Ãîðèçîíòû XXI âåêà: ìàòåìàòè÷åñêèé àíàëèç ãèïîòåçû ñèíãóëÿðíîñòè

Òðóäíî íå çàìåòèòü, ÷òî ïàòòåðí óñêîðåíèÿ òåìïîâ ðîñòà ÷èñëåííîñòè íàñåëåíèÿ ìèðà, îáíàðóæåííûé åùå â 1960 ãîäó ôîí Ôåðñòåðîì â ýìïèðè÷åñêèõ äàííûõ ïî äèíàìèêå ÷èñëåííîñòè íàñåëåíèÿ Çåìëè ìåæäó 1 è 1958 ãîäàìè í. ý., îêàçûâàåòñÿ ïðàêòè÷åñêè èäåíòè÷íûì òîìó ïàòòåðíó óñêîðåíèÿ òåìïîâ ãëîáàëüíîãî ðàçâèòèÿ, êîòîðûé ìû âûøå îáíàðóæèëè âî âðåìåííûõ ðÿäàõ êàê Ìîäèñà - Êóðöâåéëà, òàê è Ïàíîâà. Îñîáî îòìåòèì, ÷òî ñòåïåííàÿ ðåãðåññèÿ äëÿ âñåõ òðåõ ðÿäîâ äàëà çíà÷åíèå ïîêàçàòåëÿ ñòåïåíè â, êðàéíå áëèçêîå ê 1 (1,003 äëÿ ðÿäà Ìîäèñà - Êóðöâåéëà, 1,01 äëÿ ðÿäà Ïàíîâà è 0,99 ó Õ. ôîí Ôåðñòåðà äëÿ äèíàìèêè ÷èñëåííîñòè íàñåëåíèÿ ìèðà).

Îäíàêî è îáíàðóæåííàÿ êðàéíÿÿ áëèçîñòü çíà÷åíèé ïàðàìåòðà t* (à ýòî èìåííî çíà÷åíèå òî÷êè Ñèíãóëÿðíîñòè) òàêæå íå ìîæåò íå âïå÷àòëèòü: ñòåïåííàÿ ðåãðåññèÿ äàåò â êà÷åñòâå òî÷êè Ñèíãóëÿðíîñòè 2029 ãîä äëÿ ðÿäà Ìîäèñà - Êóðöâåéëà, 2027 ãîä - äëÿ ðÿäà Ïàíîâà è â òî÷íîñòè òîò æå 2027 ãîä äëÿ ðÿäà ôîí Ôåðñòåðà.

Ìû óæå ãîâîðèëè âûøå, ÷òî, êàê è â ñëó÷àå ñ óðàâíåíèÿìè (8) è (9), â óðàâíåíèè ôîí Ôåðñòåðà (13) çíà÷åíèå ñòåïåíè â çíàìåíàòåëå (0,99) ñòîëü ñëàáî îòëè÷íî îò 1, ÷òî, êàê óæå ïðåäëàãàëîñü Ñ. ôîí Õåðíåðîì (von Hoerner 1975) è Ñ. Ï. Êàïèöåé (1992; 1999), åãî öåëåñîîáðàçíî èñïîëüçîâàòü â ñëåäóþùåì óïðîùåííîì âèäå:

Ïîëó÷åííîå â ðåçóëüòàòå ýòîãî óðàâíåíèå ïîëíîñòüþ èäåíòè÷íî âûøåïðèâåäåííîìó óðàâíåíèþ (5), êîòîðîå îêàçàëîñü ñïîñîáíûì îïèñàòü ñ ÷ðåçâû÷àéíî âûñîêîé òî÷íîñòüþ îáùèé ïàòòåðí óñêîðåíèÿ òåìïîâ ãëîáàëüíîãî ðàçâèòèÿ â òå÷åíèå êàê ìèíèìóì ïîñëåäíèõ 4 ìëðä. ëåò. Îòìåòèì, ÷òî óðàâíåíèå (14) îêàçàëîñü ñïîñîáíûì îïèñàòü òðåíä äèíàìèêè ÷èñëåííîñòè íàñåëåíèÿ ìèðà (âïëîòü äî íà÷àëà 1970-õ ãîäîâ) ñ òàêîé æå ïðåäåëüíî âûñîêîé òî÷íîñòüþ, ñ êàêîé óðàâíåíèå (5) ñïîñîáíî îïèñàòü óñêîðåíèå òåìïîâ ðîñòà ãëîáàëüíîé ñëîæíîñòè (êàê ìèíèìóì çà ïîñëåäíèå 4 ìëðä. ëåò). Íèæå èìååò ñìûñë ïðîäåìîíñòðèðîâàòü ýòî äëÿ óðàâíåíèÿ ôîí Ôåðñòåðà (14).

Äåéñòâèòåëüíî, çàìåíèì â óðàâíåíèè (14) t* íà 2027 (ýòî ïðîñòî ðåçóëüòàò îêðóãëåíèÿ ïîëó÷åííîãî ôîí Ôåðñòåðîì çíà÷åíèÿ òî÷êè Ñèíãóëÿðíîñòè, 2026,87), à C - íà 215 000 [12]. Ýòî äàñò íàì âàðèàíò óðàâíåíèÿ ôîí Ôåðñòåðà - ôîí Õåðíåðà - Êàïèöû ñ îïðåäåëåííûìè ïàðàìåòðàìè:

Îáùåå ñîâïàäåíèå êðèâîé, îïèñûâàåìîé óðàâíåíèåì ôîí Ôåðñòåðà, è íàèáîëåå äåòàëüíîãî ðÿäà ýìïèðè÷åñêèõ îöåíîê âûãëÿäèò ñëåäóþùèì îáðàçîì (ñì. ðèñ. 15):

Ðèñ. 15 - Êîððåëÿöèÿ ìåæäó ýìïèðè÷åñêèìè îöåíêàìè äîëãîñðî÷íîé äèíàìèêè ÷èñëåííîñòè íàñåëåíèÿ ìèðà (â ìëí. ÷åëîâåê, 1000-1970 ãîäû) è êðèâîé, ãåíåðèðóåìîé óðàâíåíèåì ôîí Ôåðñòåðà (15)



Ïðèìå÷àíèå: ÷åðíûå ìàðêåðû ñîîòâåòñòâóþò ýìïèðè÷åñêèì îöåíêàì ÷èñëåííîñòè íàñåëåíèÿ ìèðà, ñäåëàííûì Ð. ÌàêÝâåäè è Ê. Äæîíñîì (McEvedy, Jones 1978) äëÿ 1000-1950 ãã., è ýìïèðè÷åñêèì îöåíêàì Îòäåëà íàðîäîíàñåëåíèÿ ÎÎÍ (UN Population Division 2018) äëÿ ïåðèîäà ñ 1950 ïî 1970 ãîä. Ñåðàÿ êðèâàÿ ñãåíåðèðîâàíà óðàâíåíèåì ôîí Ôåðñòåðà (15). Ôîðìàëüíûå õàðàêòåðèñòèêè ýòîé êîððåëÿöèè òàêîâû: r = 0,998; R2 = 0,996; p = 9,4 × 10-17 ˜ 1 × 10-16.

Òàêèì îáðàçîì, íà ñàìîì äåëå óðàâíåíèå (14) îêàçûâàåòñÿ ñïîñîáíûì îïèñàòü äèíàìèêó óñêîðåíèÿ ðîñòà ÷èñëåííîñòè íàñåëåíèÿ ìèðà (âïëîòü äî íà÷àëà 1970-õ ãîäîâ) òàêèì æå óäèâèòåëüíî òî÷íûì îáðàçîì, êàêèì óðàâíåíèå (5) ñïîñîáíî îïèñàòü îáùèé ïàòòåðí ãëîáàëüíîãî ìàêðîýâîëþöèîííîãî óñêîðåíèÿ çà ïîñëåäíèå 4 ìëðä. ëåò.

 êîíòåêñòå Áîëüøîé èñòîðèè î÷åíü âàæíûì ïðåäñòàâëÿåòñÿ òî îáñòîÿòåëüñòâî, ÷òî óðàâíåíèå (5), îïèñûâàþùåå óñêîðåíèå òåìïîâ ãëîáàëüíîãî ìàêðîýâîëþöèîííîãî óñêîðåíèÿ, è óðàâíåíèå (14), îïèñûâàþùåå ðîñò ÷èñëåííîñòè íàñåëåíèÿ Çåìëè, îêàçûâàþòñÿ ïîëíîñòüþ èäåíòè÷íûìè. Áîëåå òîãî, ýìïèðè÷åñêèé è ìàòåìàòè÷åñêèé àíàëèç ïîêàçûâàåò, ÷òî ìåæäó íèìè ñóùåñòâóåò î÷åíü ãëóáîêàÿ âçàèìîñâÿçü è ÷òî îíè îïèñûâàþò äâå ñòîðîíû îäíîãî ïðîöåññà (ñì.: Korotayev 2018: 106-110).

Î ôîðìóëå óñêîðåíèÿ ãëîáàëüíîé ýâîëþöèè

×åñòíî ñêàæó, ó ìåíÿ áûëè ñåðüåçíûå ñîìíåíèÿ, êîãäà ÿ âïåðâûå îçíàêîìèëñÿ ñ ðàñ÷åòàìè À. Ä. Ïàíîâà è Ò. Ìîäèñà (è ÿ íå óäèâëåí, ÷òî ó áîëüøèíñòâà èñòîðèêîâ âîçíèêàþò î÷åíü ïîõîæèå ñîìíåíèÿ, êîãäà îíè âèäÿò ýòè ðàáîòû). Ó ìåíÿ ïîÿâèëîñü ìíîãî âîçðàæåíèé îòíîñèòåëüíî òî÷íîñòè ìíîãèõ îïèñàíèé èõ «êàíîíè÷åñêèõ âåõ», àäåêâàòíîñòè èõ îòáîðà è òî÷íîñòè äàòèðîâîê. ß íà÷àë ñåðüåçíî îòíîñèòüñÿ ê ðàñ÷åòàì Ìîäèñà è Ïàíîâà, òîëüêî êîãäà ñàì ïðîàíàëèçèðîâàë äâà ñîîòâåòñòâóþùèõ âðåìåííûõ ðÿäà, èäåíòèôèöèðîâàííûõ (êàê ìû âèäåëè âûøå) ïîëíîñòüþ íåçàâèñèìî äðóã îò äðóãà äâóìÿ ðàçíûìè èññëåäîâàòåëÿìè, èñïîëüçóþùèìè ñîâåðøåííî ðàçíûå èñòî÷íèêè. Ïðè ýòîì ÿ àíàëèçèðîâàë èõ ïðè ïîìîùè ìàòåìàòè÷åñêîé ìîäåëè, êîòîðàÿ íå ïðèìåíÿëàñü íè Ìîäèñîì, íè Ïàíîâûì, è îáíàðóæèë, ÷òî îíè îïèñûâàþòñÿ íåîáû÷àéíî òî÷íûì îáðàçîì ïî÷òè èäåíòè÷íîé ìàòåìàòè÷åñêîé ãèïåðáîëè÷åñêîé ôóíêöèåé. Íà ìîé âçãëÿä, ýòî çàñòàâëÿåò ïðåäïîëàãàòü îáúåêòèâíîå íàëè÷èå äîâîëüíî ïðîñòîé ãèïåðáîëè÷åñêîé çàêîíîìåðíîñòè óñêîðåíèÿ ãëîáàëüíîãî ìàêðîýâîëþöèîííîãî ðàçâèòèÿ, íàáëþäàåìîãî íà Çåìëå â òå÷åíèå ïîñëåäíèõ 4 ìëðä. ëåò. Ýòî âïå÷àòëåíèå ñòàëî åùå áîëåå ñèëüíûì, êîãäà óðàâíåíèå, îïèñûâàþùåå êàðòèíó óñêîðåíèÿ ïëàíåòàðíîé ìàêðîýâîëþöèè â ðÿäàõ Ìîäèñà - Êóðöâåéëà è Ïàíîâà, îêàçàëîñü ïîëíîñòüþ èäåíòè÷íûì óðàâíåíèþ, íàéäåííîìó åùå â 1960 ãîäó ôîí Ôåðñòåðîì, ïîêàçàâøèì óæå òîãäà, ÷òî îíî ñïîñîáíî ñ íåîáû÷àéíîé òî÷íîñòüþ îïèñàòü ãëîáàëüíûé ïàòòåðí óñêîðåíèÿ ðîñòà ÷èñëåííîñòè íàñåëåíèÿ Çåìëè ìåæäó 1 è 1958 ãîäàìè.

