Sò͘-ha̍k

**Sò͘-ha̍k** （數學）tiāⁿ hông khoàⁿ-chò sī leh gián-kiù sò͘-liōng, kiat-kò͘, khong-kan kap piàn-tōng ê ha̍k-būn. Ū koá sò͘-ha̍k-ka kā sò͘-ha̍k tòng-chò sī kong-siat kap teng-gī thaù-koè chhui-lūn tit--tio̍h ê tì-sek.

Ta̍k ê siā-hoē leh chhek-liông thó͘-tē, chhú-lí koè-kè, ī-chhek thian-bûn sū-kiāⁿ, ... ê sî chha-put-to lóng ē-tit khoàⁿ tio̍h èng-iōng sò͘-ha̍k ê iáⁿ. Sò͘-ha̍k thong-siông sī leh chhiau-chhoē bô͘-sek, bô tiāⁿ-tio̍h ài kap èng-iōng ū tī-tāi. M̄-koh Eugene Wigner bat kóng koè "sò͘-ha̍k siuⁿ koè-thaû-á ū-haū", siōng 'sûn' ê sò͘-ha̍k tiāⁿ chhoē tio̍h siōng si̍t-chè ê èng-iōng. Hiān-chhú-sî ê chū-jiân kho-ha̍k, kang-têng, keng-chè, i-ha̍k lóng tio̍h khoàⁿ sò͘-ha̍k ū sin hoat-tián--bô.

Hi-lia̍p-gí ê μάθημα (*máthema*) piáu-sī "kho-ha̍k, tì-sek, ha̍k-si̍p", μαθηματικός (*mathematikós*) piáu-sī "kah-ì o̍h mi̍h-kiāⁿ". Eng-gí kā sò͘-ha̍k siá chò *mathematics*, kán-siá chò *maths* (Liân-pang Eng-gí), *math* (Bí-kok Eng-gí).

## Sò͘-ha̍k ê le̍k-sú
Sò͘-ha̍k ê ián-piàn ē-tit khoàⁿ chò kóng thiu-siōng ê chú-tê iah sī thêng-tō͘ ti̍t-ti̍t leh cheng-ka. Siōng chá ê thiu-siōng hoān-sè sī sò͘-ba̍k. Liáu-kái kóng 2 lia̍p phōng-kó kap 2 lia̍p kam-á ká-ná ū siáⁿ-mih sio-kâng ê só͘-chāi, sò͘-ha̍k tō khai-sí hoat-tián--a.

Sú-chêng ê jîn-lūi ē-hiáu sǹg *kū-thé* ê mi̍h-kiāⁿ, mā chai-iáⁿ beh án-choáⁿ sǹg *thiu-siōng* ê sò͘-liōng, chhiūⁿ kóng sî-kan (kang, kùi, nî, ...).  Ē-hiáu sǹg, koh lâi tō sī sǹg-siàu (chhiūⁿ ke, kiám, sêng, tû).

M̄-koh beh sǹg tio̍h-ài ē-tàng siá kap 1 thò sò͘-jī hē-thóng.  Sú-chêng ê jîn-lūi hoān-sè sī tī thô͘-kha oē sûn iah sī tī chhâ-thaû khek sûn lâi piáu-sī sò͘-liōng. Inca Tè-kok bô su-siá hē-thóng, in lī-iōng soh-á phah-kat ê hong-hoat lâi sǹg-siàu, hō-chò khipu.

Sò͘-ha̍k ê hoat-tián kap chhek-liông thó͘-tē, chhú-lí koè-kè, ī-chhek thian-bûn sū-kiāⁿ ū chin bi̍t-chhiat ê koan-hē. In-ūi ū su-iàu, sò͘-ha̍k lāi-té chhú-lí *khong-kan*, *kiat-kò͘*, *piàn-tōng* ê gián-kiù tō toè leh hoat-tián. Khah lō͘-boé, beh kái-koat sio kap kng ê bûn-tê, sò͘-ha̍k bu̍t-lí-ha̍k mā khai-sí hoat-tián.

