96:
their meanings. At the same time we adopt the principle: not to employ any of the other expressions of the discipline under consideration, unless its meaning has first been determined with the help of primitive terms and of such expressions of the discipline whose meanings have been explained previously. The sentence which determines the meaning of a term in this way is called a DEFINITION,...
112:
To a non-mathematician it often comes as a surprise that it is impossible to define explicitly all the terms which are used. This is not a superficial problem but lies at the root of all knowledge; it is necessary to begin somewhere, and to make progress one must clearly state those elements and
95:
When we set out to construct a given discipline, we distinguish, first of all, a certain small group of expressions of this discipline that seem to us to be immediately understandable; the expressions in this group we call PRIMITIVE TERMS or UNDEFINED TERMS, and we employ them without explaining
296:
of relations are primitive. (p 25) As for denotation of objects by description, Russell acknowledges that a primitive notion is involved. (p 27) The thesis of
Russell’s book is "Pure mathematics uses only a few notions, and these are logical constants." (p xxi)
202:
in Paris in 1900. The notions themselves may not necessarily need to be stated; Susan Haack (1978) writes, "A set of axioms is sometimes said to give an implicit definition of its primitive terms."
51:. Some authors refer to the latter as "defining" primitive notions by one or more axioms, but this can be misleading. Formal theories cannot dispense with primitive notions, under pain of
137:
writes: 'definition' of 'set' is less a definition than an attempt at explication of something which is being given the status of a primitive, undefined, term. As evidence, she quotes
169:
are primitive notions. Since Peano arithmetic is useful in regards to properties of the numbers, the objects that the primitive notions represent may not strictly matter.
39:
is a concept that is not defined in terms of previously-defined concepts. It is often motivated informally, usually by an appeal to
199:
141:: "A set is formed by the grouping together of single objects into a whole. A set is a plurality thought of as a unit."
601:
227:
637:
341:
253:
622:
511:
632:
593:
316:
248:
209:
77:
549:
122:
The necessity for primitive notions is illustrated in several axiomatic foundations in mathematics:
575:
491:
311:
544:
503:
419:
285:
273:
265:
105:
76:
are some primitive notions. Instead of attempting to define them, their interplay is ruled (in
40:
396:
378:
306:
277:
541:
Mechanising
Hilbert's Foundations of Geometry in Isabelle (see ref 16, re: Hilbert's take)
360:
More generally, in a formal system, rules restrict the use of primitive notions. See e.g.
8:
627:
567:
445:
336:
101:
331:
223:
205:
162:
597:
281:
130:
80:) by axioms like "For every two points there exists a line that contains them both".
563:
326:
293:
257:
195:
194:: The primitive notions will depend upon the set of axioms chosen for the system.
191:
177:
158:
144:
64:
52:
44:
392:
321:
138:
56:
474:
denotes the set of points, of lines, and the "contains" relation, respectively.
412:
269:
185:
113:
relations which are undefined and those properties which are taken for granted.
616:
483:
88:
32:
280:. (pp 18,9) Regarding relations, Russell takes as primitive notions the
173:
20:
522:
261:
134:
126:
28:
361:
151:
is a primitive notion. To assert that it exists would be an implicit
148:
488:
Introduction to Logic and the
Methodology of the Deductive Sciences
181:
16:
Concept that is not defined in terms of previously defined concepts
373:
272:
as a primitive notion. To establish sets, he also establishes
152:
48:
24:
276:
as primitive, as well as the phrase "such that" as used in
166:
566:(1900) "Logical introduction to any deductive theory" in
47:, relations between primitive notions are restricted by
176:: Typically, primitive notions are: real number, two
91:
explained the role of primitive notions as follows:
100:An inevitable regress to primitive notions in the
292:. Furthermore, logical products of relations and
614:
260:used the following notions: for class-calculus (
572:A Source Book in Mathematical Logic, 1879–1931
543:(Master's thesis). University of Edinburgh.
538:
214:point, line, plane, congruence, betweeness
548:
376:(300 B.C.) still gave definitions in his
133:is an example of a primitive notion. As
242:
62:For example, in contemporary geometry,
615:
382:, like "A line is breadthless length".
587:
200:International Congress of Philosophy
13:
14:
649:
188:, numbers 0 and 1, ordering <.
391:This axiom can be formalized in
364:for a non-logical formal system.
198:discussed this selection at the
43:and everyday experience. In an
581:
557:
532:
516:
497:
477:
385:
367:
354:
1:
347:
342:Natural semantic metalanguage
254:The Principles of Mathematics
527:The Philosophy of Set Theory
7:
512:University of Toronto Press
300:
117:
10:
654:
594:Cambridge University Press
317:Foundations of mathematics
230:the primitive notions are
212:the primitive notions are
83:
249:philosophy of mathematics
576:Harvard University Press
508:Foundations of Geometry
492:Oxford University Press
312:Foundations of geometry
274:propositional functions
504:Gilbert de B. Robinson
286:complementary relation
210:Hilbert's axiom system
106:Gilbert de B. Robinson
78:Hilbert's axiom system
638:Mathematical concepts
588:Haack, Susan (1978),
129:: The concept of the
590:Philosophy of Logics
307:Axiomatic set theory
278:set builder notation
243:Russell's primitives
228:Peano's axiom system
623:Philosophy of logic
568:Jean van Heijenoort
539:Phil Scott (2008).
