Top

Mind & Society

Published in:

14-06-2024

A human-like artificial intelligence for mathematics

Author: Santiago Alonso-Diaz

Published in: Mind & Society | Issue 1-2/2024

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

This paper provides a brief overview of findings in mathematical cognition and how a human-like AI in mathematics may look like. Then, it provides six reasons in favor of a human-like AI for mathematics: (1) human cognition, with all its limits, creates mathematics; (2) human mathematics is insightful, not merely deductive steps; (3) human cognition detects structure in the real world; (4) human cognition can tackle and detect complex problems; (5) human cognition is creative; (6) human cognition considers ethical issues. The paper provides a tentative frame to the question whether human mathematical cognition is relevant for designing an artificial intelligence that works on and creates mathematics.

previous article Behavioral economics and the nature of neoclassical paradigm

next article Toward a third-generation rational choice theory: the multiple player approach to collective action problems

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Alonso-Diaz S, Piantadosi ST, Hayden BY, Cantlon JF (2018) Intrinsic whole number bias in humans. J Exp Psychol Hum Percept Perform 44(9):1472–1481. https://doi.org/10.1037/xhp0000544CrossRefPubMed

Alonso-Díaz S, Penagos-Londoño GI (2021) The numerator bias exists in millions of real-world comparisons. Acta Psychol 213:1–9. https://doi.org/10.1016/j.actpsy.2020.103248CrossRef

Amalric M, Dehaene S (2017) Cortical circuits for mathematical knowledge: evidence for a major subdivision within the brain ’ s semantic networks. Philosophical Trans Royal Soc B 373:1–9

Amalric M, Wang L, Pica P, Figueira S, Sigman M, Dehaene S (2017) The language of geometry: fast comprehension of geometrical primitives and rules in human adults and preschoolers. PLoS Comput Biol 13(1):1–31. https://doi.org/10.1371/journal.pcbi.1005273CrossRef

Aru J, Drüke M, Pikamäe J, Larkum ME (2022) Mental navigation and the neural mechanisms of insight. Trends Neurosci 1–10. https://doi.org/10.1016/j.tins.2022.11.002

Ashcraft MH, Krause Ja (2007) Working memory, math performance, and math anxiety. Psychon Bull Rev 14(2):243–248. http://www.ncbi.nlm.nih.gov/pubmed/21707166CrossRefPubMed

Bates KE, Gilligan-Lee K, Farran EK (2021) Reimagining mathematics: the role of Mental Imagery in explaining Mathematical calculation skills in Childhood. Mind Brain Educ 15(2):189–198. https://doi.org/10.1111/mbe.12281CrossRef

Besold TR, Garcez A, d’Avila, Bader S, Bowman H, Domingos P, Hitzler P, Kühnberger K-U, Lamb L, Lowd D, Viera-Lima PM, de Penning L, Pinkas G, Poon H, Zaverucha G (2021) Neural-symbolic learning and reasoning: A survey and interpretation. In Neuro-Symbolic Artificial Intelligence: The State of the Art. IOS press

Bonacina MP (2022) Set of support, demodulation, paramodulation: a historical perspective. J Automated Reasoning. https://doi.org/10.1007/s10817-022-09628-0CrossRef

Bossaerts P, Murawski C (2017) Computational complexity and human decision-making. Trends Cogn Sci 21(12):917–929. https://doi.org/10.1016/j.tics.2017.09.005CrossRefPubMed

Braithwaite DW, Pyke AA, Siegler RS (2017) A computational model of fraction arithmetic. Psychol Rev 124(5):603–625. https://doi.org/10.1037/rev0000072CrossRefPubMed

Brunton BW, Botvinick MM, Brody CD (2013) Rats and humans can optimally accumulate evidence for decision-making. Science 340:95–98CrossRefPubMed

Callaway E (2022) What’s next for the AI protein-folding revolution. Nature, 604(April)

Carey S, Spelke E (1994) Domain-specific knowledge and conceptual change. In: Hirschfeld LA, Gelman SA (eds) Mapping the mind: domain specificity in cognition and culture. Cambridge University Press, pp 169–200

Cassenti D, Veksler V, Ritter F (eds) (2022) Editor’s Review and Introduction: Cognition-Inspired Artificial Intelligence. Topics in Cognitive Science, 1–22

