AISC Meets Natural Typography James H. Davenport Department of - PowerPoint PPT Presentation

AISC Meets Natural Typography James H. Davenport Department of Computer Science University of Bath Bath BA2 7AY England J.H.Davenport@bath.ac.uk July 31, 2008

Notation exists to be abused . . . the abuses of language without which any mathematical text threatens to be- come pedantic and even unreadable. (Bourbaki) ❜✉t s♦♠❡ ❛❜✉s❡ ✐s ♠♦r❡ ❤❛r♠❢✉❧ t❤❛♥ ♦t❤❡rs✳ 1

Notation exists to be abused . . . the abuses of language without which any mathematical text threatens to be- come pedantic and even unreadable. (Bourbaki) but some abuse is more harmful than others. 2

The trivial differences Intervals The “Anglo-saxon” way (0 , 1] and the “French” way ]0 , 1]. � ■♥✈❡rs❡ ❢✉♥❝t✐♦♥s ■s ❛r❝s✐♥ t❤❡ ✭❛❄✱ ✇❤✐❝❤❄❄✮ s✐♥❣❧❡✲✈❛❧✉❡❞ ✐♥✈❡rs❡ ❛♥❞ ❆r❝s✐♥ t❤❡ ♠✉❧t✐✲ ✈❛❧✉❡❞ ✭❆♥❣❧♦✲❙❛①♦♥✮✱ ♦r t❤❡ ❝♦♥✈❡rs❡ ✭❋r❡♥❝❤✮❄ ✐ ♦r ❥ ■♥ ♣r❛❝t✐❝❡✱ t❤✐s ❝❛✉s❡s ❧✐tt❧❡ ❝♦♥❢✉s✐♦♥ ❢♦r ❡①♣❡rts✱ s♦♠❡ ❢♦r st✉❞❡♥ts✳ 3

The trivial differences Intervals The “Anglo-saxon” way (0 , 1] and the “French” way ]0 , 1]. � Inverse functions Is arcsin the (a?, which??) single-valued inverse and Arcsin the multi- valued (Anglo-Saxon), or the converse (French)? ✐ ♦r ❥ ■♥ ♣r❛❝t✐❝❡✱ t❤✐s ❝❛✉s❡s ❧✐tt❧❡ ❝♦♥❢✉s✐♦♥ ❢♦r ❡①♣❡rts✱ s♦♠❡ ❢♦r st✉❞❡♥ts✳ 4

The trivial differences Intervals The “Anglo-saxon” way (0 , 1] and the “French” way ]0 , 1]. � Inverse functions Is arcsin the (a?, which??) single-valued inverse and Arcsin the multi- valued (Anglo-Saxon), or the converse (French)? i or j In practice, this causes little confusion for experts, some for students. 5

metric tensor Is the metric tensor for flat Min-   − 1 0 0 0 0 1 0 0     kowski space or its nega-   0 0 1 0     0 0 0 1  1 0 0 0  0 − 1 0 0     tive ? Is the temporal   0 0 − 1 0     0 0 0 − 1 variable the last, rather than the first, co-   1 0 0 0 0 1 0 0     ordinate, giving , or its   0 0 1 0     0 0 0 − 1 negative? 6

� n � , C k n , C n Binomial coefficients � ? k k ✵ ✷ N ❄ ❑♥♦✇❧❡❞❣❡ ♦❢ ❧✐♥❣✉✐st✐❝ ❝♦♥t❡①t ♠❛② ❤❡❧♣ ❞❡❝✐❞❡ t❤✐s q✉❡st✐♦♥✱ ❜✉t ✐s ❢❛r ❢r♦♠ ❝❡rt❛✐♥✳ ✏ ✑ ❈❧❡❛r❧② ✭ � ✶✮ ✭ ♣ � ✶✮ ❂ ✷ ✿ ✐t✬s ❛ q✉❛❞r❛t✐❝ r❡s✐❞✉❡ s②♠❜♦❧✦ ❭❧❡❢t✭❭❢r❛❝④✲✶⑥④♣⑥❭r✐❣❤t✮ ✐s✱ ❛❧❛s✱ ❤♦✇ ♣r♦❢❡s✲ s✐♦♥❛❧ t②♣❡s❡tt❡rs ❡♥❝♦❞❡ ✐t✳ 7

