ON WHAT THERE IS IN PHILOSOPHY OF MATHEMATICS
SOBRE LO QUE HAY EN FILOSOFÍA DE LAS MATEMÁTICAS
ICE B. RISTESKI 7RURQWR&DQDGD (PDLO LFH#VFLHQWLVWFRP
CARLOS E. GARCÍA D. 8QLYHUVLGDGGH&DOGDV8QLYHUVLGDGGH0DQL]DOHV (PDLO FDUORVJDUFLDBG#XFDOGDVHGXFR
RECIBIDO EL 30 DE NOVIEMBRE DE 2007 Y APROBADO EL 14 DE MARZO DE 2008
ABSTRACT RESUMEN
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This assertion of the potential UHDOL]DWLRQRUFRQVWUXFWLELOLW\FDQEHJLYHQWKHIRUPmn p. 165 Ice B. Risteski & Carlos E. García D. 7KHDSRGLFWLFDOWUXWKRIVXFKDQDVVHUWLRQOLHVH[DFWO\LQWKHSRWHQWLDO UHDOL]DWLRQRIQRUPDWLYHO\GLUHFWHGFDOFXODWLRQV7KLVLVJXDUDQWHHG E\WKHUHDOL]DWLRQRIWKHSUHVFULEHGFRQVWUXFWLRQWKLVPHDQVE\WKH UHDOL]DWLRQRIWKHV\QWKHVLVZKLFKLVQRWGHVFULSWLYHEXWLVQRUPDWLYHO\ GLUHFWHGi.e.DFFRUGLQJWRWKHROGYRFDEXODU\LWLVa priori:HFDQVD\ WKDWE\WKHPHDQLQJRIWKHDVVHUWLRQ LQRXUFDVHWKLVLVWKHSRWHQWLDO UHDOL]DWLRQRIDFRQVWUXFWLRQ LWVWUXWKLVGHWHUPLQHG 0HDQZKLOHZHGRQRWDELGHE\WKHZRUGVsimple proposition:HFRPELQH WKHSURSRVLWLRQE\PHDQVRIFRQQHFWLYHVQHJDWHLWSDUWLFXODUL]HLWHWF ,QZKDWOLHVDQGZKDWLVWKHWUXWKYDOXHRIWKHLQGLYLGXDOSURSRVLWLRQV RIVXFKDULFKDVVHUWLYHVSHHFK":KDWDUHWKHPHDQLQJVRIWKHVH SURSRVLWLRQV" :KLOHLQRXUVSHHFKZHDUHVDWLVÀHGZLWKDVLPSOHFRPELQDWLRQRI WKHSURSRVLWLRQVE\PHDQVRIWKHVRFDOOHGSURSRVLWLRQDOFRQQHFWLYHV LQFOXGLQJWKHQHJDWLRQ WKHPHDQLQJVDQGWKHFULWHULDRIWUXWKRIWKHVR REWDLQHGSURSRVLWLRQVDVDZKROHDUHXQGHUVWRRGZLWKWKHWZRYDOXHG FODVVLFDOFRQFHSWVRIWUXHDQGIDOVHDVSUHGLFDWHVZKLFKDUHDVVLJQHGWR WKHSURSRVLWLRQLQFRQFRUGDQFHZLWKWKHVRFDOOHGWUXWKWDEOHVRIWKH values of the individual propositions. 7KLVFRPSUHKHQVLRQLVLQFRQFRUGDQFHZLWKWKHFRPSUHKHQVLRQRIWKH VLPSOHSURSRVLWLRQVVRWKDWLWFDQEHH[WHQGHGLQDQDSSURSULDWHZD\ WRWKHXQLYHUVDOSURSRVLWLRQVDVVKRZQE\DQH[DPSOHRIWKHUHFXUVLYH DULWKPHWLFHODERUDWHGLQ E\6NROHP>@+HKDVVKRZQWKDWLQ UHFXUVLYHDULWKPHWLFDVDZKROHLWLVSRVVLEOHWRVLPXODWHVSHHFKE\ SURSRVLWLRQDOFRQQHFWLYHVRIFRPELQHGSURSRVLWLRQVWKURXJKSXUHO\ FDOFXODWHGVSHHFKRIVLPSOHSURSRVLWLRQV,QWKLVZD\DSXUHO\DULWKPHWLF ORJLFZKLFKLVGHSULYHGRIWKHGLUHFWXVHRIWKHH[LVWHQWLDODQGXQLYHUVDO TXDQWLÀHUVLVIRXQGHG 7KHIROORZLQJTXHVWLRQVUHPDLQRSHQ :KDWLVWKHPHDQLQJRIWKH XQLYHUVDODQGSDUWLFXODUSURSRVLWLRQV",QZKDWOLHVDQGZKDWLVWKHWUXWK YDOXHRIWKHVHSURSRVLWLRQV":KDWGRWKHSURSRVLWLRQV¶DWOHDVWRQHRGG QXPEHULVSHUIHFW·RU¶HDFKRGGQXPEHULVLPSHUIHFW·PHDQ",IE\DQ RQWRORJLFDODFFHQWZHXQGHUVWDQGWKHVHSURSRVLWLRQVDVDGHVFULSWLRQRI