Bagaimana Cara Menulis Matematika

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Prakata

Ini adalah esai yang subjektif, dan judulnya menyesatkan; mungkin judul yang lebih jujur adalah BAGAIMANA CARA SAYA MENULIS MATEMATIKA. Esai ini dimulai dari suatu komite di American Mathematical Society, tempat saya bekerja untuk waktu yang singkat, tetapi dengan cepat menjadi suatu proyek pribadi yang ikut melarikan diri bersama saya. Dalam upaya untuk mengendalikannya, saya meminta beberapa teman untuk membaca dan mengkritiknya. Kritik-kritik yang diberikan sangat bagus; tajam, jujur dan konstruktif; dan saling bertentangan. “Kurang bukti-bukti konkret” kata seorang; “tidak setuju bahwa contoh yang lebih konkret diperlukan” kata yang lain. “Terlalu panjang” kata seorang yang lain; “mungkin butuh lebih banyak hal lagi” kata yang lain lagi. “Ada metode tradisional (dan efektif) untuk meminimalkan rasa bosan pembuktian yang panjang, seperti memecahnya dalam beberapa lema,” kata seorang. “Salah satu hal yang yang sangat mengganggu saya, adalah kebiasaan (terutama para pemula) untuk menyajikan sebuah bukti sebagai rangkaian panjang lema, yang dinyatakan dengan rumit dan sangat membosankan” kata yang lain.

Ada satu hal yang disetujui oleh sebagian besar penasihat saya; bahwa menulis esai semacam ini pasti merupakan tugas tanpa pamrih. Penasihat 1: “Pada saat seorang matematikawan selesai menulis makalah keduanya, dia merasa yakin bahwa dia tahu bagaimana menulis makalah, dan akan bereaksi terhadap saran dengan ketidaksabaran.” Penasihat 2: “Kita semua, saya pikir, diam-diam merasa bahwa jika kita mau berusaha, kita dapat menjadi penulis yang sangat baik. Orang-orang yang cukup rendah diri dengan [tingkat] matematikanya akan merasa gusar jika kemampuan mereka untuk menulis dengan baik dipertanyakan.” Penasihat 3 menggunakan bahasa yang paling keras; dia memperingatkan bahwa karena saya tidak dapat menunjukkan kedalaman intelektual yang luar biasa dalam diskusi tentang hal terkait teknik, saya tidak boleh terkejut dengan “cemoohan yang mungkin Anda dapatkan dari beberapa kolega kami yang lebih sombong.”

Para penasihat saya adalah ahli matematika yang mapan dan terkenal. Sebuah pujian dari saya di sini tidak akan menambah apa pun pada ketenaran mereka, malahan saya mungkin salah paham, salah menempatkan, dan salah menerapkan nasihat mereka tentang prosedur mengutip maupun mengungkapkan terima kasih tanpa nama. Saya tidak lelah berterima kasih untuk itu, dan dengan sungguh-sungguh mengakui bahwa tanpa bantuan mereka, esai ini akan lebih buruk.

“hier stehe ich; ich kann nicht anders.”

Tidak ada resep dan apapun itu

Saya pikir saya bisa memberitahu seseorang bagaimana cara menulis, tetapi saya tidak bisa memikirkan siapa yang mau mendengarkannya. Kemampuan untuk berkomunikasi secara efektif, kekuatan untuk dapat dimengerti, adalah bawaan dari lahir; saya percaya, atau dalam hal apa pun, kemampuan ini sudah diperoleh sejak dini sehingga pada saat seseorang membaca nasihat saya tentang topik ini, dia kemungkinan tidak akan berubah karenanya.

Lalu, mengapa melanjutkannya? Alasan kecilnya adalah sebuah harapan bahwa apa yang saya tuliskan tadi tidak sepenuhnya benar; dan, bagaimanapun juga, saya ingin mendapatkan kesempatan untuk mencoba melakukan apa yang mungkin tidak dapat dilakukan. Alasan yang lebih praktis yaitu, bahwa dalam seni-seni yang memerlukan bakat bawaan, bahkan mereka yang berbakat, yang terlahir dengan bakat tersebut, biasanya tidak terlahir dengan pengetahuan mendalam mengenai semua trik yang ada. Beberapa esai seperti ini dapat berfungsi untuk “mengingatkan” (dalam pengertian Plato) orang-orang yang ingin menjadi dan ditakdirkan untuk menjadi penulis di masa depan tentang teknik-teknik yang dianggap berguna oleh para penulis di masa lalu.

Masalah dasar dalam menulis matematika sama dengan menulis biologi, menulis novel, atau menulis petunjuk untuk merakit harpsichord: masalahnya adalah mengomunikasikan ide. Untuk melakukannya, dan untuk melakukannya dengan jelas, Anda harus memiliki sesuatu untuk dikatakan, dan Anda harus memiliki seseorang untuk mengatakannya, Anda harus mengatur apa yang ingin Anda katakan, dan Anda harus mengaturnya sesuai urutan yang Anda inginkan, Anda harus menulisnya, menulis ulang, dan menulis ulang beberapa kali, dan Anda harus bersedia untuk berpikir keras dan bekerja keras pada detail teknis seperti diksi, notasi, dan tanda baca. Hanya itu saja yang perlu dilakukan.

Katakan sesuatu

Mungkin terlihat tidak penting untuk menegaskan bahwa untuk mengatakan sesuatu dengan baik, Anda harus memiliki sesuatu untuk dikatakan, tetapi ini bukan candaan. Banyak tulisan yang buruk, baik secara matematis maupun tidak, disebabkan oleh pelanggaran prinsip pertama tersebut. Seperti halnya ada dua alasan sebuah barisan tidak memiliki limit (tidak ada titik limit atau terlalu banyak), ada dua alasan sebuah tulisan tidak memiliki topik (tidak ada ide atau terlalu banyak).

Penyakit pertama lebih sulit untuk didiagnosa. Sulit untuk menulis banyak kata yang kosong, terutama dalam matematika, tapi itu bisa dilakukan, dan hasilnya pasti akan sulit untuk dibaca. Ada sebuah buku klasik karya Carl Theodore Heisel¹ yang bisa dijadikan contoh. Buku ini penuh dengan kata-kata yang dieja dengan benar yang dirangkai dalam tata bahasa yang baik, tetapi setelah tiga dekade melihatnya sesekali, saya masih tidak dapat membaca dua halaman berturut-turut dan membuat rangkuman satu paragraf tentang apa yang ia katakan; alasannya, menurut saya, karena ia tidak mengatakan apa-apa.

Penyakit kedua sangat umum terjadi: ada banyak buku yang melanggar prinsip dengan mencoba mengatakan terlalu banyak hal. Para guru matematika dasar di Amerika Serikat sering mengeluh bahwa semua buku kalkulus itu jelek. Ini adalah kasus yang tepat. Buku kalkulus buruk karena tidak ada mata pelajaran seperti kalkulus; kalkulus bukanlah sebuah mata pelajaran karena terdiri dari banyak mata pelajaran. Apa yang kita sebut kalkulus saat ini adalah gabungan dari sedikit logika dan teori himpunan, beberapa teori aksiomatik dari lapangan terurut lengkap, geometri analitik dan topologi, yang terakhir dalam artian “umum” (limit dan fungsi kontinu) juga dalam artian aljabar (orientasi), teori variabel-real yang terkenal (diferensiasi), manipulasi simbol kombinatorik yang disebut integrasi formal, pengantar teori ukuran dimensi-rendah, beberapa geometri diferensial, pengantar analisis klasik fungsi trigonometri, eksponensial, dan logaritmik, dan, tergantung pada banyak kertas yang tersedia dan kecenderungan pribadi penulis, beberapa persamaan diferensial yang standar, mekanika dasar, dan sejumlah kecil matematika terapan. Sulit untuk menulis buku yang bagus tentang salah satu hal di atas; campurannya, tidak mungkin.

Sebutir permata kecil dari Nelson berupa bukti bahwa fungsi harmonik terbatas adalah sebuah konstanta² dan risalah monumental oleh Dunford dan Schwartz tentang analisis fungsional³ adalah contoh tulisan matematika yang memiliki sesuatu untuk dikatakan. Karya Nelson tidak sampai setengah halaman dan Dunford-Schwartz lebih dari empat ribu kali lipatnya, tetapi jelas bahwa penulisnya memiliki gagasan yang jelas tentang apa yang ingin mereka katakan. Subjeknya diuraikan dengan jelas; [jelas bahwa itu] sebuah subjek; semuanya saling berkaitan; ada sesuatu yang ingin disampaikan.

Memiliki sesuatu untuk dikatakan adalah unsur terpenting dari tulisan yang baik – sedemikian rupa sehingga jika idenya cukup penting, karya tersebut memiliki peluang untuk menjadi abadi, meskipun tidak teratur dengan baik dan diekspresikan dengan canggung. Bukti Birkhoff tentang teorema ergodik⁴ almost maximally membingungkan, dan “surat terakhir” Vanzetti⁵ terasa berhenti tiba-tiba dan janggal, tetapi siapa pun yang membacanya pasti senang bahwa itu telah ditulis. Namun, hanya bertahan dengan prinsip pertama saja, jarang sekali berhasil dan tidak akan disukai.