Ó ìåíÿ áûëè îñíîâàíèÿ îæèäàòü, ÷òî ïëàíåòàðíîå ìàêðîýâîëþöèîííîå óñêîðåíèå çà ïîñëåäíèå 4 ìëðä. ëåò ìîæåò áûòü îïèñàíî åäèíûì ãèïåðáîëè÷åñêèì óðàâíåíèåì ñ äîñòàòî÷íî âûñîêîé òî÷íîñòüþ, òàê êàê ïðåäûäóùèå èññëåäîâàíèÿ ïîêàçàëè, ÷òî è áèîëîãè÷åñêàÿ, è ñîöèàëüíàÿ ýâîëþöèÿ ìîæåò îïèñûâàòüñÿ ñ äîñòàòî÷íî âûñîêîé òî÷íîñòüþ ïðîñòûìè ãèïåðáîëè÷åñêèìè óðàâíåíèÿìè [13], íî, äîëæåí ñêàçàòü, è ÿ áûë óäèâëåí, êîãäà îáíàðóæèë, ÷òî óñêîðåíèå çà ïîñëåäíèå 4 ìëðä. ëåò îïèñûâàåòñÿ åäèíûì ãèïåðáîëè÷åñêèì óðàâíåíèåì ñ òàêîé âûñîêîé òî÷íîñòüþ.

Òàêæå íåáåçûíòåðåñíûì ïðåäñòàâëÿåòñÿ òî îáñòîÿòåëüñòâî, ÷òî äàòû ñèíãóëÿðíîñòè äëÿ âñåõ òðåõ (î÷åíü ðàçëè÷íûõ) ðàññìîòðåííûõ íàìè âðåìåííûõ ðÿäîâ îêàçàëèñü ïðàêòè÷åñêè èäåíòè÷íûìè (2029 ãîä äëÿ ðÿäà Ìîäèñà - Êóðöâåéëà è 2027 ãîä - äëÿ ðÿäîâ Ïàíîâà è ôîí Ôåðñòåðà).

Ê èíòåðïðåòàöèè Ñèíãóëÿðíîñòè. Ìåñòî Ñèíãóëÿðíîñòè â Áîëüøîé èñòîðèè è ãëîáàëüíîé ýâîëþöèè

Íî íàñêîëüêî ñåðüåçíî ìû äîëæíû îòíîñèòüñÿ ê «ïðåäñêàçàíèþ» Ñèíãóëÿðíîñòè, ñîäåðæàùåìóñÿ â òàêèõ ìàòåìàòè÷åñêèõ ìîäåëÿõ? Ñëåäóåò ëè íàì âìåñòå ñ Ð. Êóðöâåéëîì äåéñòâèòåëüíî îæèäàòü, ÷òî ãäå-òî â ðàéîíå 2029 ãîäà ìû áóäåì èìåòü äåëî ñ óñêîðåíèåì ãëîáàëüíîãî òåõíîëîãè÷åñêîãî ðîñòà íà íåñêîëüêî ïîðÿäêîâ (÷òî äåéñòâèòåëüíî âûòåêàåò èç óðàâíåíèÿ (4), åñëè ïîíèìàòü åãî áóêâàëüíî, êàê ïðåäëàãàåò, íàïðèìåð, À. Ï. Íàçàðåòÿí [2015a; 2015á; Nazaretyan 2015; 2016; 2017; 2018])? Íàïðèìåð, ìîæåì ëè ìû èñïîëüçîâàòü òî îáñòîÿòåëüñòâî, ÷òî íàø àíàëèç ðÿäà Ìîäèñà - Êóðöâåéëà âûÿâèë Ñèíãóëÿðíîñòü â ðàéîíå 2029 ãîäà êàê óêàçàíèå íà òî, ÷òî â ðàéîíå ýòîãî âðåìåíè íàì ñëåäóåò îæèäàòü íà÷àëî äåâÿòîé ïîðîãîâîé âåõè Áîëüøîé èñòîðèè?

Îòìåòèì, ÷òî íåêîòîðûå ñïåöèàëèñòû â îáëàñòè Áîëüøîé èñòîðèè ñêëîííû îòíîñèòüñÿ ê òàêèì «ìàòåìàòè÷åñêè îáîñíîâàííûì» ïðåäñêàçàíèÿì ñîâåðøåííî ñåðüåçíî. Íàïðèìåð, À. Ï. Íàçàðåòÿí â ñòàòüå ñ ñèìïòîìàòè÷åñêèì çàãîëîâêîì «Ìåãàèñòîðèÿ è åå “çàãàäî÷íàÿ ñèíãóëÿðíîñòü”» óòâåðæäàåò ñëåäóþùåå: «Ñîëíå÷íàÿ ñèñòåìà îáðàçîâàëàñü îêîëî 4,6 ìëðä. ëåò íàçàä, à ñàìûå ïåðâûå ïðèçíàêè æèçíè íà Çåìëå íàñ÷èòûâàþò äî 4 ìëðä. ëåò. Òàêèì îáðàçîì, íàøà ïëàíåòà ñòàëà îäíîé èç (âåðîÿòíî, ìíîæåñòâà) òî÷åê, íà êîòîðûõ ëîêàëèçîâàëàñü ïîñëåäóþùàÿ ýâîëþöèÿ Ìåòàãàëàêòèêè. Õîòÿ åå óñêîðåíèå çàìå÷åíî äàâíî, â ïîñëåäíåå âðåìÿ îáíàðóæèëîñü íîâîå îáñòîÿòåëüñòâî. Àâñòðàëèéñêèé ýêîíîìèñò è èñòîðèê-ãëîáàëèñò Ã. Ñíóêñ, ðîññèéñêèé ôèçèê À. Ä. Ïàíîâ è àìåðèêàíñêèé ìàòåìàòèê Ð. Êóðöâåéë íåçàâèñèìî, ïî ðàçíûì èñòî÷íèêàì è ñ èñïîëüçîâàíèåì ðàçíîãî ìàòåìàòè÷åñêîãî àïïàðàòà ñîïîñòàâèëè âðåìåííûå èíòåðâàëû ìåæäó ãëîáàëüíûìè ôàçîâûìè ïåðåõîäàìè â áèîëîãè÷åñêîé, ïðàñîöèàëüíîé è ñîöèàëüíîé ýâîëþöèè (Ïàíîâ 2005; 2008; Kurzweil 2005; Snooks 1996; Âàéíáåðã 1981). Ðàñ÷åòû ïîêàçûâàþò, ÷òî ïåðèîäû ñîêðàùàëèñü ïî ñòðîãî óáûâàþùåé ãåîìåòðè÷åñêîé ïðîãðåññèè, òî åñòü óñêîðåíèå ýâîëþöèè íà Çåìëå ñëåäîâàëî ëîãàðèôìè÷åñêîìó çàêîíó» (Íàçàðåòÿí 2015a: 759). È äàëåå: «Ýêñòðàïîëèðîâàâ ëèíèþ ãèïåðáîëè÷åñêîãî óñêîðåíèÿ â áóäóùåå, èññëåäîâàòåëè ïðèøëè ê åäèíîäóøíîìó è åùå áîëåå øîêèðóþùåìó âûâîäó: îêîëî ñåðåäèíû XXI âåêà îíà óïèðàåòñÿ â òî÷êó ôèíàëüíîé (áîëüøîé) ñèíãóëÿðíîñòè. Êðèâàÿ çàâîðà÷èâàåò â âåðòèêàëü, òî åñòü ñêîðîñòü ýâîëþöèîííîãî ïðîöåññà ñòðåìèòñÿ ê áåñêîíå÷íîñòè, à èíòåðâàëû ìåæäó ôàçîâûìè ïåðåõîäàìè - ê íóëþ» (Òàì æå: 761; ñì. òàêæå: Nazaretyan 2017: 32).

Íàçàðåòÿí èñïîëüçóåò ìàòåìàòè÷åñêèå ðàñ÷åòû äàòû Ñèíãóëÿðíîñòè ãëîáàëüíîé ýâîëþöèîííîé ãèïåðáîëû äëÿ ïðåäñêàçàíèÿ âîçìîæíîé äàòû òîãî, ÷òî â òåðìèíîëîãèè îñíîâîïîëîæíèêà Áîëüøîé èñòîðèè Ä. Êðèñòèàíà (Christian 2008) ìîæíî áûëî áû íàçâàòü äåâÿòîé ïîðîãîâîé âåõîé Áîëüøîé èñòîðèè (Big History Threshold 9), êîòîðàÿ, ñîãëàñíî Íàçàðåòÿíó, áóäåò ñóùåñòâåííî áîëåå çíà÷èìîé, ÷åì ïðåäøåñòâóþùèå âåõè 7 («Àãðàðíàÿ ðåâîëþöèÿ») è 8 («Ìîäåðíèçàöèîííàÿ ðåâîëþöèÿ») [14].

Îäíàêî äàþò ëè ðàñ÷åòû, ïðîâåäåííûå Ïàíîâûì â 2003-2005 ãîäàõ èëè íàìè â ýòîé ñòàòüå, äåéñòâèòåëüíûå îñíîâàíèÿ îæèäàòü Ñèíãóëÿðíîñòè / íàñòóïëåíèÿ äåâÿòîé ïîðîãîâîé âåõè Áîëüøîé èñòîðèè ìåæäó 2029 è 2050 ãîäàìè? Êàê, íàâåðíîå, óæå ïîíÿòíî, ÿ ñêëîíåí äàòü íà ýòîò âîïðîñ îäíîçíà÷íî îòðèöàòåëüíûé îòâåò. Ñîáñòâåííî ãîâîðÿ, êàê ìû ìîãëè âèäåòü, äàííàÿ ñòàòüÿ ïðåäñòàâëÿåò ñîáîé, ñóäÿ ïî âñåìó, ïåðâóþ ïîïûòêó â ÿâíîì âèäå (ãîâîðÿ ñëîâàìè À. Ï. Íàçàðåòÿíà [2015à: 761]) «ýêñòðàïîëèðîâàòü ëèíèþ ãèïåðáîëè÷åñêîãî óñêîðåíèÿ â áóäóùåå» [15]. Õîòÿ Íàçàðåòÿí è óòâåðæäàåò îáðàòíîå, ïîäîáíàÿ ïîïûòêà íå ïðåäïðèíèìàëàñü Ã. Ä. Ñíóêñîì (Snooks 1996), êîòîðûé íå ïûòàëñÿ âû÷èñëÿòü êàêèå áû òî íè áûëî ìàòåìàòè÷åñêèå ñèíãóëÿðíîñòè. Íèêàêèõ ôîðìàëüíûõ ïîïûòîê «ýêñòðàïîëèðîâàòü ëèíèþ ãèïåðáîëè÷åñêîãî óñêîðåíèÿ â áóäóùåå» ñ èñïîëüçîâàíèåì êàêèõ-ëèáî ìàòåìàòè÷åñêèõ ìåòîäîâ Êóðöâåéëîì íå ïðåäïðèíèìàëîñü - óæå õîòÿ áû ïîòîìó, ÷òî îí äî ñèõ ïîð óâåðåí â òîì, ÷òî èìååò äåëî ñ ýêñïîíåíöèàëüíûì, à íå ãèïåðáîëè÷åñêèì óñêîðåíèåì. Òàêèì îáðàçîì, åäâà ëè íå åäèíñòâåííûì èññëåäîâàòåëåì, ïîïûòàâøèìñÿ ìàòåìàòè÷åñêè ðàññ÷èòàòü âðåìÿ ñèíãóëÿðíîñòè äëÿ ëèíèè óñêîðåíèÿ ïëàíåòàðíîé ýâîëþöèè, áûë À. Ä. Ïàíîâ (2004; 2005; 2006; 2008; Panov 2005; 2011; 2017) - õîòÿ ñ íåêîòîðûìè îãîâîðêàìè çäåñü ìîæíî òàêæå óïîìÿíóòü Ñ. Í. Ãðèí÷åíêî (2001; 2004; 2006; 2007 è äð.), Ò. Ìîäèñà (Modis 2002; 2003) è Ä. ËåÏóàðà (LePoire 2013; 2015).

