1575:, which puts tight restraints on what would otherwise appear to be a large class of manifolds. This (informal) usage reflects the opinion of the mathematical community: not only should such a theorem be strong in the descriptive sense (below) but it should also be definitive in its area. A theorem, result, or condition is further called
214:
An aesthetic term referring to the ability of an idea to provide insight into mathematics, whether by unifying disparate fields, introducing a new perspective on a single field, or by providing a technique of proof which is either particularly simple, or which captures the intuition or imagination as
2204:
A rhetorical shortcut made by authors who invite the reader to verify, at a glance, the correctness of a proposed expression or deduction. If an expression can be evaluated by straightforward application of simple techniques and without recourse to extended calculation or general theory, then it can
2301:
In a context not requiring rigor, this phrase often appears as a labor-saving device when the technical details of a complete argument would outweigh the conceptual benefits. The author gives a proof in a simple enough case that the computations are reasonable, and then indicates that "in general"
100:
to a new level of abstraction...if certain mathematicians could console themselves for a time with the hope that all these complicated structures were 'abstract nonsense'...the later papers of
Grothendieck and others showed that classical problems...which had resisted efforts of several generations
936:
of a random variable is said to be finite, this implies it is a non-negative real number, possibly zero. In some contexts though, for example in "a small but finite amplitude", zero and infinitesimals are meant to be excluded. When said of the value of a variable assuming values from the extended
281:
A result is called "folklore" if it is non-obvious and non-published, yet generally known to the specialists within a field. In many scenarios, it is unclear as to who first obtained the result, though if the result is significant, it may eventually find its way into the textbooks, whereupon it
236:
The beauty of a mathematical theory is independent of the aesthetic qualities...of the theory's rigorous expositions. Some beautiful theories may never be given a presentation which matches their beauty....Instances can also be found of mediocre theories of questionable beauty which are given
2152:
can be more easily proved with additional assumptions on the objects it concerns. If the proposition as stated follows from this modified one with a simple and minimal explanation (for example, if the remaining special cases are identical but for notation), then the modified assumptions are
2183:
Rather than finding underlying principles or patterns, this is a method where one would evaluate as many cases as needed to sufficiently prove or provide convincing evidence that the thing in question is true. Sometimes this involves evaluating every possible case (where it is also known as
622:. An arbitrary choice is one which is made unrestrictedly, or alternatively, a statement holds of an arbitrary element of a set if it holds of any element of that set. Also much in general-language use among mathematicians: "Of course, this problem can be arbitrarily complicated".
478:
Although ultimately every mathematical argument must meet a high standard of precision, mathematicians use descriptive but informal statements to discuss recurring themes or concepts with unwieldy formal statements. Note that many of the terms are completely rigorous in context.
240:
Mathematicians may say that a theorem is beautiful when they really mean to say that it is enlightening. We acknowledge a theorem's beauty when we see how the theorem 'fits' in its place....We say that a proof is beautiful when such a proof finally gives away the secret of the
382:
which seem to try to resemble as little as possible the honest functions which serve some purpose....Nay more, from the logical point of view, it is these strange functions which are the most general....to-day they are invented expressly to put at fault the reasonings of our
237:
brilliant, exciting expositions.... is rich in beautiful and insightful definitions and poor in elegant proofs.... remain clumsy and dull.... vied for one another in elegance of presentation and in cleverness of proof....In retrospect, one wonders what all the fuss was about.
365:. In many occasions, these can be and often are contradictory requirements, while in other occasions, the term is more deliberately used to refer to an object artificially constructed as a counterexample to these properties. A simple example is that from the definition of a
1933:
a theorem, the use of this expression in the statement of the theorem indicates that the conditions involved may be not yet known to the speaker, and that the intent is to collect the conditions that will be found to be needed in order for the proof of the theorem to go
287:
Many of the results mentioned in this paper should be considered "folklore" in that they merely formally state ideas that are well-known to researchers in the area, but may not be obvious to beginners and to the best of my knowledge do not appear elsewhere in
1619:
meeting stronger conditions. When used in this way, the stronger notion (such as "strong antichain") is a technical term with a precisely defined meaning; the nature of the extra conditions cannot be derived from the definition of the weaker notion (such as
398:
took on an enormous importance...as giving an incentive for the creation of new types of function whose properties departed completely from what intuitively seemed admissible. A celebrated example of such a so-called 'pathological' function...is
443:
The act of establishing a mathematical result using indisputable logic, rather than informal descriptive argument. Rigor is a cornerstone quality of mathematics, and can play an important role in preventing mathematics from degenerating into
1546: = 2.0870652... results in a sharp upper bound; the slightly smaller choice α = 2 fails to produce an upper bound, since then α = 8 < 3. In applied fields the word "tight" is often used with the same meaning.
1952:
It is often the case that two objects are shown to be equivalent in some way, and that one of them is endowed with additional structure. Using the equivalence, we may define such a structure on the second object as well, via
1793:
An obsolescent term which is used to announce to the reader an alternative method, or proof of a result. In a proof, it therefore flags a piece of reasoning that is superfluous from a logical point of view, but has some other
1145:
In a descriptive context, this phrase introduces a simple characterization of a broad class of objects, with an eye towards identifying a unifying principle. This term introduces an "elegant" description which holds for
2295:
A non-technique of proof mostly employed in lectures, where formal argument is not strictly necessary. It proceeds by omission of details or even significant ingredients, and is merely a plausibility argument.
1834:
arguments when passing from the base case to the induction step, and similarly, in the definition of sequences whose first few terms are exhibited as examples of the formula giving every term of the sequence.
3064:
2930:
in a category. Depending on authors, the term "maps" or the term "functions" may be reserved for specific kinds of functions or morphisms (e.g., function as an analytic term and map as a general term).
434:
found out in 1931, differentiable functions are colloquially speaking a rare exception among the continuous ones. Thus it can hardly be defended any-more to call non-differentiable continuous functions
458:) if it satisfies certain prevailing regularity properties, or if it conforms to mathematical intuition (even though intuition can often suggest opposite behaviors as well). In some occasions (e.g.,
930:
1885:
is algebraically closed if and only if it has no finite extensions". Often used in lists, as in "The following conditions are necessary and sufficient for a field to be algebraically closed...".
2326:
Usually applied to a claim within a larger proof when the proof of that claim can be produced routinely by any member of the audience with the necessary expertise, but is not so simple as to be
523:
having a property to mean "all except finitely many", despite the integers not admitting a measure for which this agrees with the previous usage. For example, "almost all prime numbers are
1773:
Accurately and precisely described or specified. For example, sometimes a definition relies on a choice of some object; the result of the definition must then be independent of this choice.
972:
125:(e.g., canonical map, canonical form, or canonical ordering). The same term can also be used more informally to refer to something "standard" or "classic". For example, one might say that
1163:. Grothendieck advised caution. The Platonic solids are so beautiful and so exceptional, he said, that one cannot assume such exceptional beauty will hold in more general situations.
31:: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in lectures, and sometimes in print, as informal shorthand for
2749:
2233:
A term which shortcuts around calculation the mathematician perceives to be tedious or routine, accessible to any member of the audience with the necessary expertise in the field;
1629:
In the context of limits, these terms refer to some (unspecified, even unknown) point at which a phenomenon prevails as the limit is approached. A statement such as that predicate
60:, using which one can employ arguments that establish a (possibly concrete) result without reference to any specifics of the present problem. For that reason, it is also known as
1416:
if it satisfies satisfactory continuity and differentiability properties, which are often context-dependent. These properties might include possessing a specified number of
806:
774:
710:
332:
1004:
742:
2780:
2688:
2483:
2578:
1061:
2616:
2198:
is an argument whereby a statement is not proved but instead illustrated by an example. If done well, the specific example would easily generalize to a general proof.
2177:
An informal computation omitting much rigor without sacrificing correctness. Often this computation is "proof of concept" and treats only an accessible special case.
2227:
Style of proof where claims believed by the author to be easily verifiable are labelled as 'obvious' or 'trivial', which often results in the reader being confused.
1919:): A Latin abbreviation, meaning "which was to be demonstrated", historically placed at the end of proofs, but less common currently, having been supplanted by the
1218:
1139:
set is true generically; however, it is usually not said that a property which holds merely on a dense set (which is not
Zariski open) is generic in this situation.
2308:
For proofs involving objects with multiple indices which can be solved by going to the bottom (if anyone wishes to take up the effort). Similar to diagram chasing.
2642:
2509:
1264:
1284:
if it satisfies hypotheses or properties, sometimes unspecified or even unknown, that are especially desirable in a given context. It is an informal antonym for
361:
way) if it either fails to conform to the generic behavior of such objects, fails to satisfy certain context-dependent regularity properties, or simply disobeys
2864:
2844:
2824:
2804:
2662:
2457:
2437:
1238:
874:
854:
834:
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The word fundamental is used to describe a theorem with a given area of mathematics considered to be the most central theorem of that particular area (e.g.
2279:, then the proof can proceed by tracing the path of elements of various objects around the diagram as successive morphisms are applied to it. That is, one
178:
A result is called "deep" if its proof requires concepts and methods that are advanced beyond the concepts needed to formulate the result. For example, the
2359:, or terms that do not typically appear in more specialized glossaries. For the terms used only in some specific areas of mathematics, see glossaries in
527:". There is a more complicated meaning for integers as well, discussed in the main article. Finally, this term is sometimes used synonymously with
2400:
is a map or morphism between objects that arises naturally from the definition or the construction of the objects being mapped against each other.
1542:. This is not sharp; the gap between the functions is everywhere at least 1. Among the exponential functions of the form α, setting α =
1929:
A condition on objects in the scope of the discussion, to be specified later, that will guarantee that some stated property holds for them. When
1719:
a condition if the establishment of that condition is the only impediment to the truth of the statement. Also used when working with members of
35:
arguments or precise ideas. Much of this uses common
English words, but with a specific non-obvious meaning when used in a mathematical sense.
3545:
3454:
3310:
3193:
1579:
than another one if a proof of the second can be easily obtained from the first but not conversely. An example is the sequence of theorems:
2162:
Mathematicians have several phrases to describe proofs or proof techniques. These are often used as hints for filling in tedious details.
1588:
1502:
Often, a mathematical theorem will establish constraints on the behavior of some object; for example, a function will be shown to have an
2875:
3583:
3168:
2360:
3008:
261:
A proof or a result is called "elementary" if it only involves basic concepts and methods in the field, and is to be contrasted with
227:, saying that for example, some topics could be written about elegantly although the mathematical content is not beautiful, and some
3578:
3118:
2881:
349:) which holds independently of any choices. Though long used informally, this term has found a formal definition in category theory.
974:
the meaning is simply "not infinite". When said of a set or a mathematical object whose main component is a set, it means that the
3103:
2982:
such that the restriction of the function to the neighbourhood can be considered as a set of functions from the neighbourhood to
2172:
1891:
Proofs sometimes proceed by enumerating several conditions whose satisfaction will together imply the desired theorem; thus, one
1561:, and still others which are more complicated. Each such usage attempts to invoke the physically intuitive notion of smoothness.
1557:
is a concept which mathematics has endowed with many meanings, from simple differentiability to infinite differentiability to
1105:. A property holds "generically" on a set if the set satisfies some (context-dependent) notion of density, or perhaps if its
265:
results which require more development within or outside the field. The concept of "elementary proof" is used specifically in
3108:
888:
2907:
of an object or a space is a property or number of the object or a space that remains unchanged under some transformations.
345:
Similar to "canonical" but more specific, and which makes reference to a description (almost exclusively in the context of
3093:
on an object is an additional set of objects or data attached to the object (e.g., relation, operation, metric, topology).
3328:
Jackson, Allyn (2004), "Comme AppelĂ© du NĂ©ant — As If
Summoned from the Void: The Life of Alexandre Grothendieck",
3005:
is, roughly, a map from some space or object to another that omits some information on the object or space. For example,
215:
to why the result it proves is true. In some occasions, the term "beautiful" can also be used to the same effect, though
3113:
1944:) are equally useful in practice; one introduces a theorem stating an equivalence of more than two statements with TFAE.
346:
2381:
is a set of ordered pairs; an element x is said to be related to another element y if and only if (x,y) are in the set.
1786:
draws repeatedly from a small pool of ideas, many of which are invoked through various lexical shorthands in practice.
3562:
463:
1292:
ought to satisfy a certain boundedness condition "for nice test functions," or one might state that some interesting
2142:
1874:
1580:
940:
352:
138:
There are two canonical proofs that are always used to show non-mathematicians what a mathematical proof is like:
2276:
2169:
Used to describe a geometrical proof that involves finding relationships between the various angles in a diagram.
2149:
1904:
1120:
1663:
A descriptive term referring to notation in which two objects are written one above the other; the upper one is
3002:
1962:
1837:
1267:
666:
625:
1349:
is onto" (i. e. surjective). Not translatable (without circumlocutions) to some languages other than
English.
543:, referring to the recurrence of a phenomenon as the limit is approached. A statement such as that predicate
2904:
1106:
408:
116:
20:
1355:
If, for some notion of substructure, objects are substructures of themselves (that is, the relationship is
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2108:
2011:
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1804:
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1503:
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362:
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74:
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2222:
1293:
1289:
505:
497:
459:
276:
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of objects and morphisms between them, if one wishes to prove some property of the morphisms (such as
194:
is irrational is usually known to be a deep result, because it requires a considerable development of
3369:
3240:
2951:
2527:
2040:
1728:
1523:
1429:
1399:
1178:
619:
524:
400:
179:
3423:
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is a projection and its restriction to a graph of a function, say, is also a projection. The terms â
779:
747:
683:
198:
before the proof can be established â even though the claim itself can be stated in terms of simple
3280:
3067:
2268:
2185:
1867:
1821:
1764:
1736:
1720:
1285:
540:
404:
209:
130:
126:
122:
1973:
and if we fix a particular isomorphism, then we may define an inner product on the other space by
1159:
Norbert A'Campo of the
University of Basel once asked Grothendieck about something related to the
1109:
satisfies some (context-dependent) notion of smallness. For example, a property which holds on a
315:
3507:
3316:
3138:
2256:
2210:
2019:
1966:
1783:
1754:
1712:
1623:
1536:
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982:
652:
155:
97:
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715:
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by inspection is to 'notice' them, or mentally check them. 'By inspection' can play a kind of
1732:
1558:
1530:
1489:
1297:
1132:
1030:
534:
509:
489:
151:
47:
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634:; the relevant argument(s) are implicit in the context. As an example, the function log(log(
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3387:
3377:
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3258:
3248:
3224:
1898:
1604:
1603:
may be added to a mathematical notion to indicate a related stronger notion; for example, a
1493:
1136:
1092:
270:
256:
187:
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which sum to Ï radians, a single straight line conforms to this definition pathologically.