Bicaralah dengan orang lain

Prinsip kedua dari menulis yang baik adalah menulis untuk seseorang. ketika Anda memutuskan untuk menulis sesuatu, tanyakan pada diri Anda sendiri siapa yang ingin Anda jangkau. Apakah Anda menulis catatan harian untuk dibaca sendiri, surat untuk teman, pengumuman hasil penelitian untuk para ahli, atau buku teks untuk mahasiswa? Masalahnya hampir sama dalam hal apa pun; yang berbeda adalah jumlah motivasi yang perlu Anda masukkan, tingkat informal yang Anda izinkan untuk diri sendiri, kerepotan menambahkan detail yang diperlukan, dan berapa kali hal-hal harus diulang. Semua tulisan dipengaruhi oleh pembaca, tetapi, diberikan pembaca, masalah seorang penulis adalah mencoba berkomunikasi dengannya sebaik mungkin.

Para penerbit tahu bahwa 25 tahun adalah usia yang cukup tua untuk sebagian besar buku matematika; untuk makalah penelitian, lima tahun (sebagai tebakan) adalah usia usang yang umum. (Tentu saja ada makalah berusia 50 tahun yang tetap hidup dan buku yang mati dalam lima tahun). Tulisan matematika bersifat fana, tentu saja, tetapi jika Anda ingin menjangkau pembaca Anda sekarang, Anda harus menulis seolah-olah untuk waktu yang lama.

Saya suka menentukan pembaca saya tidak hanya dalam pengertian yang abstrak dan luas (misalnya, ahli topologi profesional, atau mahasiswa pascasarjana tahun kedua), tetapi juga dalam pengertian yang sangat spesifik dan pribadi. Hal ini membantu saya untuk memikirkan seseorang, mungkin seseorang yang pernah berdiskusi dengan saya dua tahun yang lalu, atau mungkin seorang kolega yang ramah namun tidak mau tahu, dan kemudian mengingat dia saat saya menulis. Dalam esai ini, contohnya, saya berharap dapat menjangkau para mahasiswa matematika yang berada di awal pengerjaan tesis mereka, tetapi, pada saat yang sama, saya juga tetap memperhatikan seorang kolega yang tingkah-lakunya [saya pikir] dapat diperbaiki. Tentu saja saya berharap bahwa (a) dia akan berubah mengikuti cara saya, tetapi (b) dia tidak akan tersinggung jika dan ketika dia menyadari bahwa saya menulis untuknya.

Ada keuntungan dan kerugian dengan menyapa pembaca yang sangat spesifik. Keuntungannya adalah bahwa hal ini memudahkan saat perlu membaca pikiran [pembaca]; kerugiannya adalah bahwa Anda akan tergoda untuk memanjakan diri dengan [membuat] komentar-komentar yang sinis dan “inside jokes” yang berlebihan. Jelas sekali apa yang saya maksud dengan kerugiannya, dan itu jelas buruk; hindarilah. Kelebihannya layak mendapat penekanan lebih lanjut.

Penulis harus mengantisipasi dan menghindari kesulitan pembaca. Ketika ia menulis, ia harus terus mencoba membayangkan apa yang ada di dalam kata-kata yang sedang ditulisnya yang mungkin akan menyesatkan pembaca, dan meluruskannya. Saya akan memberikan contoh satu atau dua hal semacam ini nanti; untuk saat ini, saya tekankan bahwa mengingat pembaca tertentu tidak hanya membantu dalam aspek pekerjaan penulis, melainkan sangat penting.

Mungkin tidak perlu dikatakan, tetapi tidak ada salahnya untuk mengatakan, bahwa pembaca yang sebenarnya dijangkau mungkin sangat berbeda dari yang dimaksudkan. Tidak ada yang menjamin bahwa tujuan seorang penulis selalu sempurna. Menurut saya, lebih baik memiliki bidikan yang pasti dan mengenai sesuatu yang lain, daripada memiliki bidikan yang terlalu umum atau terlalu samar-samar, dan tidak mengenai apa pun. Bersiaplah, bidik, dan tembak, dan berharaplah bahwa Anda akan mengenai sasaran: apa yang Anda bidik sebagai target utama, tetapi mengenai beberapa sasaran lain lebih baik daripada tidak sama sekali.

Atur terlebih dahulu

Kontribusi utama yang bisa diberikan oleh seorang penulis adalah mengatur dan menyusun materi, yang meminimalkan hambatan dan memaksimalkan wawasan pembaca, serta menjaganya tetap berada di jalur yang benar tanpa gangguan yang tidak diinginkan. Apa, sih, kelebihan sebuah buku dibandingkan dengan setumpuk cetakan ulang (reprints)? Jawaban: adanya susunan yang efisien dan menyenangkan, penekanan di tempat penekanan diperlukan, ditunjukkannya [suatu] keterkaitan, dan deskripsi contoh dan contoh penyangkal (counterexample) yang menjadi dasar teori; dalam satu kata, suatu pengaturan.

Penemu sebuah ide, yang tentu saja mungkin juga merangkap sebagai yang menyampaikannya, menemukan [ide tersebut] dengan cara yang tidak efisien, nyaris secara acak. Jika tidak ada cara untuk memangkas, menyatukan, dan menyusun ulang penemuan itu, setiap siswa harus merangkumnya kembali, tidak akan ada keuntungan yang bisa diperoleh dari berdiri “on the shoulders of giants”, dan tidak akan pernah ada waktu untuk mempelajari sesuatu yang baru yang tidak diketahui oleh generasi sebelumnya.

Setelah Anda mengetahui apa yang ingin Anda sampaikan, dan kepada siapa Anda ingin menyampaikannya, langkah selanjutnya adalah membuat kerangka (outline). Menurut pengalaman saya, hal ini biasanya tidak mungkin dapat dilakukan. Idealnya adalah membuat suatu kerangka sehingga setiap diskusi heuristik (pengantar), setiap lema, setiap teorema, setiap konsekuensi, setiap komentar, dan setiap bukti, perlu disebutkan, dan lokasi semua hal ini muncul, ada dalam urutan yang benar secara logis dan dapat dicerna secara psikologis. Dalam pengaturan yang ideal, ada tempat untuk segala sesuatu dan segala sesuatu ada di tempatnya. Perhatian pembaca terjaga karena dia diberitahu sejak awal apa yang diharapkan, dan, pada saat yang sama dan dalam kontradiksi yang jelas, kejutan yang menyenangkan terus terjadi yang tidak dapat diprediksi dari definisi-definisi yang sederhana. Bagian-bagiannya pas, dan pas sekali. Lema-lema ada di situ saat dibutuhkan, dan hubungan antar teorema-teorema terlihat jelas; dan kerangka memberi tahu Anda di mana letak semua ini berada.

Saya membuat perbedaan kecil, mungkin yang tidak perlu, antara mengatur (to organize) dan menyusun (to arrange). Mengatur sebuah subjek berarti memutuskan apa judul-judul dan sub-subjudul utama, apa yang ada di dalamnya, dan apa hubungan diantara keduanya. Struktur hasil pengaturan [nantinya] berbentuk sebuah grafik, kemungkinan besar berupa pohon, tetapi hampir pasti bukan berupa rantai. Ada banyak cara untuk mengatur sebagian besar subjek, dan biasanya ada banyak cara untuk menyusun hasil dari setiap metode pengaturan dalam suatu urutan linear. Pengaturan lebih penting daripada penyusunannya, tetapi yang terakhir ini sering kali memiliki nilai psikologis.

Salah satu pujian yang paling saya sanjung kepada seorang penulis berasal dari sebuah kegagalan [saya]; saya mengacaukan suatu mata kuliah yang mengacu pada bukunya. Awalnya, ada satu bagian dari buku itu yang tidak saya sukai, dan saya melewatkannya. Tiga bagian kemudian saya membutuhkan sebuah penggalan kecil dari akhir bagian yang dihilangkan, tetapi lebih mudah untuk memberikan bukti yang berbeda. Hal yang sama terjadi beberapa kali lagi, tetapi setiap kali butuh sedikit kecerdikan dan satu atau dua konsep ad hoc untuk menambal lubang tersebut. Namun, pada bab berikutnya, sesuatu yang lain muncul karena apa yang dibutuhkan bukanlah penggalan dari bagian yang dihilangkan, melainkan fakta bahwa hasil dari bagian tersebut dapat diterapkan pada dua situasi yang sangat berbeda. Hal ini hampir tidak mungkin untuk ditambal, dan setelah itu kekacauan dengan cepat terjadi. Pengaturan di buku ini sangat ketat; segala sesuatunya ada di sana karena memang dibutuhkan; penyajiannya memiliki keterkaitan yang memudahkan dalam membaca dan memahami. Pada saat yang sama, benang-benang yang menyatukan semuanya tidak mengganggu; benang-benang tersebut hanya terlihat apabila ada bagian struktur yang dirusak.

Bahkan penulis yang paling tidak teratur pun [dapat] membuat kerangka yang kasar dan mungkin tidak tertulis; [yaitu] subjek itu sendiri, [yang] bagaimanapun juga, adalah kerangka satu-konsep dari buku. Jika Anda tahu bahwa Anda sedang menulis tentang “teori ukuran”, maka Anda memiliki kerangka dua-kata, dan itu sudah sesuatu. Kerangka bab yang bersifat tentatif adalah sesuatu yang lebih baik. Mungkin seperti ini: Saya akan menjelaskan tentang himpunan, lalu ukuran, lalu fungsi, lalu integral. Pada tahap ini, Anda mungkin ingin membuat beberapa keputusan, yang mungkin akan dibatalkan di kemudian hari; misalnya, Anda mungkin memutuskan untuk tidak memasukkan peluang, tetapi menyertakan ukuran Haar.