Èñïîëüçîâàííàÿ Ïàíîâûì ìåòîäèêà ðàñ÷åòà Ñèíãóëÿðíîñòè ñóùåñòâåííî îòëè÷àëàñü îò «ýêñòðàïîëèðîâàíèÿ ëèíèè ãèïåðáîëè÷åñêîãî óñêîðåíèÿ â áóäóùåå» (ýòî, ñêîðåå, òà ñàìàÿ ìåòîäèêà, êîòîðàÿ áûëà èñïîëüçîâàíà íàìè, à íå Ïàíîâûì); îäíàêî íåò ñîìíåíèé â òîì, ÷òî èì áûëà ïðèìåíåíà íå ìåíåå ñòðîãàÿ ìåòîäèêà ðàñ÷åòà âðåìåíè Ñèíãóëÿðíîñòè ïëàíåòàðíîé ýâîëþöèè. Íî êàêîâû áûëè ðåçóëüòàòû ýòèõ ðàñ÷åòîâ? Ïîñëå òîãî êàê À. Ä. Ïàíîâ ïðèìåíèë ñâîþ ìåòîäèêó ìàòåìàòè÷åñêîãî àíàëèçà ê âðåìåííîìó ðÿäó, íà÷èíàþùåìóñÿ ñ ôàçîâîãî ïåðåõîäà 0 («Âîçíèêíîâåíèå æèçíè íà Çåìëå») è çàêàí÷èâàþùåìóñÿ íà ôàçîâîì ïåðåõîäå 19 («Êðèçèñ è ðàñïàä ñîöèàëèñòè÷åñêîãî ëàãåðÿ, èíôîðìàöèîííàÿ ãëîáàëèçàöèÿ»), îí îáíàðóæèë, ÷òî òî÷êà ñèíãóëÿðíîñòè äëÿ åãî âðåìåííîãî ðÿäà íàõîäèòñÿ âîâñå íå «îêîëî ñåðåäèíû XXI âåêà», êàê ýòî óòâåðæäàåò À. Ï. Íàçàðåòÿí (2015à: 761), à ïðèõîäèòñÿ íà 2004 ãîä í. ý. (!) [16] (Ïàíîâ 2005: 130; Panov 2005: 222). Ïðè ýòîì Íàçàðåòÿí, êàæåòñÿ, äàæå íå çàìåòèë, ÷òî âñêîðå ïîñëå îáíàðóæåíèÿ òî÷êè Ñèíãóëÿðíîñòè Ïàíîâ çàíÿëñÿ èçó÷åíèåì ïîñòñèíãóëÿðíîãî ðàçâèòèÿ ÷åëîâå÷åñòâà (è âîîáùå âîïðîñîì î ïîñòñèíãóëÿðíûõ öèâèëèçàöèÿõ) è òåñíî ñâÿçàííûì ñ ýòèì âîïðîñîì î ãëîáàëüíîì çàìåäëåíèè òåìïîâ íàó÷íî-òåõíè÷åñêîãî ïðîãðåññà (Ïàíîâ 2009; 2013; Panov 2011; 2017).

Êàê ïèøåò Ä. ËåÏóàð, «ïðîñëåæèâàåìûå â Áîëüøîé èñòîðèè òåíäåíöèè ê óñêîðÿþùèìñÿ èçìåíåíèÿì è ðîñòó ñëîæíîñòè è ñâÿçàííàÿ ñ íèìè òåíäåíöèÿ ê óñêîðåííîìó ðîñòó ïîòðåáëåíèÿ ýíåðãèè íå ìîãóò ïðîäîëæàòüñÿ äî áåñêîíå÷íîñòè. Ìû èññëåäîâàëè ïðèçíàêè ïîòåíöèàëüíîãî çàìåäëåíèÿ òåìïîâ èçìåíåíèé â ýêîíîìèêå, òåõíîëîãèè è ñîöèàëüíîé ñôåðå. Ýòî íå îçíà÷àåò, ÷òî èçìåíåíèÿ ïðåêðàòÿòñÿ, ïðîñòî òåìïû èçìåíåíèé óæå íå áóäóò óñêîðÿòüñÿ. Ôàêòè÷åñêè ê òî÷êå ïåðåãèáà â ëîãèñòè÷åñêîé êðèâîé îáó÷åíèÿ áûëà ñäåëàíà òîëüêî ïîëîâèíà îòêðûòèé. Ïîñêîëüêó â èñòîðèè æèçíè, ÷åëîâåêà è òåõíîëîãè÷åñêîé öèâèëèçàöèè áûëî òðè îñíîâíûõ ýòàïà, ïðîäîëæåíèå ëîãèñòè÷åñêîé êðèâîé ïðåäïîëàãàåò åùå òðè ôàçû. Íàïðàâëåíèå ðàçâèòèÿ òåõíîëîãèé óêàçûâàåò íà ñëåäóþùèé ýòàï, âêëþ÷àÿ óñîâåðøåíñòâîâàííûå òåõíîëîãèè èçìåíåíèÿ ïðèðîäû ÷åëîâåêà ïîñðåäñòâîì óñîâåðøåíñòâîâàííûõ áèîòåõíîëîãèé è êîìïüþòåðíîé èíòåãðàöèè… Ñëèøêîì áûñòðîå èçìåíåíèå íå âñåãäà õîðîøî. Îíî ïðèâîäèò ê òîìó, ÷òî ýôôåêòèâíîñòü ñèñòåì ïàäàåò, ïîòîìó ÷òî ìû èìååì ìàëî äîëãîñðî÷íûõ îæèäàíèé» (LePoire 2013: 115-116).  êà÷åñòâå âàæíûõ ôàêòîðîâ íà÷àâøåãîñÿ çàìåäëåíèÿ òåìïîâ ãëîáàëüíîãî ðàçâèòèÿ ËåÏóàð íàçûâàåò «ðîñò ñåáåñòîèìîñòè ïðîèçâîäèìîé ýíåðãèè, îãðàíè÷åííûå ïðèðîäíûå ðåñóðñû, ñíèæåíèå òåìïîâ ôóíäàìåíòàëüíûõ îòêðûòèé â îáëàñòè ôèçè÷åñêèõ íàóê è íåîáõîäèìîñòü èíâåñòèöèé â îõðàíó îêðóæàþùåé ñðåäû» (Ibid.: 109).

Îòìåòèì òàêæå, ÷òî Ò. Ìîäèñ (Modis 2002; 2003; 2005; 2012) òîæå èíòåðïðåòèðóåò ìàêñèìàëüíîå óñêîðåíèå òåìïîâ ðîñòà ãëîáàëüíîé ñëîæíîñòè, âûÿâëÿåìîå èì â ðàéîíå 2000 ãîäà í. ý., èìåííî êàê òî÷êó ïåðåãèáà, ïîñëå êîòîðîé îí ïðîãíîçèðóåò ïîñëåäîâàòåëüíîå ñíèæåíèå òåìïîâ ðîñòà ãëîáàëüíîé ñëîæíîñòè. Åùå ðàíüøå ïîïûòêà ìàòåìàòè÷åñêè âûÿâèòü Ñèíãóëÿðíîñòü â ðÿäó ñîáûòèé ïëàíåòàðíîé ýâîëþöèè [17], êîòîðûå Ìîäèñ íàçâàë áû «êàíîíè÷åñêèìè âåõàìè», áûëà ïðåäïðèíÿòà â 2001 ãîäó (ò. å. âñåãî çà ãîä äî ïóáëèêàöèè åãî îñíîâîïîëàãàþùåé ñòàòüè) Ñ. Í. Ãðèí÷åíêî (2001; ñì. òàêæå: Ãðèí÷åíêî 2004; 2006; 2007; 2015; Ãðèí÷åíêî, Ùàïîâà 2017); òî÷êà Ñèíãóëÿðíîñòè áûëà èì ìàòåìàòè÷åñêè îïðåäåëåíà êàê 1981 ãîä í. ý. Îòìåòèì, ÷òî ýòî õîðîøî êîððåëèðóåò ñ èäåíòèôèêàöèåé íàìè 1973 ãîäà êàê òî÷êè ïåðåãèáà; ïîñëå êâàäðàòè÷íî-ãèïåðáîëè÷åñêèé òðåíä óñêîðåíèÿ ðîñòà ìèðîâîãî ÂÂÏ íà÷àë ìåíÿòüñÿ íà ïðÿìî ïðîòèâîïîëîæíûé òðåíä ê çàìåäëåíèþ ðîñòà ýòîãî ïîêàçàòåëÿ (Êîðîòàåâ 2006; Êîðîòàåâ, Ìàëêîâ, Õàëòóðèíà 2007; Êîðîòàåâ è äð. 2010; Êîðîòàåâ, Áîæåâîëüíîâ 2010; Àêàåâ è äð. 2014; Ñàäîâíè÷èé è äð. 2014; Êîðîòàåâ, Áèëþãà 2016; Khaltourina, Korotayev 2007). Âñå ýòî õîðîøî ïîäòâåðæäàåòñÿ ðàñòóùèì îáúåìîì äàííûõ, ñâèäåòåëüñòâóþùèõ î íà÷àëå äîëãîñðî÷íîé òåíäåíöèè ê çàìåäëåíèþ òåìïîâ íàó÷íî-òåõíè÷åñêîãî è ýêîíîìè÷åñêîãî ðîñòà (ñì., íàïðèìåð: Êðûëîâ 1999; 2002; 2007; Ïàíîâ 2009; 2013; Àêàåâ 2010; Êîðîòàåâ, Ìàëêîâ, Õàëòóðèíà 2007; Êîðîòàåâ, Áîæåâîëüíîâ 2010; Êîðîòàåâ è äð. 2010; Êîðîòàåâ, Áèëþãà 2016; Huebner 2005; Khaltourina, Korotayev 2007; Maddison 2007; Modis 2002; 2005; 2012; Gordon 2012; Teulings, Baldwin 2014; Piketty 2014; LePoire 2005; 2009; 2013; 2015; 2016; Summers 2016; Cervellati et al. 2017; Taylor, Tyers 2017; Jones 2018; Popoviæ 2018 etc.).

Èòàê, íàñêîëüêî ñåðüåçíî ìû äîëæíû îòíîñèòüñÿ ê «ïðåäñêàçàíèþ» ñèíãóëÿðíîñòè, ñîäåðæàùåìóñÿ â ãèïåðáîëè÷åñêèõ ìàòåìàòè÷åñêèõ ìîäåëÿõ ãëîáàëüíîãî ðàçâèòèÿ? Èëè - ñëåäóåò ëè íàì âìåñòå ñ Êóðöâåéëîì îæèäàòü, ÷òî ãäå-òî â ðàéîíå 2029 ãîäà ìû áóäåì ðåàëüíî èìåòü äåëî ñ óñêîðåíèåì ãëîáàëüíîãî òåõíîëîãè÷åñêîãî ðîñòà íà íåñêîëüêî ïîðÿäêîâ (÷òî äåéñòâèòåëüíî âûòåêàåò èç óðàâíåíèÿ (4), åñëè ïîíèìàòü åãî áóêâàëüíî)?

