- 問題
- 解説
- 第1段落
- How do mathematicians solve problems?
- There have been few rigorous scientific studies of this question.
- Modern educational research, based on cognitive science, largely focuses on education up to high school level.
- Some studies address the teaching of undergraduate mathematics, but those are relatively few.
- There are significant differences between learning and teaching existing mathematics and creating new mathematics.
- Many of us can play a musical instrument, but far fewer can compose a concerto or even write a pop song.
- 第2段落
- When it comes to creativity at the highest levels, much of what we know — or think we know — comes from introspection.
- We ask mathematicians to explain their thought processes, and seek general principles.
- One of the first serious attempts to find out how mathematicians think was Jacques Hadamard’s The Psychology of Invention in the Mathematical Field, first published in 1945.
- Hadamard interviewed leading mathematicians and scientists of his day and asked them to describe how they thought when working on difficult problems.
- What emerged very strongly was the vital role of what for lack of a better term must be described as intuition.
- Some feature of the subconscious mind guided their thoughts.
- Their most creative insights did not arise through step by step logic, but by sudden, wild leaps.
- 第3段落
- One of the most detailed descriptions of this apparently illogical approach to logical questions was provided by the French mathematician Henri Poincaré, one of the leading figures of the late nineteenth and early twentieth centuries.
- Poincaré was adamant that conscious logic was only part of the creative process.
- Yes, there were times when it was indispensable: deciding what the problem really was, systematically verifying the answer.
- But in between, Poincaré felt that his brain was often working on the problem without telling him, in ways that he simply could not fathom.
- 第4段落
- His outline of the creative process distinguished three key stages: preparation, incubation, and illumination.
- Preparation consists of conscious logical efforts to pin the problem down, make it precise, and attack by conventional methods.
- This stage Poincaré considered essential; it gets the subconscious going and provides raw materials for it to work with.
- Incubation takes place when you stop thinking about the problem and go off and do something else.
- The subconscious now starts combining ideas with each other, often quite wild ideas, until light starts to dawn.
- With luck, this leads to illumination: your subconscious taps you on the shoulder and the proverbial light bulb goes off in your mind.
- 第5段落
- This kind of creativity is like walking a tightrope.
- On the one hand, you won’t solve a difficult problem unless you make yourself familiar with the area to which it seems to belong — along with many other areas which may or may not be related, just in case they are.
- On the other hand, if all you do is get trapped into standard ways of thinking, which others have already tried, fruitlessly, then you will be stuck in a mental swamp and discover nothing new.
- So the trick is to know a lot, integrate it consciously, put your brain in gear for weeks… and then set the question aside.
- The intuitive part of your mind then goes to work, rubs ideas against each other to see whether the sparks fly, and notifies you when it has found something.
- This can happen at any moment: Poincaré suddenly saw how to solve a problem that had been bugging him for months when he was stepping off a bus.
- Archimedes famously worked out how to test metal to see if it were gold when he was having a bath.
- 第1段落
問題
次の文章の下線をほどこした部分(1)~(3)を和訳しなさい。
How do mathematicians solve problems? There have been few rigorous scientific studies of this question. Modern educational research, based on cognitive science, largely focuses on education up to high school level. (1)Some studies address the teaching of undergraduate mathematics, but those are relatively few. There are significant differences between learning and teaching existing mathematics and creating new mathematics. Many of us can play a musical instrument, but far fewer can compose a concerto or even write a pop song.
When it comes to creativity at the highest levels, much of what we know — or think we know — comes from introspection. We ask mathematicians to explain their thought processes, and seek general principles. One of the first serious attempts to find out how mathematicians think was Jacques Hadamard’s The Psychology of Invention in the Mathematical Field, first published in 1945. Hadamard interviewed leading mathematicians and scientists of his day and asked them to describe how they thought when working on difficult problems. (2)What emerged very strongly was the vital role of what for lack of a better term must be described as intuition. Some feature of the subconscious mind guided their thoughts. Their most creative insights did not arise through step by step logic, but by sudden, wild leaps.