Caviola S, Colling LJ, Mammarella IC, Szűcs D (2020) Predictors of mathematics in primary school: Magnitude comparison, verbal and spatial working memory measures. Dev Sci 23(6):1–19. https://doi.org/10.1111/desc.12957CrossRef

Chater N, Tenenbaum JB, Yuille A (2006) Probabilistic models of cognition: conceptual foundations. Trends Cogn Sci 10(7):287–291CrossRefPubMed

Chen Q, Verguts T (2010) Beyond the mental number line: a neural network model of number-space interactions. Cogn Psychol 60(3):218–240. https://doi.org/10.1016/j.cogpsych.2010.01.001CrossRefPubMed

Chen L, Zaharia M, Zou J (2023) How is ChatGPT’s behavior changing over time?http://arxiv.org/abs/2307.09009

Cheyette SJ, Piantadosi ST (2020) A unified account of numerosity perception. Nat Hum Behav 4(12):1265–1272. https://doi.org/10.1038/s41562-020-00946-0CrossRefPubMed

Clarke S, Beck J (2021) The number sense represents (rational) numbers. Behav Brain Sci 44(e178):1–62. https://doi.org/10.1017/S0140525X21000571CrossRef

Corneli J, Martin U, Murray-Rust D, Nesin R, G., Pease A (2019) Argumentation Theory for Mathematical Argument. In Argumentation (Vol. 33, Issue 2). Springer Netherlands. https://doi.org/10.1007/s10503-018-9474-x

Cragg L, Gilmore C (2014) Skills underlying mathematics: the role of executive function in the development of mathematics proficiency. Trends Neurosci Educ 3(2):63–68. https://doi.org/10.1016/j.tine.2013.12.001CrossRef

Cvencek D, Brečić R, Gaćeša D, Meltzoff AN (2021) Development of Math attitudes and Math Self-Concepts: gender differences, implicit–explicit dissociations, and relations to Math Achievement. Child Dev 92(5):e940–e956. https://doi.org/10.1111/cdev.13523CrossRefPubMed

Davies A, Veličković P, Buesing L, Blackwell S, Zheng D, Tomašev N, Tanburn R, Battaglia P, Blundell C, Juhász A, Lackenby M, Williamson G, Hassabis D, Kohli P (2021) Advancing mathematics by guiding human intuition with AI. Nature 600(7887):70–74. https://doi.org/10.1038/s41586-021-04086-xCrossRefPubMedPubMedCentral

Davis E (2019) The Use of Deep Learning for Symbolic Integration: A Review of (Lample and Charton, 2019). ArXiv Preprint, 1–7. http://arxiv.org/abs/1912.05752

de Marchi S, Page SE (2014) Agent-based models. Annu Rev Polit Sci 17:1–20. https://doi.org/10.1146/annurev-polisci-080812-191558CrossRef

de Smedt B (2022) Individual differences in mathematical cognition: a Bert’s eye view. Curr Opin Behav Sci 46:101175. https://doi.org/10.1016/j.cobeha.2022.101175CrossRef

Degen J, Tanenhaus MK (2014) Processing Scalar implicature: a constraint-based Approach. Cogn Sci Early View 1–44. https://doi.org/10.1111/cogs.12171

Dehaene S (2009) Origins of mathematical intuitions: the case of arithmetic. Ann N Y Acad Sci 1156:232–259. https://doi.org/10.1111/j.1749-6632.2009.04469.xCrossRefPubMed

Dehaene S, Changeux J-P (1993) Development of Elementary Numerical abilities: a neuronal model. J Cogn Neurosci 5(4):390–407CrossRefPubMed

Dehaene S, Mehler J (1992) Cross-linguistic regularities in the frequency of number words. Cognition 43:1–29. http://www.sciencedirect.com/science/article/pii/001002779290030LCrossRefPubMed

Dehaene S, Izard V, Pica P, Spelke E (2006) Core knowledge of geometry in an amazonian indigene group. Science 311(5759):381–384. https://doi.org/10.1126/science.1121739CrossRefPubMed