Êàê óæå ïîíÿòíî, ÿ ñêëîíåí äàòü íà ýòîò âîïðîñ îäíîçíà÷íî îòðèöàòåëüíûé îòâåò. È â çàêëþ÷åíèå ïðèâåäó åùå îäèí àðãóìåíò. Íà ìîé âçãëÿä, îòðèöàòåëüíûé îòâåò íà ýòîò âîïðîñ âûòåêàåò, íàïðèìåð, èç èçâåñòíûõ íàì ýìïèðè÷åñêèõ äàííûõ ïî äèíàìèêå ÷èñëåííîñòè íàñåëåíèÿ ìèðà è óñòàíîâëåííûõ ê íàñòîÿùåìó âðåìåíè ìåõàíèçìîâ ýòîé äèíàìèêè. Êàê ìû ïîìíèì, ôîðìóëà ãèïåðáîëè÷åñêîãî ðîñòà ÷èñëåííîñòè íàñåëåíèÿ Çåìëè, îòêðûòàÿ Õ. ôîí Ôåðñòåðîì, èäåíòè÷íà ôîðìóëå óñêîðåíèÿ òåìïîâ ãëîáàëüíîãî ðàçâèòèÿ, ïðîñëåæèâàåìîãî â ðÿäàõ Ìîäèñà - Êóðöâåéëà è Ïàíîâà, è îíà õàðàêòåðèçóåòñÿ ïàðàìåòðîì ñèíãóëÿðíîñòè (2027 ãîä í. ý.), êîòîðûé ïðîñòî èäåíòè÷åí äëÿ ôîðìóëû óñêîðåíèÿ â ðÿäó Ïàíîâà è èìååò ðàçíèöó âñåãî ëèøü â 2 ãîäà äëÿ ðÿäà Ìîäèñà - Êóðöâåéëà. Îäíàêî êàêîâû îñíîâàíèÿ îæèäàòü, ÷òî ê ïÿòíèöå 13 íîÿáðÿ 2026 ãîäà òåìïû ïðèðîñòà íàñåëåíèÿ ìèðà óâåëè÷àòñÿ íà íåñêîëüêî ïîðÿäêîâ, êàê ýòî ïîäðàçóìåâàåò óðàâíåíèå ôîí Ôåðñòåðà? Îòâåò íà âîïðîñ ÿñåí: íåò íèêàêèõ îñíîâàíèé ýòîãî îæèäàòü. Íåñîìíåííî, êîãäà ôîí Ôåðñòåð è åãî êîëëåãè äàâàëè ñâîåé ñòàòüå (von Foerster et al. 1960) çíàìåíèòîå íàçâàíèå «Ñóäíûé äåíü: ïÿòíèöà, 13 íîÿáðÿ 2026 ãîäà», îíè âîâñå íå èìåëè â âèäó, ÷òî íàñåëåíèå Çåìëè â ýòîò äåíü äåéñòâèòåëüíî ìîæåò ñòàòü áåñêîíå÷íûì (õîòÿ «ýñõàòîëîãè÷åñêèå» âûâîäû èç ãèïåðáîëè÷åñêèõ ìîäåëåé ðîñòà Ìèð-Ñèñòåìû áåç êàêèõ-ëèáî äîñòàòî÷íûõ íà òî îñíîâàíèé äåëàþòñÿ äî ñèõ ïîð [ñì., íàïðèìåð: Johansen, Sornette 2001]). Èç ýòîé ñòàòüè, ñêîðåå, âûòåêàë ïðÿìî ïðîòèâîïîëîæíûé ïðîãíîç - íàáëþäàâøèéñÿ âïëîòü äî 1960 ãîäà íà ïðîòÿæåíèè ìíîãèõ âåêîâ ãèïåðáîëè÷åñêèé ðîñò ìèðîâîãî íàñåëåíèÿ äîëæåí èñïûòàòü â ñàìûå áëèæàéøèå ãîäû ðàäèêàëüíóþ òðàíñôîðìàöèþ è ñìåíèòüñÿ ïðèíöèïèàëüíî èíûì òèïîì äåìîãðàôè÷åñêîé ìàêðîäèíàìèêè. Îòìåòèì, ÷òî ýòîò ïðîãíîç ñòàë áëåñòÿùå îïðàâäûâàòüñÿ âñåãî ëèøü ÷åðåç íåñêîëüêî ëåò ïîñëå ïóáëèêàöèè ñòàòüè ôîí Ôåðñòåðà è åãî êîëëåã (Êîðîòàåâ, Ìàëêîâ, Õàëòóðèíà 2007). Ìèð-Ñèñòåìà íà÷àëà âûõîäèòü èç ðåæèìà ñ îáîñòðåíèåì (Îíè æå 2008: 99; ñì. òàêæå: Êîðîòàåâ, Õàëòóðèíà 2009; Çèíüêèíà è äð. 2016; Çèíüêèíà, Êîðîòàåâ 2017; Korotayev 2008: 154).

Äåéñòâèòåëüíî, ñ íà÷àëà 1970-õ ãîäîâ êðèâàÿ ðîñòà ÷èñëåííîñòè íàñåëåíèÿ ìèðà ñòàëà âñå áîëüøå îòêëîíÿòüñÿ îò ãèïåðáîëè÷åñêîé òðàåêòîðèè (ñì., íàïðèìåð: Êàïèöà 1999; Êîðîòàåâ, Ìàëêîâ, Õàëòóðèíà 2005à; 2007; Êîðîòàåâ 2015; 2016; Kapitza 2003; 2006; 2010; Livi-Bacci 2012; Korotayev, Malkov, Khaltourina 2006a; 2006b; Korotayev, Goldstone, Zinkina 2015; Grinin, Korotayev 2015), è çà ïîñëåäíèå äåñÿòèëåòèÿ îíà ïðèíÿëà îïðåäåëåííî ëîãèñòè÷åñêóþ ôîðìó - òåíäåíöèÿ ê ãèïåðáîëè÷åñêîìó óñêîðåíèþ ñìåíèëàñü òåíäåíöèåé ê ëîãèñòè÷åñêîìó çàìåäëåíèþ.

 íåêîòîðûõ îòíîøåíèÿõ âïîëíå ìîæíî ñêàçàòü, ÷òî ôîí Ôåðñòåð îòêðûë ñèíãóëÿðíîñòü ãëîáàëüíîé äåìîãðàôè÷åñêîé èñòîðèè; îí îáíàðóæèë, ÷òî ÷åëîâå÷åñêàÿ Ìèð-Ñèñòåìà ïðèáëèæàëàñü ê ñèíãóëÿðíîìó ïåðèîäó â ñâîåé èñòîðèè, êîãäà òåíäåíöèÿ ãèïåðáîëè÷åñêîãî óñêîðåíèÿ, êîòîðîé îíà ñëåäîâàëà ìíîãî òûñÿ÷åëåòèé (à ïî ìíåíèþ íåêîòîðûõ, äàæå íåñêîëüêî ìèëëèîíîâ ëåò), áóäåò çàìåíåíà íà ïðîòèâîïîëîæíûé òðåíä ê çàìåäëåíèþ. Èìåííî ÷åðåç ýòîò ñèíãóëÿðíûé ïåðèîä ìû ñåé÷àñ è ïðîõîäèì. Ïðîöåññû è ìåõàíèçìû äàííîãî ðàçâîðîòà òðåíäîâ ê íàñòîÿùåìó âðåìåíè î÷åíü òùàòåëüíî èçó÷åíû è èçâåñòíû êàê «ãëîáàëüíûé äåìîãðàôè÷åñêèé ïåðåõîä» (Êàïèöà 1999; 2007; Ïîäëàçîâ 2001; 2017; Ðîìàí÷óê, Ìåäâåäåâà 2009; Êîðîòàåâ 2015; Kapitza 2003; 2006; 2010; Korotayev, Goldstone, Zinkina 2015; Podlazov 2017). Ïðè ýòîì îñîáîå âíèìàíèå çäåñü ñòîèò îáðàòèòü íà òî îáñòîÿòåëüñòâî, ÷òî â ñëó÷àå ñ ãëîáàëüíîé äåìîãðàôè÷åñêîé ýâîëþöèåé ïåðåõîä îò ãèïåðáîëè÷åñêîãî óñêîðåíèÿ ê ëîãèñòè÷åñêîìó çàìåäëåíèþ íà÷àëñÿ çà íåñêîëüêî äåñÿòèëåòèé äî äàòû Ñèíãóëÿðíîñòè, ìàòåìàòè÷åñêè âû÷èñëåííîé ôîí Ôåðñòåðîì.

Íà ìîé âçãëÿä, èìåþòñÿ îñíîâàíèÿ óòâåðæäàòü, ÷òî çàìåäëåíèå ñêîðîñòè ãëîáàëüíîãî ìàêðîýâîëþöèîííîãî ðàçâèòèÿ óæå íà÷àëîñü - è íà÷àëîñü îíî òàêæå çà íåñêîëüêî äåñÿòèëåòèé äî òîé òî÷êè Ñèíãóëÿðíîñòè, êîòîðóþ ìîæíî ìàòåìàòè÷åñêè âûÿâèòü â ãëîáàëüíûõ ýâîëþöèîííûõ ðÿäàõ Ìîäèñà - Êóðöâåéëà è Ïàíîâà.



Çàêëþ÷åíèå

Èòàê, ïðîâåäåííûé íàìè àíàëèç ïîçâîëÿåò ïðåäïîëàãàòü íàëè÷èå äîñòàòî÷íî ñòðîãèõ ìàêðîýâîëþöèîííûõ çàêîíîìåðíîñòåé (îïèñûâàþùèõ ýâîëþöèþ ñëîæíîñòè íà íàøåé ïëàíåòå çà ïîñëåäíèå íåñêîëüêî ìèëëèàðäîâ ëåò), êîòîðûå ìîãóò óäèâèòåëüíî òî÷íî îïèñûâàòüñÿ êðàéíå ïðîñòûìè ìàòåìàòè÷åñêèìè ôóíêöèÿìè. Âìåñòå ñ òåì ýòîò àíàëèç çàñòàâëÿåò ïðåäïîëàãàòü, ÷òî â ðàéîíå òî÷êè Ñèíãóëÿðíîñòè íåò îñíîâàíèÿ âñëåä çà Êóðöâåéëîì îæèäàòü íåâèäàííîãî (íà ìíîãî ïîðÿäêîâ) óñêîðåíèÿ òåìïîâ òåõíîëîãè÷åñêîãî ðàçâèòèÿ; èìåþòñÿ áîëüøèå îñíîâàíèÿ èíòåðïðåòèðîâàòü ýòó òî÷êó êàê èíäèêàòîð çîíû ïåðåãèáà, ïîñëå ïðîõîæäåíèÿ êîòîðîé òåìïû ãëîáàëüíîé ýâîëþöèè áóäóò ñèñòåìàòè÷åñêè â äîëãîñðî÷íîé ïåðñïåêòèâå çàìåäëÿòüñÿ.



Ëèòåðàòóðà

1. Àêàåâ, À. À. 2010. Ôóíäàìåíòàëüíûå ïðåäåëû ýêîíîìè÷åñêîãî ðîñòà è ïîòðåáëåíèÿ. Â: Õàëòóðèíà, Ä. À., Êîðîòàåâ, À. Â., Çèíüêèíà, Þ. Â. (ðåä.), Ñèñòåìíûé ìîíèòîðèíã ãëîáàëüíûõ è ðåãèîíàëüíûõ ðèñêîâ. Ì.: URSS, ñ. 12-30.

2. Àêàåâ, A. À., Êîðîòàåâ, À. Â., Ìàëêîâ, Ñ. Þ. 2014. Ñîâðåìåííàÿ ñèòóàöèÿ è êîíòóðû áóäóùåãî. Â: Êîðîòàåâ, À. Â., Ìàëêîâ, Ñ. Þ., Àêàåâ, À. À. (îòâ. ðåä.), Êîìïëåêñíûé ñèñòåìíûé àíàëèç, ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå è ïðîãíîçèðîâàíèå ðàçâèòèÿ ñòðàí ÁÐÈÊÑ. Ïðåäâàðèòåëüíûå ðåçóëüòàòû. Ì.: Êðàñàíä/URSS, ñ. 10-31.

3. Àíòèñåðè, Ä., Ðåàëå, Äæ. 2001. Çàïàäíàÿ ôèëîñîôèÿ îò èñòîêîâ äî íàøèõ äíåé. Àíòè÷íîñòü, Ñðåäíåâåêîâüå. ÑÏá.: Ïåòðîïîëèñ.