� n � , C k n , C n Binomial coefficients � ? k k 0 ∈ N ? Knowledge of linguistic context may help decide this question, but is far from certain. ✏ ✑ ❈❧❡❛r❧② ✭ � ✶✮ ✭ ♣ � ✶✮ ❂ ✷ ✿ ✐t✬s ❛ q✉❛❞r❛t✐❝ r❡s✐❞✉❡ s②♠❜♦❧✦ ❭❧❡❢t✭❭❢r❛❝④✲✶⑥④♣⑥❭r✐❣❤t✮ ✐s✱ ❛❧❛s✱ ❤♦✇ ♣r♦❢❡s✲ s✐♦♥❛❧ t②♣❡s❡tt❡rs ❡♥❝♦❞❡ ✐t✳ 8

� n � , C k n , C n Binomial coefficients � ? k k 0 ∈ N ? Knowledge of linguistic context may help decide this question, but is far from certain. � − 1 � Clearly ( − 1) ( p − 1) / 2 : it’s a quadratic residue p symbol! ❭❧❡❢t✭❭❢r❛❝④✲✶⑥④♣⑥❭r✐❣❤t✮ ✐s✱ ❛❧❛s✱ ❤♦✇ ♣r♦❢❡s✲ s✐♦♥❛❧ t②♣❡s❡tt❡rs ❡♥❝♦❞❡ ✐t✳ 9

� n � , C k n , C n Binomial coefficients � ? k k 0 ∈ N ? Knowledge of linguistic context may help decide this question, but is far from certain. � − 1 � Clearly ( − 1) ( p − 1) / 2 : it’s a quadratic residue p symbol! \left(\frac{-1}{p}\right) is, alas, how profes- sional typesetters encode it. 10

“this has the usual mathematical meaning” (1) Mathematics a 1 ∪ a 2 ∪ a 3 L A T X a_1 \cup a_2 \cup a_3 E OpenMath <OMS name="union" cd="set1"/> MathML <apply> <union/> <i> a 1 </i>...</apply> 11

“this has the usual mathematical meaning” (2) Mathematics � { a 1 , a 2 , a 3 } L A T X \bigcup \{a_1,a_2,a_3\} E OpenMath <OMS name="big union" cd="set3"/> or <OMS name="apply to list" cd="fns2"/> MathML <apply> <union/> <bvar>i</bvar> <domain ...> <set> <i> a 1 </i>...</set> 12

“this has the usual mathematical meaning” (3) Mathematics � 3 i =1 a i L A T X \bigcup_{i=1}^3 a_i E OpenMath big union on make list MathML <apply> <union/> <bvar>i</bvar> <lowlimit>... 13

Pq and friends (1) ❩ ✉ ✵ ♣q ✷ ✭ t ✮❞ t Pq✭ ✉ ✮ ❂ ✭✶✻ ✿ ✷✺ ✿ ✶✮ ✭✇❤❡r❡ ♣q ✷ ✭ t ✮ ♠❡❛♥s ♣q✭ t ✮ ✷ ✱ ❛♥❞ ♥♦t ♣ ✁ q ✷ ✮ ❙❤♦rt ❢♦r ✶✷ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠ ❩ ✉ ✵ s♥ ✷ ✭ t ✮❞ t❀ ❙♥✭ ✉ ✮ ❂ s✐♥❝❡ ♣❀ q❀ ✷ ❢ s❀ ❝❀ ♥❀ ❞ ❣ ✳ ❇✉t ✇❤❡♥ q ❂ s ✒ ✓ ❩ ✉ ♣q ✷ ✭ t ✮ � Pq✭ ✉ ✮ ❂ ❞ t � ✿ ✵ 14

Pq and friends (1) � u 0 pq 2 ( t )d t Pq( u ) = (16 . 25 . 1) ✭✇❤❡r❡ ♣q ✷ ✭ t ✮ ♠❡❛♥s ♣q✭ t ✮ ✷ ✱ ❛♥❞ ♥♦t ♣ ✁ q ✷ ✮ ❙❤♦rt ❢♦r ✶✷ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠ ❩ ✉ ✵ s♥ ✷ ✭ t ✮❞ t❀ ❙♥✭ ✉ ✮ ❂ s✐♥❝❡ ♣❀ q❀ ✷ ❢ s❀ ❝❀ ♥❀ ❞ ❣ ✳ ❇✉t ✇❤❡♥ q ❂ s ✒ ✓ ❩ ✉ ♣q ✷ ✭ t ✮ � Pq✭ ✉ ✮ ❂ ❞ t � ✿ ✵ 15