Dehaene S, Izard V, Spelke E, Pica P (2008) Log or linear? Distinct intuitions of the number scale in western and amazonian indigene cultures. Science 320:1217–1220. https://doi.org/10.1126/science.1156540CrossRefPubMedPubMedCentral

Dehaene S, al Roumi F, Lakretz Y, Planton S, Sablé-Meyer M (2022) Symbols and mental programs: a hypothesis about human singularity. Trends Cogn Sci 26(9):751–766. https://doi.org/10.1016/j.tics.2022.06.010CrossRefPubMed

Desender K, Sasanguie D (2022) Math anxiety relates positively to metacognitive insight into mathematical decision making. Psychol Res 86(3):1001–1013. https://doi.org/10.1007/s00426-021-01511-8CrossRefPubMed

Devlin KJ (2000) The math gene: how mathematical thinking evolved and why numbers are like gossip. Basic Books

Devlin D, Moeller K, Reynvoet B, Sella F (2022) A critical review of number order judgements and arithmetic: what do order verification tasks actually measure ? Cogn Dev 64(October). https://doi.org/10.1016/j.cogdev.2022.101262

Dillon MR, Huang Y, Spelke ES (2013) Core foundations of abstract geometry. Proceedings of the National Academy of Sciences, 110(35), 14191–14195. https://doi.org/10.1073/pnas.1312640110

Donlan C, Cowan R, Newton EJ, Lloyd D (2007) The role of language in mathematical development: evidence from children with specific language impairments. Cognition 103(1):23–33. https://doi.org/10.1016/j.cognition.2006.02.007CrossRefPubMed

Eisenkolb VM, Held LM, Utzschmid A, Lin XX, Krieg SM, Meyer B, Gempt J, Jacob SN (2023) Human acute microelectrode array recordings with broad cortical access, single-unit resolution, and parallel behavioral monitoring. Cell Rep 42(5):112467. https://doi.org/10.1016/j.celrep.2023.112467CrossRefPubMedPubMedCentral

Fauconnier G, Turner M (2008) The way we think: conceptual blending and the mind’s hidden complexities. Basic Books

Feigenson L, Carey S, Hauser M (2002) The representations underlying infants’ choice of more: object files Versus Analog magnitudes. Psychol Sci 13(2):150–156. https://doi.org/10.1111/1467-9280.00427CrossRefPubMed

Finell J, Sammallahti E, Korhonen J, Eklöf H, Jonsson B (2022) Working Memory and its mediating role on the relationship of Math anxiety and Math Performance: a Meta-analysis. Front Psychol 12(January):1–14. https://doi.org/10.3389/fpsyg.2021.798090CrossRef

Fischer MH (2012) A hierarchical view of grounded, embodied, and situated numerical cognition. Cogn Process 13:161–164. https://doi.org/10.1007/s10339-012-0477-5CrossRef

Floridi L, Chiriatti M (2020) GPT-3: its nature, scope, limits, and consequences. Mind Mach 30(4):681–694. https://doi.org/10.1007/s11023-020-09548-1CrossRef

Fodor J (1975) The language of thought. Harvard University Press

Fontanari L, Gonzalez M, Vallortigara G, Girotto V (2014) Probabilistic cognition in two indigenous mayan groups. Proc Natl Acad Sci 111(48):17075–17080CrossRefPubMedPubMedCentral

Franco JP, Doroc K, Yadav N, Bossaerts P, Murawski C (2022) Task-independent metrics of computational hardness predict human cognitive performance. Sci Rep 12(1):1–15. https://doi.org/10.1038/s41598-022-16565-wCrossRef

Genest W, Stauffer WR, Schultz W (2016) Utility functions predict variance and skewness risk preferences in monkeys. Proc Natl Acad Sci USA 113(30):8402–8407. https://doi.org/10.1073/pnas.1602217113CrossRefPubMedPubMedCentral

Gessen M (2011) Perfect Rigour: a genius and the Mathematical breakthrough of the Century. Icon Books

Gigerenzer G, Brighton H (2009) Homo Heuristicus: why biased minds make better inferences. Top Cogn Sci 1(1):107–143. https://doi.org/10.1111/j.1756-8765.2008.01006.xCrossRefPubMed

