1841:
14:481-540. The paper contains presentation of point-free system of geometry originating from
Whitehead's ideas and based on Lesniewski's mereology. It also briefly discusses the relation between point-free and point-based systems of geometry. Basic properties of mereological structures are given as
1044:↔ ∀z. Unlike the case with inclusion spaces, connection theory enables defining "non-tangential" inclusion, a total order that enables the construction of abstractive classes. Gerla and Miranda (2008) argue that only thus can mereotopology unambiguously define a
228:
347:
419:
756:
574:
1247:
938:
1474:
1161:
660:
1374:
1310:
802:
Intuitively, an abstractive class defines a geometrical entity whose dimensionality is less than that of the inclusion space. For example, if the inclusion space is the
492:
1952:
279:
1099:
829:
with this defect repaired. Simons did not repair this defect, instead proposing in a footnote that the reader do so as an exercise. The primitive relation of
1788:
1777:
1759:
1541:, and models of such algebras cannot distinguish connection from overlap. It is doubtful whether either fact is faithful to Whitehead's intent.
2044:
1806:
1490:
Following the verbal description of each axiom is the identifier of the corresponding axiom in Casati and Varzi (1999). Their system
1986:
174:
1866:. Routledge. Chpt. 10, on "prototopology," discusses Whitehead's systems and is strongly influenced by the unpublished writings of
293:
2103:
2098:
1938:
2069:
1022:
365:
1609:
are, but for numbering, those of Def. 2.1 in Gerla and
Miranda (2008) (see also Gerla (1995)). The identifiers of the form
817:
Inclusion-based point-free geometry (henceforth "point-free geometry") is essentially an axiomatization of Simons's system
1945:
980:
notion of "contact" between two regions, resulting in a primitive "connection relation" between events. Connection theory
976:
A different approach was proposed in
Whitehead (1929), one inspired by De Laguna (1922). Whitehead took as primitive the
2108:
985:
885:
93:
1867:
1538:
92:
Whitehead did not set out his theories in a manner that would satisfy present-day canons of formality. The two formal
2076:
2062:
1915:
696:
164:." Assuming that equality, denoted by the infix operator "=", is part of the background logic, the binary relation
1592:
See
Kneebone (1963), chpt. 13.5, for a gentle introduction to Whitehead's theory. Also see Lucas (2000), chpt. 10.
520:
2037:
1859:
1187:
1826:
112:. No axiom requires more than three quantified variables; hence a translation of first-order theories into
1961:
1616:, included in the verbal description of each axiom, refer to the corresponding axiom in Simons (1987: 83).
1015:
902:
878:
422:
117:
105:
101:
77:
65:
1393:
1125:
606:
1977:
581:
96:
described in this entry were devised by others in order to clarify and refine
Whitehead's theories. The
1914:. Cambridge Univ. Press. 2004 paperback, Prometheus Books. Being the 1919 Tarner Lectures delivered at
834:
37:
1996:
1923:
1503:
1258:
589:
503:
350:
242:
97:
2113:
2030:
1560:
1056:
795:
by inclusion. Moreover, there does not exist a region included in all of the regions included in
282:
1326:
1274:
444:
2123:
2118:
858:
255:
109:
1072:
1045:
825:
formalizes a theory of
Whitehead whose axioms are not made explicit. Point-free geometry is
807:
113:
46:
42:
1910:
2016:
1887:
811:
803:
129:
1171:
137:
1834:
2092:
1555:
1005:
238:
81:
54:
1821:
1534:
762:
977:
965:
866:
792:
499:
20:
1894:: 423-454. Translated as Hurley, P.J., 1979, "The relational theory of space,"
1729:
Grzegorczyk (1960) proposed a similar theory, whose motivation was primarily
1817:
1550:
1109:
997:
585:
145:
73:
50:
1690:
below are, but for numbering, those of Def. 3.1 in Gerla and
Miranda (2008)
1708:
Presumably this is Casati and Varzi's (1999) "Internal Part" predicate, IP
1930:
1730:
1502:, and is essentially due to Clarke (1981). Any mereotopology can be made
996:
is a proper fragment of the theories proposed by Clarke, who noted their
940:
Hence inclusion-based point-free geometry would be a proper extension of
430:
Given any two regions, there exists a region that includes both of them.