One of the most detailed descriptions of this apparently illogical approach to logical questions was provided by the French mathematician Henri Poincaré, one of the leading figures of the late nineteenth and early twentieth centuries. Poincaré was adamant that conscious logic was only part of the creative process. Yes, there were times when it was indispensable: deciding what the problem really was, systematically verifying the answer. But in between, Poincaré felt that his brain was often working on the problem without telling him, in ways that he simply could not fathom.
His outline of the creative process distinguished three key stages: preparation, incubation, and illumination. Preparation consists of conscious logical efforts to pin the problem down, make it precise, and attack by conventional methods. This stage Poincaré considered essential; it gets the subconscious going and provides raw materials for it to work with. Incubation takes place when you stop thinking about the problem and go off and do something else. The subconscious now starts combining ideas with each other, often quite wild ideas, until light starts to dawn. With luck, this leads to illumination: your subconscious taps you on the shoulder and the proverbial light bulb goes off in your mind.
This kind of creativity is like walking a tightrope. (3)On the one hand, you won’t solve a difficult problem unless you make yourself familiar with the area to which it seems to belong — along with many other areas which may or may not be related, just in case they are. On the other hand, if all you do is get trapped into standard ways of thinking, which others have already tried, fruitlessly, then you will be stuck in a mental swamp and discover nothing new. So the trick is to know a lot, integrate it consciously, put your brain in gear for weeks… and then set the question aside. The intuitive part of your mind then goes to work, rubs ideas against each other to see whether the sparks fly, and notifies you when it has found something. This can happen at any moment: Poincaré suddenly saw how to solve a problem that had been bugging him for months when he was stepping off a bus. Archimedes famously worked out how to test metal to see if it were gold when he was having a bath.
解説
第1段落
How do mathematicians solve problems?
How do mathematiciansS solveV problemsO?
どのように数学者Sは問題Oを解くVのか?
数学者はどのように問題を解くのか?
There have been few rigorous scientific studies of this question.
There have beenV few rigorous scientific studiesS of this question.
この質問の少ない厳しい科学の研究SがあってきたV。
この質問に対する厳密な科学的研究は, ほとんどなされてこなかった。
Modern educational research, based on cognitive science, largely focuses on education up to high school level.
- up to Aは多義イディオムで, 「A[地点]まで, Aの責任で, A[悪事]を企んで」を押さえておく。特にIt is up to you「それはあなた次第だ」は頻出。
Modern educational researchS, based on cognitive science, largely focuses onV educationO up to high school level.
認知科学に基づいた, 近代の教育的な研究Sは, 大部分は高校レベルまでの教育Oに注目するV。
認知科学に基づいた現代的な教育調査は, 主に高校レベルまでの教育に焦点を当てている。
Some studies address the teaching of undergraduate mathematics, but those are relatively few.
- undergraduateは大学1~4年生(学部生)のことで, 卒業してgraduateになると, 大学院生のことを指す。
Some studiesS1 addressV1 the teachingO1 of undergraduate mathematics, but thoseS2 areV2 relatively fewC2.
いくつかの研究S1は学部学生の数学の教えO1を取り組むV1が, それらS2は比較的少数C2だ。
大学の学部生レベルの数学教育を扱う研究もあるが, その数は比較的少ない。
There are significant differences between learning and teaching existing mathematics and creating new mathematics.
There areV significant differencesS between learning and teaching existing mathematicsA andand creating new mathematicsB.
存在する数学を学ぶことと教えることAと, 新しい数学を作ることBの間に, 重大な違いSがあるV。
今ある数学を教えたり学んだりすることと, 新しい数学を想像することの間には, 非常に大きな違いがある。
Many of us can play a musical instrument, but far fewer can compose a concerto or even write a pop song.
Many of usS1 can playV1 a musical instrumentO1, but far fewerS2 can composeV21 a concertoO21 or even writeV22 a pop songO22.
私たちの多くS1は楽器O1を弾けるV1が, ずっとより少数S2はコンチェルトO21を編集しV21たり, ポップソングO22を書くことさえできるV22。
楽器を弾ける人は多いが, ほとんどの人は, 協奏曲はもちろん, 現代風の流行曲さえ作曲することができない。
第2段落
When it comes to creativity at the highest levels, much of what we know — or think we know — comes from introspection.