Ada suatu aspek di mana persiapan garis besar dapat memakan waktu bertahun-tahun, atau, paling tidak, berminggu-minggu. Bagi saya, biasanya ada waktu yang lama, antara saat menyenangkan pertama kali ketika saya membayangkan gagasan untuk menulis sebuah buku, dan saat menyakitkan pertama kali saat saya harus duduk dan mulai menulis. Sementara itu, sembari saya melanjutkan pekerjaan sehari-hari, saya melamun tentang proyek [menulis buku] baru, dan, ketika ide muncul di benak saya, saya mencatatnya di secarik kertas dan menaruhnya di dalam selembar map. Satu “ide” dalam hal ini mungkin berupa bidang matematika yang saya rasa harus dimasukkan, atau mungkin sebuah notasi; mungkin sebuah bukti, mungkin sebuah kata deskriptif yang tepat, atau mungkin sebuah candaan yang, saya harap, tidak akan gagal namun akan menghidupkan, menekankan, dan mencontohkan apa yang ingin saya katakan. Ketika saat yang menyakitkan itu akhirnya tiba, setidaknya saya sudah memiliki map; bermain Solitaire dengan sobekan kertas-kertas bisa sangat membantu dalam mempersiapkan kerangka.

Dalam mengatur naskah tulisan, pertanyaan tentang apa yang harus disertakan hampir tidak lebih penting daripada apa yang harus diabaikan; terlalu banyak detail bisa sama mengecewakannya dengan tidak ada sama sekali. The last dotting of the last i, in the manner of the old-fashioned Cours d'Analyse in general and Bourbaki in particular, gives satisfaction to the author who understands it anyway and to the helplessly weak student who never will; bagi sebagian besar pembaca yang serius, hal itu lebih buruk daripada tidak berguna. Jantung dari matematika terbuat dari contoh-contoh konkret dan masalah-masalah konkret. Teori-teori umum yang besar biasanya merupakan renungan yang didasarkan pada wawasan yang kecil namun mendalam; wawasan itu sendiri berasal dari kasus-kasus khusus yang konkret. Pesan moralnya adalah akan jauh lebih baik jika Anda mengatur pekerjaan Anda di sekitar contoh-contoh utama dan contoh-contoh penyangkalnya. Pengamatan bahwa sebuah bukti dapat membuktikan sesuatu yang sedikit lebih umum daripada yang ditujukan, sering kali dapat diserahkan kepada pembaca. Saat ketika pembaca membutuhkan bimbingan yang berpengalaman, adalah dalam menemukan hal-hal yang tidak dibuktikan oleh bukti tersebut; apa contoh-contoh penyangkal yang sesuai dan ke mana kita akan melangkah dari sini?

Pikirkan tentang alfabet

Setelah Anda memiliki suatu rencana pengaturan, sebuah kerangka, yang mungkin tidak bagus namun itu terbaik yang bisa Anda lakukan, Anda hampir siap untuk mulai menulis. Satu-satunya hal lain yang saya sarankan untuk Anda lakukan pertama kali adalah menginvestasikan satu atau dua jam untuk memikirkan alfabet; Anda akan menemukan bahwa hal ini akan menyelamatkan Anda dari sakit kepala nantinya.

Huruf-huruf yang digunakan untuk menunjukkan konsep yang akan Anda bahas, perlu dipikirkan dan dirancang dengan hati-hati. Notasi yang baik dan konsisten dapat menjadi bantuan yang luar biasa, dan saya menyarankan (untuk para penulis artikel juga, tetapi terutama untuk para penulis buku) agar notasi tersebut dirancang di awal. Saya membuat tabel besar dengan banyak huruf, dengan banyak jenis huruf, baik untuk huruf besar maupun huruf kecil, dan saya mencoba untuk mengantisipasi semua ruang, grup, vektor, fungsi, titik, permukaan, ukuran, dan apa pun yang cepat atau lambat perlu dibaptis. Notasi yang buruk dapat membuat eksposisi yang baik menjadi buruk dan eksposisi yang buruk menjadi lebih buruk; keputusan ad hoc tentang notasi, mendefinisikan di tengah-tengah penyampaian, hampir pasti menghasilkan notasi yang buruk.

Notasi yang baik memiliki harmoni dalam urutan abjad dan menghindari disonansi. Contoh: $ax + by$ atau $a_1x_1 + a_2x_2$ lebih disukai daripada $ax_1 + bx_2$. Atau: jika Anda harus menggunakan $\sum$ untuk penjumlahan dan himpunan indeks, pastikan Anda tidak akan menulis $\sum_{\sigma\in\sum} a_\sigma$. Seirama dengan itu: mungkin sebagian besar pembaca tidak akan menyadari bahwa Anda menggunakan $\vert z\vert < \epsilon$ di bagian atas halaman dan $z\;\epsilon\;U$ di bagian bawah, tetapi itu adalah jenis agak-disonansi yang menyebabkan perasaan tidak enak secara keseluruhan. Cara mengatasinya mudah dan semakin lama semakin diterima secara universal: $\in$ digunakan untuk keanggotaan dan $\epsilon$ untuk penggunaan ad hoc.

Matematika memiliki akses ke alfabet yang dapat tak terbatas (misalnya $x$, $x'$, $x''$, $x'''$, $\dots$), tetapi dalam praktiknya, hanya sebagian kecil saja dari alfabet tersebut yang dapat digunakan. Salah satu alasannya adalah bahwa kemampuan manusia untuk membedakan antara simbol-simbol jauh lebih terbatas daripada kemampuannya untuk memahami simbol-simbol baru; alasan lainnya adalah kebiasaan buruk untuk membekukan (membakukan) huruf. Beberapa analis lawas akan berbicara tentang “ruang-$xyz$”, yang berarti, menurut saya, ruang Euklides 3-dimensi, ditambah dengan konvensi bahwa sebuah titik pada ruang tersebut akan selalu dilambangkan dengan “$(x,y,z)$”. Ini buruk: dengan “membekukan” $x$, dan $y$, dan $z$, yaitu, melarang penggunaannya dalam konteks lain, pada saat yang sama akan membuatnya tidak mungkin (atau, dalam hal apa pun, tidak konsisten) untuk menulis, katakanlah, “$(a, b, c)$” ketika “$(x, y, z)$” sedang tidak bisa digunakan. Versi modern dari kebiasaan ini juga ada, dan tidak lebih baik. Contoh: matriks dengan “properti $L$” – sebutan beku dan tidak berarti.

Ada cara lain yang canggung dan tidak membantu dalam menggunakan huruf: “kompleks CW” dan “grup CCR” adalah contohnya. Salah satu keingintahuan yang berkaitan adalah menggunakan huruf dengan cara yang tidak dapat digunakan, seperti yang terjadi pada Lefschetz.⁶ Di sana $x^p_i$ adalah sebuah rantai berdimensi $p$ (tika bawahnya – subscript – sebagai indeks), sedangkan $x^i_p$ adalah sebuah ko-rantai berdimensi $p$ (dan tika atasnya – superscript – juga sebagai indeks). Pertanyaan: apa yang dimaksud dengan $x^2_3$?

Seiring perkembangan sejarah, semakin banyak simbol yang dibekukan. Contoh standarnya adalah $e$, $i$ dan $\pi$, dan tentu saja, 0, 1, 2, 3, $\dots$ (Siapa yang berani menulis “Misalkan 6 adalah sebuah grup.”?) Beberapa huruf lainnya hampir dibekukan: banyak pembaca akan tersinggung jika “$n$” digunakan untuk menyatakan sebuah bilangan kompleks, “$\epsilon$” untuk sebuah bilangan bulat positif, dan “$z$” untuk sebuah ruang topologi. (Mimpi buruk para matematikawan adalah barisan “$n_\epsilon$” yang cenderung menuju $0$ ketika $\epsilon$ membesar tak terbatas).

Moral: jangan menambah kekakuan. Pikirkan tentang alfabet. Memang merepotkan, tetapi itu sepadan. Untuk menghemat waktu dan masalah di kemudian hari, pikirkan tentang alfabet selama satu jam sekarang; lalu mulailah menulis.

Menulis secara berputar

Cara terbaik untuk mulai menulis, mungkin satu-satunya cara, adalah menulis secara spiral. Menurut aturan ini, bab-bab ditulis dan ditulis ulang dengan urutan 1, 2, 1, 2, 3, 1, 2, 3, 4, dst. Anda merasa sudah tahu cara menulis Bab 1, tetapi setelah Anda menyelesaikannya dan melanjutkan ke Bab 2, Anda akan menyadari bahwa Anda dapat melakukan pekerjaan yang lebih baik di Bab 2 jika Anda mengerjakan Bab 1 dengan cara yang berbeda. Tidak ada cara lain yang dapat membantu Anda selain kembali, mengerjakan Bab 1 dengan cara yang berbeda, mengerjakan Bab 2 dengan lebih baik, dan kemudian masuk ke Bab 3. Dan, tentu saja, Anda tahu apa yang akan terjadi: Bab 3 akan menunjukkan kelemahan Bab 1 dan 2, dan tidak ada bantuan untuk itu… dst, dst, dst. Ini adalah ide yang jelas, dan sering kali tidak dapat dihindari, tetapi dapat membantu penulis di masa depan untuk mengetahui sebelumnya apa yang akan dia hadapi, dan dapat membantunya untuk mengetahui bahwa fenomena yang sama akan terjadi tidak hanya pada bab, tetapi juga pada bagian, paragraf, kalimat, dan bahkan kata-kata.