Giurfa M, Marcout C, Hilpert P, Thevenot C, Rugani R (2022) An insect brain organizes numbers on a left-to-right mental number line. Proc Natl Acad Sci USA 119(44):e2203584119. https://doi.org/10.1073/pnas.2203584119CrossRefPubMedPubMedCentral

Gordon R, De Morais S, Whitelock D, E., Mukarram A (2022) Mapping components of verbal and visuospatial working memory to mathematical topics in seven- to fifteen-year-olds. Br J Educ Psychol 92(1):1–18. https://doi.org/10.1111/bjep.12440CrossRefPubMed

Hadamard J (1945) The psychology of invention in the mathematical field. Princeton University Press

Halmos PR (1968) Mathematics as a creative art. Am Sci 56(4):375–389. https://doi.org/10.1080/01421590802082536CrossRef

Hauser MD, Chomsky N, Fitch WT (2002) The faculty of language: what is it, who has it, and how did it evolve? Science 298(5598):1569–1579. https://doi.org/10.1017/CBO9780511817755.002CrossRefPubMed

Hawes Z, Ansari D (2020) What explains the relationship between spatial and mathematical skills? A review of evidence from brain and behavior. Psychonomic Bull Rev 27(3):465–482. https://doi.org/10.3758/s13423-019-01694-7CrossRef

Hélie S, Pizlo Z (2022) When is psychology research useful in Artificial Intelligence? A case for reducing computational complexity in Problem solving. Top Cogn Sci 00:1–15. https://doi.org/10.1111/tops.12572CrossRef

Hélie S, Sun R (2015) Cognitive architectures and agents. In J. Kacprzyk & W. Pedrycz (Eds.), Springer Handbook of Computational Intelligence (pp. 683–696). Springer. https://doi.org/10.1007/978-3-662-43505-2_36

Howe CQ, Purves D (2005a) Natural-scene geometry predicts the perception of angles and line orientation. Proceedings of the National Academy of Sciences, 102(4), 1228–1233. https://doi.org/10.1073/pnas.0409311102

Howe CQ, Purves D (2005b) The Müller-Lyer illusion explained by the statistics of image-source relationships. Proc Natl Acad Sci USA 102(4):1234–1239. https://doi.org/10.1073/pnas.0409314102CrossRefPubMedPubMedCentral

Huber S, Nuerk H, Willmes K, Moeller K (2016) A General Model Framework for Multisymbol Number Comparison. Psychol Rev 123(6):667–695CrossRefPubMed

Hunt TE, Maloney EA (2022) Appraisals of previous math experiences play an important role in math anxiety. Ann N Y Acad Sci 1–12. https://doi.org/10.1111/nyas.14805

Hurst M, Cordes S (2016) Rational-number comparison across notation: fractions, Decimals, and whole numbers. J Exp Psychol Hum Percept Perform 42(2):281–293CrossRefPubMed

Jacob SN, Vallentin D, Nieder A (2012) Relating magnitudes: the brain’s code for proportions. Trends Cogn Sci 16(3):157–166. https://doi.org/10.1016/j.tics.2012.02.002CrossRefPubMed

Jobin A, Ienca M, Vayena E (2019) The global landscape of AI ethics guidelines. Nat Mach Intell 1(9):389–399. https://doi.org/10.1038/s42256-019-0088-2CrossRef

Kidd C, Piantadosi ST, Aslin RN (2012) The Goldilocks effect: human infants allocate attention to visual sequences that are neither too simple nor too complex. PLoS ONE 7(5):1–8. https://doi.org/10.1371/journal.pone.0036399CrossRef

Kotseruba I, Tsotsos JK (2020) 40 Years of Cognitive Architectures: Core Cognitive Abilities and Practical Applications. In Artificial Intelligence Review (Vol. 53, Issue 1). Springer Netherlands. https://doi.org/10.1007/s10462-018-9646-y

Laird JE, Lebiere C, Rosenbloom PS (2017) A standard model of the mind: toward a common computational framework across artificial intelligence, cognitive science, neuroscience, and robotics. AI Magazine 38(4):13–26. https://doi.org/10.1609/aimag.v38i4.2744CrossRef