337:Notion (philosophy)
102:theory of knowledge
332:Mathematical logic
224:Euclidean geometry
206:Euclidean geometry
163:successor function
633:Concepts in logic
510:, 4th ed., p. 8,
294:relative products
282:converse relation
192:Axiomatic systems
178:binary operations
104:was explained by
645:
607:
606:
585:
579:
564:Alessandro Padoa
561:
555:
554:
552:
536:
530:
520:
514:
501:
495:
481:
475:
389:
383:
371:
365:
358:
327:Logical constant
258:Bertrand Russell
196:Alessandro Padoa
159:Peano arithmetic
145:Naive set theory
53:infinite regress
45:axiomatic theory
37:primitive notion
653:
652:
648:
647:
646:
644:
643:
642:
613:
612:
611:
610:
604:
596:, p. 245,
586:
582:
562:
558:
550:10.1.1.218.9262
537:
533:
521:
517:
502:
498:
482:
478:
461:
443:
411:
404:
393:predicate logic
390:
386:
372:
368:
359:
355:
350:
322:Logical atomism
303:
247:In his book on
245:
165:and the number
139:Felix Hausdorff
120:
86:
57:regress problem
17:
12:
11:
5:
651:
641:
640:
635:
630:
625:
609:
608:
602:
580:
556:
531:
515:
496:
476:
459:
441:
409:
402:
384:
366:
352:
351:
349:
346:
345:
344:
339:
334:
329:
324:
319:
314:
309:
302:
299:
270:set membership
244:
241:
240:
239:
232:point, segment
221:
203:
189:
186:multiplication
172:Arithmetic of
170:
156:
142:
119:
116:
115:
114:
98:
97:
85:
82:
33:formal systems
15:
9:
6:
4:
3:
2:
650:
639:
636:
634:
631:
629:
626:
624:
621:
620:
618:
605:
603:9780521293297
599:
595:
591:
584:
577:
573:
569:
565:
560:
551:
546:
542:
535:
528:
524:
519:
513:
509:
505:
500:
493:
489:
485:
484:Alfred Tarski
480:
473:
469:
465:
458:
454:
450:
447:
440:
436:
432:
428:
424:
421:
417:
414:
408:
401:
398:
394:
388:
381:
380:
375:
370:
363:
357:
353:
343:
340:
338:
335:
333:
330:
328:
325:
323:
320:
318:
315:
313:
310:
308:
305:
304:
298:
295:
291:
287:
283:
279:
275:
271:
267:
263:
259:
256:
255:
250:
237:
233:
229:
225:
222:
219:
215:
211:
207:
204:
201:
197:
193:
190:
187:
183:
179:
175:
171:
168:
164:
160:
157:
154:
150:
146:
143:
140:
136:
132:
128:
125:
124:
123:
111:
110:
109:
107:
103:
94:
93:
92:
90:
89:Alfred Tarski
81:
79:
75:
71:
67:
66:
60:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
589:
583:
571:
559:
540:
534:
526:
518:
507:
499:
487:
479:
471:
467:
463:
456:
452:
448:
438:
434:
430:
426:
422:
415:
406:
399:
387:
377:
369:
356:
289:
252:
246:
235:
231:
217:
213:
174:real numbers
121:
99:
87:
73:
69:
63:
61:
36:
18:
288:of a given
264:), he used
21:mathematics
628:Set theory
617:Categories
523:Mary Tiles
490:, p. 118,
462:)", where
348:References
262:set theory
135:Mary Tiles
127:Set theory
29:philosophy
545:CiteSeerX
362:MU puzzle
268:, taking
266:relations
218:incidence
149:empty set
55:(per the
41:intuition
379:Elements
301:See also
226:: Under
208:: Under
182:addition
118:Examples
74:contains
570:(1967)
529:, p. 99
525:(2004)
506:(1959)
486:(1946)
84:Details
600:
578:118–23
547:
470:, and
374:Euclid
236:motion
234:, and
216:, and
161:: The
147:: The
72:, and
49:axioms
31:, and
153:axiom
65:point
25:logic
598:ISBN
395:as "
284:and
184:and
167:zero
70:line
35:, a
290:xRy
131:set
59:).
19:In
619::
592:,
574:,
466:,
444:)
429:.
418:.
251:,
180::
108::
68:,
27:,
23:,
553:.
494:.
472:C
468:L
464:P
460:2
457:x
455:,
453:y
451:(
449:C
446:∧
442:1
439:x
437:,
435:y
433:(
431:C
427:L
425:∈
423:y
420:∃
416:P
413:∈
410:2
407:x
405:,
403:1
400:x
397:∀
238:.
220:.
155:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.