Tû-khì éng-koè chiah-ê gián-kiù ê chú-tê, sò͘-ha̍k kaù taⁿ mā iû-goân it-ti̍t leh chhòng-sin.

## Sûn sò͘-ha̍k kap èng-iōng sò͘-ha̍k
Ū oh kái-koat ê būn-tê tō su-iàu sò͘-ha̍k. Éng-koè chhiūⁿ siong-gia̍p, thó͘-tē chhek-liông, thian-bûn-ha̍k; hiān-chhú-sî toā-pō͘-hūn ùi chū-jiân kho-ha̍k tit tio̍h lêng-kám, chiâⁿ chē sò͘-ha̍k-ka mā sī bu̍t-lí-ha̍k-ka. Chhiūⁿ í-chá Newton hoat-tián bî-chek-hun, Richard Feynman hoat-tián Feynman kèng-chek-hun, chit-má thaù-koè chhui-lí kap tùi bu̍t-lí ê chhim-ji̍p liáu-kái, tō hoat-tián chhut hiân-lūn (*string theory*). 

Ū būn-tê beh kái-koat, gián-kiù siong-koan ê sò͘-ha̍k liáu-aū, tō ē-tit it-ti̍t èng-iōng chiah-ê sò͘-ha̍k. Kā ū kiōng-tông ki-chhó͘ ê sò͘-ha̍k khioh chò-hoé, chi̍t-koá koan-liām ē-tit tha̍h chò-hoé. 19 sè-kí ê sî, tō án-ne kā sò͘-ha̍k hun-chò èng-iōng sò͘-ha̍k kap sûn sò͘-ha̍k. 

Sò͘-ha̍k hō͘ lâng kám-kak iu-ngá, hoān-sè i ê pún-chit tō sī bí-ha̍k, hoān-sè i pun-sin ū 1 chióng bí-kám, m̄-koh che lóng chin oh bêng-khak piáu-ta̍t. Kán-tan-sèng kap phó͘-phiàn-sèng sī sò͘-ha̍k 2 ê chin tiong-iàu ê te̍k-sek. Chiah-ê khoàⁿ--khí-lâi taù bē chò-hoé ê sèng-chit ū-tang-sî-á ē-tàng kiat-ha̍p, chhiūⁿ kóng kā chi̍t-koá bô kāng hun-ki ê sò͘-ha̍k it-thé-hoà, iah sī hoat-tián ē-sái kiōng-tông kè-sǹg ê ke-si. Sûn sò͘-ha̍k chho͘-chho͘ khoàⁿ ká-ná kan-taⁿ i ê bí-kám ū koá kè-ta̍t, kan-taⁿ sò͘-ha̍k-ka ū chhù-bī. M̄-koh gián-kiù chi̍t-ē chìn-tián liáu-aū, sûn sò͘-ha̍k tiāⁿ ē pìⁿ-chiâⁿ èng-iōng sò͘-ha̍k.

## Sò͘-ha̍k hû-hō
Sò͘-ha̍k ê bûn-chiuⁿ bô kài hó tha̍k. Stephen Hawking 1988 chhut ê *[[Sî-kan kán-sú]]* (*A Brief History of Time*) lāi-té kan-taⁿ ū 1 tiâu sò͘-ha̍k kong-sek, in-ūi chhut-pán-siong kóng múi 1 tiâu kong-sek tō ē kā siau-lō͘ kàng chi̍t-poàⁿ.

Sò͘-ha̍k-ka chin giâm-keh iau-kiû ài kā siuⁿ beh piáu-ta̍t--ê siá chheng-chhó, chóng--sī sò͘-ha̍k iáu-sī ká-ná hoat-lu̍t tiâu-bûn hiah pháiⁿ tha̍k.  Sò͘-ha̍k-ka khok-chhiong chū-jiân gí-giân, ēng tēng kah chin bêng-khak ê sò͘-ha̍k bêng-sû, sò͘-ha̍k hû-hō kap bûn-hoat (gí-hoat) lâi ta̍t kaù giâm-keh ê bo̍k-phiau. Ū chi̍t-koá bêng-sû chioh ēng phó͘-thong ê bêng-sû, m̄-koh ì-sù bô tī-tāi, chhiūⁿ khoân (*ring*), kûn (*group*), lūi (*category*); ū-ê bêng-sû sī sò͘-ha̍k choan-iōng--ê, chhiūⁿ homotopy, Hilbert khong-kan. Thiaⁿ kóng Henri Poincaré hông soán ji̍p-khì Hoat-kok Gián-kiù-īⁿ (*Académie Française*) tō sī beh chhiáⁿ i lâi tēng-gī *[[automorphe]]*.