Pq and friends (1) � u 0 pq 2 ( t )d t Pq( u ) = (16 . 25 . 1) (where pq 2 ( t ) means pq( t ) 2 , and not p · q 2 ) ❙❤♦rt ❢♦r ✶✷ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠ ❩ ✉ ✵ s♥ ✷ ✭ t ✮❞ t❀ ❙♥✭ ✉ ✮ ❂ s✐♥❝❡ ♣❀ q❀ ✷ ❢ s❀ ❝❀ ♥❀ ❞ ❣ ✳ ❇✉t ✇❤❡♥ q ❂ s ✒ ✓ ❩ ✉ ♣q ✷ ✭ t ✮ � Pq✭ ✉ ✮ ❂ ❞ t � ✿ ✵ 16

Pq and friends (1) � u 0 pq 2 ( t )d t Pq( u ) = (16 . 25 . 1) (where pq 2 ( t ) means pq( t ) 2 , and not p · q 2 ) Short for 12 equations of the form � u 0 sn 2 ( t )d t, Sn( u ) = since p, q, ∈ { s, c, n, d } . ❇✉t ✇❤❡♥ q ❂ s ✒ ✓ ❩ ✉ ♣q ✷ ✭ t ✮ � Pq✭ ✉ ✮ ❂ ❞ t � ✿ ✵ 17

Pq and friends (1) � u 0 pq 2 ( t )d t Pq( u ) = (16 . 25 . 1) (where pq 2 ( t ) means pq( t ) 2 , and not p · q 2 ) Short for 12 equations of the form � u 0 sn 2 ( t )d t, Sn( u ) = since p, q, ∈ { s, c, n, d } . But when q = s � u pq 2 ( t ) − 1 d t − 1 � � Pq( u ) = u. t 2 0 18

Pq and friends (2) pq( u ) = pr( u ) (16 . 3 . 4) qr( u ) ✭❡①❝❡♣t t❤❛t ❤❡r❡ t❤❡r❡ ✐s ♥♦ ❞✐st✐♥❝t♥❡ss ❛s✲ s✉♠♣t✐♦♥✱ ❜✉t ♣♣ ✐s t♦ ❜❡ t❛❦❡♥ ❛s t❤❡ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ ✶✮✳ 19

Pq and friends (2) pq( u ) = pr( u ) (16 . 3 . 4) qr( u ) (except that here there is no distinctness as- sumption, but pp is to be taken as the constant function 1). 20

Juxtaposition (1) ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❚❤✐s ✐s ❡♥❝♦❞❡❞ ❛s ✫■♥✈✐s✐❜❧❡❚✐♠❡s❀ ✐♥ ▼❛t❤▼▲✳ ❚❤✐s ♦♥❧② ❛♣♣❧✐❡s t♦ ✐t❛❧✐❝ ❧❡t✲ t❡rs✿ ❥✉①t❛♣♦s❡❞ r♦♠❛♥ ❧❡tt❡rs ❝♦♥st✐t✉t❡ ❛ s✐♥❣❧❡ ❧❡①❡♠❡✱ ❛s ✐♥ s✐♥ ♦r ♣q✳ ✭❋✉♥❝t✐♦♥✮ ❆♣♣❧✐❝❛t✐♦♥ s✐♥ ① ♦t❤❡r✇✐s❡ s✐♥✭ ① ✮✳ s✐♥✭ ① ✰ ② ✮ ✐s ❞✐☛❡r❡♥t ❢r♦♠ ✷✭ ① ✰ ② ✮✱ ❛♥❞ ❢ ✭ ① ✰ ② ✮ ✐s ❄❄ ❚❤✐s ✐s ❡♥❝♦❞❡❞ ❛s ✫❆♣♣❧②❋✉♥❝t✐♦♥❀ ✳ ❆❞❞✐t✐♦♥ ✹ ❝♦✉❧❞ ♦t❤❡r✇✐s❡ ❜❡ r❡♥❞❡r❡❞ ❛s ✹✰ ✳ ❚❤✐s ✐s ✭♥♦✇✮ ❡♥❝♦❞❡❞ ❛s ✫■♥✈✐s✐❜❧❡P❧✉s❀ ✳ 21