Langkah pertama dalam proses menulis, menulis ulang, dan mengulang menulis ulang, adalah menulis. Setelah menentukan subjek, pembaca, dan kerangka (dan, jangan lupa, alfabet), mulailah menulis, dan jangan biarkan apa pun menghentikan Anda. Tidak ada motivasi yang lebih baik untuk dapat menulis buku yang baik daripada [membayangkan membaca] buku yang buruk. Setelah Anda memiliki draf pertama, yang ditulis secara spiral, berdasarkan subjek, ditujukan untuk pembaca dan didukung oleh kerangka sedetail mungkin, maka buku Anda sudah lebih dari separuhnya selesai.

Aturan spiral mencakup sebagian besar penulisan ulang dan mengulang proses menulis ulang yang dilakukan dalam sebuah buku (sebagian besar, tetapi tidak semua). Pada draf pertama setiap bab, saya sarankan agar Anda menumpahkan isi hati Anda, menulis dengan cepat, melanggar semua aturan, menulis dengan rasa benci atau bangga, menjadi sinis, bingung, menjadi “lucu” jika perlu, menjadi tidak jelas, menjadi tidak-gramatikal – teruslah menulis. Akan tetapi, ketika Anda harus menulis ulang, dan sesering apa pun hal itu diperlukan, jangan mengubahnya, tetapi tulis ulang. Sangat menggoda untuk menggunakan pulpen merah untuk menunjukkan [semua] penyisipan, penghapusan, dan permutasi [yang diperlukan], tetapi menurut pengalaman saya, hal ini akan menyebabkan kesalahan besar. Melawan ketidaksabaran manusia, dan melawan keberpihakan yang terlalu manusiawi, yang dirasakan setiap orang ketika membaca kata-katanya sendiri, pulpen merah adalah senjata yang terlalu lemah. Anda dihadapkan pada draf pertama yang tidak akan bisa diterima oleh pembaca mana pun kecuali diri Anda sendiri; Anda harus tidak kenal ampun terhadap segala macam perubahan, dan, terutama, terhadap penghilangan kata secara besar-besaran. Menulis ulang berarti menulis lagi – setiap kata.

Saya tidak bermaksud secara harfiah bahwa, dalam sebuah buku yang terdiri dari 10 bab, Bab 1 harus ditulis sebanyak sepuluh kali, tetapi yang saya maksudkan adalah sekitar tiga atau empat kali. Kemungkinan besar Bab 1 harus ditulis ulang, secara harfiah, tepat setelah Bab 2 selesai, dan, kemungkinan besar, setidaknya sekali lagi, di suatu saat setelah Bab 4. Jika beruntung, Anda hanya perlu menulis Bab 9 sekali saja.

Deskripsi praktik yang saya lakukan sendiri mungkin lebih menjelaskan jumlah total penulisan ulang yang saya bicarakan. Setelah draf pertama selesai ditulis secara spiral, saya biasanya menulis ulang seluruh buku, dan kemudian menambahkan alat bantu baca yang bersifat mekanis namun sangat diperlukan (seperti daftar prasyarat, kata pengantar, indeks, dan daftar isi). Selanjutnya, saya menulis ulang lagi, kali ini dengan mesin tik, atau setidaknya, dengan sangat rapi dan indah sehingga juru ketik yang tidak terlatih secara matematis dapat menggunakan versi ini (yang bisa dianggap sebagai versi ketiga) untuk menyiapkan naskah “final” tanpa kesulitan. Penulisan ulang pada versi ketiga ini sangat minim; biasanya terbatas pada perubahan yang mempengaruhi satu kata saja, atau, dalam kasus terburuk, satu kalimat. Versi ketiga adalah versi pertama yang dilihat orang lain. Saya meminta teman untuk membacanya, istri saya membacanya, murid-murid saya mungkin membaca sebagian, dan yang terbaik dari semuanya, seorang ahli tingkat junior, yang dibayar dengan pantas untuk melakukan pekerjaan dengan baik, membacanya dan didorong untuk tidak bersikap sopan dalam kritik-kritiknya. Perubahan-perubahan yang diperlukan pada versi ketiga, jika beruntung, dapat dilakukan dengan pulpen merah; jika tidak beruntung, perubahan-perubahan tersebut akan menyebabkan sepertiga halaman diketik ulang. Naskah “final” didasarkan pada versi ketiga yang telah disunting, dan, setelah jadi, naskah tersebut dibaca, dibaca ulang, dikoreksi, dan dikoreksi ulang. Kira-kira dua tahun setelah dimulai (dua tahun kerja, yang mungkin lebih dari dua tahun kalender), buku tersebut dikirim ke penerbit. Kemudian dimulailah jenis kerja keras yang lain, tetapi itu cerita lain.

Archimedes mengajarkan kita bahwa sejumlah kecil yang ditambahkan ke sendirinya, berulang-ulang, akan menjadi jumlah yang besar (atau, dalam istilah pepatah, every little bit helps). Dalam hal menyelesaikan sebagian besar pekerjaan di dunia, dan, khususnya, dalam hal menulis buku, saya percaya bahwa kebalikan dari ajaran Archimedes juga benar: satu-satunya cara untuk menulis buku yang besar adalah terus menulis sedikit demi sedikit, setiap hari, tanpa terkecuali, tanpa hari libur. Teknik yang baik, untuk membantu kestabilan laju produksi Anda, adalah berhenti setiap hari dengan menyiapkan pompa semangat untuk hari berikutnya. Apa yang akan Anda mulai dengan besok? Apa isi dari bagian berikutnya; apa judulnya? (Saya sarankan agar Anda mencari judul yang singkat untuk setiap bagian, sebelum atau sesudah [isi bagian itu] ditulis, bahkan jika Anda tidak berencana untuk mencetak judul bagian. Tujuannya adalah untuk menguji seberapa baik bagian itu direncanakan: jika Anda tidak dapat menemukan judul, alasannya mungkin karena bagian itu tidak memiliki satu subjek yang menyatu). Terkadang saya menulis kalimat pertama untuk besok di hari ini; beberapa penulis memulai hari ini dengan merevisi dan menulis ulang halaman terakhir, atau lebih, dari pekerjaan hari kemarin. Bagaimanapun, akhiri setiap sesi kerja dengan sebuah semangat; berikan alam bawah sadar Anda sesuatu yang mantap untuk disantap di antara sesi. Sungguh mengejutkan betapa Anda bisa membodohi diri Anda sendiri dengan cara seperti itu; teknik pump-priming sudah cukup untuk mengatasi inersia alami manusia ketika menghadapi pekerjaan kreatif.

Selalu rapikan

Bahkan jika rencana pengaturan (!organization) awal Anda [dirasa] mendetail dan bagus (dan terlebih ketika tidak), pekerjaan mahapenting untuk mengatur materi tidak akan berhenti ketika tahap menulis dimulai; itu akan berjalan seiring penulisan dan bahkan setelahnya.

Rencana menulis secara spiral berjalan beriringan dengan rencana mengatur secara spiral, sebuah rencana yang sering (mungkin selalu) dapat diterapkan dalam menulis matematika. Ceritanya seperti ini. Mulai dengan apapun yang Anda pilih sebagai konsep dasar – katakan ruang vektor – dan lakukan dari situ: apa motivasinya, tulis definisinya, beri contoh-contoh dan contoh-contoh penyangkalnya. Itu Subbab 1. Di Subbab 2, perkenalkan konsep terkait yang ingin dipelajari – misal ketergantungan linear – dan lakukan dari situ: apa motivasinya, tulis definisinya, beri contoh-contoh dan contoh-contoh penyangkalnya, lalu, ini bagian yang penting, review kembali Subbab 1, sebisa mungkin sepenuhnya, dari sudut pandang Subbab 2. Sebagai contoh, apa contoh-contoh himpunan bergantung dan bebas linear yang mudah diakses langsung dari contoh-contoh ruang vektor yang diperkenalkan Subbab 1? (Disini, omong-omong, adalah alasan nyata lainnya mengapa menulis spiral diperlukan: Anda mungkin berpikir, di Subbab 2, bahwa Anda lupa menuliskannya sebagai contoh di Subbab 1.) Di Subbab 3 perkenalkan konsep selanjutnya (yang tentunya apapun itu perlu perencanaan yang matang, dan, seringkali, [menghasilkan] perubahan sudut pandang yang mendasar yang membuat sekali lagi, menulis spiral adalah prosedur yang benar), dan, setelah cukup membersihkannya, review kembali Subbab 1 dan 2 dari sudut pandang konsep baru [tadi]. [Cara] ini berhasil, dengan hasil yang ciamik. Ini [proses yang] mudah dilakukan, seru untuk dilakukan, enak dibaca, dan para pembaca dimudahkan dari pengaturan yang kokoh, bahkan ketika mereka tidak peduli untuk mengecek dan melihat dimana satu hal datang dan ditopang oleh hal-hal lain.