4. Áàëàøîâà, Í. À., Ñàâ÷åíêî, Â. À., Ñàæèåíêî, Å. Â., Íàçàðåòÿí, À. Ï. 2017. Ìåãàèñòîðèÿ è ãëîáàëüíûå âûçîâû XXI âåêà: ñèíåðãåòè÷åñêàÿ ìîäåëü. Èñòîðè÷åñêàÿ ïñèõîëîãèÿ è ñîöèîëîãèÿ èñòîðèè 10(1): 193-212.

5. Áèãàí, Ä. 2004. Ïëàíåòà ÷åëîâåêîîáðàçíûõ.  ìèðå íàóêè 11: 68-77.

6. Áîðèñêîâñêèé, Ï. È.

7. 1970. Àøåëüñêàÿ êóëüòóðà. Â: Ïðîõîðîâ, À. Ì. (ãë. ðåä.), Áîëüøàÿ ñîâåòñêàÿ ýíöèêëîïåäèÿ. Ò. 2. Ì.: Ñîâåòñêàÿ ýíöèêëîïåäèÿ, ñ. 471.

8. 1974a. Îëäîâàé. Â: Ïðîõîðîâ, À. Ì. (ãë. ðåä.), Áîëüøàÿ ñîâåòñêàÿ ýíöèêëîïåäèÿ. Ò. 18. Ì.: Ñîâåòñêàÿ ýíöèêëîïåäèÿ, ñ. 369.

9. 1974á. Ìóñòüåðñêàÿ êóëüòóðà. Â: Ïðîõîðîâ, À. Ì. (ãë. ðåä.), Áîëüøàÿ ñîâåòñêàÿ ýíöèêëîïåäèÿ. Ò. 17. Ì.: Ñîâåòñêàÿ ýíöèêëîïåäèÿ, ñ. 134.

10. 1978. Øåëëüñêàÿ êóëüòóðà. Â: Ïðîõîðîâ, À. Ì. (ãë. ðåä.), Áîëüøàÿ ñîâåòñêàÿ ýíöèêëîïåäèÿ. Ò. 29. Ì.: Ñîâåòñêàÿ ýíöèêëîïåäèÿ, ñ. 377.

11. Âàéíáåðã, Ñ. 1981. Ïåðâûå òðè ìèíóòû. Ñîâðåìåííûé âçãëÿä íà ïðîèñõîæäåíèå Âñåëåííîé. Ì.: Ýíåðãîèçäàò.

12. Âîíã, Ê. 2003. Ó êîëûáåëè Homo sapiens. Â ìèðå íàóêè 11: 9-10.

13. Ãàëèìîâ, Ý. Ì. 2001. Ôåíîìåí æèçíè: ìåæäó ðàâíîâåñèåì è íåëèíåéíîñòüþ. Ïðîèñõîæäåíèå è ïðèíöèïû ýâîëþöèè. Ì.: URSS.

14. Ãðèí÷åíêî, Ñ. Í. 2001. Ñîöèàëüíàÿ ìåòàýâîëþöèÿ ×åëîâå÷åñòâà êàê ïîñëåäîâàòåëüíîñòü øàãîâ ôîðìèðîâàíèÿ ìåõàíèçìîâ åãî ñèñòåìíîé ïàìÿòè. Ýëåêòðîííûé æóðíàë «Èññëåäîâàíî â Ðîññèè» 145: 1652-1681.

15. 2004. Ñèñòåìíàÿ ïàìÿòü æèâîãî (êàê îñíîâà åãî ìåòàýâîëþöèè è ïåðèîäè÷åñêîé ñòðóêòóðû). Ì.: Ìèð.

16. 2006. Èñòîðèÿ ×åëîâå÷åñòâà ñ èíôîðìàòèêî-êèáåðíåòè÷åñêèõ ïîçèöèé: ïðîáëåìû ïåðèîäèçàöèè. Â: Ãðèíèí, Ë. Å., Êîðîòàåâ, À. Â., Ìàëêîâ, Ñ. Þ., (îòâ. ðåä.), Èñòîðèÿ è Ìàòåìàòèêà: Ïðîáëåìû ïåðèîäèçàöèè èñòîðè÷åñêèõ ìàêðîïðîöåññîâ. Ì.: ÊîìÊíèãà, ñ. 38-52.

17. 2007. Ìåòàýâîëþöèÿ (ñèñòåì íåæèâîé, æèâîé è ñîöèàëüíî-òåõíîëîãè÷åñêîé ïðèðîäû). Ì.: ÈÏÈ ÐÀÍ.

18. 2015. Ìîäåëèðîâàíèå: èíäóêòèâíîå è äåäóêòèâíîå. Ïðîáëåìû èñòîðè÷åñêîãî ïîçíàíèÿ. Ì.: ÈÂÈ ÐÀÍ, ñ. 95-101.

19. Ãðèí÷åíêî, Ñ. Í., Ùàïîâà, Þ. Ë. 2017. Ïàëåîàíòðîïîëîãèÿ, õðîíîëîãèÿ è ïåðèîäèçàöèÿ àðõåîëîãè÷åñêîé ýïîõè: ÷èñëîâàÿ ìîäåëü. Ïðîñòðàíñòâî è Âðåìÿ 1(27): 72-82.

20. Äüÿêîíîâ, È. Ì. 1994. Ïóòè èñòîðèè. Îò äðåâíåéøåãî ÷åëîâåêà äî íàøèõ äíåé. Ì.: Âîñò. ëèò-ðà.

21. Çàâàðçèí, Ã. À. 2003. Ñòàíîâëåíèå ñèñòåìû áèîãåîõèìè÷åñêèõ öèêëîâ. Ïàëåîíòîëîãè÷åñêèé æóðíàë 6: 16-24.

22. Çàéöåâ, À. È. 2001. Èç íàñëåäèÿ À. È. Çàéöåâà. Ò. 1. Êóëüòóðíûé ïåðåâîðîò â Ãðåöèè VIII-V ââ. äî í. ý. ÑÏá.: Ïåòðîïîëèñ.

23. Çèíüêèíà, Þ. Â., Êîðîòàåâ, À. Â. 2017. Ñîöèàëüíî-äåìîãðàôè÷åñêîå ðàçâèòèå ñòðàí Òðîïè÷åñêîé Àôðèêè: êëþ÷åâûå ôàêòîðû ðèñêà, ìîäèôèöèðóåìûå óïðàâëÿþùèå ïàðàìåòðû, ðåêîìåíäàöèè. Ì.: Ëåíàíä/URSS.

24. Çèíüêèíà, Þ. Â., Øóëüãèí, Ñ. Ã., Êîðîòàåâ, À. Â. 2016. Ýâîëþöèÿ ãëîáàëüíûõ ñåòåé. Çàêîíîìåðíîñòè, òåíäåíöèè, ìîäåëè. Ì.: Ëåíàíä/URSS.

25. Êàïèöà, Ñ. Ï. 1992. Ìàòåìàòè÷åñêàÿ ìîäåëü ðîñòà íàñåëåíèÿ ìèðà. Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå 4(6): 65-79.

26. 1996. Ôåíîìåíîëîãè÷åñêàÿ òåîðèÿ ðîñòà íàñåëåíèÿ Çåìëè. Óñïåõè ôèçè÷åñêèõ íàóê 166(1): 63-80.

27. 1999. Ñêîëüêî ëþäåé æèëî, æèâåò è áóäåò æèòü íà çåìëå. Ì.: Íàóêà.

28. 2007. Äåìîãðàôè÷åñêèé ïåðåõîä è áóäóùåå ÷åëîâå÷åñòâà. Âåñòíèê Åâðîïû 21: 7-16.

29. Êåëëåð, Á. Ì. 1975. Ïàëåîçîéñêàÿ ãðóïïà (ýðà). Â: Ïðîõîðîâ, À. Ì. (ãë. ðåä.), Áîëüøàÿ ñîâåòñêàÿ ýíöèêëîïåäèÿ. Ò. 19. Ì.: Ñîâåòñêàÿ ýíöèêëîïåäèÿ, ñ. 103-104.

30. Êîðîòàåâ, À. Â. 2006. Ïåðèîäèçàöèÿ èñòîðèè Ìèð-Ñèñòåìû è ìàòåìàòè÷åñêèå ìàêðîìîäåëè ñîöèàëüíî-èñòîðè÷åñêèõ ïðîöåññîâ. Â: Ãðèíèí, Ë. Å., Êîðîòàåâ, À. Â., Ìàëêîâ, Ñ. Þ. (îòâ. ðåä.), Èñòîðèÿ è Ìàòåìàòèêà: Ïðîáëåìû ïåðèîäèçàöèè èñòîðè÷åñêèõ ìàêðîïðîöåññîâ. Ì.: URSS, ñ. 116-167.

31. 2007. Ìàêðîäèíàìèêà óðáàíèçàöèè Ìèð-Ñèñòåìû: êîëè÷åñòâåííûé àíàëèç. Â: Ìàëêîâ, Ñ. Þ., Ãðèíèí, Ë. Å., Êîðîòàåâ, À. Â. (ðåä.), Èñòîðèÿ è Ìàòåìàòèêà: Ìàêðîèñòîðè÷åñêàÿ äèíàìèêà îáùåñòâà è ãîñóäàðñòâà. Ì.: ÊîìÊíèãà/URSS, ñ. 21-39.

32. 2010à. Êîìïàêòíûå ìàòåìàòè÷åñêèå ìîäåëè äîëãîñðî÷íîãî ðàçâèòèÿ Ìèð-Ñèñòåìû. Óðàëüñêèé èñòîðè÷åñêèé âåñòíèê 3: 15-24.

33. 2010á. Êîìïàêòíûå ìàòåìàòè÷åñêèå ìîäåëè ðàçâèòèÿ Ìèð-Ñèñòåìû. Â: Àêàåâ, À. À., Êîðîòàåâ, À. Â., Ìàëèíåöêèé, Ã. Ã. (îòâ. ðåä.), Ïðîãíîç è ìîäåëèðîâàíèå êðèçèñîâ è ìèðîâîé äèíàìèêè. Ì.: ËÊÈ/URSS, ñ. 70-92.

34. 2015. Ãëîáàëüíûé äåìîãðàôè÷åñêèé ïåðåõîä è ôàçû äèâåðãåíöèè - êîíâåðãåíöèè öåíòðà è ïåðèôåðèè Ìèð-Ñèñòåìû. Âåñòíèê Èíñòèòóòà ýêîíîìèêè Ðîññèéñêîé àêàäåìèè íàóê 1: 149-162.

35. 2016. Âåëèêàÿ äèâåðãåíöèÿ è âåëèêàÿ êîíâåðãåíöèÿ êàê ôàçû ïðîöåññà ìîäåðíèçàöèè. Ãåîãðàôèÿ ìèðîâîãî ðàçâèòèÿ: ñá. íàó÷. òð. Âûï. 3. Ì.: Òîâ. íàó÷íûõ èçäàíèé ÊÌÊ, ñ. 49-60.

36. Êîðîòàåâ, À. Â., Áèëþãà, Ñ. Ý. 2016. Î íåêîòîðûõ ñîâðåìåííûõ òåíäåíöèÿõ ìèðîâîãî ýêîíîìè÷åñêîãî ðàçâèòèÿ. Âåñòíèê Èíñòèòóòà ýêîíîìèêè Ðîññèéñêîé àêàäåìèè íàóê 4: 20-39.

37. Êîðîòàåâ, À. Â., Áîæåâîëüíîâ, Þ. Â. 2010. Íåêîòîðûå îáùèå òåíäåíöèè ýêîíîìè÷åñêîãî ðàçâèòèÿ Ìèð-Ñèñòåìû. Â: Àêàåâ, À. À., Êîðîòàåâ, À. Â., Ìàëèíåöêèé, Ã. Ã., Ïðîãíîç è ìîäåëèðîâàíèå êðèçèñîâ è ìèðîâîé äèíàìèêè. Ì.: ËÊÈ/URSS, ñ. 161-171.