## Sò͘-ha̍k kap kho-ha̍k
Carl Friedrich Gauss kóng sò͘-ha̍k sī kho-ha̍k ê lú-ông. Sò͘-ha̍k bu̍t-lí-ha̍k-ka Leon M. Lederman khau-sé kóng bu̍t-lí-ha̍k-ka kan-taⁿ sūn-thàn sò͘-ha̍k-ka, sò͘-ha̍k-ka kan-taⁿ sūn-thàn Siōng-tè, sui-bóng chiah-nī kò͘-khiam ê sò͘-ha̍k-ka chin oh chhoē.

## Sò͘-ha̍k gián-kiù ê hoān-ûi
Chá-kî ê sò͘-ha̍k ūi tio̍h siong-gia̍p, chhek-liông, ī-chhek thian-bûn hiān-siōng, hoat-tián kiat-kò͘, khong-kan, piàn-tōng ê gián-kiù, chhiūⁿ tāi-sò͘ (*algebra*), kí-hô-ha̍k (*geometry*), hun-sek (*analysis*). Aū--lâi, tau̍h-tau̍h koh hoat-tián kaù lô-chek, khah kán-tan ê ki-chhó͘ hē-thóng (*foundations*), kap khah si̍t-iōng ê èng-iōng sò͘-ha̍k (*applied mathematics*).

## Sò͘-ha̍k chú-iàu ê gián-kiù
Chia lia̍t--ê sī khah chú-iàu ê gián-kiù tê-ba̍k. Khah oân-chéng ê gián-kiù chú-tê chhiáⁿ chham-khó sò͘-ha̍k chú-tê lia̍t-toaⁿ.

### Sò͘-liōng (''Quantity'')
Leh chhú-lí sò͘-ba̍k, chi̍p-ha̍p, toā-sè (*size*) chit lūi ê būn-tê, kap chaú-chhoē chhú-lí ê hong-hoat.

:

:Sò͘-ba̍k – Chū-jiân-sò͘ – Chéng-sò͘ – pí-sò͘ – Si̍t-sò͘ – Ho̍k-cha̍p-sò͘ – Hypercomplex numbers – Quaternions – Octonions – Sedenions – Hyperreal numbers – Surreal numbers – Ordinal numbers – Cardinal numbers – ''p''-adic numbers – Integer sequences – Sò͘-ha̍k tiāⁿ-sò͘ – Number names – Bû-hān – ki-té (Base)]]

### Piàn-tōng (''Change'')
Leh chhú-lí sò͘-ba̍k kap hâm-sò͘ ê piàn-hoà.

:

:Sǹg-siàu (*Arithmetic*) – Bî-chek-hun – Hiòng-liōng bî-chek-hun  –  Hun-sek (*Analysis*) – Bî-hun hong-têng (*Differential equations*) – Tōng-thài hē-thóng (*Dynamical systems*) – Hūn-tūn lí-lūn (*Chaos theory*) – Hâm-sò͘ lia̍t-toaⁿ

### Kiat-kò͘ (''Structure'')
Leh chhú-lí toā-sè (*size*), tùi-thīn (*symmetry*) kap chi̍t-koá sò͘-ha̍k kiat-kò͘ (*mathematical structure*) ê būn-tê.