The historical novelist’s plots and subplots and the detective story writer’s hints and clues all have their mathematical analogues. To make the point by way of an example: much of the theory of metric spaces could be developed as a “subplot” in a book on general topology, in unpretentious comments, parenthetical asides, and illustrative exercises. Such an organization would give the reader more firmly founded motivation and more insight than can be obtained by inexorable generality, and with no visible extra effort. As for clues: a single word, first mentioned several chapters earlier than its definition, and then re-mentioned, with more and more detail each time as the official treatment comes closer and closer, can serve as an inconspicuous, subliminal preparation for its full-dress introduction. Such a procedure can greatly help the reader, and, at the same time, make the author’s formal work much easier, at the expense, be sure, of greatly increasing the thought and preparation that goes into his informal prose writing. It’s worth it. If you work eight hours to save five minutes of the reader’s time, you have saved over 80 man-hours for each 1000 readers, and your name will be deservedly blessed down the corridors of many mathematics buildings. But remember: for an effective use of subplots and clues, something very like the spiral plan of organization is indispensable.

The last, least, but still very important aspect of organization that deserves mention here is the correct arrangement of the mathematics from the purely logical point of view. There is not much that one mathematician can teach another about that, except to warn that as the size of the job increases, its complexity increases in frightening proportion. At one stage of writing a 300-page book, I had 1000 sheets of paper, each with a mathematical statement on it, a theorem, a lemma, or even a minor comment, complete with proof. The sheets ere numbered, any which way. My job was to indicate on each sheet the numbers of the sheets whose statement must logically come before, and then to arrange the sheets in linear order so that no sheet comes after one on which it’s mentioned. That problem had, apparently, uncountably many solutions; the difficulty was to pick one that was as efficient and pleasant as possible.

Tulis bahasa Inggris yang baik

Everything I’ve said so far has to do with writing in the large, global sense; it is time to turn to the local aspects of the subject.

Why shouldn’t an author spell “continuous” as “continous”? There is no chance at all that it will be misunderstood, and it is one letter shorter, so why not? The answer that probably everyone would agree on, even the most libertarian among modern linguists, is that whenever the “reform” is introduced it is bound to cause distraction, and therefore a waste of time, and the “saving” is not worth it. A random example such as this one is probably not convincing; more people would agree that an entire book written in reformed spelling, with, for instance, “izi” for “easy” is not likely to be an effective teaching instrument for mathematics. Whatever the merits of spelling reform may be, words that are misspelled according to currently accepted dictionary standards detract from the good a book can do: they delay and distract the reader, and possibly confuse or anger him.

The reason for mentioning spelling is not that it is a common danger or a serious one for most authors, but that it serves to illustrate and emphasize a much more important point. I should like to argue that it is important that mathematical books (and papers, and letters, and lectures) be written in good English style, where good means “correct” according to currently and commonly accepted public standards. (French, Japanese, or Russian authors please substitute “French”, “Japanese”, or “Russian” for “English”.) I do not mean that the style is to be pedantic, or heavy-handed, or formal, or bureaucratic, or flowery, or academic jargon. I do mean that it should be completely unobtrusive, like good background music for a movie, so that the reader may proceed with no conscious or unconscious blocks caused by the instrument of communication and not its content.

Good English style implies correct grammar, correct choice of words, correct punctuation, and, perhaps above all, common sense. There is a difference between “that” and “which”, and “less” and “fewer” are not the same, and a good mathematical author must know such things. The reader may not be able to define the difference, but a hundred pages of colloquial misusage, or worse, has a cumulative abrasive effect that the author surely does not want to produce. Fowler⁷, Roger⁸, and Webster⁹ are next to Dunford-Schwartz on my desk; they belong in a similar position on every author’s desk. It is unlikely that a single missing comma will convert a correct proof into a wrong one, but consistent mistreatment of such small things has large effects.

The English language can be a beautiful and powerful instrument for interesting, clear, and completely precise information, and I have faith that the same is true for French or Japanese or Russian. It is just as important for an expositor to familiarize himself with that instrument as for a surgeon to know his tools. Euclid can be explained in bad grammar and bad diction, and a vermiform appendix can be removed with a rusty pocket knife, but the victim, even if he is unconscious of the reason for his discomfort, would surely prefer better treatment that that.

All mathematicians, even very young students very near the beginning of their mathematical learning, know that mathematics has a language of its own (in fact it is one), and an author must have thorough mastery of the grammar and vocabulary of that language as well as of the vernacular. There is no Berlitz course for the language of mathematics; apparently the only way to learn it is to live with it for years. What follows is not, it cannot be, a mathematical analogue of Fowler, Roget, and Webster, but it may perhaps serve to indicate a dozen or two of the thousands of items that those analogues would contain.

Kejujuran adalah kebijakan terbaik

The purpose of using good mathematical language is, of course, to make the understanding of the subject easy for the reader, and perhaps even pleasant. The style should be good not in the sense of flashy brilliance, but good in the sense of perfect unobtrusiveness. The purpose is to smooth the reader’s way, to anticipate his difficulties and to forestall them. Clarity is what’s wanted, not pedantry; understanding, not fuss.

The emphasis in the preceding paragraph, while perhaps necessary, might seem to point in an undesirable direction, and I hasten to correct a possible misinterpretation. While avoiding pedantry and fuss, I do not want to avoid rigor and precision; I believe that these aims are reconcilable. I do not mean to advise a young author to be ever so slightly but very very cleverly dishonest and to gloss over difficulties. Sometimes, for instance, there may be no better way to get a result than a cumbersome computation. In that case it is the author’s duty to carry it out, in public; the best he can do to alleviate it is to extend his sympathy to the reader by some phrase such as “unfortunately the only known proof is the following cumbersome computation”.

Here is the sort of thing I mean by less than complete honesty. At a certain point, having proudly proved a proposition $p$, you fell moved to say: “Note, however, that $p$ does not imply $q$”, and then, thinking that you’ve done a good expository job, go happily on to other things. Your motives may be perfectly pure, but the reader fell cheated just the same. If he knew all about the subject, he wouldn’t be reading you; for him the non-implication is, quite likely, unsupported. Is it obvious? (Say so.) Will a counterexample be supplied later? (Promise it now.) Is it a standard but for present purposes irrelevant part of the literature? (Give a reference.) Or, horrible dictu, do you merely mean that you have tried to derive $q$ from $p$, you failed, and you don’t in fact know whether $p$ implies $q$? (Confess immediately!) In any event: take the reader into your confidence.

There is nothing wrong with the often derided “obvious” and “easy to see”, but there are certain minimal rules to their use. Surely when you wrote that something was obvious, you thought it was. When, a month, or two months, or six months later, you picked up the manuscript and re-read it, did you still think that something was obvious? (A few months’ ripening always improves manuscripts.) When you explained it to a friend, or to a seminar, was the something at issue accepted as obvious? (Or did someone question it and subside, muttering, when you reassured him? Did your assurance consist of demonstration or intimidation?) The obvious answers to these rhetorical questions area among the rules that should control the use of “obvious”. There is another rule, the major one, and everybody knows it, the one whose violation is the most frequent source of mathematical error: make sure that the “obvious” is true.

It should go without saying that you are not setting out to hide facts from the reader; you are writing to uncover them. What I am saying now is that you should not hide the status of your statements and your attitude toward them either. Whenever you tell him something, tell him where it stands: this has been proved, that hasn’t, this will be proved, that won’t. Emphasize the important and minimize the trivial. There are many good reasons for making obvious statements every now and then; the reason for saying that they are obvious is to put them in proper perspective for the uninitiate. Even if your saying so makes an occasional reader angry at you, a good purpose is served by your telling him how you view the matter. But, of course, you must obey the rules. Don’t let the reader down; he wants to believe in you. Pretentiousness, bluff, and concealment may not get caught out immediately, but most readers will soon sense that there is something wrong, and they will blame neither the facts nor themselves, but, quite properly, the author. Complete honesty makes for greatest clarity.

Tentang yang tak relevan dan yang mudah

Sometimes a proposition can be so obvious that it needn’t even be called obvious and still the sentence that announces it is bad exposition, bad because it makes for confusion, misdirection, delay. I mean something like this: “If $R$ is a commutative semisimple ring with unit and if $x$ and $y$ are in $R$, then $x^2 − y^2 = (x−y)(x+y)$.” The alert reader will ask himself what semisimplicity and a unit have to do with what he had always thought was obvious. Irrelevant assumptions wantonly dragged in, incorrect emphasis, or even just the absence of correct emphasis can wreak havoc.

Just as distracting as an irrelevant assumption and the cause of just as much wasted time is an author’s failure to gain the reader’s confidence by explicitly mentioning trivial cases and excluding them if need be. Every complex number is the product of a non-negative number and a number of modulus $1$. That is true, but the reader will feel cheated and insecure if soon after first being told that fact (or being reminded of it on some other occasion, perhaps preparatory to a generalization being sprung on him) he is not told that there is something fishy about $0$ (the trivial case). The point is not that failure to treat the trivial cases separately may sometimes be a mathematical error; I am not just saying “do not make mistakes”. The point is that insistence on legalistically correct but insufficiently explicit explanations (“The statement is correct as it stands – what else do you want ?”) is misleading, bad exposition, bad psychology. It may also be almost bad mathematics. If, for instance, the author is preparing to discuss the theorem that, under suitable hypotheses, every linear transformation is the product of a dilatation and a rotation, the his ignoring of $0$ in the 1-dimensional case leads to the reader’s misunderstanding of the behavior of singular linear transformations in the general case.

This may be the right place to say a few words about the statements of theorems: there, more than anywhere else, irrelevancies must be avoided.