216:
57:
53:
2621:
2488:
2485:; in other words, it is a binary relation but with the specification of the ambient sets
1309:
A function (which in mathematics is generally defined as mapping the elements of one set
1243:
3373:
3244:
1514:) if it cannot be made more restrictive without failing in some cases. For example, for
1077:
Qualifies anything that is sufficiently precise to be translated straightforwardly in a
3263:
2849:
2829:
2809:
2789:
2647:
2442:
2422:
2404:
2234:
1223:
1160:
1102:
883:
859:
839:
819:
547:
is satisfied by arbitrarily large values, can be expressed in more formal notation by
3572:
3534:
3392:
3071:
2397:
1970:
1616:
1571:
if it deduces restrictive results from general hypotheses. One celebrated example is
1485:
1441:
1078:
431:
266:
199:
195:
3511:
1591:, each of which is stronger than the last; another is that a sharp upper bound (see
306:
A term regarding statements. If a statement holds false, then it is said to exhibit
3320:
1958:
1672:
1082:
493:
447:
144:
93:
1633:
holds for sufficiently large values, can be expressed in more formal notation by â
3468:
3354:
2939:
2356:
2272:
975:
501:
101:
of talented mathematicians, could be solved in terms of...complicated concepts.
3495:
2885:
2290:
1930:
1907:
of an object; the object exists, and furthermore, no other such object exists.
1417:
1342:
483:
427:
24:
3503:
3139:"The Elementary Proof of the Prime Number Theorem: An Historical Perspective"
1888:
need to show (NTS), required to prove (RTP), wish to show, want to show (WTS)
1184:
Most often, these refer simply to the left-hand or the right-hand side of an
3382:
3302:
2346:
to a known statement, or is a simple special case of a more general concept.
2343:
1608:
1150:" objects. Exceptions to this description may be mentioned explicitly, as "
1110:
613:
513:
3272:
3253:
882:
When said of the value of a variable assuming values from the non-negative
519:
of the real numbers with measure zero. One can also speak of "almost all"
3401:
2215:
2209:. It is also applied to solving equations; for example to find roots of a
1940:
Often several equivalent conditions (especially for a definition, such as
1595:
above) is a stronger result than a non-sharp one. Finally, the adjective
269:, where it usually refers to a proof that does not resort to methods from
2927:
1808:
1731:
is (categorical) isomorphism; for example, "The tensor product in a weak
1696:
1688:
1684:
1185:
1124:
1025:
933:
662:
366:
203:
38:
Some phrases, like "in general", appear below in more than one section.
1383:
979:
520:
357:
An object behaves pathologically (or, somewhat more broadly used, in a
228:
3285:
Proc. Tenth Annual
Structure in Complexity Theory Conference (SCT'95)
2407:
of an object is some standard or universal way to express the object.
1910:
1753:
that are integer multiples of Ï." This can also apply to limits: see
1535: = 2.7182818..., gives an upper bound on the values of the
1481:
516:
438:
32:
28:
426:
Note for that latter quote that as the differentiable functions are
2342:. A concept is trivial if it holds by definition, is an immediate
642:
becomes larger than 100"; in this context, "eventually" means "for
27:
of specialist and technical terms. It also has a certain amount of
2153:
introduced with this phrase and the altered proposition is proved.
1705:
370:
16:
Collection of commonly used phrases found in mathematical fields
3059:{\displaystyle \mathbb {R} ^{2}\to \mathbb {R} ,(x,y)\mapsto x}
2107:
for this last algebra. By transport of structure, we obtain a
1814:
1450:(Respectively) A convention to shorten parallel expositions. "
121:
A reference to a standard or choice-free presentation of some
1101:
but is used particularly for concepts outside the purview of
231:
or proofs are beautiful but may be written about inelegantly.
2970:. It has typically the property that, for almost all points
1432:. Informally, this term is sometimes used synonymously with
334:
is compact if and only if it is bounded? This is chicanery!"
77:' — a subject then called 'general abstract nonsense'!
2355:
This section features terms used across different areas in
932:
the meaning is usually "not infinite". For example, if the
378:
Since half a century we have seen arise a crowd of bizarre
191:
1270:
and rvalue: an RHS is primitive, and an LHS is derivative.
1266:
on the RHS. Occasionally, these are used in the sense of
1711:
An extension to mathematical discourse of the notions of
1611:
satisfying certain additional conditions, and likewise a
3424:"Some Trends in Modern Mathematics and the Fields Medal"
1420:, with the function and its derivatives exhibiting some
1024:, the intended variant is implicit. As an example, the
1830:
In the context of proofs, this phrase is often seen in
1071:
for which the value of the sequence is in the interval.
3283:(1995), "A personal view of average-case complexity",
2219:
role: the answer or solution simply clicks into place.
1877:
it is necessary and sufficient that it have no finite
1745:
To assume the value 0. For example, "The function sin(
925:{\displaystyle \mathbb {R} _{\geq 0}\cup \{\infty \},}
3011:
2852:
2832:
2812:
2792:
2757:
2696:
2670:
2650:
2624:
2586:
2554:
2491:
2465:
2445:
2425:
1363:
requires the objects to be different. For example, a
1246:
1226:
1194:
1033:
985:
943:
891:
862:
842:
822:
782:
750:
718:
686:
318:
3482:(1977), "The phenomenology of mathematical beauty",
630:
In the context of limits, this is shorthand meaning
1484:(resp. triangles) have 4 sides (resp. 3 sides); or
3533:
3058:
2858:
2838:
2818:
2798:
2774:
2743:
2682:
2656:
2636:
2610:
2572:
2503:
2477:
2451:
2431:
1258:
1232:
1212:
1055:
998:
966:
924:
868:
848:
828:
800:
768:
736:
704:
452:An object is well-behaved (in contrast with being
326:
3207:Numerous examples can be found in (Mac Lane
250:, pp.173–174, pp.181–182)
3231:(1942), "Natural Isomorphisms in Group Theory",
1067:(1/2, 3/2), because there are arbitrarily large
1013:In the context of limits, this is shorthand for
587:"can be made" arbitrarily large, corresponds to
143:—The proof that there are infinitely many
3446:Infinitesimal methods for mathematical analysis
1980:
1157:
396:
376:
285:
234:
136:
90:
71:
2323:the proof is left as an exercise to the reader
3443:Pinto, J. Sousa (2004), Hoskins, R.F. (ed.),
1496:has a finite (resp. countable) open subcover.
