{"id":"https://openalex.org/W3168982779","doi":"https://doi.org/10.3390/sym13061002","title":"The Brain and the New Foundations of Mathematics","display_name":"The Brain and the New Foundations of Mathematics","publication_year":2021,"publication_date":"2021-06-03","ids":{"openalex":"https://openalex.org/W3168982779","doi":"https://doi.org/10.3390/sym13061002","mag":"3168982779"},"language":"en","primary_location":{"id":"doi:10.3390/sym13061002","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13061002","pdf_url":"https://www.mdpi.com/2073-8994/13/6/1002/pdf?version=1622802179","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/13/6/1002/pdf?version=1622802179","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5002462532","display_name":"Alexey V. Melkikh","orcid":"https://orcid.org/0000-0003-1674-1981"},"institutions":[{"id":"https://openalex.org/I130823665","display_name":"Ural Federal University","ror":"https://ror.org/00hs7dr46","country_code":"RU","type":"education","lineage":["https://openalex.org/I130823665"]},{"id":"https://openalex.org/I4210089467","display_name":"Institute of Physics and Technology","ror":"https://ror.org/005n3yy14","country_code":"RU","type":"facility","lineage":["https://openalex.org/I1313323035","https://openalex.org/I4210089467"]}],"countries":["RU"],"is_corresponding":true,"raw_author_name":"Alexey V. Melkikh","raw_affiliation_strings":["Institute of Physics and Technology, Ural Federal University, Yekaterinburg 620002, Russia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Physics and Technology, Ural Federal University, Yekaterinburg 620002, Russia","institution_ids":["https://openalex.org/I130823665","https://openalex.org/I4210089467"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5002462532"],"corresponding_institution_ids":["https://openalex.org/I130823665","https://openalex.org/I4210089467"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2227},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2227},"fwci":0.2601,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.63906708,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":95},"biblio":{"volume":"13","issue":"6","first_page":"1002","last_page":"1002"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12805","display_name":"Cognitive Science and Mapping","score":0.8697999715805054,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12805","display_name":"Cognitive Science and Mapping","score":0.8697999715805054,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11445","display_name":"Origins and Evolution of Life","score":0.8418999910354614,"subfield":{"id":"https://openalex.org/subfields/3103","display_name":"Astronomy and Astrophysics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12946","display_name":"Fractal and DNA sequence analysis","score":0.7555000185966492,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/axiom","display_name":"Axiom","score":0.6107056140899658},{"id":"https://openalex.org/keywords/mathematical-structure","display_name":"Mathematical structure","score":0.5638576149940491},{"id":"https://openalex.org/keywords/axiom-of-choice","display_name":"Axiom of choice","score":0.5578587055206299},{"id":"https://openalex.org/keywords/object","display_name":"Object (grammar)","score":0.5410268902778625},{"id":"https://openalex.org/keywords/constructive","display_name":"Constructive","score":0.5303849577903748},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5224282145500183},{"id":"https://openalex.org/keywords/consistency","display_name":"Consistency (knowledge bases)","score":0.4975304901599884},{"id":"https://openalex.org/keywords/equivalence","display_name":"Equivalence (formal languages)","score":0.49082884192466736},{"id":"https://openalex.org/keywords/qubit","display_name":"Qubit","score":0.4734575152397156},{"id":"https://openalex.org/keywords/statement","display_name":"Statement (logic)","score":0.43293577432632446},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.38473132252693176},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.362152636051178},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.3464726209640503},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.31305912137031555},{"id":"https://openalex.org/keywords/process","display_name":"Process (computing)","score":0.28705939650535583},{"id":"https://openalex.org/keywords/set-theory","display_name":"Set theory","score":0.26653045415878296},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.2558065354824066},{"id":"https://openalex.org/keywords/quantum","display_name":"Quantum","score":0.2282729148864746},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.19595640897750854},{"id":"https://openalex.org/keywords/mathematics-education","display_name":"Mathematics