Lake BM, Ullman TD, Tenenbaum JB, Gershman SJ (2017) Building machines that learn and think like people. Behav Brain Sci 40:1–72. https://doi.org/10.1017/S0140525X16001837CrossRef

Lakoff G, Nuñez R (2000) Where mathematics comes from: how the embodied mind brings mathematics into being. Basic Books

Lample G, Charton F (2019) Deep Learning for Symbolic Mathematics. http://arxiv.org/abs/1912.01412

Leibovich T, Katzin N, Harel M, Henik A (2016) From ‘ sense of number ’ to ‘ sense of magnitude ’ – the role of continuous magnitudes in numerical cognition. Behav Brain Sci 1:1–62

Leikin R, Lev M (2013) Mathematical creativity in generally gifted and mathematically excelling adolescents: what makes the difference? ZDM - Int J Math Educ 45(2):183–197. https://doi.org/10.1007/s11858-012-0460-8CrossRef

Lewkowycz A, Andreassen A, Dohan D, Dyer E, Michalewski H, Ramasesh V, Slone A, Anil C, Schlag I, Gutman-Solo T, Wu Y, Neyshabur B, Gur-Ari G, Misra V (2022) Solving Quantitative Reasoning Problems with Language Models. ArXiv Preprint, 1–54. http://arxiv.org/abs/2206.14858

Loetscher T, Bockisch CJ, Nicholls MER, Brugger P (2010) Eye position predicts what number you have in mind. Curr Biol 20(6):R264–R265CrossRefPubMed

Lu L, Jin P, Pang G, Zhang Z, Karniadakis GE (2021) Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nat Mach Intell 3(3):218–229. https://doi.org/10.1038/s42256-021-00302-5CrossRef

Lucangeli D, Cornoldi C (1997) Mathematics and Metacognition: what is the nature of the relationship? Math Cognition 3(2):121–139. https://doi.org/10.1080/135467997387443CrossRef

Mandel DR (2014) Do framing effects reveal irrational choice? J Exp Psychol Gen 143(3):1185–1198. https://doi.org/10.1037/a0034207CrossRefPubMed

Mann EL (2006) Creativity: the essence of mathematics. J Educ Gifted 30(2):236–260. https://doi.org/10.4219/jeg-2006-264CrossRef

Matthews PG, Lewis MR, Hubbard EM (2015) Individual Differences in Nonsymbolic Ratio Processing Predict Symbolic Math performance. Psychol Sci, 0956797615617799

Mcclelland JL (2022) Capturing advanced human cognitive abilities with deep neural networks. Trends Cogn Sci 1–4. https://doi.org/10.1016/j.tics.2022.09.018

Mistry PK, Strock A, Liu R, Young G, Menon V (2023) Learning-induced reorganization of number neurons and emergence of numerical representations in a biologically inspired neural network. Nat Commun 14(1):1–21. https://doi.org/10.1038/s41467-023-39548-5CrossRef

Moyer R, Landauer T (1967) Time required for judgements of numerical inequality. Nature 215:1519–1520CrossRefPubMed

Murawski C, Bossaerts P (2016) How humans solve complex problems: the case of the Knapsack Problem. Sci Rep 6(September):1–10. https://doi.org/10.1038/srep34851CrossRef

Nakai T, Nishimoto S (2023) Artificial neural network modelling of the neural population code underlying mathematical operations. NeuroImage 270:119980. https://doi.org/10.1016/j.neuroimage.2023.119980CrossRefPubMed

Nawaz MS, Malik M, Li Y, Sun M, Lali MIU (2019) A Survey on Theorem Provers in Formal Methods. http://arxiv.org/abs/1912.03028

Nieder A (2016a) Representing something out of nothing: the dawning of Zero. Trends Cogn Sci 20(11):830–842. https://doi.org/10.1016/j.tics.2016.08.008CrossRefPubMed

Nieder A (2016b) The neuronal code for number. Nat Rev Neurosci 17(6):366–382. https://doi.org/10.1038/nrn.2016.40CrossRefPubMed

Nieder A, Dehaene S (2009) Representation of number in the brain. Annu Rev Neurosci 32:185–208. https://doi.org/10.1146/annurev.neuro.051508.135550CrossRefPubMed