1276:A mathematical object is colloquially called
967:{\displaystyle \mathbb {N} \cup \{\infty \},}
539:Notions which arise mostly in the context of
190:were found. On the other hand, the fact that
186:â was once thought to be a deep result until
8:
3188:, Oxford science publications, p. 119,
1440:are not to be confused with the notion of a
958:
952:
916:
910:
2133:
1288:. For example, one might conjecture that a
1131:, one says that a property of points on an
294:
107:
2782:. In other words, it is a special kind of
1986:be a finite-dimensional vector space over
1436:, below. These imprecise uses of the word
3391:
3381:
3292:
3262:
3252:
3028:
3027:
3018:
3014:
3013:
3010:
2851:
2831:
2811:
2791:
2756:
2695:
2669:
2649:
2623:
2585:
2553:
2490:
2464:
2444:
2424:
2145:(WLOG, WOLOG, WALOG), we may assume (WMA)
1245:
1225:
1193:
1047:
1032:
990:
984:
945:
944:
942:
898:
894:
893:
890:
861:
841:
821:
781:
749:
717:
685:
430:in the space of continuous functions, as
320:
319:
317:
3414:Categories for the Working Mathematician
3208:
389:
182:â originally proved using techniques of
83:
73:introduced the very abstract idea of a '
3129:
2283:elements around the diagram, or does a
1169:
643:
500:to speak of. For example, "almost all
488:A shorthand term for "all except for a
166:
1842:A minor variant on "if and only if"; "
1675:, the total space is often said to be
1097:This term has similar connotations as
1015:
3074:â are also synonyms for a projection.
2255:commonly reserved for jokes (puns on
2057:)....It extends to an isomorphism of
417:
7:
3082:
2994:
2915:
2896:
2538:
2519:
2389:
2370:
1626:, suitably small, sufficiently close
469:can also be used to the same effect.
247:
3449:, Horwood Publishing, p. 246,
2876:List of theorems called fundamental
2459:is a subset of a Cartesian product
2018:....There is an isomorphism of the
1937:the following are equivalent (TFAE)
1702:, as in "bringing a term upstairs".
1492:) spaces are ones where every open
1127:) is said to hold generically. In
3536:The Seventeen Provers of the World
2361:Category:Glossaries of mathematics
2275:) which can be stated in terms of
1735:is associative and unital up to a
987:
955:
913:
14:
3119:Category:Mathematical terminology
2882:Fundamental Theorem of Arithmetic
2744:{\displaystyle (a,b),(a,b')\in f}
1151:
454:
3473:, The Science Press, p. 435
3104:Glossary of areas of mathematics
2173:back-of-the-envelope calculation
2143:without (any) loss of generality
712:can be written as a composition
632:for sufficiently large arguments
1749:) vanishes for those values of
1691:is occasionally referred to as
1296:should be computable "for nice
1179:left-hand side, right-hand side
401:the one provided by Weierstrass
3467:(1913), Halsted, Bruce (ed.),
3163:. Cambridge University Press.
3109:List of mathematical constants
3050:
3047:
3035:
3024:
2978:, there is a neighbourhood of
2732:
2715:
2709:
2697:
2605:
2587:
2564:
1811:of the statement to be proved.
1515:
1444:, which is rigorously defined.
1402:word is also non-jargon for a
1147:
1044:
1034:
801:{\displaystyle h\colon B\to C}
792:
769:{\displaystyle g\colon A\to B}
760:
705:{\displaystyle f\colon A\to C}
696:
575:. The statement that quantity
312:"What do you mean a subset of
293:Russell Impagliazzo (
106:Michael Monastyrsky (
1:
3522:(1991), Kandall, G.A. (ed.),
3422:Monastyrsky, Michael (2001),
1969:; if one of them is given an
1800:(BWOC), or "for, if not, ..."
66:generalized abstract nonsense
3532:Wiedijk, Freek, ed. (2006),
3114:List of mathematical symbols
2826:, there is a unique element
2316:
2230:clearly, can be easily shown
2061:to the localized algebra Sym
1803:The rhetorical prelude to a
1345:; it may even be said that "
661:referring to composition of
327:{\displaystyle \mathbb {R} }
129:is the "canonical proof" of
82:Saunders Mac Lane (
3563:Encyclopedia of Mathematics
2132:Igor Shafarevich (
1020:and its relatives; as with
999:{\displaystyle \aleph _{0}}
3600:
3470:The Foundations of Science
3362:Proc. Natl. Acad. Sci. USA
3233:Proc. Natl. Acad. Sci. USA
2873:
1518:non-negative real numbers
1359:), then the qualification
737:{\displaystyle f=h\circ g}
246:Gian-Carlo Rota (
3584:Glossaries of mathematics
3211:), for example on p. 100.
2690:subject to the condition
2683:{\displaystyle A\times B}
2664:of the Cartesian product
2478:{\displaystyle A\times B}
1442:regular topological space
474:Descriptive informalities
416:J. Sousa Pinto (
388:Henri Poincaré (
262:
62:general abstract nonsense
42:Philosophy of mathematics
3579:Mathematical terminology
3526:, vol. IV, Springer
3355:"The PNAS way back then"
2573:{\displaystyle f:A\to B}
1957:. For example, any two
1921:Halmos end-of-proof mark
1905:existence and uniqueness
1866:". For example, "For a
1838:necessary and sufficient
1567:A theorem is said to be
1168:Allyn Jackson (
1056:{\displaystyle (-1)^{n}}
978:of the set is less than
221:elegance of presentation
165:Freek Wiedijk (
150:—The proof of the
131:the infinitude of primes
3496:10.1023/A:1004930722234
3383:10.1073/pnas.94.12.5983
3303:10.1109/SCT.1995.514853
2866:that corresponds to it.
2786:where given an element
2611:{\displaystyle (A,B,f)}
2511:used in the definition.
1917:Quod erat demonstrandum
1798:by way of contradiction
1782:The formal language of
1715:. A statement is true
1581:Fermat's little theorem
1394:that is different from
1375:that is different from
1313:to elements of another
21:language of mathematics
3254:10.1073/pnas.28.12.537
3159:Boyd, Stephen (2004).
3091:mathematical structure
3060:
2860:
2840:
2820:
2800:
2776:
2745:
2684:
2658:
2638:
2612:
2574:
2505:
2479:
2453:
2433:
2139:
2109:linear algebraic group
1955:transport of structure
1948:transport of structure
1895:just these statements.
1805:proof by contradiction
1679:, with the base space
1613:strongly regular graph
1260:
1234:
1214:
1175:
1135:that holds on a dense
1057:
1000:
968:
926:
870:
850:
830:
802:
770:
738:
706:
510:algebraic real numbers
423:
395:
363:mathematical intuition
328:
300:
282:ceases to be folklore.
253:
219:distinguished between
172:
113:
89:
3431:Can. Math. Soc. Notes
3061:
2861:
2841:
2821:
2801:
2777:
2746:
2685:
2659:
2639:
2613:
2580:is an ordered triple
2575:
2506:
2480:
2454:
2434:
2302:the proof is similar.