education","score":0.132038414478302},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.12147855758666992}],"concepts":[{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.6107056140899658},{"id":"https://openalex.org/C130327152","wikidata":"https://www.wikidata.org/wiki/Q748349","display_name":"Mathematical structure","level":2,"score":0.5638576149940491},{"id":"https://openalex.org/C151797676","wikidata":"https://www.wikidata.org/wiki/Q179692","display_name":"Axiom of choice","level":4,"score":0.5578587055206299},{"id":"https://openalex.org/C2781238097","wikidata":"https://www.wikidata.org/wiki/Q175026","display_name":"Object (grammar)","level":2,"score":0.5410268902778625},{"id":"https://openalex.org/C2778701210","wikidata":"https://www.wikidata.org/wiki/Q28130034","display_name":"Constructive","level":3,"score":0.5303849577903748},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5224282145500183},{"id":"https://openalex.org/C2776436953","wikidata":"https://www.wikidata.org/wiki/Q5163215","display_name":"Consistency (knowledge bases)","level":2,"score":0.4975304901599884},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.49082884192466736},{"id":"https://openalex.org/C203087015","wikidata":"https://www.wikidata.org/wiki/Q378201","display_name":"Qubit","level":3,"score":0.4734575152397156},{"id":"https://openalex.org/C2777026412","wikidata":"https://www.wikidata.org/wiki/Q2684591","display_name":"Statement (logic)","level":2,"score":0.43293577432632446},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.38473132252693176},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.362152636051178},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.3464726209640503},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.31305912137031555},{"id":"https://openalex.org/C98045186","wikidata":"https://www.wikidata.org/wiki/Q205663","display_name":"Process (computing)","level":2,"score":0.28705939650535583},{"id":"https://openalex.org/C153046414","wikidata":"https://www.wikidata.org/wiki/Q12482","display_name":"Set theory","level":3,"score":0.26653045415878296},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2558065354824066},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.2282729148864746},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.19595640897750854},{"id":"https://openalex.org/C145420912","wikidata":"https://www.wikidata.org/wiki/Q853077","display_name":"Mathematics education","level":1,"score":0.132038414478302},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.12147855758666992},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390/sym13061002","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13061002","pdf_url":"https://www.mdpi.com/2073-8994/13/6/1002/pdf?version=1622802179","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:elar.urfu.ru:10995/102847","is_oa":true,"landing_page_url":"https://www.scopus.com/inward/record.uri?eid=2-s2.0-85108109851&doi=10.3390%2fsym13061002&partnerID=40&md5=fd9f611056427509293e01ff5dbdf1a6","pdf_url":"https://www.mdpi.com/2073-8994/13/6/1002/pdf","source":{"id":"https://openalex.org/S4406922546","display_name":"Electronic Archive of Ural Federal University (ELAR UrFU)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:doaj.org/article:9b244b0984d948ed9307faafbf954688","is_oa":true,"landing_page_url":"https://doaj.org/article/9b244b0984d948ed9307faafbf954688","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 13, Iss 6, p 1002 (2021)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/13/6/1002/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym13061002","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym13061002","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13061002","pdf_url":"https://www.mdpi.com/2073-8994/13/6/1002/pdf?version=1622802179","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3168982779.pdf","grobid_xml":"https://content.openalex.org/works/W3168982779.grobid-xml"},"referenced_works_count":91,"referenced_works":["https://openalex.org/W73629738","https://openalex.org/W401986301","https://openalex.org/W628877253","https://openalex.org/W851845004","https://openalex.org/W1510667201","https://openalex.org/W1522489781","https://openalex.org/W1567975588","https://openalex.org/W1574216169","https://openalex.org/W1708393443","https://openalex.org/W1847033326","https://openalex.org/W1966058685","https://openalex.org/W1976701173","https://openalex.org/W1978443256","https://openalex.org/W1984891974","https://openalex.org/W1985156157","https://openalex.org/W1989757574","https://openalex.org/W1991697528","https://openalex.org/W1993466133","https://openalex.org/W1998146115","https://openalex.org/W2009385420","https://openalex.org/W2012396586","https://openalex.org/W2015680479","https://openalex.org/W2019954639","https://openalex.org/W2022367308","https://openalex.org/W2025833343","https://openalex.org/W2031430431","https://openalex.org/W2040879131","https://openalex.org/W2044046116","https://openalex.org/W2044413922","https://openalex.org/W2045744264","https