WKHDFWXDOVHWRIWKHQXPEHUVWKHQWKHPHDQLQJVRIWKHVHSURSRVLWLRQV DUHGHVFULSWLRQVRIVWDWHVRIWKHREMHFWVLQWKLVVHW7KHFRQGLWLRQRIWUXWK GHWHUPLQHGE\WKLVPHDQLQJFRUUHVSRQGVWRWKHGHVFULSWLRQRIWKHVWDWH RIWKHREMHFWV7KHXQFHUWDLQW\ZKLFKDULVHVIURPWKLVXQGHUVWDQGLQJLV 166 'LVFXVLRQHV)LORVyÀFDV$xR 1(QHUR²-XQLRSS ON WHAT THERE IS IN PHILOSOPHY OF MATHEMATICS DOUHDG\GHVLJQDWHG0RUHRYHUWKLVFRQGLWLRQRIWUXWKRIVRPHSURSRVLWLRQ GRHVQRWPDNHVLPXOWDQHRXVO\SRVVLEOHDFULWHULRQIRULWVWUXWK ,IWKHSRVLWLYHQHVVRIWKHSXUHPDWKHPDWLFVLVLPSRUWDQWIRUXVKHUH ZHDOUHDG\NQRZZKHUHLWVVRXUFHOLHV6RPHZRUNLVEHLQJGRQHRQ WKHIRXQGDWLRQRIDPDWKHPDWLFVZLWKDVVHUWLYHVSHHFKEXWIUHHRI RQWRORJLFDODFFHQW7KLVPDWKHPDWLFVi.e.PDWKHPDWLFDOORJLFLQVRPH VHQVHDOUHDG\SUHFHGHVWKHRQHDOUHDG\GHVFULEHGIUHHO\VSHDNLQJLQ WKHVDPHVHQVHLQZKLFKVSHHFKSUHFHGHVHQXPHUDWLRQ 7KXVZHVHHKRZFRQVWUXFWLYLVPGHWHUPLQHVSXUHPDWKHPDWLFVDQG ZKHUHLQFRQFRUGDQFHZLWKWKLVGHWHUPLQDWLRQÀQGVLWVSRVLWLYHQHVV :HDOVRVHHWKDWWKHFRQVWUXFWLYLVWLFDOO\XQGHUVWRRGSXUHPDWKHPDWLFV GRHVQRWSUHVXSSRVHDORJLF But the constructivistic reconstruction of the entire classical mathematics LVQRWSRVVLEOH&RQVWUXFWLYLVPÀQGVSRVLWLYHQHVVLQWKHSXUHPDWKHPDWLFV ZKLFKGRHVQRWRYHUODSZLWKWKHVWDQGDUGFODVVLFDOPDWKHPDWLFV7R PDQ\SHRSOHWKLVVHHPVWREHDWRRELJVDFULÀFH The term formalismLVXVHGLQWZRVHQVHV4XLWHRIWHQLWPHDQVD QRPLQDOLVWLFDWWLWXGHWRZDUGVWKHH[LVWHQFHRIWKHPDWKHPDWLFDOREMHFWV +HUHZHZLOOXVHLWLQDQRWKHUVHQVH)RUXV DQGIRURWKHUVOLNHYRQ 1HXPDQQ>@ IRUPDOLVPZLOOEHDV\QRQ\PIRU+LOEHUW·VPDWKHPDWLFDO foundation of mathematics since this foundation requires in one of its VWHSVDPHWKRGLFDOFRPSUHKHQVLRQRIWKHQRPLQDOLVWLFIRUPDOLVP:H VKDOOH[SODLQZKDWIRXQGDWLRQZHDUHWDONLQJDERXW Hilbert understands the constructivistic criticism of the classical PDWKHPDWLFVDVQRQSRVLWLYHPDWKHPDWLFV6WLOOZKDWKHFRQVLGHUV SRVLWLYHPDWKHPDWLFVLVMXVWDSDUWRIWKHFRQVWUXFWLYLVWLFDOO\XQGHUVWRRG SXUHPDWKHPDWLFVWKHVRFDOOHGÀQLWLVWSDUW%XWXQOLNHWKHFRQVWUXFWLYLVWV RULQWXLWLRQLVWVKHGRHVQRWZDQWWRUHMHFWWKDWSDUWRIWKHFODVVLFDO PDWKHPDWLFVZKLFKE\WKHFRQVWUXFWLYLVWLFUHFRQVWUXFWLRQLVQRWMXVWLÀHG DVDSRVLWLYHVFLHQFH+LVSURJUDPLVDSURJUDPRIMXVWLÀFDWLRQRIWKH FRQVWUXFWLYLVWLFDOO\UHMHFWHGQRQSRVLWLYHPDWKHPDWLFVE\WKHVRFDOOHG method of ideal elements. 