:

:Thiu-siōng tāi-sò͘ – Sò͘-lūn – Tāi-sò͘ kí-hô-ha̍k –  Kûn-lūn (*Group theory*) – Monoids –  Hun-sek (*Analysis*) – Topology – Soàⁿ-sèng tāi-sò͘ (*Linear algebra*) – Tô͘-lūn (*Graph theory*) – Universal tāi-sò͘ (*Universal algebra*) – Category theory – Order theory – Measure theory

### Khong-kan koan-hē (''Spatial relations'')
Ē-tàng khoàⁿ ê sò͘-ha̍k.

:

:Topology – Kí-hô-ha̍k – Saⁿ-kak-hoat – Tāi-sò͘ kí-hô-ha̍k – Bî-hun kí-hô-ha̍k – Bî-hun topology (*Differential topology*) – Tāi-sò͘ topology (*Algebraic topology*) – Soàⁿ-sèng tāi-sò͘ – Chhùi-hêng kí-hô-ha̍k (*Fractal geometry*)

### lî-sàn sò͘-ha̍k (''Discrete mathematics'')
Leh chhú-lí kan-taⁿ ē-tàng 1 ê 1 ê sǹg ê mi̍h-á.

:

:Combinatorics – Native chi̍p-ha̍p-lūn (*Naive set theory*) – Kè-sǹg lí-lūn (*Theory of computation*) – Bi̍t-bé-ha̍k (*Cryptography*) – Tô͘-lūn (*Graph theory*)

### Èng-iōng sò͘-ha̍k (''Applied mathematics'')
Leh kái-koat hiān-si̍t būn-tê ê sò͘-ha̍k.

: Sò͘-ha̍k bu̍t-lí-ha̍k (*Mathematical physics*) – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Ki-lu̍t – Thóng-kè-ha̍k – Financial mathematics – Game theory – Mathematical biology – Bi̍t-bé-ha̍k (*Cryptography*) – Chu-sìn lí-lūn (*Information theory*)

### Tēng-lí (''Theorems'')
Che lia̍t chi̍t-koá m̄-sī sò͘-ha̍k-ka mā ē kám-kak sim-sek ê tēng-lí. Oân-chéng ê chu-liāu chhiáⁿ chham-khó Tēng-lí lia̍t-toaⁿ.

: Ti̍t-kak tēng-lí (*Pythagorean theorem*) – Fermat choè-aū tēng-lí (*Fermat's last theorem*) – Gödel's incompleteness theorems – Cantor's diagonal argument – 4 sek tēng-lí (*Four color theorem*) – Zorn's lemma – Euler's identity – Church-Turing thesis– Riemann hypothesis – Continuum hypothesis –  Central limit theorem – Sǹg-siàu ki-pún tēng-lí (*Fundamental theorem of arithmetic*) – Tāi-sò͘ ki-pún tēng-lí (*Fundamental theorem of algebra*) – Bî-chek-hun ki-pún tēng-lí (*Fundamental theorem of calculus*) – Fundamental theorem of projective geometry – Gauss-Bonnet theorem.

### Chhai-chhek (''Conjectures'')
Che lia̍t chi̍t-koá tng-leh gián-kiù ê būn-tê. Oân-chéng ê chu-liāu chhiáⁿ chham-khó Chhai-chhek lia̍t-toaⁿ.

:Goldbach Chhai-chhek – Twin prime conjecture – Collatz conjecture – Poincaré conjecture – classification theorems of surfaces –  P=NP

Lēng-goā, ū-ê sò͘-ha̍k-ka jīn-ûi continuum hypothesis kap ZFC bô tī-tāi, bián chhap i; ū-ê khah chek-ke̍k leh gián-kiù.

### Ki-chân kap hong-hoat (''Foundations and methods'')
Leh su-khó sò͘-ha̍k ê pún-chit kap sò͘-ha̍k beh án-choáⁿ gián-kiù.

:Philosophy of mathematics – Mathematical intuitionism – Mathematical constructivism – Sò͘-ha̍k ê ki-chân (Foundations of mathematics) – Chi̍p-ha̍p-lūn – Hû-hō lô-chek (*Symbolic logic*) – Bô͘-hêng-lūn (Model theory) – Category theory – Lô-chek – Reverse Mathematics – Sò͘-ha̍k hû-hō-pió (*Table of mathematical symbols*)