69:
32:
28:
988:
that distills the first 12 of
Whitehead's 31 assumptions into 6 axioms,
837:. The theory of Whitehead (1919) has a single primitive binary relation
1845:
Grzegorczyk, A., 1960, "Axiomatizability of geometry without points,"
1019:
1830:, Frankfurt / Lancaster, ontos verlag, Process Thought X1 & X2.
1798:
De Laguna, T., 1922, "Point, line and surface as sets of solids,"
1316:
Given any two regions, there is a region connected to both of them.
1004:, feature both inclusion and topological primitives, are called
1934:
1742:
For an advanced and detailed discussion of systems related to
223:{\displaystyle x<y\leftrightarrow (x\leq y\land x\not =y).}
1818:
Inclusion and
Connection in Whitehead's Point-free Geometry
1771:
Parts and places: the structures of spatial representation
342:{\displaystyle (x\leq z\land z\leq y)\rightarrow x\leq y.}
1903:
An
Enquiry Concerning the Principles of Natural Knowledge
1811:
Handbook of incidence geometry: buildings and foundations
168:, denoted by the infix operator "<", is defined as:
414:{\displaystyle (x\leq y\land y\leq x)\rightarrow x=y.}
1396:
1329:
1277:
1190:
1128:
1075:
905:
699:
609:
523:
447:
368:
296:
258:
177:
1854:
Mathematical Logic and the Foundation of Mathematics
80:" between events. Whitehead's purposes were as much
2054:
2008:
1968:
877:establishes that inclusion, unlike Proper Part, is
666:Proper Parts Principle. If all the proper parts of
104:variables in this entry should be taken as tacitly
1468:
1368:
1304:
1241:
1155:
1093:
932:
750:
654:
568:
486:
413:
341:
273:
222:
1720:. This definition combines their (4.8) and (3.1).
1380:All regions have at least two unconnected parts.
806:, then the corresponding abstractive classes are
1890:, 1916, "La Theorie Relationiste de l'Espace,"
1835:Full development of Tarski's geometry of solids
1778:A calculus of individuals based on 'connection'
1510:, without risking paradox or triviality. Hence
124:Inclusion-based point-free geometry (mereology)
1946:
1833:Gruszczynski R., and Pietruszczak A., 2008, "
751:{\displaystyle \forall z\rightarrow x\leq y.}
116:is possible. Each set of axioms has but four
100:for both theories consists of "regions." All
8:
884:Point-free geometry is closely related to a
569:{\displaystyle x<y\rightarrow \exists z.}
64:Point-free geometry was first formulated by
1873:Roeper, P., 1997, "Region-Based Topology,"
1533:Biacino and Gerla (1991) showed that every
1953:
1939:
1931:
1395:
1328:
1276:
1242:{\displaystyle \forall z\rightarrow x=y.}
1189:
1127:
1074:
904:
698:
608:
522:
446:
367:
295:
257:
176:
1827:Handbook of Whiteheadian Process Thought
592:has neither an upper nor a lower bound.
1905:. Cambridge Univ. Press. 2nd ed., 1925.
1576:
1253:All regions have proper parts, so that
144:relation that is a standard feature in
76:, but of "events" and of an "extension
1809:" in Buekenhout, F., Kantor, W. eds.,
1526:, suggested by chapter 2 of part 4 of
1018:, binary "connection," denoted by the
108:; hence all axioms should be taken as
1892:Revue de Metaphysique et de Morale 23
1793:Notre Dame Journal of Formal Logic 26
1782:Notre Dame Journal of Formal Logic 22
140:"≤", which corresponds to the binary
7:
1769:Casati, R., and Varzi, A. C., 1999.