7KH H[SODQDWLRQ RI WKH EDVLF LGHDV RI +LOEHUW·V SURJUDP FDQ EH IRXQG ZLWK PRUH GHWDLOV LQ 3UDZLW]>@ 167 Ice B. Risteski & Carlos E. García D. 7KHÀUVWSUREOHPRIWKDWSURJUDPLVWKHIRUPDOL]DWLRQRIWKHVWDQGDUG classical mathematics. The main results of that problem have been DOUHDG\REWDLQHGE\WKHORJLFLVWVWUDQVIRUPLQJPDWKHPDWLFVLQWRD V\VWHPRISURSRVLWLRQVRIWKHIRUPp implies q,ZKHUHp represents DFRQMXQFWLRQRISRVWXODWHGPDWKHPDWLFDODVVHUWLRQVDQGq are their TXDVLDOJHEUDLFDOO\GHGXFHGFRQVHTXHQFHV%XWZKLOHORJLFLVPVHHNV DORJLFDOLQWHUSUHWDWLRQRIWKHSURSRVLWLRQVRIWKDWV\VWHP+LOEHUW DFFHSWVRQO\DÀQLWHLQWHUSUHWDWLRQ,QVLPSOHUZRUGV+LOEHUWJLYHVD PHDQLQJRQO\WRWKHVHSURSRVLWLRQVRIWKHV\VWHPVZKLFKKDYHDVLPSOH FRPSXWDWLRQDOVHQVHIRULQVWDQFHWKHSURSRVLWLRQVRIWKHIRUPmm p$OORWKHUSURSRVLWLRQVZKLFKFDQQRWEHVLPSO\H[SODLQHGDUHQRW LQWHUSUHWHGDWDOOQRUJLYHQDQ\PHDQLQJE\+LOEHUW6WULFWO\VSHDNLQJ IRUKLPDQGKLVVFKRROWKH\KDYHQRVHQVHDWDOO7KH\DUHV\PEROV ZKLFKPHDQQRWKLQJi.e.WKH\DUHXQGHUVWRRGDVQRPLQDOLVWLFVLJQV 7KHXQLQWHUSUHWHGSURSRVLWLRQVRIWKHV\VWHPVDUHLGHDOHOHPHQWVZKLFK FDQEHIRUPDOO\ORJLFDOFRQVHTXHQFHVRIWKHLQWHUSUHWHGSURSRVLWLRQV IURPZKLFKDJDLQRWKHUXQLQWHUSUHWHGSURSRVLWLRQVFDQEHGHGXFHG IRUPDOO\ORJLFDOO\DQGDWODVWDOVRLQWHUSUHWHGRQHV)RUPDOO\WKHVHDUH PHPEHURIHTXDOULJKWVRIWKHGHGXFWLYHV\VWHP7KHGHGXFWLYHV\VWHP DVDZKROHLVMXVWLÀHGE\DPDWKHPDWLFDOSURRIRIFRQVLVWHQF\ZKLFK PXVWSURYLGHWKHIROORZLQJ each interpreted proposition which has a simple ÀQLWLVWPHDQLQJDQGZKLFKFDQEHSURYHGLQWKHIUDPHZRUNRIWKHV\VWHPHYHQ by using ideal elements, according to its meaning is true. 7KHLGHDOHOHPHQWVZKLFKKDYHQRVHQVHDQGZKLFKVKRXOGEHJLYHQ QRPHDQLQJVHUYHWRSURYHPRUHTXLFNO\DQGPRUHVLPSO\WKHÀQLWLVW propositions. ,QWKLVSURJUDPDPRPHQWZKLFKGLVWLQJXLVKHVLWIURPWKHQRPLQDOLVWLFDOO\ XQGHUVWRRGIRUPDOLVPVKRXOGEHQRWHG7KLVLVWKHPRPHQWRIFKRLFH RUEHWWHUWRVD\FRQVWUXFWLRQRIWKHV\VWHPZKLFKLVMXVWLÀHG2QWKHRQH KDQGIRUWKDWMXVWLÀFDWLRQWREHDWWKHVDPHWLPHDGHVLUHGMXVWLÀFDWLRQ RIWKHFODVVLFDOPDWKHPDWLFVWKHV\VWHPLWVHOIVKRXOGEHDIRUPDOL]DWLRQ RIWKHFODVVLFDOPDWKHPDWLFVDQGWKLVFDQEHWKHFDVHRQO\LIWKH FODVVLFDOPDWKHPDWLFVLVLWVLQWHUSUHWDWLRQ7KXVDWWKHYHU\PRPHQW RIIRUPDOL]DWLRQWKHV\VWHPPXVWEHLQWHUSUHWHGLIWKHMXVWLÀFDWLRQRI the classical mathematics is important and not the formal result about WKHIRUPDOV\VWHP2QWKHRWKHUKDQGWKHYHU\MXVWLÀFDWLRQLVIRXQG E\DPHWKRGLFDOIRUJHWWLQJRIWKDWLQWHUSUHWDWLRQ7KHÀQDO+LOEHUW·V FRQFOXVLRQVKRXOGEHWKHIROORZLQJ We understand the classical mathematics but are not positive about some of its parts. 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