### Sò͘-ha̍k-sú kap sò͘-ha̍k-ka
Chhiáⁿ chham-khó sò͘-ha̍k-sú lia̍t-toaⁿ

: Sò͘-ha̍k-sú – Sò͘-ha̍k nî-pió – Sò͘-ha̍k-ka – Fields Chióng – Abel Prize – Millennium Prize Problems (Clay Math Prize) – International Mathematical Union – Mathematics competitions – Lateral thinking – Mathematical abilities and gender issues

### Sò͘-ha̍k ê hûn-iáⁿ
:Sò͘-ha̍k kap kiàn-tio̍k – Sò͘-ha̍k kap kaù-io̍k – Sò͘-ha̍k kap im-ga̍k

## Tùi sò͘-ha̍k ê gō͘-kái
Sò͘-ha̍k m̄ sī 1 ê sím-mih būn-tê lóng kái-koat tiāu ê hē-thóng, i lāi-té iáu chin chē ài gián-kiù ê būn-tê.

Ké-sò͘-ha̍k khoàⁿ--khí-lâi chhin-chhiūⁿ sò͘-ha̍k, i bô tī ha̍k-su̍t-kài chiap-siū kàm-tok, ū-sî-á koh si sò͘-ha̍k-ka pìⁿ--ê. I thong-siông sī leh thó-lūn khah ū-miâ ê būn-tê, m̄-koh i koh m̄ sī ēng lâng chèng-bêng koè ê lí-lūn chò ki-chân lâi chèng-bêng. Ké-sò͘-ha̍k kap sò͘-ha̍k tō ká-ná ké-kho-ha̍k kap kho-ha̍k chhan-chhiūⁿ. Chit khoán tāi-chì ē lâi hoat-seng ê goân-in ū:

* bô hù-ha̍p giâm-keh ê iau-kiû;
* in-ūi bó͘ chi̍t-koá in-sò͘, bô chun-chiàu sò͘-ha̍k lūn-bûn ài seng keng-koè tông-liâu sím-cha, tī ha̍k-su̍t kî-khan hoat-piáu, ... chiah-ê têng-sū;
* tùi bûn-hiàn bô kài liáu-kái.
## Chham-khó

* Benson, Donald C., *The Moment of Proof: Mathematical Epiphanies*, Oxford University Press, USA; New Ed edition (December 14, 2000). ISBN 0-19-513919-4.
* Boyer, Carl B., *A History of Mathematics*, Wiley; 2 edition (March 6, 1991). ISBN 0-471-54397-7. — A concise history of mathematics from the Concept of Number to contemporary Mathematics.
* Courant, R. and H. Robbins, *What Is Mathematics? : An Elementary Approach to Ideas and Methods*, Oxford University Press, USA; 2 edition (July 18, 1996). ISBN 0-19-510519-2.
* Davis, Philip J. and Hersh, Reuben, *[[The Mathematical Experience]]*. Mariner Books; Reprint edition (January 14, 1999). ISBN 0-395-92968-7.— A gentle introduction to the world of mathematics.
* Eves,  Howard, *An Introduction to the History of Mathematics*, Sixth Edition, Saunders, 1990, ISBN 0-03-029558-0.
* Gullberg, Jan, *Mathematics—From the Birth of Numbers*. W. W. Norton & Company; 1st edition (October 1997). ISBN 0-393-04002-X. — An encyclopedic overview of mathematics presented in clear, simple language.
expanded version of a Soviet mathematics encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM, and online http://eom.springer.de/default.htm.
* Jourdain, Philip E. B., *The Nature of Mathematics*, in *The World of Mathematics*, James R. Newman, editor, Dover, 2003, ISBN 0-486-43268-8.
* Kline, Morris, *Mathematical Thought from Ancient to Modern Times*, Oxford University Press, USA; Paperback edition (March 1, 1990). ISBN 0-19-506135-7.
* 
* *[[The Oxford Dictionary of English Etymology]]*, 1983 reprint. ISBN 0-19-861112-9.
* Pappas, Theoni, *The Joy Of Mathematics*, Wide World Publishing; Revised edition (June 1989). ISBN 0-933174-65-9.
* Peterson, Ivars, *Mathematical Tourist, New and Updated Snapshots of Modern Mathematics*, Owl Books, 2001,  ISBN 0-8050-7159-8.
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