933:{\displaystyle x\leq y\lor y\leq x.}
1469:{\displaystyle \exists y\exists z.}
1156:{\displaystyle Cxy\rightarrow Cyx.}
779:. Given some inclusion space S, an
655:{\displaystyle \exists y\exists z.}
148:theories. The intuitive meaning of
49:are set out below, one grounded in
1764:Notre Dame Journal of Formal Logic
1625:Gerla and Miranda 2008: Def. 4.1).
1448:
1403:
1397:
1330:
1278:
1191:
700:
616:
610:
536:
448:
14:
1875:Journal of Philosophical Logic 26
1816:--------, and Miranda A., 2008, "
1758:Biacino L., and Gerla G., 1991, "
972:Connection theory (mereotopology)
16:Geometric theory based on regions
1514:extends the atomless variant of
84:as scientific and mathematical.
1864:Conceptual Roots of Mathematics
1000:character. Theories that, like
2070:Contemporary Whitehead Studies
1896:Philosophy Research Archives 5
1665:. Dover reprint, 1979. P. 423.
1460:
1442:
1430:
1424:
1412:
1409:
1360:
1336:
1296:
1284:
1224:
1221:
1209:
1197:
1138:
733:
730:
718:
706:
646:
622:
560:
542:
533:
478:
454:
396:
393:
369:
324:
321:
297:
214:
190:
187:
1:
1800:The Journal of Philosophy 19
865:asserts that Proper Part is
2140:
2077:Whitehead Research Project
2063:Center for Process Studies
1882:Parts: A Study in Ontology
1839:Bulletin of Symbolic Logic
1824:and Will Desmond, (eds.),
1674:In chapter 2 of part 4 of
1369:{\displaystyle \exists z.}
1305:{\displaystyle \exists y.}
487:{\displaystyle \exists z.}
128:The fundamental primitive
1813:. North-Holland: 1015-31.
960:}), were it not that the
899:, and the totality axiom
861:of Proper Part. Simons's
2038:Tensor product of graphs
1652:Kneebone (1963), p. 346.
1537:of Clarke's theory is a
588:do not exist. Hence the
274:{\displaystyle x\leq x.}
1921:--------, 1979 (1929).
1776:Clarke, Bowman, 1981, "
1518:by means of the axioms
118:existential quantifiers
2104:History of mathematics
2099:Alfred North Whitehead
1962:Alfred North Whitehead
1856:. Dover reprint, 2001.
1789:Individuals and Points
1583:Whitehead (1919, 1920)
1470:
1370:
1306:
1243:
1157:
1095:
1094:{\displaystyle \ Cxx.}
1036:can now be defined as
934:
869:and so corresponds to
752:
656:
570:
488:
415:
343:
275:
224:
106:universally quantified
66:Alfred North Whitehead
2045:Theory of gravitation
1978:Principia Mathematica
1911:The Concept of Nature
1884:. Oxford Univ. Press.
1760:Connection Structures
1471:
1371:
1307:
1244:
1158:
1096:
935:
787:of regions such that
753:
657:
571:
489:
416:
344:
276:
225:
68:, not as a theory of
1852:Kneebone, G., 1963.
1807:Pointless Geometries
1746:, see Roeper (1997).
1663:Set Theory and Logic
1661:Stoll, R. R., 1963.
1496:strong mereotopology
1394:
1327:
1275:
1188:
1126:
1073:
903:
835:strict partial order
697:
670:are proper parts of
607:
521:
445:
366:
294:
256:
175:
94:first-order theories
2109:Mathematical axioms
2024:Point-free geometry
1997:Process and Reality
1924:Process and Reality
1676:Process and Reality
1528:Process and Reality
891:, whose axioms are
98:domain of discourse
25:point-free geometry
2031:Process philosophy
1880:Simons, P., 1987.
1805:Gerla, G., 1995, "
1634:Simons (1987: 83)
1561:Pointless topology
1466:
1366:
1302:
1239:
1153:
1091:
1014:has one primitive
986:first-order theory
964:relation "≤" is a
930:
886:dense linear order
849: <
833:is Proper Part, a
748:
652:
566:
484:
411:
339:
271:
220:
110:universal closures
2086:
2085:
1078:
781:abstractive class
136:, denoted by the
59:connection theory
47:axiomatic systems
2131:
2079:
2072:
2065:
2047:
2040:
2033:
2026:
2019:
2001:
1989:
1982:
1955:
1948:
1941:
1932:
1908:--------, 1920.