Noveck IA, Reboul A (2008) Experimental pragmatics: a Gricean turn in the study of language. Trends Cogn Sci 12(11):425–431. https://doi.org/10.1016/j.tics.2008.07.009CrossRefPubMed

Nuñez R, Fias W (2015) Ancestral mental number lines: what is the evidence ? Cogn Sci 1–5. https://doi.org/10.1111/cogs.12296

Núñez R (2017) Is there really an evolved capacity for number? Trends Cogn Sci 21(6):409–424. https://doi.org/10.1016/j.tics.2017.03.005CrossRefPubMed

O’Neil C (2016) Weapons of math destruction. Broadway Books

Obersteiner A, van Dooren W, van Hoof J, Verschaffel L (2013) The natural number bias and magnitude representation in fraction comparison by expert mathematicians. Learn Instruction 28:64–72. https://doi.org/10.1016/j.learninstruc.2013.05.003CrossRef

Padoa-Schioppa C, Rustichini A (2014) Rational attention and adaptive coding: a puzzle and a solution. Am Econ Rev 104(5):507–513. https://doi.org/10.1257/aer.104.5.507CrossRefPubMedPubMedCentral

Parkes DC, Wellman MP (2015) Economic reasoning and artificial intelligence. Science 349(6245):267–272CrossRefPubMed

Pease A, Lawrence J, Budzynska K, Corneli J, Reed C (2017) Lakatos-style collaborative mathematics through dialectical, structured and abstract argumentation. Artif Intell 246:181–219. https://doi.org/10.1016/j.artint.2017.02.006CrossRef

Peterson EG, Cohen J (2019) A case for domain-specific curiosity in Mathematics. Educational Psychol Rev 31(4):807–832. https://doi.org/10.1007/s10648-019-09501-4CrossRef

Piantadosi ST (2016) A rational analysis of the approximate number system. Psychon Bull Rev 23(3):877–886. https://doi.org/10.3758/s13423-015-0963-8CrossRefPubMedPubMedCentral

Piantadosi ST, Tenenbaum JB, Goodman ND (2012) Bootstrapping in a language of thought: a formal model of numerical concept learning. Cognition 123(2):199–217. https://doi.org/10.1016/j.cognition.2011.11.005CrossRefPubMed

Piantadosi ST, Tenenbaum JB, Goodman ND (2016) The logical primitives of thought: empirical foundations for compositional cognitive models. Psychol Rev 123(4):392CrossRefPubMed

Piazza M, Pica P, Izard V, Spelke ES, Dehaene S (2013) Education enhances the acuity of the nonverbal approximate number system. Psychol Sci 24(6):1037–1043. https://doi.org/10.1177/0956797612464057CrossRefPubMed

Raghubar KP, Barnes MA, Hecht SA (2010) Working memory and mathematics: a review of developmental, individual difference, and cognitive approaches. Learn Individual Differences 20(2):110–122. https://doi.org/10.1016/j.lindif.2009.10.005CrossRef

Rahwan I, Cebrian M, Obradovich N, Bongard J, Bonnefon JF, Breazeal C, Crandall JW, Christakis NA, Couzin ID, Jackson MO, Jennings NR, Kamar E, Kloumann IM, Larochelle H, Lazer D, McElreath R, Mislove A, Parkes DC, Pentland A (2019) Machine behaviour. Nature 568(7753):477–486 ‘Sandy,’ … Wellman,. https://doi.org/10.1038/s41586-019-1138-yCrossRefPubMed

Rau MA (2017) Conditions for the effectiveness of multiple visual representations in enhancing STEM learning. Educational Psychol Rev 29(4):717–761. https://doi.org/10.1007/s10648-016-9365-3CrossRef

Razeghi Y, Logan RL, Gardner M, Singh S (2022) Impact of Pretraining Term Frequencies on Few-Shot Reasoning. http://arxiv.org/abs/2202.07206

Ritter S, Anderson JR, Koedinger KR, Corbett A (2007) Cognitive tutor: Applied research in mathematics education. Psychon Bull Rev 14(2):249–255CrossRefPubMed

Rudy SH, Brunton SL, Proctor JL, Kutz JN (2017) Data-driven discovery of partial differential equations. Sci Adv 3(4):1–7. https://doi.org/10.1126/sciadv.1602614CrossRef