The first question is where the theorem should be stated, and my answer is: first. Don’t ramble on a leisurely way, not telling the reader where you are going, and then suddenly announce “Thus we have proved that…”. The reader can pay closer attention to the proof if he knows what you are proving, and he can see better where the hypotheses are used if he knows in advance what they are. (The rambling approach frequently leads to the “hanging” theorem, which I think is ugly. I mean something like: “Thus we have proved

Theorem 2 …”.

The indentation, which is after all a sort of invisible punctuation mark, makes a jarring separation in the sentence, and, after the reader has collected his wits and caught on to the trick that was played on him, it makes an undesirable separation between the statement of the theorem and its official label.)

This is not to say the theorem is to appear with no introductory comments, preliminary definitions, and helpful motivations. All that comes first; the statement comes next; and the proof comes last. The statement of the theorem should consist of one sentence whenever possible: a simple implication, or, assuming that some universal hypotheses were stated before and are still in force, a simple declaration. Leave the chit-chat out: “Without loss of generality we may assume…” and “Moreover it follows from Theorem 1 that…” do not belong in the statement of a theorem.

Ideally the statement of a theorem is not only one sentence, but a short one at that. Theorems whose statement fills almost a whole page (or more!) are hard to absorb, harder than they should be; they indicate that author did not think the material through and did not organize it as he should have done. A list of eight hypotheses (even if carefully so labelled) and a list of six conclusions do not a theorem make; they are a badly expounded theory. Are all the hypotheses needed for each conclusion? If the answer is no, the badness of the statement is evident; if the answer is yes, then the hypotheses probably describe a general concept that deserves to be isolated, named, and studied.

Lakukan dan jangan ulangi

One important rule of good mathematical style calls for repetition and another calls for its avoidance.

By repetition in the first sense I do not mean the saying of the same thing several times in different words. What I do mean, in the expression of a precise subject such as mathematics, is the word-for-word repetition of a phrase, or even many phrases, with the purpose of emphasizing a slight change in a neighboring phrase. If you have defined something, or stated something, or proved something in Chapter 1, and if in Chapter 2 you want to treat a parallel theory or a more general one, it is a big help to the reader if you use the same words in the same order for as long as possible, and then, with a proper roll of drums, emphasize the difference. The roll of drums is important. It is not enough to list six adjectives in one definition, and re-list five of them, with a diminished sixth, in the second. That’s the thing to do, but what helps is to say, in addition: “Note that the first five conditions in the definitions of p and q are the same; what makes them different is the weakening of the sixth.”

Often in order to be able to make such an emphasis in Chapter 2 you’ll have to go back to Chapter 1 and rewrite what you thought you had already written well enough, but this time so that its parallelism with the relevant part of Chapter 2 is brought out by the repetition device. This is another illustration of why the spiral plan of writing is unavoidable, and it is another aspect of what I call the organization of the material.

The preceding paragraphs describe an important kind of mathematical repetition, the good kind; there are two other kinds, which are bad.

One sense in which repetition is frequently regarded as a device of good teaching is that the oftener you say the same thing, in exactly the same words, or else with slight differences each time, the more likely you are to drive the point home. I disagree. The second time you say something, even the vaguest reader will dimly recall that there was a first time, and he’ll wonder if what he is now learning is exactly the same as what he should have learned before, or just similar but different. (If you tell him “I am now saying exactly what I fist said on p. 3”, that helps.) Even the dimmest such wonder is bad. Anything is bad that unnecessarily frightens, irrelevantly amuses, or in any other way distracts. (Unintended double meanings are the woe of many an author’s life.) Besides, good organization, and, in particular, the spiral plan of organization discussed before is a substitute for repetition, a substitute that works much better.

Another sense in which repetition is bad is summed up in the short and only partially inaccurate precept: never repeat a proof. If several steps in the proof of Theorem 2 bear a very close resemblance to parts of the Theorem 1, that’s signal that something may be less than completely understood. Other symptoms of the same disease are: “by the same technique (or method, or device, or trick) as in the proof of Theorem 1…”, or, brutally, “see the proof of Theorem 1”. When that happens the chances are very good that there is a lemma that is worth finding, formulating, and proving, a lemma which both Theorem 1 and Theorem 2 are more easily and more clearly deduced.

Kata ganti ‘kita’ tidak selalu buruk

One aspect of expository style that frequently bothers beginning authors is the use of the editorial “we”, as opposed to the singular “I”, or the neutral “one”. It is matters like this that common sense is most important. For what it’s worth, I present here my recommendation.

Since the best expository style is the least obtrusive one, I tend nowadays to prefer the neutral approach. That does not mean using “one” often, or ever; sentences like “one has thus proved that…” are awful. It does mean the complete avoidance or first person pronouns in either singular or plural. “Since $p$, it follows that $q$.” “This implies $p$.” “An application of $p$ to $q$ yields $r$.” Most (all ?) mathematical writing is (should be?) factual; simple declarative sentences are the best for communicating facts.

A frequently effective and time-saving device is the use of the imperative. “To find $P$, multiply $q$ by $r$.” “Given $p$, put $q$ equal to $r$.” (Two digressions about “given”. (1) Do not use it when it means nothing. Example: “For any given $p$ there is a $q$.” (2) Remember that it comes from an active verb and resist the temptation to leave it dangling. Example: Not “Given $p$, there is a $q$”, but “Given $p$, find $q$”.)

There is nothing wrong with the editorial “we”, but if you like it, do not misuse it. Let “we” mean “the author and the reader” (or “the lecturer and the audience”). Thus, it is fine to say “Using Lemma 2 we can generalize Theorem 1”, or “Lemma 3 gives us a technique for proving Theorem 4”. It is not good to say “Our work on this result was done in 1969” (unless the voice is that of two authors, or more, speaking in unison), and “We thank our wife for her help with the typing” is always bad.

The use of “I”, and especially its overuse, sometimes has a repellent effect, as arrogance or ex-cathedra preaching, and, for that reason, I like to avoid it whenever possible. In short notes, obviously in personal historical remarks, and, perhaps, in essays such as this, it has it place.

Gunakan kata dengan benar

The next smallest units of communication, after the whole concept, the major chapters, the paragraphs, and the sentences are the words. The preceding section about pronouns was about words, in a sense, although, in a more legitimate sense, it was about global stylistic policy. What I am now going to say is not just “use words correctly”; that should go without saying. What I do mean to emphasize is the need to think about and use with care the small words of common sense and intuitive logic, and the specifically mathematical words (technical terms) that can have a profound effect on mathematical meaning.

The general rule is to use the words of logic and mathematics correctly. The emphasis, as in the case of sentence-writing, is not encouraging pedantry; I am not suggesting a proliferation of technical terms with hairline distinctions among them. Just the opposite; the emphasis is on craftsmanship so meticulous that it is not only correct, but unobtrusively so.

Here is a sample: “Prove that any complex number is the product of a non-negative number and a number of modulus $1$.” I have had students who would have offered the following proof: “$−4i$ is a complex number, and it is the product of $4$, which is non-negative, and $−i$, which has modulus $1$; q.e.d.” The point is that in everyday English “any” is an ambiguous word; depending on context, it may hint at an existential quantifier (“have you any wool?”), “if anyone can do it, he can”) or a universal one (“any number can play”). Conclusion: never use “any” in mathematical writing. Replace it by “each” or “every”, or recast the whole sentence.

One way to recast the sample sentence of the preceding paragraph is to establish the convention that all “individual variables” range over the set of complex number and then write something like

I recommend against it. The symbolism of formal logic is indispensable in the discussion of the logic of mathematics, but used as a means of transmitting ideas from one mortal to another it becomes a cumbersome code. The author had to code his thoughts in it (I deny that anybody thinks in terms of $\exists$, $\forall$, $\wedge$, and the like), and the reader has to decode what the author wrote; both steps are a waste of time and an obstruction to understanding. Symbolic presentation, in the sense of either the modern logician or the classical epsilontist, is something that machines can write and few but machines can read.

So much for “any”. Other offenders, charged with lesser crimes, are “where”, and “equivalent”, and “if … then … if … then”. “Where” is usually a sign of a lazy afterthought that should have been thought through before. “If $n$ is sufficiently large, then $\vert a_n\vert < \epsilon$, where $\epsilon$ is a preassigned positive number”; both disease and cure are clear. “Equivalent” for theorems is logical nonsense. (By “theorem” I mean a mathematical truth, something that has been proved. A meaningful statement can be false, but a theorem cannot; “a false theorem” is self-contradictory). What sense does it make to say that the completeness of $L^2$ is equivalent to the representation theorem for linear functionals on $L^2$? What is meant is that the proofs of both theorems are moderately hard, but once one of them has been proved, either one, the other can be proved with relatively much less work. The logically precise word “equivalent” is not a good word for that. As for “if … then … if … then”, that is just a frequent stylistic bobble committed by quick writers and rued by slow readers. “If $p$, then if $q$, then $r.$” Logically all is well $(p \Longrightarrow (q \Longrightarrow r))$, but psychologically it is just another pebble to stumble over, unnecessarily. Usually all that is needed to avoid it is to recast the sentence, but no universally good recasting exists; what is best depends on what is important in the case at hand. It could be “If $p$ and $q,$ then $r,$” or “In the presence of $p$, the hypotheses $q$ implies the conclusion $r$”, or many other versions.

Gunakan istilah teknis dengan benar

The examples of mathematical diction mentioned so far were really logical matters. To illustrate the possibilities of the unobtrusive use of precise language in the everyday sense of the working mathematician, I briefly mention three examples: function, sequence, and contain.