://openalex.org/W2049640053","https://openalex.org/W2054195362","https://openalex.org/W2054741405","https://openalex.org/W2055514745","https://openalex.org/W2056343541","https://openalex.org/W2058259421","https://openalex.org/W2063535080","https://openalex.org/W2086119999","https://openalex.org/W2088443095","https://openalex.org/W2091752660","https://openalex.org/W2093214980","https://openalex.org/W2094716966","https://openalex.org/W2097212070","https://openalex.org/W2100738443","https://openalex.org/W2106616453","https://openalex.org/W2112571042","https://openalex.org/W2120713972","https://openalex.org/W2128314959","https://openalex.org/W2130862594","https://openalex.org/W2139757190","https://openalex.org/W2170285198","https://openalex.org/W2185839818","https://openalex.org/W2328141028","https://openalex.org/W2400660831","https://openalex.org/W2413604966","https://openalex.org/W2485587655","https://openalex.org/W2489517180","https://openalex.org/W2531040576","https://openalex.org/W2567719667","https://openalex.org/W2605990666","https://openalex.org/W2611357879","https://openalex.org/W2759098914","https://openalex.org/W2804863524","https://openalex.org/W2891506141","https://openalex.org/W2897443919","https://openalex.org/W2907176537","https://openalex.org/W2916132439","https://openalex.org/W2917356768","https://openalex.org/W2922717123","https://openalex.org/W2963487520","https://openalex.org/W2964016551","https://openalex.org/W2964214846","https://openalex.org/W2980514527","https://openalex.org/W2980659848","https://openalex.org/W2998554535","https://openalex.org/W3010963950","https://openalex.org/W3016069020","https://openalex.org/W3031087307","https://openalex.org/W3038243199","https://openalex.org/W3039508055","https://openalex.org/W3106216700","https://openalex.org/W3152901149","https://openalex.org/W4205308139","https://openalex.org/W4232076615","https://openalex.org/W4245373813","https://openalex.org/W4246962530","https://openalex.org/W4250573589","https://openalex.org/W4250703642","https://openalex.org/W4253671811","https://openalex.org/W4255721438","https://openalex.org/W6639827933"],"related_works":["https://openalex.org/W2031083694","https://openalex.org/W1504849543","https://openalex.org/W2377770720","https://openalex.org/W94411513","https://openalex.org/W4210343663","https://openalex.org/W2373903136","https://openalex.org/W172273113","https://openalex.org/W1488339321","https://openalex.org/W3101095055","https://openalex.org/W4389147218"],"abstract_inverted_index":{"Many":[0],"concepts":[1],"in":[2,78,111,171,195,212],"mathematics":[3,20,38,94,112,168,187],"are":[4,11,138],"not":[5],"fully":[6],"defined,":[7],"and":[8,32,55,74,132,134,214],"their":[9],"properties":[10],"implicit,":[12],"which":[13,43,101,161,193],"leads":[14],"to":[15,42,190],"paradoxes.":[16],"New":[17],"foundations":[18],"of":[19,27,30,37,61,66,70,81,93,98,124,142,147,152,155,186,208],"were":[21],"formulated":[22],"based":[23,202],"on":[24,76,107,203],"the":[25,67,79,91,96,118,143,153,158,172,204,215],"concept":[26],"innate":[28],"programs":[29],"behavior":[31],"thinking.":[33],"The":[34,145,166,184],"basic":[35],"axiom":[36,146],"is":[39,86,95,113,149,169,181,188,201],"proposed,":[40],"according":[41],"any":[44,71,175],"mathematical":[45,72,84,128,176],"object":[46,85,177],"has":[47],"a":[48,58,64,83,89,122,150],"physical":[49,156],"carrier.":[50],"This":[51],"carrier":[52],"can":[53,102,162],"store":[54],"process":[56],"only":[57,103],"finite":[59],"amount":[60],"information.":[62],"As":[63,88],"result":[65],"D-procedure":[68],"(encoding":[69],"objects":[73],"operations":[75,106],"them":[77],"form":[80],"qubits),":[82],"digitized.":[87],"consequence,":[90],"basis":[92],"interaction":[97],"brain":[99],"qubits,":[100],"implement":[104],"arithmetic":[105],"numbers.":[108],"A":[109],"proof":[110],"an":[114],"algorithm":[115],"for":[116],"finding":[117],"correct":[119],"statement":[120],"from":[121],"list":[123],"already-existing":[125],"statements.":[126],"Some":[127],"paradoxes":[129],"(e.g.,":[130],"Banach\u2013Tarski":[131],"Russell)":[133],"Smale\u2019s":[135],"18th":[136],"problem":[137],"solved":[139],"by":[140],"means":[141],"D-procedure.":[144],"choice":[148,159],"consequence":[151],"equivalence":[154],"states,":[157],"among":[160],"be":[163],"made":[164],"randomly.":[165],"proposed":[167],"constructive":[170],"sense":[173],"that":[174],"exists":[178],"if":[179],"it":[180],"physically":[182],"realized.":[183],"consistency":[185],"due":[189],"directed":[191],"evolution,":[192],"results":[194],"effective":[196],"structures.":[197],"Computing":[198],"with":[199],"qubits":[200],"nontrivial":[205],"quantum":[206],"effects":[207],"biologically":[209],"important":[210],"molecules":[211],"neurons":[213],"brain.":[216]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1}],"updated_date":"2026-07-22T07:51:19.307946","created_date":"2025-10-10T00:00:00"}