- introspectionは知らなくてよいが, 接頭辞intro-から, 雰囲気で読む。
When it comes to creativity at the highest levels, muchS of what weS’ knowV’1 — or thinkV’2 we know — comesV from introspection.
最も高いレベルでの創造性ということになると, 私たちS’が知っているV’1, または私たちが知っていると考えているV’2ことのほとんどSはintrospectionから来るV。
最高水準の創造性については, 私たちが知っている, もしくはそう思っていることの大部分が, 自己反省から生じる。
We ask mathematicians to explain their thought processes, and seek general principles.
WeS askV1 mathematiciansO1 to explain their thought processesto do, and seekV2 general principlesO2.
私たちSは数学者O1に彼らの試行プロセスを説明するto doように頼みV1, 一般の原理O2を求めるV2。
私たちは, 数学者に思考の過程を説明してもらい, 普遍的に成り立つ法則を追い求めている。
One of the first serious attempts to find out how mathematicians think was Jacques Hadamard’s The Psychology of Invention in the Mathematical Field, first published in 1945.
OneS of the first serious attempts to find out how mathematiciansS’ thinkV’ wasV Jacques Hadamard’s The Psychology of Invention in the Mathematical FieldC, first published in 1945.
どのように数学者S’が考えるV’かということを発見するための最初の真面目な試みの一つSは, 最初に1945に出版された, Jacques HadamardのThe psychology of Invention in the Mathematical FieldCだった。
数学者の思考方法を理解しようとする最初の本格的な試みの一つは, 1945年にジャック・アダマールによってはじめて出版された, 「数学における発明の心理」だった。
Hadamard interviewed leading mathematicians and scientists of his day and asked them to describe how they thought when working on difficult problems.
HadamardS interviewedV1 leading mathematicians and scientistsO1 of his day and askedV2 themO2 to describeto do how theyS’ thoughtV’ when working onV” difficult problemsO”.
HadamardSは彼の日のリードする数学者や科学者O1にインタビューしV1て, 彼らO2にどのように彼らS’が難しい問題O”に取り組んでいるV”ときに考えるV’かということを述べるto doように頼んだV2。
アダマールは, 当時の優れた数学者や科学者を面談し, 難しい問題に取り組む際の思考方法を述べてもらった。
What emerged very strongly was the vital role of what for lack of a better term must be described as intuition.
What emergedS’ very strongly wasV the vital roleC of what for lack of a better term must be describedV” as intuition.
とても強く出現したV’ものは, より良い用語の欠落のために, 直感として述べられる必要のあるV”ものの重要な役割Cだった。
はっきりと明らかになったことは, 良い用語がないので直感としか言いようのないものが果たす, 重要な役割だった。
Some feature of the subconscious mind guided their thoughts.
- Aが単数形のとき, some A = certain A「あるA, 特定のA」
- subconsciousは, sub + conscious。接頭辞のsub-は, subleader, subtitleなどから, 一つ下の意味があることを知っていれば, subconsciousは「潜在意識下の」と訳せる。
Some featureS of the subconscious mind guidedV their thoughtsO.
subconscious mindのある特徴Sが, 彼らの思考OをガイドしたV。
潜在意識下にある何らかの特徴が, 彼らの思考を導いていた。
Their most creative insights did not arise through step by step logic, but by sudden, wild leaps.
- butの並列は, 何と何が並列されているかを確定することは容易だが, 意味を取り違えないように注意。but以下は主語と動詞が無いので, 前半を使って復元してみる。ここで, Their most creative insights did not arise by sudden, wild leapsとやってしまうと間違い。notは基本的には省略されない。よって, Their most creative insights arose by sudden, wild leapsと復元しないといけない。
Their most creative insightsS did not ariseV through step by step logic, but by sudden, wild leaps.