1901:--------, 1919.
1747:
1740:
1734:
1727:
1721:
1712:↔ (x≤y)∧(C
1706:
1700:
1697:
1691:
1684:
1678:
1672:
1666:
1659:
1653:
1650:
1644:
1643:Whitehead (1919)
1641:
1635:
1632:
1626:
1623:
1617:
1599:
1593:
1590:
1584:
1581:
1485:connection space
1475:
1473:
1472:
1467:
1375:
1373:
1372:
1367:
1311:
1309:
1308:
1303:
1248:
1246:
1245:
1240:
1162:
1160:
1159:
1154:
1100:
1098:
1097:
1092:
1076:
1023:predicate letter
939:
937:
936:
931:
757:
755:
754:
749:
661:
659:
658:
653:
586:universal region
575:
573:
572:
567:
493:
491:
490:
485:
420:
418:
417:
412:
348:
346:
345:
340:
280:
278:
277:
272:
239:partially orders
233:The axioms are:
229:
227:
226:
221:
114:relation algebra
31:whose primitive
2139:
2138:
2134:
2133:
2132:
2130:
2129:
2128:
2089:
2088:
2087:
2082:
2075:
2068:
2061:
2050:
2043:
2036:
2029:
2022:
2017:Inert knowledge
2015:
2004:
1994:
1985:
1975:
1964:
1959:
1916:Trinity College
1888:Whitehead, A.N.
1787:------, 1985, "
1755:
1750:
1741:
1737:
1728:
1724:
1707:
1703:
1698:
1694:
1685:
1681:
1673:
1669:
1660:
1656:
1651:
1647:
1642:
1638:
1633:
1629:
1624:
1620:
1600:
1596:
1591:
1587:
1582:
1578:
1574:
1569:
1547:
1539:Boolean algebra
1392:
1391:
1325:
1324:
1273:
1272:
1186:
1185:
1124:
1123:
1071:
1070:
1032:is included in
1006:mereotopologies
974:
901:
900:
804:Euclidean plane
793:totally ordered
771:inclusion space
695:
694:
678:is included in
605:
604:
519:
518:
443:
442:
364:
363:
292:
291:
254:
253:
173:
172:
130:binary relation
126:
90:
53:, the other in
17:
12:
11:
5:
2137:
2135:
2127:
2126:
2121:
2116:
2111:
2106:
2101:
2091:
2090:
2084:
2083:
2081:
2080:
2073:
2066:
2058:
2056:
2052:
2051:
2049:
2048:
2041:
2034:
2027:
2020:
2012:
2010:
2006:
2005:
2003:
2002:
1992:
1991:
1990:
1972:
1970:
1966:
1965:
1960:
1958:
1957:
1950:
1943:
1935:
1929:
1928:
1919:
1906:
1899:
1885:
1878:
1871:
1857:
1850:
1843:
1831:
1814:
1803:
1796:
1785:
1774:
1767:
1754:
1751:
1749:
1748:
1735:
1722:
1701:
1692:
1679:
1667:
1654:
1645:
1636:
1627:
1618:
1594:
1585:
1575:
1573:
1570:
1568:
1565:
1564:
1563:
1558:
1553:
1546:
1543:
1498:) consists of
1477:
1476:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1385:
1384:
1377:
1376:
1365:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1341:
1338:
1335:
1332:
1318:
1317:
1313:
1312:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1266:
1265:
1250:
1249:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1179:
1178:
1164:
1163:
1152:
1149:
1146:
1143:
1140:
1137:
1134:
1131:
1117:
1116:
1102:
1101:
1090:
1087:
1084:
1081:
1064:
1063:
973:
970:
929:
926:
923:
920:
917:
914:
911:
908:
759:
758:
747:
744:
741:
738:
735:
732:
729:
726:
723:
720:
717:
714:
711:
708:
705:
702:
688:
687:
663:
662:
651:
648:
645:
642:
639:
636:
633:
630:
627:
624:
621:
618:
615:
612:
598:
597:
582:atomic regions
577:
576:
565:
562:
559:
556:
553:
550:
547:
544:
541:
538:
535:
532:
529:
526:
512:
511:
500:densely orders
495:
494:
483:
480:
477:
474:
471:
468:
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459:
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453:
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436:
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201:
198:
195:
192:
189:
186:
183:
180:
138:infix operator
125:
122:
89:
88:Formalizations
86:
15:
13:
10:
9:
6:
4:
3:
2:
2136:
2125:
2122:
2120:
2117:
2115:
2112:
2110:
2107:
2105:
2102:
2100:
2097:
2096:
2094:
2078:
2074:
2071:
2067:
2064:
2060:
2059:
2057:
2053:
2046:
2042:
2039:
2035:
2032:
2028:
2025:
2021:
2018:
2014:
2013:
2011:
2007:
1999:
1998:
1993:
1988:
1984:
1983:
1980:
1979:
1974:
1973:
1971:
1967:
1963:
1956:
1951:
1949:
1944:
1942:
1937:
1936:
1933:
1927:. Free Press.