Rugani R, Vallortigara G, Priftis K, Regolin L (2015) Number-space mapping in the newborn chick resembles humans ’ mental number line. Science 347(6221):534–536CrossRefPubMed

Rule JS, Tenenbaum JB, Piantadosi ST (2020) The child as Hacker. Trends Cogn Sci 24(11):900–915. https://doi.org/10.1016/j.tics.2020.07.005CrossRefPubMedPubMedCentral

Rustichini A, Conen KE, Cai X, Padoa-schioppa C (2017) Optimal coding and neuronal adaptation in economic decisions. Nat Commun 8(1):1–14. https://doi.org/10.1038/s41467-017-01373-yCrossRef

Sablé-Meyer M, Fagot J, Caparos S, van Kerkoerle T, Amalric M, Dehaene S (2021) Sensitivity to geometric shape regularity in humans and baboons: a putative signature of human singularity. Proc Natl Acad Sci USA 118(16):1–10. https://doi.org/10.1073/pnas.2023123118CrossRef

Santens S, Verguts T (2011) The size congruity effect: is bigger always more? Cognition 118(1):94–110. https://doi.org/10.1016/j.cognition.2010.10.014CrossRefPubMed

Schneider W, Artelt C (2010) Metacognition and mathematics education. ZDM - Int J Math Educ 42(2):149–161. https://doi.org/10.1007/s11858-010-0240-2CrossRef

Schneider E, Maruyama M, Dehaene S, Sigman M (2012) Eye gaze reveals a fast, parallel extraction of the syntax of arithmetic formulas. Cognition 125(3):475–490. https://doi.org/10.1016/j.cognition.2012.06.015CrossRefPubMed

Shaki S, Petrusic WM (2005) On the mental representation of negative numbers: context-dependent SNARC effects with comparative judgments. Psychon Bull Rev 12(5):931–937CrossRefPubMed

Simon H (1997) Models of bounded rationality. MIT Press

Spelke ES (2011) Natural number and natural geometry. In: Dehaene S, Brannon EM (eds) Space, Time, and number in the brain: searching for the foundations of Mathematical Thought. Academic, pp 287–317

Spelke ES, Lee SA (2012) Core systems of geometry in animal minds. Philosophical Trans Royal Soc B 367:2784–2793. https://doi.org/10.1098/rstb.2012.0210CrossRef

Sterling L, Bundy A, Byrd L, O’Keefe R, Silver B (1989) Solving symbolic equations with PRESS. J Symbolic Comput 7(1):71–84. https://doi.org/10.1016/S0747-7171(89)80007-0CrossRef

Stoianov I, Zorzi M (2012) Emergence of a visual number sense in hierarchical generative models. Nat Neurosci 15(2):194–196. https://doi.org/10.1038/nn.2996CrossRefPubMed

Szkudlarek E, Brannon EM (2017) Does the approximate number System serve as a Foundation for Symbolic Mathematics? Lang Learn Dev 13(2):171–190. https://doi.org/10.1080/15475441.2016.1263573CrossRefPubMedPubMedCentral

Téglás E, Girotto V, Gonzalez M, Bonatti LL (2007) Intuitions of probabilities shape expectations about the future at 12 months and beyond. Proceedings of the National Academy of Sciences, 104(48), 19156–19159. https://doi.org/10.1073/pnas.0700271104

Tenenbaum JB, Kemp C, Griffiths TL, Goodman ND (2011) How to grow a mind: statistics, structure, and abstraction. Science 331(6022):1279–1285. https://doi.org/10.1126/science.1192788CrossRefPubMed

Testolin A, Zou WY, McClelland JL (2020) Numerosity discrimination in deep neural networks: initial competence, developmental refinement and experience statistics. Dev Sci 23(5):1–13. https://doi.org/10.1111/desc.12940CrossRef

Thagard P, Stewart TC (2011) The AHA! Experience: Creativity through emergent binding in neural networks. Cogn Sci 35(1):1–33. https://doi.org/10.1111/j.1551-6709.2010.01142.xCrossRefPubMed

Turing AM (1936) On computable numbers, with an application to the Entscheidungsproblem. J Math 58(5):345–363