彼らの最も創造的な洞察Sは, ステップバイステップのロジックを通じて生じなかったV, しかし, 突然の, ワイルドな飛躍によって。
彼らの最も創造的な洞察は, 段階的な論理の積み上げではなく, 突飛で大胆な飛躍によって, 生じていた。
第3段落
One of the most detailed descriptions of this apparently illogical approach to logical questions was provided by the French mathematician Henri Poincaré, one of the leading figures of the late nineteenth and early twentieth centuries.
OneS of the most detailed descriptions of this apparently illogical approach to logical questions was providedV by the French mathematician Henri Poincaré, one of the leading figures of the late nineteenth and early twentieth centuries.
論理的な質問に対するこの一見すると論理的でないアプローチの最も詳細化された説明の一つSは, フランスの数学者Henri Poincaréによって提供されたV。つまり, 19世紀後期と初期の20世紀のリードしている人物の一人。
論理的な問いに対する, この一見非論理的に思える取り組み方の, 最も詳細な説明の一つは, 19世紀後半と20世紀前半を代表する人物の一人である, フランスの数学者, アンリ・ポアンカレによってなされた。
Poincaré was adamant that conscious logic was only part of the creative process.
- adamantは知らなくてよいが, 用法を見ると, be adamant that S Vのようになっているので, be sure that S Vのようなものだと仮定して何となくで読むと良い。
PoincaréS wasV adamantC that conscious logicS’ wasV’ only partC’ of the creative process.
PoincaréSは, 意識のロジックS’がただ創造的なプロセスの一部V’だということをadamantCだった。
ポアンカレは, 意識的な論理思考は, 創造的な思考過程の一部にすぎないということを強く確信していた。
Yes, there were times when it was indispensable: deciding what the problem really was, systematically verifying the answer.
Yes, there wereV timesS when itS’ wasV’ indispensableC’: deciding what the problemS” really wasV”, systematically verifying the answer.
はい, それS’が不可欠C’であるようなときSがあるV。つまり, 問題S”が本当は何かということを決めること, システマチックに答えを証明すること。
もちろん, 論理的思考が不可欠な場合もあった。例えば, 問題が実際には何であるかを決定する際や, 体系的に答えを証明する際など。
But in between, Poincaré felt that his brain was often working on the problem without telling him, in ways that he simply could not fathom.
- fathomは知らなくてよい。
But in between, PoincaréS feltV that his brainS’ was often working onV’ the problemO’ without telling him, in ways that heS” simply could not fathomV”.
しかし間の中で, PoincaréSは彼の脳S’はしばしば, 彼S”が単にfathom出来ないV”ような方法で, 彼に伝えることなしに問題O’に取り組んでいるV’ということを感じていたV。
しかし, 論理的思考と創造的思考の間で, 自分の脳が無意識に, 全く理解できないやり方で, 問題に取り組んでいることがあると, ポアンカレは感じていた。
第4段落
His outline of the creative process distinguished three key stages: preparation, incubation, and illumination.
- incubation「孵化, 培養」は出来れば知っておきたいが, 準備と解明の間の中間段階だということが文脈からわかるので, 知らなければ適当に訳せばよい。
His outlineS of the creative process distinguishedV three key stagesO: preparation, incubation, and illumination.
創造的なプロセスの彼のアウトラインSは, 3つのキーステージOを見分けたV。つまり, 準備, incubation, イルミネーション。
想像的過程についての彼の概要は, 3つの鍵となる段階に分かれていた。つまり, 準備段階, 孵化段階, 解明段階である。
Preparation consists of conscious logical efforts to pin the problem down, make it precise, and attack by conventional methods.
- consist of A = be composed of A = comprise A「Aから構成される」, consist in A = lie in A「Aに在る」
PreparationS consists ofV conscious logical effortsO to pin the problem down, make it precise, and attack by conventional methods.
準備Sは, 問題をpin downして, それを正確にして, 伝統的なメソッドによってアタックするための意識的な論理的な努力Oから成るV。
準備段階は, 問題を特定し, それを正確な形にし, 従来の方法で取り組む意識的な論理的努力から成る。
This stage Poincaré considered essential; it gets the subconscious going and provides raw materials for it to work with.