38. Êîðîòàåâ, À. Â., Êîìàðîâà, Í. Ë., Õàëòóðèíà, Ä. À. 2007. Çàêîíû èñòîðèè. Âåêîâûå öèêëû è òûñÿ÷åëåòíèå òðåíäû. Äåìîãðàôèÿ. Ýêîíîìèêà. Âîéíû. Ì.: ÊîìÊíèãà/URSS.

39. Êîðîòàåâ, À. Â., Ìàëêîâ, À. Ñ., Õàëòóðèíà, Ä. À. 2005à. Çàêîíû èñòîðèè. Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå èñòîðè÷åñêèõ ìàêðîïðîöåññîâ. Äåìîãðàôèÿ. Ýêîíîìèêà. Âîéíû. Ì.: ÊîìÊíèãà/ URSS.

40. 2005á. Êîìïàêòíàÿ ìàòåìàòè÷åñêàÿ ìàêðîìîäåëü òåõíèêî-ýêîíîìè÷åñêîãî è äåìîãðàôè÷åñêîãî ðàçâèòèÿ Ìèð-Ñèñòåìû (1-1973 ãã.). Â: Êîðîòàåâ, À. Â., Ìàëêîâ, Ñ. Þ. (ðåä.), Èñòîðèÿ è ñèíåðãåòèêà: Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå ñîöèàëüíîé äèíàìèêè. Ì.: URSS/ÊîìÊíèãà, ñ. 6-48.

41. 2007. Çàêîíû èñòîðèè: Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå ðàçâèòèÿ Ìèð-Ñèñòåìû. Äåìîãðàôèÿ, ýêîíîìèêà, êóëüòóðà. Ì.: ÊîìÊíèãà/URSS.

42. 2008. Êîìïàêòíàÿ ìàòåìàòè÷åñêàÿ ìîäåëü ýêîíîìè÷åñêîãî è äåìîãðàôè÷åñêîãî ðàçâèòèÿ Ìèð-Ñèñòåìû (1-1973 ãã.). Ýêîíîìèêà è ìàòåìàòè÷åñêèå ìåòîäû 44(4): 90-101.

43. Êîðîòàåâ, À. Â., Ìàðêîâ, À. Â. 2006. Ìåõàíèçì ãèïåðáîëè÷åñêîãî ðîñòà â áèîëîãè÷åñêèõ è ñîöèàëüíûõ ñèñòåìàõ. Ôèëîñîôñêèå íàóêè 11: 138-141.

44. Êîðîòàåâ, À. Â., Õàëòóðèíà, Ä. À. 2009. Ñîâðåìåííûå òåíäåíöèè ìèðîâîãî ðàçâèòèÿ. Ì.: ËÈÁÐÎÊÎÌ/URSS.

45. Êîðîòàåâ, À. Â., Õàëòóðèíà, Ä. À., Ìàëêîâ, À. Ñ., Áîæåâîëüíîâ, Þ. Â., Êîáçåâà, Ñ. Â., Çèíüêèíà, Þ. Â. 2010. Çàêîíû èñòîðèè. Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå è ïðîãíîçèðîâàíèå ìèðîâîãî è ðåãèîíàëüíîãî ðàçâèòèÿ. 3-å èçä., èñïð. è äîï. Ì.: ËÊÈ/URSS.

46. Êðèíã, Ä., Äóðäà, Ä. Ä. 2004. Äåíü, êîãäà ìèð áûë ñîææåí. Â ìèðå íàóêè 3: 56-63.

47. Êðûëîâ, Î. Â. 1999. Áóäåò ëè êîíåö íàóêè. Ðîññèéñêèé õèìè÷åñêèé æóðíàë 43(6): 96-106.

48. 2002. Äèíàìèêà ðàçâèòèÿ õèìè÷åñêîé íàóêè. Ðîññèéñêèé õèìè÷åñêèé æóðíàë 46(3): 96-99.

49. 2007. Ñîâðåìåííàÿ íàóêà: áëèçêèé êîíåö èëè çàâåðøåíèå î÷åðåäíîãî ýòàïà? Ðîññèéñêèé õèìè÷åñêèé æóðíàë 51(3): 71-78.

50. Êýððîëë, Ð. 1992. Ïàëåîíòîëîãèÿ è ýâîëþöèÿ ïîçâîíî÷íûõ. Ò. 1. Ì.: Ìèð.

51. 1993à. Ïàëåîíòîëîãèÿ è ýâîëþöèÿ ïîçâîíî÷íûõ. Ò. 2. Ì.: Ìèð.

52. 1993á. Ïàëåîíòîëîãèÿ è ýâîëþöèÿ ïîçâîíî÷íûõ. Ò. 3. Ì.: Ìèð.

53. Ëîïàòèí, Í. Â. 1983. Äðåâíèå áèîñôåðû è ãåíåçèñ ãîðþ÷èõ èñêîïàåìûõ. Ïàëåîíòîëîãèÿ è ýâîëþöèÿ áèîñôåðû. Òðóäû XXV ñåññèè Âñåñîþçíîãî ïàëåîíòîëîãè÷åñêîãî îáùåñòâà. Âûï. 6. Ë.: ÀÍ ÑÑÑÐ, ñ. 63-67.

54. Ìàðêîâ, A. Â., Àíèñèìîâ, Â. À., Êîðîòàåâ, À. Â. 2010. Âçàèìîñâÿçü ðàçìåðà ãåíîìà è ñëîæíîñòè îðãàíèçìà â ýâîëþöèîííîì ðÿäó îò ïðîêàðèîò ê ìëåêîïèòàþùèì. Ïàëåîíòîëîãè÷åñêèé æóðíàë 4: 3-14.

55. 2011. Ãèïåðýêñïîíåíöèàëüíûé ðîñò ìèíèìàëüíîãî ðàçìåðà ãåíîìà â ýâîëþöèîííîì ðÿäó îò ïðîêàðèîò ê ìëåêîïèòàþùèì. Â: Ãðèíèí, Ë. Å., Èëüèí, È. Â., Êîðîòàåâ, À. Â., Ìàðêîâ, À. Â. (ðåä.), Ýâîëþöèÿ: äèñêóññèîííûå àñïåêòû ãëîáàëüíûõ ýâîëþöèîííûõ ïðîöåññîâ. Ì.: ËÊÈ, ñ. 113-154.

56. Ìàðêîâ, À. Â., Êîðîòàåâ, À. Â. 2007. Äèíàìèêà ðàçíîîáðàçèÿ ôàíåðîçîéñêèõ ìîðñêèõ æèâîòíûõ ñîîòâåòñòâóåò ìîäåëè ãèïåðáîëè÷åñêîãî ðîñòà. Æóðíàë îáùåé áèîëîãèè 68(1): 3-18.

57. 2008à. Ãèïåðáîëè÷åñêèé ðîñò ðàçíîîáðàçèÿ ìîðñêîé è êîíòèíåíòàëüíîé áèîò ôàíåðîçîÿ è ýâîëþöèÿ ñîîáùåñòâ. Æóðíàë îáùåé áèîëîãèè 69(3): 175-194.

58. 2008á. Ãèïåðáîëè÷åñêèé ðîñò áèîðàçíîîáðàçèÿ â ôàíåðîçîå îáúÿñíÿåòñÿ ðîñòîì ñëîæíîñòè è óñòîé÷èâîñòè ñîîáùåñòâ. Ñîâðåìåííûå ïðîáëåìû áèîëîãè÷åñêîé ýâîëþöèè: ìàòåðèàëû III Ìåæäóíàðîäíîé êîíôåðåíöèè ê 130-ëåòèþ ñî äíÿ ðîæäåíèÿ Í. È. Âàâèëîâà è 110-ëåòèþ ñî äíÿ îñíîâàíèÿ Ãîñóäàðñòâåííîãî Äàðâèíîâñêîãî ìóçåÿ. Ì.: Èçä-âî Ãîñóäàðñòâåííîãî Äàðâèíîâñêîãî ìóçåÿ, ñ. 278-323.

59. 2008â. Äèíàìèêà ðàçíîîáðàçèÿ ìîðñêîé è êîíòèíåíòàëüíîé áèîòû ôàíåðîçîÿ ñîîòâåòñòâóåò ìîäåëè ãèïåðáîëè÷åñêîãî ðîñòà. Ãåîáèîñôåðíûå ñîáûòèÿ è èñòîðèÿ îðãàíè÷åñêîãî ìèðà: òåçèñû äîêëàäîâ LIV ñåññèè Ïàëåîíòîëîãè÷åñêîãî îáùåñòâà ïðè ÐÀÍ (7-11 àïðåëÿ 2008 ã., Ñàíêò-Ïå- òåðáóðã). ÑÏá.: Ïàëåîíòîëîãè÷åñêîå îáùåñòâî ÐÀÍ, ñ. 108?110.

60. 2009à. Ãèïåðáîëè÷åñêèé ðîñò â æèâîé ïðèðîäå è îáùåñòâå. Ì.: ËÈÁÐÎÊÎÌ/URSS.

61. 2009á. Î ïðè÷èíàõ óñêîðåíèÿ ðîñòà ðàçíîîáðàçèÿ ìîðñêîé è êîíòèíåíòàëüíîé áèîòû ôàíåðîçîÿ: ôàêòû è ìîäåëè. Ñèíòåòè÷åñêàÿ òåîðèÿ ýâîëþöèè: ñîñòîÿíèå, ïðîáëåìû, ïåðñïåêòèâû: ïðîãðàììà è òåçèñû äîêëàäîâ Ìåæäóíàðîäíîé íàó÷íîé êîíôåðåíöèè, ïîñâÿùåííîé 200-ëåòèþ ñî äíÿ ðîæäåíèÿ ×àðëçà Äàðâèíà è 150-ëåòèþ âûõîäà êíèãè «Ïðîèñõîæäåíèå âèäîâ ïóòåì åñòåñòâåííîãî îòáîðà èëè ñîõðàíåíèå áëàãîïðèÿòñòâóåìûõ ïîðîä â áîðüáå çà æèçíü». Ëóãàíñê: Ýëòîí-2, ñ. 37-39.

62. Ìóðàòîâ, Ì. Â., Âàõðàìååâ, Â. À. 1974. Ìåçîçîéñêàÿ ãðóïïà (ýðà). Â: Ïðîõîðîâ, À. Ì. (ãë. ðåä.), Áîëüøàÿ ñîâåòñêàÿ ýíöèêëîïåäèÿ. Ò. 16. Ì.: Ñîâåòñêàÿ ýíöèêëîïåäèÿ, ñ. 6-8.

63. Íàçàðåòÿí, À. Ï. 2004. Öèâèëèçàöèîííûå êðèçèñû â êîíòåêñòå Óíèâåðñàëüíîé èñòîðèè. 2-å èçä., ïåðåðàá. è äîï. Ì.: Ïåð ñå. 2005. Çíàåò ëè èñòîðèÿ ñîñëàãàòåëüíîå íàêëîíåíèå? (Ìåãàèñòîðè÷åñêèé âçãëÿä íà àëüòåðíàòèâíûå ìîäåëè). Ôèëîñîôñêèå íàóêè 2: 7-18.

64. 2009. Ñìûñëîîáðàçîâàíèå êàê ãëîáàëüíàÿ ïðîáëåìà ñîâðåìåííîñòè: ñèíåðãåòè÷åñêèé âçãëÿä. Âîïðîñû ôèëîñîôèè 5: 3-19.

65. 2013. Ñåðåäèíà XXI âåêà: çàãàäêà ñèíãóëÿðíîñòè. Ôèëîñîôñêèå íàóêè 9: 15-24.

66. 2014. «Íàöèîíàëüíàÿ èäåÿ»: Ðîññèÿ â ãëîáàëüíûõ ñöåíàðèÿõ XXI âåêà. Èñòîðè÷åñêàÿ ïñèõîëîãèÿ è ñîöèîëîãèÿ èñòîðèè 7(1): 75-91.