1671:. For example, in a
1506:. The constraint is
1412:A function is called
1294:topological invariant
1290:differential operator
1261:
1235:
1215:
1213:{\displaystyle x=y+1}
1063:is frequently in the
1058:
1001:
969:
927:
871:
851:
831:
803:
771:
739:
707:
403:....This function is
329:
3287:, pp. 134â147,
3281:Impagliazzo, Russell
3009:
2952:multivalued function
2850:
2830:
2810:
2790:
2775:{\displaystyle b=b'}
2755:
2694:
2668:
2648:
2622:
2584:
2552:
2528:mathematical diagram
2489:
2463:
2443:
2423:
1875:algebraically closed
1820:An abbreviation for
1729:equivalence relation
1708:, modulo, mod out by
1660:upstairs, downstairs
1524:exponential function
1504:upper or lower bound
1244:
1224:
1192:
1031:
983:
941:
889:
860:
840:
820:
780:
748:
716:
684:
620:universal quantifier
618:A shorthand for the
316:
180:prime number theorem
3374:1997PNAS...94.5983M
3245:1942PNAS...28..537E
3186:Elementary Geometry
3161:Convex Optimization
3146:Columbia University
3068:idempotent operator
2962:is a function from
2637:{\displaystyle A,B}
2618:consisting of sets
2504:{\displaystyle A,B}
2269:commutative diagram
2252:complete intuition
2186:proof by exhaustion
2094: ⊗
2071: ⊗
2053: ⊗
1903:A statement of the
1822:logical equivalence
1737:natural isomorphism
1721:equivalence classes
1573:Donaldson's theorem
1259:{\displaystyle y+1}
496:", when there is a
123:mathematical object
3524:Algebraic Geometry
3410:Mac Lane, Saunders
3351:Mac Lane, Saunders
3229:Mac Lane, Saunders
3184:Roe, John (1993),
3137:Goldfeld, Dorian.
3056:
2966:to the subsets of
2926:between sets or a
2856:
2836:
2816:
2796:
2772:
2741:
2680:
2654:
2634:
2608:
2570:
2501:
2475:
2449:
2429:
2257:complete induction
2211:quadratic equation
2020:polynomial algebra
1923:, a square sign â.
1755:Vanish at infinity
1713:modular arithmetic
1624:sufficiently large
1589:Lagrange's theorem
1537:quadratic function
1256:
1230:
1210:
1129:algebraic geometry
1123:of countably many
1053:
996:
964:
922:
866:
846:
826:
798:
766:
734:
702:
644:sufficiently large
324:
156:square root of two
98:algebraic geometry
3547:978-3-540-30704-4
3520:Shafarevich, Igor
3456:978-1-898563-99-0
3368:(12): 5983â5985,
3312:978-0-8186-7052-7
3225:Eilenberg, Samuel
3195:978-0-19-853456-3
2859:{\displaystyle B}
2839:{\displaystyle b}
2819:{\displaystyle A}
2799:{\displaystyle a}
2657:{\displaystyle f}
2452:{\displaystyle B}
2432:{\displaystyle A}
2085: = det(
1975:factoring through
1926:sufficiently nice
1778:Proof terminology
1733:monoidal category
1430:Hölder continuity
1341:") only if it is
1282:sufficiently nice
1233:{\displaystyle x}
1133:algebraic variety
1087:formal definition
1081:. For example. a
1016:arbitrarily large
869:{\displaystyle h}
849:{\displaystyle g}
829:{\displaystyle B}
816:any (and all) of
535:arbitrarily large
225:beauty of concept
188:elementary proofs
48:abstract nonsense
3591:
3550:
3539:
3527:
3514:
3480:Rota, Gian-Carlo
3474:
3459:
3438:
3428:
3417:
3404:
3395:
3385:
3359:
3337:
3323:
3296:
3275:
3266:
3256:
3212:
3205:
3199:
3198:
3181:
3175:
3174:
3156:
3150:
3149:
3143:
3134:
3065:
3063:
3062:
3057:
3031:
3023:
3022:
3017:
2985:
2981:
2977:
2973:
2969:
2965:
2961:
2957:
2922:A synonym for a
2865:
2863:
2862:
2857:
2845:
2843:
2842:
2837:
2825:
2823:
2822:
2817:
2805:
2803:
2802:
2797:
2781:
2779:
2778:
2773:
2771:
2750:
2748:
2747:
2742:
2731:
2689:
2687:
2686:
2681:
2663:
2661:
2660:
2655:
2643:
2641:
2640:
2635:
2617:
2615:
2614:
2609:
2579:
2577:
2576:
2571:
2510:
2508:
2507:
2502:
2484:
2482:
2481:
2476:
2458:
2456:
2455:
2450:
2438:
2436:
2435:
2430:
2403:2. A
2396:1. A
2196:proof by example
2158:Proof techniques
2137:
2118:) isomorphic to
1977:the isomorphism.
1899:one and only one
1879:field extensions
1807:, preceding the
1763:The converse of
1723:, especially in
1605:strong antichain
1564:strong, stronger
1428:above), such as
1390:is a divisor of
1367:subset of a set
1265:
1263:
1262:
1257:
1239:
1237:
1236:
1231:
1219:
1217:
1216:
1211:
1173:
1074:formal, formally
1062:
1060:
1059:
1054:
1052:
1051:
1005:
1003:
1002:
997:
995:
994:
973:
971:
970:
965:
948:
937:natural numbers
931:
929:
928:
923:
906:
905:
897:
875:
873:
872:
867:
855:
853:
852:
847:
835:
833:
832:
827:
807:
805:
804:
799:
775:
773:
772:
767:
743:
741:
740:
735:
711:
709:
708:
703:
665:. If for three
609:
570:
421:
393:
333:
331:
330:
325:
323:
298:
271:complex analysis
251:
184:complex analysis
170:
111:
87:
3599:
3598:
3594:
3593:
3592:
3590:
3589:
3588:
3569:
3568:
3558:
3548:
3531:
3518:
3478:
3465:Poincare, Henri
3463:
3457:
3442:
3426:
3421:
3408:
3357:
3349:
3327:
3313:
3294:10.1.1.678.8930
3279:
3239:(12): 537â543,
3223:
3220:
3215:
3206:
3202:
3196:
3183:
3182:
3178:
3171:
3158:
3157:
3153:
3141:
3136:
3135:
3131:
3127:
3100:
3086:
3081:
3012:
3007:
3006:
2998:
2993:
2983:
2979:
2975:
2971:
2967:
2963:
2959:
2955:
2947:
2935:
2919:
2914:
2900:
2895:
2878:
2871:
2848:
2847:
2828:
2827:
2808:
2807:
2788:
2787:
2764:
2753:
2752:
2724:
2692:
2691:
2666:
2665:
2646:
2645:
2620:
2619:
2582:
2581:
2550:
2549:
2542:
2537:
2523:
2518:
2487:
2486:
2461:
2460:
2441:
2440:
2421:
2420:
2412:
2393:
2388:
2379:binary relation
2374:
2369:
2353:
2263:diagram chasing
2223:by intimidation
2160:
2138:
2131:
2126:
2102:
2093:
2080:
2066:
2048:
2038:
2009:
1998:
1942:normal subgroup
1780:
1725:category theory
1667:and the lower,
1585:Euler's theorem
1480:. For example,
1404:proper morphism
1371:is a subset of
1325:" (instead of "
1242:
1241:
1240:on the LHS and
1222:
1221:
1190:
1189:
1188:; for example,
1174:
1172:, p.1197)
1167:
1161:Platonic solids
1118:
1043:
1029:
1028:
986:
981:
980:
939:
938:
892:
887:
886:
858:
857:
838:
837:
818:
817:
778:
777:
746:
745:
714:
713:
682:
681:
659:category theory
588:
583:) depending on
548:
476:
422:
415:
394:
387:
347:transformations
314:
313:
299:
292:
252:
245:
239:
238:
217:Gian-Carlo Rota
171:
164:
161:
112:
105:
88:
81:
58:category theory
54:tongue-in-cheek
44:
17:
12:
11:
5:
3597:
3595:
3587:
3586:
3581:
3571:
3570:
3567:
3566:
3557:
3554:
3553:
3552:
3546:
3540:, BirkhÀuser,
3529:
3516:
3490:(2): 171â182,
3476:
3461:
3455:
3440:
3419:
3406:
3347:
3325:
3311:
3277:
3219:
3216:
3214:
3213:
3200:
3194:
3176:
3170:978-0521833783
3169:
3151:
3128:
3126:
3123:
3122:
3121:
3116:
3111:
3106:
3099:
3096:
3095:
3094:
3087:
3084:
3080:
3077:
3076:
3075:
3055:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3030:
3026:
3021:
3016:
2999:
2996:
2992:
2989:
2988:
2987:
2948:
2945:
2943:
2936:
2933:
2931:
2920:
2917:
2913:
2910:
2909:
2908:
2901:
2898:
2894:
2891:
2890:
2889:
2874:Main article:
2872:
2869:
2867:
2855:
2835:
2815:
2795:
2784:correspondence
2770:
2767:
2763:
2760:
2740:
2737:
2734:
2730:
2727:
2723:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2679:
2676:
2673:
2653:
2633:
2630:
2627:
2607:
2604:
2601:
2598:
2595:
2592:
2589:
2569:
2566:
2563:
2560:
2557:
2543:
2540:
2536:
2533:
2532:
2531:
2524:
2521:
2517:
2514:
2513:
2512:
2500:
2497:
2494:
2474:
2471:
2468:
2448:
2428:
2417:correspondence
2413:
2411:correspondence
2410:
2408:
2405:canonical form
2401:
2394:
2391:
2387:
2384:
2383:
2382:
2375:
2372:
2368:
2365:
2352:
2349:
2348:
2347:
2336:
2331:
2324:
2321:
2312:
2309:
2306:
2303:
2299:
2296:
2293:
2288:
2265:
2260:
2253:
2250:
2231:
2228:
2225:
2220:
2202:
2199:
2192:
2189:
2181:
2178:
2175:
2170:
2167:
2159:
2156:
2155:
2154:
2146:
2129:
2122:
2103:)....We write
2098:
2089:
2076:
2062:
2044:
2025:
2000:
1994:
1979:
1978:
1950:
1945:
1938:
1935:
1927:
1924:
1913:
1908:
1901:
1896:
1889:
1886:
1840:
1835:
1828:
1825:
1824:of statements.
1818:
1815:if and only if
1812:
1801:
1795:
1791:
1779:
1776:
1775:
1774:
1771:
1768:
1761:
1758:
1743:
1740:
1709:
1703:
1661:
1658:
1627:
1621:
1599:or the adverb
1565:
1562:
1552:
1547:
1500:
1497:
1473:and also that
1466:)" means that
1448:
1445:
1424:property (see
1410:
1407:
1353:
1350:
1307:
1304:
1274:
1271:
1255:
1252:
1249:
1229:
1209:
1206:
1203:
1200:
1197:
1182:
1165:
1156:
1155:
1143:
1140:
1116:
1103:measure theory
1095:
1090:
1075:
1072:
1050:
1046:
1042:
1039:
1036:
1011:
1008:
993:
989:
963:
960:
957:
954:
951:
947:
921:
918:
915:
912:
909:
904:
901:
896:
884:extended reals
880:
877:
865:
845:
825:
814:factor through
797:
794:
791:
788:
785:
765:
762:
759:
756:
753:
733:
730:
727:
724:
721:
701:
698:
695:
692:
689:
655:
653:factor through
650:
628:
623:
616:
611:
537:
532:
508:" because the
506:transcendental
486:
475:
472:
471:
470:
450:
445:
441:
439:rigor (rigour)
436:
413:
409:differentiable
385:
375:
374:
355:
350:
343:
337:
336:
322:
304:
290:
284:
283:
279:
274:
259:
243:
233:
232:
212:
207:
176:
162:
160:
159:
148:
140:
135:
134:
127:Euclid's proof
119:
103:
79:
70:
69:
50:
43:
40:
15:
13:
10:
9:
6:
4:
3:
2:
3596:
3585:
3582:
3580:
3577:
3576:
3574:
3565:
3564:
3560:
3559:
3555:
3549:
3543:
3538:
3537:
3530:
3525:
3521:
3517:
3513:
3509:
3505:
3501:
3497:
3493:
3489:
3485:
3481:
3477:
3472:
3471:
3466:
3462:
3458:
3452:
3448:
3447:
3441:
3436:
3432:
3425:
3420:
3415:
3411:
3407:
3403:
3399:
3394:
3389:
3384:
3379:
3375:
3371:
3367:
3363:
3356:
3352:
3348:
3345:
3341:
3335:
3331:
3326:
3322:
3318:
3314:
3308:
3304:
3300:
3295:
3290:
3286:
3282:
3278:
3274:
3270:
3265:
3260:
3255:
3250:
3246:
3242:
3238:
3234:
3230:
3226:
3222:
3221:
3217:
3210:
3204:
3201:
3197:
3191:
3187:
3180:
3177:
3172:
3166:
3162:
3155:
3152:
3147:
3140:
3133:
3130:
3124:
3120:
3117:
3115:
3112:
3110:
3107:
3105:
3102:
3101:
3097:
3092:
3088:
3083:
3078:
3073:
3072:forgetful map
3069:
3053:
3044:
3041:
3038:
3032:
3019:
3004:
3000:
2995:
2990:
2954:â from a set
2953:
2949:
2944:
2941:
2937:
2932:
2929:
2925:
2921:
2916:
2911:
2906:
2902:
2897:
2892:
2887:
2883:
2879:
2877:
2868:
2853:
2833:
2813:
2793:
2785:
2768:
2765:
2761:
2758:
2738:
2735:
2728:
2725:
2721:
2718:
2712:
2706:
2703:
2700:
2677:
2674:
2671:
2651:
2644:and a subset
2631:
2628:
2625:
2602:
2599:
2596:
2593:
2590:
2567:
2561:
2558:
2555:
2548:
2544:
2539:
2534:
2529:
2525:
2520:
2515:
2498:
2495:
2492:
2472:
2469:
2466:
2446:
2426:
2418:
2414:
2409:
2406:
2402:
2399:
2398:canonical map
2395:
2390:
2385:
2380:
2376:
2371:
2366:
2364:
2362:
2358:
2351:Miscellaneous
2350:
2345:
2341:
2337:
2335:
2332:
2329:
2325:
2322:
2319:
2318:
2313:
2310:
2307:
2304:
2300:
2297:
2294:
2292:
2289:
2286:
2285:diagram chase
2282:
2278:
2274:
2270:
2266:
2264:
2261:
2258:
2254:
2251:
2248:
2244:
2240:
2236:
2232:
2229:
2226:
2224:
2221:
2218:
2217:
2212:
2208:
2207:by inspection
2205:be evaluated
2203:
2201:by inspection
2200:
2197:
2193:
2190:
2187:
2182:
2179:
2176:
2174:
2171:
2168:
2166:angle chasing
2165:
2164:
2163:
2157:
2151:
2147:
2144:
2141:
2140:
2136:, p.12)
2135:
2128:
2125:
2121:
2117:
2113:
2110:
2106:
2101:
2097:
2092:
2088:
2084:
2079:
2074:
2070:
2065:
2060:
2056:
2052:
2047:
2042:
2037:
2033:
2029:
2024:
2021:
2017:
2013:
2008:
2004:
1997:
1993:
1989:
1985:
1976:
1972:
1971:inner product
1968:
1964:
1960:
1959:vector spaces
1956:
1951:
1949:
1946:
1943:
1939:
1936:
1932:
1928:
1925:
1922:
1918:
1914:
1912:
1909:
1906:
1902:
1900:
1897:
1894:
1893:needs to show
1890:
1887:
1884:
1880:
1876:
1872:
1869:
1865:
1862:if (only if)
1861:
1857:
1853:
1849:
1845:
1841:
1839:
1836:
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1806:
1802:
1799:
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1789:
1788:
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1766:
1762:
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1718:
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1682:
1678:
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1662:
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1652:
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1640:
1636:
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1628:
1625:
1622:
1620:"antichain").