- consider A B = consider A as B = consider A to be B「AをBと見なす」
- provide A with B = provide B for A「AにBを供給する」
- get O doing「Oを~の状態にする」
This stageO1 PoincaréS1 consideredV1 essentialC1; itS2 getsV21 the subconsciousO21 goingC21 and providesV22 raw materialsB forfor itA to work with.
PoincaréS1はこのステージO1を本質的C1だとみなしたV1。それS2は潜在意識O21をgoingC21にしV21て, それAに一緒にworkすべき生の材料Bを供給するV22。
この準備段階こそ, ポアンカレは最も重要だとみなした。準備段階によって, 潜在意識が働くようになり, 取り組むべき最初の素材が潜在意識に与えられると考えた。
Incubation takes place when you stop thinking about the problem and go off and do something else.
- go off = explode「爆発する」だが, ここでは明らかに違うので, go「行く」 + off「離れて」と考える。
IncubationS takes placeV when youS’ stopV’1 thinkingO’1 about the problem and go offV’2 and doV’3 something elseO’3.
incubationSは, あなたS’が問題について考えるこO’1とを止めV’1て, go offしV’2て, 他の何かO’3をするV’3ときに起こるV。
孵化段階は, 問題について考えることを止めて, 一旦離れ, 何か他のことをしているときに生じる。
The subconscious now starts combining ideas with each other, often quite wild ideas, until light starts to dawn.
- combine A with B「AとBを結びつける」
- S V, until S’ V’は直訳すると「S’ V’するまでS Vする」となるが, 時系列を考えると, 「S Vして, それからS’ V’する」と前から訳せる。
The subconsciousS now startsV combining ideasO with each other, often quite wild ideas, until lightS’ starts to dawnV’.
潜在意識Sは今それぞれとアイデアを結びつけることOを始めるV。しばしばかなりワイルドなアイデア。ライトS’が夜明け始めるV’まで。
潜在意識は今度は, 様々な考えをお互いに結び付け始める。それらはの中にはかなり大胆な考えも含まれているかもしれない。その後, 解決への明かりが灯りはじめる。
With luck, this leads to illumination: your subconscious taps you on the shoulder and the proverbial light bulb goes off in your mind.
- proverb = saying「ことわざ」
With luck, thisS1 leads toV1 illuminationO1: your subconsciousS2 tapsV2 youO2 on the shoulder and the proverbial light bulbS3 goes offV3 in your mind.
幸運とともに, これS1はイルミネーションO1まで導くV1。あなたの潜在意識S2が肩の上であなたO2をタップしV2て, ことわざ的なライトバルブS2があなたの心の中で爆発するV2。
もし運がよければ, これは解明段階につながる。潜在意識があなたの肩を叩き, 例の電球が頭の中で光る。
第5段落
This kind of creativity is like walking a tightrope.
This kind of creativityS isV like walking a tightrope.
この種の創造性Sは, tightropeを歩くようなものだ。
このような創造性は, 綱渡りをすることに似ている。
On the one hand, you won’t solve a difficult problem unless you make yourself familiar with the area to which it seems to belong — along with many other areas which may or may not be related, just in case they are.
- you won’t V1 unless you V2は, 直訳すると「V2しない限りV1しないだろう」となるが, 対偶を意識すると, 「V1するためにはV2しないといけない」と解釈できる。この方が前から訳せて楽なので, 特に文が長い場合は有効。
On the one hand, youS won’t solveV a difficult problemO unless youS’ makeV’ yourselfO’ familiar with the areaC’ to which itS” seems to belongV” — along with many other areas which may or may not be relatedV”’, just in case theyS”” areV””.
一方で, あなたSは難しい問題Oを解かないだろうV。あなたS’があなた自身O’をそれS”が属していると思われるV”ようなエリアに精通している状態C’にしV’ない限り。関係しているかしていないかもしれないV”’ような多くの他のエリアと一緒に。それらS””が~であるV””場合に備えて。
一方では, 難しい問題を解くためには, それが属していると思われる領域はもちろん, 他の多くの関係ないかもしれない領域についても, もし関係していた場合に備えて, 詳しくならなければいけないだろう。
On the other hand, if all you do is get trapped into standard ways of thinking, which others have already tried, fruitlessly, then you will be stuck in a mental swamp and discover nothing new.