1926:
1925:
1920:
1917:
1913:
1912:
1907:
1904:
1900:
1897:
1893:
1889:
1886:
1883:
1879:
1876:
1872:
1869:
1868:David Bostock
1865:
1861:
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1757:
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1752:
1745:
1739:
1736:
1732:
1726:
1723:
1719:
1715:
1711:
1705:
1702:
1699:Clarke (1981)
1696:
1693:
1689:
1683:
1680:
1677:
1671:
1668:
1664:
1658:
1655:
1649:
1646:
1640:
1637:
1631:
1628:
1622:
1619:
1615:
1614:
1608:
1604:
1598:
1595:
1589:
1586:
1580:
1577:
1571:
1566:
1562:
1559:
1557:
1556:Mereotopology
1554:
1552:
1549:
1548:
1544:
1542:
1540:
1536:
1531:
1529:
1525:
1521:
1517:
1513:
1509:
1505:
1501:
1497:
1493:
1488:
1486:
1482:
1463:
1457:
1454:
1451:
1445:
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1433:
1427:
1421:
1418:
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1400:
1390:
1387:
1386:
1383:
1379:
1378:
1363:
1357:
1354:
1351:
1348:
1345:
1342:
1339:
1333:
1323:
1320:
1319:
1315:
1314:
1299:
1293:
1290:
1287:
1281:
1271:
1268:
1267:
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1260:
1256:
1252:
1251:
1236:
1233:
1230:
1227:
1218:
1215:
1212:
1206:
1203:
1200:
1194:
1184:
1181:
1180:
1177:
1173:
1169:
1166:
1165:
1150:
1147:
1144:
1141:
1135:
1132:
1129:
1122:
1119:
1118:
1115:
1111:
1107:
1104:
1103:
1088:
1085:
1082:
1079:
1069:
1066:
1065:
1062:
1058:
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879:antisymmetric
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1822:Michel Weber
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1773:. MIT Press.
1770:
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141:
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41:rather than
36:
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1981:(1910–1913)
1847:Synthese 12
1766:32: 242-47.
1731:topological
1686:The axioms
1601:The axioms
1479:A model of
1172:extensional
978:topological
966:total order
867:irreflexive
841:defined as
783:is a class
166:Proper Part
160:is part of
33:ontological
21:mathematics
2093:Categories
1898:: 712-741.
1877:: 251-309.
1849:: 228-235.
1572:References
777:Definition
351:transitive
237:Inclusion
35:notion is
2114:Mereology
1802:: 449-61.
1784:: 204-18.
1551:Mereology
1449:¬
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1279:∃
1225:→
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1057:reflexive
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134:inclusion
74:spacetime
51:mereology
2124:Topology
2119:Ontology
2009:Concepts
1987:glossary
1862:, 2000.
1795:: 61-75.
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1545:See also
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1567:Notes
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1500:C1-C3
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