invention: Foundations, Implementations, Social Aspects and Applications, Cognitive Technologies. Springer. [16] Cook, S. A. and Reckhow, R. A. (1979). The relative efficiency of propositional proof systems. The Journal of Symbolic Logic, 44(01): 36–50. [17] Crandall, R. and Pomerance, C. (2006). Prime numbers: a computational perspective, volume 182. Springer. [18] Darwiche, A. and Marquis, P. (2002). A knowledge compilation map. Journal of Artificial Intelligence Research, 17(1): 229–264.

[19] Défourneaux G., Bourely C., and Peltier N. (1998). Semantic generalizations for proving and disproving conjectures by analogy. Journal of Automated Reasoning, 20(1-2): 27–45. [20] Dirac, P. (1938). A new notation for quantum mechanics. Mathematical Proceedings of the Cambridge Philosophical Society, 35(3): 416–418. [21] Eisenbud, D. (1995). Commutative Algebra with a View Toward Algebraic Geometry. Springer. [22] Fauconnier, G. and Turner, M. (2003). The Way We Think. Basic Books. [23] Feferman, S. (1988). Hilbert’s program relativized; proof-theoretical and foundational reductions. The Journal of Symbolic Logic, 53(02): 364–384. [24] Feferman, S. (1989). Finitary inductively presented logics. Studies in Logic and the Foundations of Mathematics, 127: 191–220. [25] Flach, P. A. (2010). Encyclopedia of Machine Learning, chapter First-Order Logic, pages 410– 415. Springer. [26] Fleuriot J., Hofner P., McIver A., and Smaill A. (2013). ATx´ 12/WInG´ 12: Joint proceedings of the workshops on automated theory exploration and on invariant generation, 6th international joint conference on automated reasoning 2012, easychair proceedings (epic volume 17). Vol. 1. [27] Fleuriot J., Obua S., Scott P., and Aspinall D. (2014). Proofpeer: Collaborative theorem proving. http://arxiv.org/abs/1404.6186. [28] G. Lakoff and R. Núñez (2000). Where Mathematics Comes From. Basic Books. [29] Gallego E., Gómez-Ramírez D., and Vélez, J. D. (2016). The direct summand conjecture for some bi-generated extensions and an asymptotic version of koh’s conjecture. Beiträge zur Algebra und Topologie, pages 1–16. [30] Ganesalingam, M. and Gowers, W. T. (2016). A fully automatic theorem prover with human-style output. Journal of Automated Reasoning, pages 1–39. [31] Gómez-Ramírez, D. (2015). Conceptual blending as a creative meta-generator of mathematical concepts: Prime ideals and dedekind domains as a blend. C3GI at UNILOG 2015, Workshop on Computational Creativity, Concept Invention, and General Intelligence. Tarek B. Besold et al., editors. Publications of the Institute of Cognitive Sciences, PICS series, University of Osnabrück Vol. 2. [32] Gómez-Ramírez, D. (2015). Conceptual blending as a creative meta-generator of mathematical concepts: Prime ideals and dedekind domains as a blend. In Besold T. R., Kühnberger K.-U., Schorlemmer M., and Smaill A., editors, Proceedings of the 4th International Workshop on Computational Creativity, Concept Invention, and General Intelligence (C3GI), volume 02-2015 of Publications of the Institute of Cognitive Science, pages 1–11. Institute of Cognitive Science. [33] Gómez-Ramírez, D. (2016). Formal Conceptual Blending as a Fundamental Logic Mechanism for highly Abstract Mathematical Invention: The Beginning of a Meta-generation of Fields and Galois Theory. Submitted. See attached preprint UnpubArticleGenerationBlending.pdf. [34] Gómez-Ramírez, D. and Smaill A. (2017). Formal conceptual blending in the (co-)invention of pure mathematics. In Confalonieri, R., Pease, A., and Schorlemmer M. et al. editors, Concept Invention: Foundations, Implementations, Social Aspects and Applications, Cognitive Technologies. Springer. [35] Grothendieck, A. and Dieudonné, J. (1971). Eléments de Géométrie Algébrique I. Springer. [36] Harrison, J. (2009). Handbook of practical logic and automated reasoning. Cambridge University Press. [37] Hartshorne, R. (1977). Algebraic Geometry. Springer. [38] Hoffman, K. and Kunze, R. (1971). Linear Algebra. Prentice-Hall. [39] Fleuriot J., Maclean E., Smaill A., and Winterstein, D. (2014). Reinventing the complex numbers. volume 1 of PICS. Workshop at ECAE, Prague. Workshop on Computational Creativity, Concept