- all S have to do is (to) V「Sがすべきことの全てはVすることだ」の定型表現は, isのあとはtoを省略して動詞の原形も来ることができる。この表現と同じようなことが起こっている。
- stuckの原形はstick。stick to A = cling to A「Aに固執する」の受身形。
- swamp「沼」は必須ではないが出来れば知っておきたい。
- fruitlesslyは, 「フルーツがないように(実りなく)」と訳せるが, 他の部分との兼ね合いが難しいのでうまい訳し方は色んな例文を見ると良い。意外とよく出てくる単語なので注意。
On the other hand, if all you doS’ isV’ get trappedC’ into standard ways of thinking, which othersS” have already triedV”, fruitlessly, then youS will be stuckV1 in a mental swamp and discoverV2 nothing newO2.
他方で, もしあなたのするすべてS’が標準的な思考の方法の中へget trappedすることC’で, そしてそれは他人S”がすでに挑戦してきV”て, fruitlesslyに, ならばそれならあなたSはメンタルのswampの中にstickされV1て, 新しい無O2を発見するだろうV2。
他方で, すでに他の人が試してきて, 成果がなかったような標準的な思考方法にとらわれるだけでは, 精神的な泥沼にはまり, 新しいことは何も見つからないだろう。
So the trick is to know a lot, integrate it consciously, put your brain in gear for weeks… and then set the question aside.
So the trickS isV to know a lot, integrate it consciously, put your brain in gear for weeks… and then set the question aside.
だから, トリックは, たくさん知って, 意識的にそれを統合して, 何週間もの間ギアの中にあなたの脳を置いて, …, それからわきに問題をセットすることだ。
したがって, 秘訣は, 多くのことを学び, それらを意識的に組み合わせ, 何週間も脳を働かせ, そして, 問題を一旦忘れることだ。
The intuitive part of your mind then goes to work, rubs ideas against each other to see whether the sparks fly, and notifies you when it has found something.
The intuitive partS of your mind then goes to workV1, rubsV2 ideasO2 against each other to see whether the sparksS’ flyV’, and notifiesV3 youO3 when itS” has foundV” somethingO”.
あなたの心の直感的な部分Sはそれから働きに行っV1て, スパークS’が飛ぶV’かどうかということを見るためにそれぞれに対してアイデアO2をこすっV2て, それS”が何かO”を見つけたV”ときにあなたO2に知らせるV2。
それから, 頭の中の直感が機能し始め, 様々な考えをお互いにすり合わせて火花が飛び散るかどうかを確かめ, 何か分かった時に知らせてくれる。
This can happen at any moment: Poincaré suddenly saw how to solve a problem that had been bugging him for months when he was stepping off a bus.
ThisS1 can happenV1 at any moment: PoincaréS2 suddenly sawV2 how to solve a problemO2 that had been buggingV’ himO’ for months when heS” was stepping offV” a busO”.
これS1は任意の瞬間で起こりうるV1。PoincaréS2は, 彼S”がバスO”から降りるV”ときに, 数か月間彼O’をbugしてきたV’ような問題の解き方O2を見たV2。
これは, どんなときにも起こる可能性がある。ポアンカレは, バスから降りている最中に突然, 彼を何か月間も悩ませてきた問題の解決方法をひらめいた。
Archimedes famously worked out how to test metal to see if it were gold when he was having a bath.
- if節は, seeの目的語になっているので, 名詞節。「~かどうか(ということ)」。
ArchimedesS famously worked outV how to test metalO to see if itS’ wereV’ goldC’ when heS” was havingV” a bathO”.
ArchimedesSは有名に, 彼S”が風呂O”に入っていたV”ときに, それS’が金C’であるかどうかということを見るための金属のテスト方法Oを理解したV。
有名な話だが, アルキメデスは入浴中に, ある金属が金であるかどうかを確かめる方法を思いついた。
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