I belong to the school that believes that functions and their values are sufficiently different that the distinction should be maintained. No fuss is necessary, or at least no visible, public fuss; just refrain from saying things like “the function $z^2+1$ is even”. It takes a little longer to say “the function $f$ defined by $f(z)=z^2+1$ is even”, or, what is from many points of view preferable, “the function $z\to z^2+1$ is even”, but it is good habit that sometimes save the reader (and the author) from serious blunder and that always makes for smoother reading.

“Sequence” always means “function whose domain is the set of natural numbers”. When an author writes “the union of sequence of measurable sets is measurable” he is guiding the reader’s attention to where it doesn’t belong. The theorem has nothing to do with the firstness of the first set, the secondness of the second, and so on; the sequence is irrelevant. The correct statement is that “the union of a countable set of measurable sets is measurable” (or, if a different emphasis is wanted, “the union of a countably infinite set of measurable sets is measurable”). The theorem that “the limit of a sequence of measurable functions is measurable” is a very different thing; there “sequence” is correctly used. If a reader knows what a sequence is, if he feels the definition in his bones, then the misuse of the word will distract him and slow his reading down, if ever so slightly; if he doesn’t really know, then the misuse will seriously postpone his ultimate understanding.

“Contain” and “include” are almost always used as synonyms, often by the same people who carefully coach their students that $\in$ and $\subset$ are not the same thing at all. It is extremely unlikely that the interchangeable use of contain and include will lead to confusion. Still, some years ago I started an experiment, and I am still trying it: I have systematically and always, in spoken word and written, used “contain” for $\in$ and “include” for $\subset$. I don’t say that I have proved anything by this, but I can report that (a) it is very easy to get used to, (b) it does no harm whatever, and (c) I don’t think that anybody ever noticed it. I suspect, but that is not likely to be provable, that this kind of terminological consistency (with no fuss made about it) might nevertheless contribute to the reader’s (and listener’s) comfort.

Consistency, by the way, is a major virtue and its opposite is a cardinal sin in exposition. Consistency is important in language, in notation, in references, in typography – it is important everywhere, and its absence can cause anything from mild irritation to severe misinformation.

My advice about the use of words can be summed up as follows. (1) Avoid technical terms, and especially the creation of new ones, whenever possible. (2) Think hard about the new ones that you must create; consult Roget; and make them as appropriate as possible. (3) Use the old ones correctly and consistently, but with a minimum of obtrusive pedantry.

Hindari simbol

Everything said about words applies, mutatis mutandis, to the even smaller units of mathematical writing, the mathematical symbols. The best notation is no notation; whenever it is possible to avoid the use of a complicated alphabetic apparatus, avoid it. A good attitude to the preparation of written mathematical exposition is to pretend that it is spoken. Pretend that you are explaining the subject to a friend on a long walk in the woods, with no paper available; fall back on symbolism only when it is really necessary.

A corollary to the principle that the less there is of notation the better it is, and in analogy with the principle of omitting irrelevant assumptions, avoid the use of irrelevant symbols. Example: “On a compact space every real-valued continuous function $f$ is bounded.” What does the symbol “$f$” contribute to the clarity of that statement? Another example: “If $0\leq\lim_n\alpha_n^{1/n}=\rho<1$, then $\lim_n\alpha_n=0.$” What does “$\rho$” contribute here? The answer is the same in both cases (nothing) but the reasons for the presence of the irrelevant symbols may be different. In the first case “$f$ “ may be just a nervous habit; in the second case “$\rho$” is probably a preparation for the proof. The nervous habit is easy to break. The other is harder, because it involves more work for the author. Without the “$\rho$” in the statement, the proof will take a half line longer; it will have to begin with something like “Write $\rho=\lim_n\alpha_n^{1/n}.$” The repetition (of “$\lim_n\alpha_n^{1/n}$”) is worth the trouble; both statement and proof read more easily and more naturally.

A showy way to say “use no superfluous letters” is to say “use no letter only once”. What I am referring to here is what logicians would express by saying “leave no variable free”. In the example above, the one about continuous functions, “$f$” was a free variable. The best way to eliminate is to convert it to free from bound. Most mathematicians would do that by saying “If $f$ is a real-valued continuous function on a compact space, then $f$ is bounded.” Some logicians would insist on pointing out that “$f$” is still free in the new sentence (twice), and technically they would be right. To make it bound, it would be necessary to insert “for all $f$” at some grammatically appropriate point, but the customary way mathematicians handle the problem is to refer (tacitly) to the (tacit) convention that every sentence preceded by all the universal quantifiers that are needed to convert all its variables into bound ones.

The rule of never leaving a free variable in a sentence, like many of the rules I am stating, is sometimes better to break than to obey. The sentence, after all, is an arbitrary unit, and if you want a free “$f$” dangling in one sentence so that you may refer to it in a later sentence in, say, the same paragraph, I don’t think you should necessarily be drummed out of the regiment. The rule is essentially sound, just the same, and while it may be bent sometimes, it does not deserve to be shattered into smithereens.

There are other symbolic logical hairs that can lead to obfuscation, or, at best, temporary bewilderment, unless they are carefully split. Suppose, for an example, that somewhere you have displayed the relation

as, say, a theorem proved about some particular $f$. If, later, you run across another function $g$ with what looks like the same property, you should resist the temptation to say “$g$ also satisfies $(1)$”. That’s logical and alphabetical nonsense. Say instead “$(1)$ remains satisfied if $f$ is replaced by $g$”, or, better, give $(1)$ a name (in this case it has a customary one) and say “$g$ also belongs to $L_2(0, 1)$”.

What about “inequality $(*)$”, or “equation $(7)$”, or “formula $\text{(iii)}$”; should all displays be labelled or numbered? My answer is no. Reason: just as you shouldn’t mention irrelevant assumptions or name irrelevant concepts, you also shouldn’t attach irrelevant labels. Some small part of the reader’s attention is attracted to the label, and some small part of his mind will wonder why the label is there. If there is a reason, then the wonder serves a healthy purpose by way of preparation, with no fuss, for a future reference to the same idea; if there is no reason, then the attention and the wonder are wasted.

It’s good to be stingy in the use of labels, but parsimony also can be carried to extremes. I do not recommend that you do what Dickson once did¹⁰. On p.89 he says: “Then … we have $(1)$ …” – but p.89 is the beginning of a new chapter, and happens to contain no display at all, let alone one bearing the label $(1)$. The display labelled (1) occurs on p. 90, overleaf, and I never thought of looking for it there. That trick gave me a helpless and bewildered five minutes. When I finally saw the light, I felt both stupid and cheated, and I have never forgiven Dickson.

One place where cumbersome notation quite often enters is in mathematical induction. Sometimes it is unavoidable. More often, however, I think that indicating the step from 1 to 2 and following it by an airy “and so on” is as rigorously unexceptionable as the detailed computation, and much more understandable and convincing. Similarly, a general statement about $n\times n$ matrices is frequently best proved not by the exhibition of many aij ‘s, accompanied by triples of dots laid out in rows and columns and diagonals, but by the proof of a typical (say $3\times3$) special case.

There is a pattern in all these injunctions about the avoidance of notation. The point is that the rigorous concept of a mathematical proof can be taught to a stupid computing machine in one way only, but to a human being endowed with geometric intuition, with daily increasing experience, and with the impatient inability to concentrate on repetitious detail for every long, that way is a bad way. Another illustration of this is a proof that consists of a chain of expressions separated by equal signs. Such a proof is easy to write. The author starts from the first equation, makes a natural substitution to get the second, collects terms, permutes, inserts and immediately cancels an inspired factor, and by steps such as these proceeds till he gets the last equation. This is, once again, coding, and the reader is forced not only to learn as he goes, but, at the same time, to decode as goes. The double effort is needless. By spending another ten minutes writing a carefully worded paragraph, the author can save each of his readers half an hour and a lot of confusion. The paragraph should be a recipe for action, to replace the unhelpful code that merely reports the result of the act and leaves the reader to guess how they were obtained. The paragraph would say something like this: “For the proof, first substitute $p$ for $q$, then collect terms, permute factors, and finally, insert and cancel a factor $r$.

A familiar trick of bad teaching is to begin a proof by saying: “Given $\epsilon$, let $\delta$ be $\left(\frac{\epsilon}{3M^2+2}\right)^{1/2}$”. This is the traditional backward proof-writing of classical analysis. It has the advantage of being easily verifiable by a machine (as opposed to understandable by a human being), and it has the dubious advantage that something at the end comes out to be less than $\epsilon$, instead of less than, say, $\left(\frac{(3M^2+7)\epsilon}{24}\right)^{1/3}$. The way to make the human reader’s task less demanding is obvious: write the proof forward. Start, as the author always starts, by putting something less than $\epsilon$, and then do what needs to be done – multiply by $3M^2 + 7$ at the right time and divide by 24 latter, etc., etc. – till you end up with what you end up with. Neither arrangement is elegant, but the forward one is graspable and rememberable.

Gunakan simbol dengan benar

There is not much harm that can be done with non-alphabetical symbols, but there too consistency is good and so is the avoidance of individually unnoticed but collectively abrasive abuses. Thus, for instance, it is good to use a symbol so consistently that its verbal translation is always the same. It is good, but it is probably impossible; nonetheless it’s a better aim than no aim at all. How are we to read “$\in$”: as the verb phrase “is in” or as the preposition “in”? Is it correct to say: “For $x\in A$, we have $x\in B$,” or “If $x\in A$, then $x\in B$”? I strongly prefer the latter (always read “$\in$” as “is in”) and I doubly deplore the former (both usages occur in the same sentence). It’s easy to write and it’s easy to read “For $x$ in $A$, we have $x\in B$”; all dissonance and all even momentary ambiguity is avoided. The same is true for “$\subset$” even though the verbal translation is longer, and even more true for “$\leq$”. A sentence such as “Whenever a positive number is $\leq 3$, its square is $\leq 9$” is ugly.