1618:
1617:regular graph
1614:
1610:
1606:
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1594:
1590:
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1570:
1566:
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1439:
1435:
1431:
1427:
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1393:
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1316:
1312:
1308:
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1180:
1177:
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1164:
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1138:
1134:
1130:
1126:
1122:
1115:
1112:
1108:
1104:
1100:
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1091:
1088:
1084:
1080:
1079:formal system
1076:
1073:
1070:
1066:
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1037:
1027:
1023:
1019:
1017:
1012:
1009:
1006:
991:
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949:
935:
919:
907:
902:
899:
885:
881:
878:
863:
843:
823:
815:
811:
795:
789:
786:
783:
763:
757:
754:
751:
731:
728:
725:
722:
719:
699:
693:
690:
687:
679:
675:
671:
668:
664:
660:
656:
654:
651:
648:
645:
641:
637:
633:
629:
627:
624:
621:
617:
615:
612:
608:
604:
600:
596:
592:
586:
582:
578:
574:
568:
564:
560:
556:
552:
546:
542:
538:
536:
533:
530:
526:
522:
518:
515:
511:
507:
503:
499:
495:
491:
487:
485:
482:
481:
480:
473:
468:
465:
462:), the term "
461:
457:
456:
451:
449:
446:
442:
440:
437:
435:pathological.
433:
429:
425:
424:
419:
412:
410:
406:
402:
391:
384:
381:
372:
368:
364:
360:
356:
354:
351:
348:
344:
342:
339:
338:
335:
309:
305:
302:
301:
296:
289:
280:
278:
275:
272:
268:
267:number theory
264:
260:
258:
255:
254:
249:
242:
230:
226:
222:
218:
213:
211:
208:
205:
201:
200:number theory
197:
196:real analysis
193:
189:
185:
181:
177:
174:
173:
168:
157:
153:
152:irrationality
149:
146:
145:prime numbers
142:
141:
139:
132:
128:
124:
120:
118:
115:
114:
109:
102:
99:
96:] raised
95:
85:
78:
76:
67:
63:
59:
56:reference to
55:
51:
49:
46:
45:
41:
39:
36:
34:
30:
26:
22:
3561:
3556:Bibliography
3535:
3523:
3487:
3483:
3469:
3445:
3434:
3430:
3413:
3365:
3361:
3333:
3329:
3284:
3236:
3232:
3203:
3185:
3179:
3160:
3154:
3145:
3132:
2354:
2339:
2327:
2315:
2305:index battle
2284:
2280:
2246:
2238:
2214:
2206:
2195:
2161:
2148:Sometimes a
2123:
2119:
2115:
2111:
2104:
2099:
2095:
2090:
2086:
2082:
2077:
2072:
2068:
2063:
2058:
2054:
2050:
2045:
2035:
2031:
2027:
2022:
2015:
2006:
2002:
1995:
1991:
1987:
1983:
1981:
1974:
1961:of the same
1954:
1916:
1892:
1882:
1870:
1863:
1859:
1855:
1851:
1847:
1843:
1781:
1770:well-defined
1760:weak, weaker
1750:
1746:
1727:, where the
1716:
1699:
1692:
1680:
1676:
1673:fiber bundle
1668:
1664:
1654:
1650:
1646:
1642:
1638:
1634:
1630:
1600:
1596:
1592:
1576:
1568:
1554:
1543:
1539:
1531:
1526:
1519:
1511:
1507:
1477:
1474:
1470:
1467:
1463:
1459:
1455:
1451:
1437:
1433:
1425:
1421:
1413:
1395:
1391:
1387:
1386:of a number
1380:
1376:
1372:
1368:
1364:
1360:
1346:
1338:
1334:
1330:
1326:
1322:
1318:
1314:
1310:
1300:
1286:pathological
1281:
1277:
1158:
1152:pathological
1137:Zariski open
1121:intersection
1113:
1098:
1086:
1083:formal proof
1068:
1021:
1014:
813:
809:
677:
673:
669:
646:
639:
635:
631:
606:
602:
598:
594:
590:
584:
580:
576:
572:
571:. See also
566:
562:
558:
554:
550:
544:
528:
502:real numbers
494:measure zero
477:
466:
455:Pathological
453:
448:well-behaved
397:
377:
358:
353:pathological
311:
307:
286:
235:
224:
220:
169:, p.2)
137:
94:Grothendieck
91:
72:
65:
61:
37:
18:
3330:AMS Notices
2946:multivalued
2940:mathematics
2934:mathematics
2870:fundamental
2419:from a set
2357:mathematics
2338:Similar to
2273:injectivity
2180:brute force
2150:proposition
1931:working out
1697:denominator
1559:analyticity
1510:(sometimes
1418:derivatives
976:cardinality
812:is said to
383:fathers....
359:degenerated
241:theorem....
23:has a vast
3573:Categories
3416:, Springer
3218:References
3003:projection
2997:projection
2886:Arithmetic
2298:in general
2291:handwaving
2191:by example
1967:isomorphic
1852:sufficient
1827:in general
1700:downstairs
1681:downstairs
1669:downstairs
1655:eventually
1555:Smoothness
1400:overloaded
1343:surjective
1181:(LHS, RHS)
1142:in general
1107:complement
1099:almost all
1022:eventually
1010:frequently
657:A term in
640:eventually
626:eventually
573:frequently
484:almost all
444:fallacies.
405:continuous
257:elementary
25:vocabulary
3504:0039-7857
3437:(2 and 3)
3289:CiteSeerX
3085:structure
3051:↦
3025:→
2958:to a set
2905:invariant
2899:invariant
2736:∈
2675:×
2565:→
2470:×
2439:to a set
2392:canonical
2344:corollary
2311:obviously
2039:onto the
1990:....Let (
1963:dimension
1881:" means "
1858:" means "
1848:necessary
1832:induction
1794:interest.