Invention, and General Intelligence. Tarek B. Besold et al. editors. Publications of the Institute of Cognitive Sciences, PICS series, University of Osnabrück. [40] Krumnack U., Gust H., Kühnberger K., and Schwering, A. (2008). The re-representation problem in a logic-based framework for analogy making. In Fox, D. and Gomes, C. editors, Proceedings of the Twenty-Third AAAI Conference on Artificial Intelligence (AAAI-2008), pages 1462–1463. AAAI Press. [41] Krumnack U., Mossakowski T., and Maibaum T. (2014). What is a derived signature morphism? In Diaconescu R., Codescu M. et al. editors, WADT 2014: 22nd International Workshop on Algebraic Development Techniques, pages 50–51. [42] Lightstone, A. H. (1978). Mathematical Logic: An Introduction to Model Theory, chapter Propositional Calculus, pages 11–29. Springer. [43] Martinez M., Abdel-Fattah A. M. H., Krumnack U., Gómez-Ramírez, D., Smaill, A., Besold, T. R., Pease, A., Schmidt, M., Guhe, M., and Kühnberger, K.-U. (2016). Theory blending: extended algorithmic aspects and examples. Annals of Mathematics and Artificial Intelligence, pages 1–25. [44] Mendelson, E. (2010). Introduction to Mathematical Logic; (5rd Ed.). Chapman& Hall/CRC, Boca Raton, Fl, USA. [45] Morel, F. and Voevodsky, V. (1999). A 1-homotopy theory of schemes. Publications Mathématiques de l’IHES, 90(1): 45–143. [46] Negri S., Von Plato J., and Ranta A. (2008). Structural proof theory. Cambridge University Press. [47] Novaes, C. D. (2012). Formal languages in logic: A philosophical and cognitive analysis. Cambridge University Press. [48] Novaes, C. D. (2013). Mathematical reasoning and external symbolic systems. Logique et Analyse, 56(221). [49] Pastre, D. (1993). Automated theorem proving in mathematics. Annals of Mathematics and Artificial Intelligence, 8(3): 425–447. [50] Presburger, M. (1929). Über die vollständigkeit eines gewissen systems der arithmetik ganzer zahlen welchem die addition als einzige operation hervortritt. Sprawozdanie z I Kongresu Matematyków Krajów S´llowia´nskich, Warszawa, pages 92–101. [51] Rescorla, M. (2015). The computational theory of mind. The Stanford Encyclopedia of Philosophy. [52] Ribenboim, P. (2012). The new book of prime number records. Springer. [53] Rota, G.-C. (1997). The phenomenology of mathematical proof. In Indiscrete Thoughts, pages 134–150. Springer. [54] Schorlemmer M., Smaill A., Kühnberger K.-U., Kutz O., Colton S., Cambouropoulos E., and Pease A. COINVENT: Towards a computational concept invention theory. In 5th International Conference on Computational Creativity (ICCC). [55] Schwering A., Krumnack U., Kühnberger K.-U., and Gust H. (2009). Syntactic principles of heuristic driven theory projection. Cognitive Systems Research, 10(3): 251–269. [56] Subramani, K. (2005). Tractable fragments of presburger arithmetic. Theory of Computing Systems, 38. [57] T. Mossakowski, Mäder C., and Codescu M. (2014). Hets user guide version 0.99,. [58] Villemaire, C. M. R. (1996). Presburger arithmetic and recognizability of sets of natural numbers by automata: New proofs of cobham’s and semenov’s theorems. Annals of Pure and Applied Logic, 77. [59] Weil, A. (1979). De la métaphysique aux mathématiques. Sciences, pages 408–412. [60] Wiles, A. (1995). Modular elliptic curves and fermat last theorem. Annals of Mathematics, 142: 443–551. [61] Ye, F. (2011). Strict finitism and the logic of mathematical applications, volume 355. Springer.