67. 2015à. Ìåãàèñòîðèÿ è åå «çàãàäî÷íàÿ ñèíãóëÿðíîñòü». Âåñòíèê Ðîññèéñêîé àêàäåìèè íàóê 85(8): 755-764.

68. 2015á. Íåëèíåéíîå áóäóùåå. Ìåãàèñòîðèÿ, ñèíåðãåòèêà, êóëüòóðíàÿ àíòðîïîëîãèÿ è ïñèõîëîãèÿ â ãëîáàëüíîì ïðîãíîçèðîâàíèè. Ì.: Àðãàìàê-Ìåäèà.

69. Ïàíîâ, À. Ä. 2004. Àâòîìîäåëüíûé àòòðàêòîð ñîöèàëüíî-áèîëîãè÷åñêîé ýâîëþöèè íà Çåìëå è ãèïîòåçà ñàìîñîãëàñîâàííîãî ãàëàêòè÷åñêîãî ïðîèñõîæäåíèÿ æèçíè. Áþëëåòåíü Íàó÷íî-êóëüòóðíîãî öåíòðà SETI Àêàäåìèè êîñìîíàâòèêè èì. Ê. Ý. Öèîëêîâñêîãî 7(24): 4-21.

70. 2005. Ñèíãóëÿðíàÿ òî÷êà èñòîðèè. Îáùåñòâåííûå íàóêè è ñîâðåìåííîñòü 1: 122-137.

71. 2006. Ñèíãóëÿðíîñòü Äüÿêîíîâà. Â: Ãðèíèí, Ë. Å., Êîðîòàåâ, À. Â., Ìàëêîâ, Ñ. Þ. (îòâ. ðåä.), Èñòîðèÿ è Ìàòåìàòèêà: Ïðîáëåìû ïåðèîäèçàöèè èñòîðè÷åñêèõ ìàêðîïðîöåññîâ. Ì.: ÊîìÊíèãà, ñ. 31-37.

72. 2008. Óíèâåðñàëüíàÿ ýâîëþöèÿ è ïðîáëåìà ïîèñêà âíåçåìíîãî ðàçóìà (SETI). Ì.: ËÊÈ/URSS.

73. 2009. Íàóêà êàê ÿâëåíèå ýâîëþöèè. Â: Ãðèíèí, Ë. Å., Ìàðêîâ, À. Â., Êîðîòàåâ, À. Â. (ðåä.), Ýâîëþöèÿ: êîñìè÷åñêàÿ, áèîëîãè÷åñêàÿ, ñîöèàëüíàÿ. Ì.: ËÈÁÐÎÊÎÌ, ñ. 99-127.

74. 2013. Ìàêðîýâîëþöèÿ è íàóêà. Íàóêîâåä÷åñêèå èññëåäîâàíèÿ. Ì.: ÐÀÍ, Èí-ò èíôîðìàöèè ïî îáùåñòâåííûì íàóêàì, ñ. 215-256.

75. Ïîäëàçîâ, À. Â. 2000. Òåîðåòè÷åñêàÿ äåìîãðàôèÿ êàê îñíîâà ìàòåìàòè÷åñêîé èñòîðèè. Ì.: ÈÏÌ ÐÀÍ.

76. 2001. Îñíîâíîå óðàâíåíèå òåîðåòè÷åñêîé äåìîãðàôèè è ìîäåëü ãëîáàëüíîãî äåìîãðàôè÷åñêîãî ïåðåõîäà. Ì.: ÈÏÌ ÐÀÍ.

77. 2002. Òåîðåòè÷åñêàÿ äåìîãðàôèÿ. Ìîäåëè ðîñòà íàðîäîíàñåëåíèÿ è ãëîáàëüíîãî äåìîãðàôè÷åñêîãî ïåðåõîäà. Â: Ìàëèíåöêèé, Ã. Ã., Êóðäþìîâ, Ñ. Ï., Íîâîå â ñèíåðãåòèêå. Âçãëÿä â òðåòüå òûñÿ÷åëåòèå. Ì.: Íàóêà, ñ. 324-345.

78. 2017. Òåîðèÿ ãëîáàëüíîãî äåìîãðàôè÷åñêîãî ïðîöåññà. Âåñòíèê Ðîññèéñêîé àêàäåìèè íàóê 6: 520-531.

79. Ðîçàíîâ, À. Þ. 1986. ×òî ïðîèçîøëî 600 ìèëëèîíîâ ëåò íàçàä. Ì.: Íàóêà.

80. 2003. Èñêîïàåìûå áàêòåðèè, ñåäèìåíòîãåíåç è ðàííèå ñòàäèè ýâîëþöèè áèîñôåðû. Ïàëåîíòîëîãè÷åñêèé æóðíàë 6: 41-49.

81. Ðîçàíîâ, À. Þ., Çàâàðçèí, Ã. À. 1997. Áàêòåðèàëüíàÿ ïàëåîíòîëîãèÿ. Âåñòíèê ÐÀÍ 67(3): 241-245.

82. Ðîìàí÷óê, À. À., Ìåäâåäåâà, Î. Â. 2009. Ãëîáàëüíûé äåìîãðàôè÷åñêèé ïåðåõîä è åãî áèîëîãè÷åñêèå ïàðàëëåëè. Â: Ãðèíèí, Ë. Å., Ìàðêîâ, À. Â., Êîðîòàåâ, À. Â. (ðåä.), Ýâîëþöèÿ: êîñìè÷åñêàÿ, áèîëîãè÷åñêàÿ, ñîöèàëüíàÿ. Ì.: ËÈÁÐÎÊÎÌ, ñ. 244-269.

83. Ñàäîâíè÷èé, Â. À., Àêàåâ, À. À., Êîðîòàåâ, À. Â., Ìàëêîâ, Ñ. Þ. 2014. Êîìïëåêñíîå ìîäåëèðîâàíèå è ïðîãíîçèðîâàíèå ðàçâèòèÿ ñòðàí ÁÐÈÊÑ â êîíòåêñòå ìèðîâîé äèíàìèêè. Ì.: Íàóêà.

84. Ôåäîíêèí, Ì. À. 2003. Ñóæåíèå ãåîõèìè÷åñêîãî áàçèñà æèçíè è ýâêàðèîòèçàöèÿ áèîñôåðû: ïðè÷èííàÿ ñâÿçü. Ïàëåîíòîëîãè÷åñêèé æóðíàë 6: 33-37.

85. Ôîóëè, Ð. 1990. Åùå îäèí íåïîâòîðèìûé âèä. Ýêîëîãè÷åñêèå àñïåêòû ýâîëþöèè ÷åëîâåêà. Ì.: Ìèð.

86. Öèðåëü, Ñ. Â. 2018. Big History è Singularity êàê ìåòàôîðû, ãèïîòåçû è ïðîãíîç (ðóêîïèñü ñòàòüè).

87. Øàíöåð, Å. Â. 1973. Êàéíîçîéñêàÿ ãðóïïà (ýðà). Â: Ïðîõîðîâ, À. Ì. (ãë. ðåä.), Áîëüøàÿ ñîâåòñêàÿ ýíöèêëîïåäèÿ. Ò. 11. Ì.: Ñîâåòñêàÿ ýíöèêëîïåäèÿ.

88. ßñïåðñ, Ê. 1991. Ñìûñë è íàçíà÷åíèå èñòîðèè. Ì.: Ïîëèòèçäàò.

A. H. 1975. Mesozoic era. In The New Encyclopedia Britannica. 15th ed. Vol. 11. Chicago: Encyclopedia Britannica, Inc., pp. 1013-1017.

A. P. 1975. Cenozoic era. In The New Encyclopedia Britannica. 15th ed. Vol. 3. Chicago: Encyclopedia Britannica, pp. 1079-1083.

89. Adams, H. 1969 [1909]. The Rule of Phase Applied to History. In Adams, H., The Degradation of the Democratic Dogma. New York, NY: Harper & Row, pp. 267-311.

90. Alvarez, L. W., Alvarez, W., Asaro, F., Michel, H. V. 1980. Extraterrestrial Cause for the Cretaceous-Tertiary Extinction. Science 208(4448): 1095-1108.

91. Barrow, J. D., Silk, J. 1980. The Structure of the Early Universe. Scientific American 242(4): 118-128.

92. Begun, D. R. 2003. Planet of the Apes. Scientific American 289(2): 64-73.

93. Burenhult, G. (ed.) 1993. The First Humans: Human Origins and History to 10,000 BC. San Francisco: Harper.

94. Callaghan, V., Miller, J., Yampolskiy, R., Armstrong, S. 2017. Technological Singularity. Dordrecht: Springer.

95. Carroll, R. L. 1988. Vertebrate Paleontology and Evolution. New York: W. H. Freeman and Co.

96. Cervellati, M., Sunde, U., Zimmermann, K. F. 2017. Demographic Dynamics and Long-run Development: Insights for the Secular Stagnation Debate. Journal of Population Economics 30(2): 401-432.

97. Christian, D. 2008. Big History: The Big Bang, Life on Earth, and the Rise of Humanity. Chantilly, VA: The Teaching Company.

98. Eden, A. H., Moor, J. H., Søraker, J. H., Steinhart, E. (eds.) 2012. Singularity Hypotheses: A Scientific and Philosophical Assessment. Berlin: Springer.

99. Foerster, H. von, Mora, P. M., Amiot, L. W. 1960. Doomsday: Friday, 13 November, AD 2026. Science 132(3436): 1291-1295.

100. Gordon, R. J. 2012. Is US Economic Growth Over? Faltering Innovation Confronts the Six Headwinds. Cambridge, MA: National Bureau of Economic Research.

101. Grinin, L., Korotayev, A. 2015. Great Divergence and Great Convergence. A Global Perspective. New York: Springer.

102. Grinin, L. E, Markov, A. V., Korotayev, A. V.

103. 2013. On Similarities between Biological and Social Evolutionary Mechanisms: Mathematical Modeling. Cliodynamics 4(2): 185-228.

104. 2014. Mathematical Modeling of Biological and Social Evolutionary Macrotrends. In Grinin, L., Korotayev, A. (eds.), History & Mathematics: Trends and Cycles. Volgograd: `Uchitel' Publishing House, pp. 9-48.

105. 2015. Modeling of Biological and Social Phases of Big History. In Grinin, L. E., Korotayev, A. V. (eds.), Evolution: From Big Bang to Nanorobots. Volgograd: `Uchitel' Publishing House, pp. 111-150.

106. Heidmann, J. 1989. Cosmic Odyssey. Cambridge: Cambridge University Press.

107. Hoerner, S. J. von. 1975. Population Explosion and Interstellar Expansion. Journal of the British Interplanetary Society 28: 691-712.

108. Huebner, J. 2005. A Possible Declining Trend for Worldwide Innovation. Technological Forecasting and Social Change 72(8): 980-986.

109. J. B. W. 1975. Paleozoic Era, Upper. The New Encyclopedia Britannica. 15th ed. Vol. 13. Chicago: Encyclopedia Britannica, pp. 921-930.

110. Jaspers, K. 1955. Von Ursprung und Ziel der Geschichte. Frankfurt/Main: Fisher Bucherei.

111. Johansen, A., Sornette, D. 2001. Finite-Time Singularity in the Dynamics of the World Population and Economic Indices. Physica A 294(3-4): 465-502.

112. Johanson, D., Edgar, B. 1996. From Lucy to Language. New York: Simon and Schuster.

113. Jones, C. 2018. Aging, Secular Stagnation and the Business Cycle. N. p.: International Monetary Fund.

114. Jones, S. (ed.). 1994. The Cambridge Encyclopedia of Human Evolution. Cambridge: Cambridge University Press.

115. Kapitza, S. P.

116. 1996. The Phenomenological Theory of World Population Growth. Physics-uspekhi 39(1): 57-71.

117. 2003. The Statistical Theory of Global Population Growth. Formal Descriptions of Developing Systems. Dordrecht: Springer, pp. 11-35.

118. 2006. Global Population Blow-up and After. Hamburg: Global Marshall Plan Initiative.

119. 2010. On the Theory of Global Population Growth. Physics-uspekhi 53(12): 1287-1296.