Not only paragraphs, sentences, words, letters, and mathematical symbols, but even the innocent looking symbols of standard prose can be the source of blemishes and misunderstanding; I refer to punctuation marks. A couple of examples will suffice. First: an equation, or inequality, or inclusion, or any other mathematical clause is, in its informative content, equivalent to a clause in ordinary language, and, therefore, it demands just as much to be separated from its neighbors. In other words: punctuate symbolic sentences just as would verbal ones. Second: don’t overwork a small punctuation mark such as a period or a comma. They are easy for the reader to overlook, and the oversight causes backtracking, confusion, delay. Example: “Assume that $a\in X$. $X$ belongs to the class $C$, …”. The period between the two $X$’s is overworked, and so is this one: “Assume that $X$ vanishes. $X$ belongs to the class $C$, …”. A good general rule is: never start a sentence with a symbol. If you insist on starting the sentence with a mention of the thing the symbol denotes, put the appropriate word in apposition, thus: “The set $X$ belongs to the class $C$, … “.

The overworked period is no worse than the overworked comma. Not “For invertible $X$, $X^∗$ also is invertible”, but “For invertible $X$, the adjoint $X^∗$ also is invertible”. Similarly, not “Since $p\neq 0$, $p\in U$”, but “Since $p\neq 0$, it follows that $p\in U$”. Even the ordinary “If you don’t like it, lump it” (or, rather, its mathematical relatives) is harder to digest that the stuffy-sounding “If you don’t like it, then lump it”; I recommend “then” with “if” in all mathematical contexts. The presence of “then”can never confuse; its absence can.

A final technicality that can serve as an expository aid, and should be mentioned here, is in a sense smaller than even the punctuation marks, it is conspicuous aspect of the printed page. What I am talking about is the layout, the architecture, the appearance of the page itself, of all the pages. Experience with writing, or perhaps even with fully conscious and critical reading, should give you a feeling for how what you are now writing will look when it’s printed. If it looks like solid prose, it will have a forbidding, sermony aspect; if it looks like computational hash, with a page full of symbols, it will have a frightening, complicated aspect. The golden mean is golden. Break it up, but not too small; use prose, but not too much. Intersperse enough displays to give the eye a chance to help the brain; use symbols, but in the middle of enough prose to keep the mind from drowning in a morass of suffixes.

Semua komunikasi adalah eksposisi

I said before, and I’d like for emphasis again, that the differences among books, articles, lectures, and letters (and whatever other means of communication you can think of) are smaller than the similarities.

When you are writing a research paper, the role of the “slips of paper” out of which a book outline can be constructed might be played by the theorems and the proofs that you have discovered; but the game of solitaire that you have to play with them is the same.

A lecture is a little different. In the beginning a lecture is an expository paper; you plan it and write it the same way. The difference is that you must keep the difficulties of oral presentation in mind. The reader of a book can let his attention wander, and later, when he decides to, he can pick up the thread, with nothing lost except his own time; a member of a lecture audience cannot do that. The reader can try to prove your theorems for himself, and use your exposition as a check on his work; the hearer cannot do that. The reader’s attention span is short enough; the hearer’s is much shorter. If computations are unavoidable, a reader can be subjected to them; a hearer must never be. Half the art of good writing is the art of omission; in speaking, the art of omission is nine-tenths of the trick. These differences are not large. To be sure, even a good expository paper, read out loud, would make an awful lecture – but not worse than some I have heard.

The appearance of the printed page is replaced, for a lecture, by the appearance of the blackboard, and the author’s imagined audience is replaced for the lecturer by live people; these are big differences. As for the blackboard: it provides the opportunity to make something grow and come alive in a way that is not possible with the printed page. (Lecturers who prepare a blackboard, cramming it full before they start speaking, are unwise and unkind to audiences.) As for live people: they provide an immediate feedback that every author dreams about but can never have.

The basic problem of all expository communication are the same; they are the ones I have been describing in this essay. Content, aim and organization, plus the vitally important details of grammar, diction, and notation – they, not showmanship, are the essential ingredients of good lectures, as well good books.

Pertahankan gaya Anda

Smooth, consistent, effective communications has enemies; they are called editorial assistants or copyreaders.

An editor can be a very great help to a writer. Mathematical writers must usually live without this help, because the editor of a mathematical book must be a mathematician, and there are very few mathematical editors. The ideal editor, who must potentially understand every detail of the author’s subject, can give the author an inside but nonetheless unbiased view of the work that the author himself cannot have. The ideal editor is the union of the friend, wife, student, and expert junior-grade whose contribution to writing I described earlier. The mathematical editors of book series and journals don’t even come near to the ideal. Their editorial work is but a small fraction of their life, whereas to be a good editor is a full-time job. The ideal mathematical editor does not exist; the friend-wife-etc. combination is only an almost ideal substitute.

The editorial assistant is a full-time worker whose job is to catch your inconsistencies, your grammatical slips, your errors of diction, your misspellings – everything that you can do wrong, short of the mathematical content. The trouble is that the editorial assistant does not regard himself as an extension of the author, and he usually degenerates into a mechanical misapplier of mechanical rules. Let me give some examples.

I once studied certain transformations called “measure-preserving”. (Note the hyphen: it plays an important role, by making a single word, an adjective, out of tow words.) Some transformations pertinent to that study failed to deserve the name; their failure was indicated, of course, by the prefix “non”. After a long sequence of misunderstood instructions, the printed version spoke of a “nonmeasure preserving transformation”. That is nonsense, of course, amusing nonsense, but, as such, it is distracting and confusing nonsense.

A mathematician friend reports that in the manuscript of a book of his he wrote something like “$p$ or $q$ holds according as $x$ is negative or positive”. The editorial assistant changed that to “$p$ or $q$ holds according as $x$ positive or negative”, on the grounds that it sounds better that way. That could be funny if it weren’t sad, and, of course, very very wrong.

A common complaint of anyone who has ever discussed quotation marks with the enemy concerns their relation to other punctuation. There appears to be an international typographical decree according to which a period or a comma immediately to the right of a quotation is “ugly”. (As here: the editorial assistant would have changed that to “ugly.” if I had let him.) From the point of view of the logical mathematician (and even more the mathematical logician) the decree makes no sense; the comma or period should come where the logic of the situation forces it to come. Thus,

He said:”The comma is ugly.”

Here, clearly, the period belongs inside the quote; the two situations are different and no inelastic rule can apply to both.

Moral: there are books on “style” (which frequently means typographical conventions), but their mechanical application by editorial assistants can be harmful. If you want to be an author, you must be prepared to defend your style; go forearmed into the battle.

Stop

The battle against copyreaders is the author’s last task, but it’s not the one that most authors regards as the last. The subjectively last step comes just before: it is to finish the book itself – to stop writing. That’s hard.

There is always something left undone, always either something more to say, or a better way to say something, or, at the very least, a disturbing vague sense that the perfect addition or improvement is just around the corner, and the dread that its omission would be everlasting cause for regret. Even as I write this, I regret that I did not include a paragraph or two on the relevance of euphony and prosody to mathematical exposition. Or, hold a minute!, surely I cannot stop without a discourse on the proper naming of concepts (why “commutator” is good and “set of first category” is bad) and the proper way to baptize theorems (why “the closed graph theorem” is good and “the Cauchy-Buniakowski-Schwarz theorem” is bad). And what about that sermonette that I haven’t been able to phrase satisfactorily about following a model. Choose someone, I was going to say, whose writing can touch you and teach you, and adapt and modify his style to fit your personality and your subject – surely I must get that said somehow.

There is no solution to this problem except the obvious one; the only way to stop is to be ruthless about it. you can postpone the agony a bit, and you should do so, by proofreading, by checking the computations, by letting the manuscript ripen, and then by reading the whole thing over in a gulp, but you won’t to stop any more then than before.

When you’ve written everything you can think of, take a day or two to read over the manuscript quickly and to test it for the obvious major points that would first strike a stranger’s eye. Is the mathematics good, is the exposition interesting, is the language clear, is the format pleasant and easy to read? Then proofread and check the computations; that’s an obvious piece of advice, and no one needs to be told how to do it. “Ripening” is easy to explain but not always easy to do: it means to put the manuscript out of sight and try to forget it off a few months. When you have done all that, and then re-read the whole work form a rested point of view, you have done all you can. Don’t wait and hope for one more result, and don’t keep on polishing. Even if you do get that result or do remove that sharp corner, you’ll only discover another mirage just ahead.

To sum it all up: begin at the beginning, go on till you come to the end, and then, with no further ado, stop.

Akhir kata

I have come to the end of all the advice on mathematical writing that I can compress into one essay. The recommendations I have been making are based partly on what I do, more on what I regret not having done, and most on what I wish others had done for me. You may criticize what I’ve said on many grounds, but I ask that a comparison of my present advice with my past actions not be one of them. Do, please, as I say, and not as I do, and you’ll do better. Then rewrite this essay and tell the next generation how to do better still.

Referensi