1689:numerator
1637: : â
1609:antichain
1516:arbitrary
1357:reflexive
1148:arbitrary
1125:open sets
1038:−
1018:arguments
988:ℵ
956:∞
950:∪
914:∞
908:∪
900:≥
793:→
787::
761:→
755::
729:∘
697:→
691::
663:morphisms
614:arbitrary
593: : â
553: : â
514:countable
380:functions
308:chicanery
303:chicanery
117:canonical
3512:44064821
3484:Synthese
3412:(1998),
3353:(1997),
3273:16588584
3098:See also
2928:morphism
2924:function
2769:′
2751:implies
2729:′
2547:function
2541:function
2277:elements
2267:Given a
2130:â
2081:, where
1934:through.
1809:negation
1695:and the
1693:upstairs
1685:fraction
1683:. In a
1677:upstairs
1665:upstairs
1645: :
1601:strongly
1577:stronger
1529:, where
1490:Lindelöf
1409:regular
1398:. This
1379:, and a
1186:equation
1166:â
1154:" cases.
1065:interval
1026:sequence
934:variance
597: :
561: :
531:, below.
521:integers
460:analysis
414:â
407:but not
386:â
367:triangle
291:â
277:folklore
244:â
229:theorems
204:geometry
163:â
104:â
80:â
75:category
33:rigorous
3402:9177152
3370:Bibcode
3338:(Parts
3336:(9, 10)
3321:2154064
3264:1078535
3241:Bibcode
3070:â and â
2522:diagram
2340:clearly
2334:trivial
2328:obvious
2317:clearly
2247:Ă©vident
2239:obvious
2235:Laplace
2216:gestalt
2041:algebra
1512:optimal
1488:(resp.
1486:compact
1482:squares
1462:(resp.
1454:(resp.
1438:regular
1414:regular
1384:divisor
1093:generic
808:, then
667:objects
529:generic
512:form a
498:measure
369:having
341:natural
210:elegant
154:of the
3544:
3510:
3502:
3453:
3400:
3390:
3319:
3309:
3291:
3271:
3261:
3192:
3167:
2373:binary
2281:chases
2243:French
1911:Q.E.D.
1873:to be
1854:) for
1790:aliter
1765:strong
1742:vanish
1687:, the
1607:is an
1597:strong
1569:strong
1550:smooth
1522:, the
1434:smooth
1381:proper
1365:proper
1361:proper
1352:proper
1333:" or "
1298:spaces
1268:lvalue
879:finite
856:, and
680:a map
676:, and
541:limits
517:subset
464:smooth
432:Banach
428:meagre
371:angles
288:print.
29:jargon
3508:S2CID
3427:(PDF)
3393:33670
3358:(PDF)
3317:S2CID
3142:(PDF)
3125:Notes
2237:used
2012:basis
2010:be a
1868:field
1817:(iff)
1784:proof
1717:up to
1706:up to
1615:is a
1593:sharp
1508:sharp
1499:sharp
1494:cover
1447:resp.
1337:into
1321:onto
1111:dense
744:with
92:[
3542:ISBN
3500:ISSN
3451:ISBN
3398:PMID
3342:and
3307:ISBN
3269:PMID
3209:1998
3190:ISBN
3165:ISBN
2938:See
2884:for
2526:See
2314:See
2134:1991
2014:for
1982:Let
1965:are
1426:nice
1422:nice
1306:onto
1278:nice
1273:nice
1220:has
1170:2004
1085:, a
776:and
605:) â„
504:are
418:2004
390:1913
295:1995
263:deep
248:1977
223:and
202:and
175:deep
167:2006
108:2001
84:1997
19:The
3492:doi
3488:111
3388:PMC
3378:doi
3299:doi
3259:PMC
3249:doi
2974:of
2950:A "
2918:map
2903:An
2846:of
2806:of
2043:Sym
2026:1â€
2001:1â€
1846:is
1458:)
1329:to
1280:or
638:))
525:odd
492:of
490:set
64:or
3575::
3506:,
3498:,
3486:,
3435:33
3433:,
3429:,
3396:,
3386:,
3376:,
3366:94
3364:,
3360:,
3346:).
3344:II
3334:51
3332:,
3315:,
3305:,
3297:,
3267:,
3257:,
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3237:28
3235:,
3227:;
3144:.
3089:A
3001:A
2888:).
2545:A
2415:A
2377:A
2363:.
2259:).
2249:).
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2194:A
2188:).
2120:GL
2112:GL
2034:â€
2030:,
2005:â€
1739:."
1641:â„
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557:â„
310:.
52:A
3551:.
3528:.
3515:.
3494::
3475:.
3460:.
3439:.
3418:.
3405:.
3380::
3372::
3340:I
3324:.
3301::
3276:.
3251::
3243::
3173:.
3148:.
3079:S
3054:x
3048:)
3045:y
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3039:x
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3029:R
3020:2
3015:R
2991:P
2986:.
2984:B
2980:x
2976:B
2972:x
2968:B
2964:A
2960:B
2956:A
2942:.
2912:M
2893:I
2854:B
2834:b
2814:A
2794:a
2766:b
2762:=
2759:b
2739:f
2733:)
2726:b
2722:,
2719:a
2716:(
2713:,
2710:)
2707:b
2704:,
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2698:(
2678:B
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2652:f
2632:B
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2626:A
2606:)
2603:f
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2597:B
2594:,
2591:A
2588:(
2568:B
2562:A
2559::
2556:f
2535:F
2530:.
2516:D
2499:B
2496:,
2493:A
2473:B
2467:A
2447:B
2427:A
2386:C
2367:B
2330:.
2320:.
2287:.
2241:(
2127:.
2124:n
2116:V
2114:(
2105:k
2100:j
2096:e
2091:i
2087:e
2083:D
2078:D
2075:)
2073:V
2069:V
2067:(
2064:k
2059:k
2055:V
2051:V
2049:(
2046:k
2036:n
2032:j
2028:i
2023:k
2016:V
2007:n
2003:i
1999:)
1996:i
1992:e
1988:k
1984:V
1915:(
1883:K
1871:K
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1860:A
1856:B
1850:(
1844:A
1767:.
1757:.
1751:x
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1475:B
1471:X
1468:A
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1460:X
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1406:.
1396:n
1392:n
1388:n
1377:S
1373:S
1369:S
1347:f
1339:B
1335:A
1331:B
1327:A
1323:B
1319:A
1315:B
1311:A
1301:X
1254:1
1251:+
1248:y
1228:x
1208:1
1205:+
1202:y
1199:=
1196:x
1146:"
1119:(
1117:ÎŽ
1114:G
1089:.
1069:n
1049:n
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1041:1
1035:(
1007:.
992:0
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959:}
953:{
946:N
920:,
917:}
911:{
903:0
895:R
876:.
864:h
844:g
824:B
810:f
796:C
790:B
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758:A
752:g
732:g
726:h
723:=
720:f
700:C
694:A
688:f
678:C
674:B
670:A
647:x
636:x
610:.
607:y
603:x
601:(
599:f
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579:(
577:f
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273:.
206:.
192:Ï
158:.
147:.
133:.
110:)
86:)
68:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.