Udrescu SM, Tegmark M (2020) AI Feynman: a physics-inspired method for symbolic regression. Sci Adv 6(16):1–16. https://doi.org/10.1126/sciadv.aay2631CrossRef

van Rooij I, Blokpoel M, Kwisthout J, Wareham T (2019) Cognition and intractability: a guide to classical and parameterized complexity analysis. Cambridge University Press

Villani C (2015) Birth of a theorem: a mathematical adventure. Farrar, Straus and Giroux

Viteri S, DeDeo S (2022) Epistemic phase transitions in mathematical proofs. Cognition 225(March):105120. https://doi.org/10.1016/j.cognition.2022.105120CrossRefPubMed

Vo VA, Li R, Kornell N, Pouget A, Cantlon JF (2014) Young Children Bet on their Numerical skills Metacognition in the Numerical Domain. Psychol Sci 25(9):1712–1721CrossRefPubMed

Voevodsky V (2010) What if Current Foundations of Mathematics are Inconsistent? Institute for Advanced Study Video Lectures. https://www.ias.edu/video/voevodsky-80th

Wang P (2007) Three fundamental misconceptions of Artificial Intelligence. J Experimental Theoretical Artif Intell 19(3):249–268. https://doi.org/10.1080/09528130601143109CrossRef

Welleck S, West P, Cao J, Choi Y (2022) Symbolic brittleness in sequence models: on systematic generalization in Symbolic Mathematics. Proc AAAI Conf Artif Intell 36(8):8629–8637. https://doi.org/10.1609/aaai.v36i8.20841CrossRef

Wilkes Mv (1977) Babbage as a computer pioneer. Historia Mathematica 4(4):415–440. https://doi.org/10.1016/0315-0860(77)90079-9CrossRef

Winkler-Rhoades N, Carey S, Spelke ES (2013) Two-year-old children interpret abstract, purely geometric maps. Dev Sci 16(3):365–376. https://doi.org/10.1111/desc.12038CrossRefPubMedPubMedCentral

Yu X, Liu K, Wang Y, Yang X, Yang J (2022) Differential contributions of phonological processing and visual- spatial abilities to four basic arithmetic operations in primary school children. Curr Psychol, 1–13

Zhou H, Nova A, Larochelle H, Courville A, Neyshabur B, Sedghi H (2022) Teaching Algorithmic Reasoning via In-context Learning. http://arxiv.org/abs/2211.09066

Zorzi M, Priftis K, Umiltà C (2002) Neglect disrupts the mental number line. Nature 417:138CrossRefPubMed

Title: A human-like artificial intelligence for mathematics
Author: Santiago Alonso-Diaz
Publication date: 14-06-2024
Publisher: Springer Berlin Heidelberg
Published in: Mind & Society / Issue 1-2/2024
Print ISSN: 1593-7879
Electronic ISSN: 1860-1839
DOI: https://doi.org/10.1007/s11299-024-00304-x

Toward a third-generation rational choice theory: the multiple player approach to collective action problems

Open Access

Willingness to pay more health taxes? The relevance of personality traits and situational effects

Open Access

Behavioral economics and the nature of neoclassical paradigm

Open Access

Springer Professional

A human-like artificial intelligence for mathematics

Abstract

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Other articles of this Issue 1-2/2024

Toward a third-generation rational choice theory: the multiple player approach to collective action problems

Willingness to pay more health taxes? The relevance of personality traits and situational effects

Behavioral economics and the nature of neoclassical paradigm

The fundamentally uncertain economic agent: Brian J. Loasby’s growth of knowledge approach to the psychology of human action

Moral decay, inequality, and the perception of corruption: the reproduction of bribery as a social norm

The rationality of recalcitrant emotions in weak judgmentalism

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Toward a third-generation rational choice theory: the multiple player approach to collective action problems

Willingness to pay more health taxes? The relevance of personality traits and situational effects

Behavioral economics and the nature of neoclassical paradigm

The fundamentally uncertain economic agent: Brian J. Loasby’s growth of knowledge approach to the psychology of human action

Moral decay, inequality, and the perception of corruption: the reproduction of bribery as a social norm

The rationality of recalcitrant emotions in weak judgmentalism