120. Khaltourina, D. A., Korotayev, A. V. 2007. A Modified Version of a Compact Mathematical Model of the World System Economic, Demographic, and Cultural Development. In Dmitriev, M. G., Petrov, A. P. (eds.), Mathematical Modeling of Social and Economic Dynamics. Moscow: RUDN, pp. 274-277.

121. Khaltourina, D., Korotayev, A., Malkov, A. 2006. A Compact Macromodel of the World System Demographic and Economic Growth, 1-1973 CE. Cybernetics and Systems 1: 330-335.

122. Korotayev, A. 2005. A Compact Macromodel of World System Evolution. Journal of World-Systems Research 11(1): 79-93.

123. 2006a. The World System History Periodization and Mathematical Models of Socio-historical Processes. In Grinin, L., de Munck, V. C., Korotayev, A. (eds.), History & Mathematics: Analyzing and Modeling Global Development. Moscow: KomKniga/URSS, pp. 39-98.

124. 2006b. The World System Urbanization Dynamics: A Quantitative Analysis. In Turchin, P., Grinin, L., de Munck, V. C., Korotayev, A. (eds.), History & Mathematics: Historical Dynamics and Development of Complex Societies. Moscow: KomKniga/URSS, pp. 44-62.

125. 2007a. Compact Mathematical Models of World System Development, and How they Can Help Us to Clarify Our Understanding of Globalization Processes. In Modelski, G., Devezas, T., Thompson, W. R., Globalization as Evolutionary Process: Modeling Global Change. London: Routledge, pp. 133-160.

126. 2007b. Secular Ñycles and Millennial Trends: A Mathematical Model. In Dmitriev, M., Petrov, A. (eds.), Mathematical Modeling of Social and Econo- mic Dynamics. Moscow: RUDN, pp. 118-125.

127. 2008. Globalization and Mathematical Modeling of Global Development. In Grinin, L. E., Beliaev, D. D., Korotayev, A. V. (eds.), Hierarchy and Power in The History Of Civilizations: Political Aspects Of Modernity. Moscow: LIBROCOM/URSS, pp. 225-240.

128. 2009. Compact Mathematical Models of the World System Development and Their Applicability to the Development of Local Solutions in Third World Countries. In Sheffield, J. (ed.), Systemic Development: Local Solutions in a Global Environment. Litchfield Park, AZ: ISCE Publishing, pp. 103?116.

129. 2012. Globalization and Mathematical Modeling of Global Development. Globalistics and Globalization Studies 1: 148-158.

130. 2013. Globalization and Mathematical Modeling of Global Evolution. In Grinin, L. E., Korotayev, À. V. (eds.), Evolution: Development within Big History, Evolutionary and World-System Paradigms. Volgograd: `Uchitel' Publishing House, pp. 69-83.

131. 2018. The 21st Century Singularity and its Big History Implications: A Re-analysis. Journal of Big History II(3): 71-118.

132. Korotayev, A., Goldstone, J., Zinkina, J. 2015. Phases of Global Demographic Transition Correlate with Phases of the Great Divergence and Great Convergence. Technological Forecasting and Social Change 95: 163-169.

133. Korotayev, A., Khaltourina, D. 2006. Introduction to Social Macrodynamics: Secular Cycles and Millennial Trends in Africa. Moscow: KomKniga/URSS.

134. Korotayev, A., Malkov, A., Khaltourina, D. 2006a. Introduction to Social Macrodynamics: Compact Macromodels of the World System Growth. Moscow: KomKniga/URSS.

135. 2006b. Introduction to Social Macrodynamics: Secular Cycles and Millennial Trends. Moscow: KomKniga/URSS.

136. Korotayev, A., Malkov, A. 2016. A Compact Mathematical Model of the World System Economic and Demographic Growth, 1 CE - 1973 CE. International Journal of Mathematical Models and Methods in Applied Sciences 10: 200-209.

137. Korotayev, A., Malkov, S. 2012. Mathematical Models of the World-system Development. Routledge Handbook of World-Systems Analysis. London: Routledge, pp. 158-161.

138. Korotayev, A. V., Markov, A. V. 2014. Mathematical Modeling of Biological and Social Phases of Big History. In Grinin, L., Baker, D., Quaedackers, E., Korotayev, A. (eds.), Teaching & Researching Big History: Exploring a New Scholarly Field. Volgograd: `Uchitel' Publishing House, pp. 188-219.

139. 2015. Mathematical Modeling of Biological and Social Phases of Big History. Globalistics and Globalization Studies 4: 319-343.

140. Korotayev, A., Zinkina, J. 2017. Systemic Boundary Issues in the Light of Mathematical Modeling of World-system Evolution. Journal of Globalization Studies 8(1): 78-96.

141. Kremer, M. 1993. Population Growth and Technological Change: One Million B.C. to 1990. The Quarterly Journal of Economics 108: 681-716.

142. Kurzweil, R. 2001. The Law of Accelerating Returns. KurzweilAI.net 3-7-2001. URL: http://www.kurzweilai.net/articles/art0134.html?printable=1.

143. 2005. The Singularity is Near: When Humans Transcend Biology. New York: Viking Penguin.

144. LePoire, D. J. 2005. Application of Logistic Analysis to the History of Physics. Technological Forecasting and Social Change 72(4): 471-479.

145. 2009. Exploration of Connections between Energy Use and Leadership Transitions. In Thompson, W. R. (ed.), Systemic Transitions: Past, Present, and Future. New York: Palgrave Macmillan, pp. 205-220.

146. 2013. Potential Economic and Energy Indicators of Inflection in Complexity. In Grinin, L. E., Korotayev, À. V. (eds.), Evolution: Development within Big History, Evolutionary and World-System Paradigms. Volgograd: `Uchitel' Publishing House, pp. 108-118.

147. 2015. Interpreting “Big History” as Complex Adaptive System Dynamics with Nested Logistic Transitions in Energy Flow and Organization. Emergence: Complexity and Organization 17(1): 107-121.

148. 2016. Exploring Temporal Patterns in Big History Dynamics. KronoScope 16(2): 229-249.

149. Livi-Bacci, M. 2012. A Concise History of World Population. Chichester: Wiley-Blackwell.

150. Maddison, A. 2001. Monitoring the World Economy: A Millennial Perspective. Paris: OECD.

151. 2007. Contours of the World Economy, 1-2030. Oxford: Oxford University Press.

152. Markov, A. V., Anisimov, V. A., Korotayev, A. V. 2010. Relationship between Genome Size and Organismal Complexity in the Lineage Leading from Prokaryotes to Mammals. Paleontological Journal 44(4): 363-373.

153. Markov, A. V., Korotayev, A. V. 2007. Phanerozoic Marine Biodiversity Follows a Hyperbolic Trend. Palaeoworld 16(4): 311-318.

154. 2008. Hyperbolic Growth of Marine and Continental Biodiversity through the Phanerozoic and Community Evolution. Zhurnal obshchei biologii 69(3): 175-194.

155. McEvedy, C., Jones, R. 1978. Atlas of World Population History. New York, NY: Facts on File.

156. Modis, T. 2002. Forecasting the Growth of Complexity and Change. Technological Forecasting and Social Change 69(4): 377-404.

157. 2003. The Limits of Complexity and Change. The Futurist 37(3): 26-32.

158. 2005. Discussion of Huebner Article Comments by Theodore Modis. Technological Forecasting and Social Change 72: 987-988.

159. 2012. Why the Singularity cannot Happen. In Eden, A. H., Moor, J. H., Søraker, J. H., Steinhart, E. (eds.), Singularity Hypothesis: A Scientific and Philosophical Assessment. Berlin: Springer, pp. 311-346.

160. Nazaretyan, A. P. 2003. Power and Wisdom: Toward a History of Social Behavior. Journal of the Theory of Social Behaviour 33(4): 405-425.

161. 2005. Big (Universal) History Paradigm: Versions and Approaches. Social Evolution & History 4(1): 61-86.

162. 2015. Megahistory and its Mysterious Singularity. Herald of the Russian Academy of Sciences 85(4): 352-361.

163. 2016. Non-linear Futures: The “Mysterious Singularity” in View of Mega-History. Between Past Orthodoxies and the Future of Globalization. Contemporary Philosophical Problems. Boston: Brill-Rodopi, pp. 171-191.

164. 2017. Mega-History and the Twenty-first Century Singularity Puzzle. Social Evolution & History 16(1): 31-52.

165. 2018. The Polyfurcation Century: Does the Evolution on Earth have a Cosmological Relevance? Journal of Big History 2(1): 27-41.

166. Orgel, L. E. 1998. The Origin of Life - How Long did it Take? Origins of Life and Evolution of the Biosphere 28: 91-96.

167. Panov, A. D. 2005. Scaling Law of the Biological Evolution and the Hypothesis of the Self-consistent Galaxy Origin of Life. Advances in Space Research 36(2): 220-225.

168. 2011. Post-singular Evolution and Post-singular Civilizations. In Grinin, L. E., Korotayev, À. V., Rodrigue, B. H. (eds.), Evolution: A Big History Perspective. Volgograd: `Uchitel' Publishing House, 2011, pp. 212-231.

169. 2017. Singularity of Evolution and Post-Singular Development. In Grinin, L., Korotayev, A., Rodrigue, B. (eds.), From Big Bang to Galactic Civilizations. A Big History Anthology. Vol. III. The Ways that Big History Works: Cosmos, Life, Society and Our Future. Delhi: Primus Books, pp. 370-402.

170. Piketty, T. 2014. Capital in the Twenty-First Century. Cambridge, MA: The Belknap Press of Harvard University Press.

171. Podlazov, A. V. 2017. A Theory of the Global Demographic Process. Herald of the Russian Academy of Sciences 87(3): 256-266.

172. Popoviæ, M. 2018. Technological Progress, Globalization, and Secular Stagnation. Journal of Central Banking Theory and Practice 7(1): 59-100.

173. Ranj, B. 2016. Google's Chief Futurist Ray Kurzweil Thinks We Could Start Living Forever by 2029. Business Insider April 20. URL: http://www.techinsider.io/googles-chief-futurist-thinks-we-could-start-living-forever-by-2029-2016-4.

174. Sagan, S. 1989. The Dragons of Eden: Speculations on the Evolution of Human Intelligence. New York: Ballantine Books.

175. Shanahan, M. 2015. The Technological Singularity. Cambridge, MA: MIT Press.

176. Schopf, J. W. (ed.). 1991. Major Events in the History of Life. Boston: Jones and Bartlett Publishers.

177. Snooks, G. D. 1996. The Dynamic Society: Exploring the Sources of Global Change. London: Routledge.

178. 2005. Big History or Big Theory? Uncovering the Laws of Life. Social Evolution & History 4(1): 160-188.

179. Summers, L. H. 2016. The Age of Secular Stagnation: What it is and What to Do about it. Foreign Affairs 95(2): 2-9.

180. T. K. 1975. Paleozoic Era, Lower. The New Encyclopedia Britannica. 15th ed. Vol. 13. Chicago: Encyclopedia Britannica, Inc., pp. 916-920.

181. Taagepera, R. 1976. Crisis around 2005 AD? A Technology-population Interaction Model. General Systems 21: 137-138.

182. Taylor, G., Tyers, R. 2017. Secular Stagnation: Determinants and Consequences for Australia. Economic Record 93(303): 615-650.

183. Teulings, C., Baldwin, R. (eds.). 2014. Secular Stagnation: Facts, Causes, and Cures. London: CEPR.

184. Tsirel, S. V. 2004. On the Possible Reasons for the Hyperexponential Growth of the Earth Population. In Dmitriev, M., Petrov, A. (eds.), Mathematical Modeling of Social and Economic Dynamics. Moscow: Russian State Social University, pp. 367-369.

185. UN Population Division. 2018. United Nations Population Division Database. New York: United Nations. URL: http://www.un.org/esa/population.

186. Wood, B. 1992. Origin and Evolution of the Genus Homo. Nature 355: 783-790.

Ðàçìåùåíî íà Allbest.ru

...