ریاضیات

از ویکی‌پدیا، دانشنامهٔ آزاد
پرش به ناوبری پرش به جستجو
فارسیEnglish
اقلیدس
اقلیدس (در حالی که کولیس در دست دارد)، ریاضیدان یونانی قرن سوم قبل از میلاد، این حد از جزئیات بر اساس تصور رافائل از ''مکتب آتن'' می باشد.

ریاضیات به مطالعه مباحثی چون کمیت (نظریه اعداد)،[۱] ساختار (جبر)،[۲] فضا (هندسه)،[۱] و تغییرات (آنالیز ریاضی) می پردازد.[۳][۴][۵] در حقیقت تعریفی جهان شمول که همه بر سر آن توافق داشته باشند برای ریاضیات وجود ندارد.

ریاضیدانان به دنبال الگوهایی هستند که بتوان از آن ها استفاده کرده و حدس های جدیدی را فرموله کرد؛ آن ها درستی یا غلطی حدس ها را توسط اثبات ریاضیاتی نشان می دهند. هرگاه ساختار های ریاضی مدل های خوبی از پدیده های جهان واقعی باشند، استدلال ریاضی می تواند بینش یا پیشبینی هایی برای طبعیت ارائه کند. علم ریاضیات با استفاده از تجرید و منطق از مفاهیمی چون شمردن، محاسبه و اندازه گیری و مطالعه ی نظام مند اشکال و حرکات اشیاء فیزیکی بوجود آمد. ریاضیات کاربردی از زمانی که انسان نوشتن را آموخت، به عنوان فعالیتی بشری وجود داشته است. تحقیقات مورد نیاز برای حل مسائل ریاضی ممکن است سال ها یا حتی قرن ها به درازا بینجامد.

استدلال های استوار ابتدا در ریاضیات یونان باستان ظاهر شدند، بخصوص در اثر عناصر اقلیدس. از زمان کار های تحقیقاتی جوزپه پئانو (۱۸۵۸-۱۹۳۲)، دیوید هیلبرت (۱۸۶۲-۱۹۴۳) و دیگران بر روی دستگاه اصول موضوعه ای در پایان قرن نوزدهم میلادی، روش تحقیقاتی ریاضیدانان به این شکل درآمده که آن ها حقایق را با استدلال استوار از مجموعه منتخبی از اصول موضوعه و تعاریف بدست می آورند. روند پیشرفت ریاضیات تا زمان رنسانس سرعت نسبتاً آرامی داشت، تا زمانی نوآوری های ریاضیاتی با کشفیات علمی برهمکنش کرده و منجر به افزایش سریع نرخ اکتشافات ریاضی گشت که تا به امروز نیز ادامه دارد [۶].

ریاضیات در بسیاری از زمینه ها مثل علوم طبیعی، مهندسی، پزشکی، اقتصاد و علوم اجتماعی یک علم ضروری می باشد. شاخه های کاملاً جدیدی در ریاضیات بوجود آمده اند مثل نظریهٔ بازی ها. ریاضیدانان در ریاضیات محض (مطالعه ریاضی به هدف کشف هرچه بیشتر راز های خود آن) بدون این که هیچ گونه هدف کاربردی در ذهن داشته باشند به تحقیقات می پردازند، در حالی که کاربرد های عملی یافته های آن ها معمولاً بعد ها کشف می شود.[۷]

تاریخچه[ویرایش]

مقاله اصلی: تاریخ ریاضیات

لوح ریاضیاتی بابلیان
لوح ریاضیاتی بابلیان، پلیمپتون ۳۲۲، مربوط به ۱۸۰۰ قبل از میلاد
روش افنا در تقریب عدد پی توسط ارشمیدس
ارشمیدس از روش افنا برای تقریب مقدار عدد پی استفاده کرد.
سیستم عددی استفاده شده در دستنویس بخشالی
سیستم عددی استفاده شده در دستنویس بخشالی (مربوط به ریاضیات هند) که بر می گردد به قرن دوم قبل از میلاد و قرن دوم پس از میلاد.

تاریخ ریاضیات را می توان به عنوان دنباله ای از تجرید سازی های فزاینده دید. اولین قابلیت تجرید سازی که در بسیاری از حیوانات مشترک است،[۸] احتمالاً مفهوم عدد می باشد: فهم این مطلب که مجموعه دو سیب و مجموعه دو پرتقال (به عنوان مثال) با هم اشتراکی دارند، و آن کمیت تعدادشان است.

همانطور که شواهد بر روی چوب‎خط نشان می دهد، مردم پیشاتاریخ می توانستند اشیاء فیزیکی را بشمرند و توانایی شمردن اشیاء تجریدی مثل روز، فصل و سال را نیز داشتند.[۹]

شواهد مربوط به ریاضیات پیچیده تر تا ۳۰۰۰ قبل میلاد مشاهده نشده، زمانی که بابلی ها و مصری ها شروع به استفاده از حساب، جبر و هندسه برای محاسبات مربوط به مالیات و دیگر مفاهیم اقتصادی، و ساخت و ساز یا نجوم کردند.[۱۰] قدیمی ترین متون ریاضیاتی مربوط به میان‎رودان و مصر می شود که به ۲۰۰۰-۱۸۰۰ قبل از میلاد باز می گردد. بسیاری از متون اولیه سه تایی های فیثاغورثی را ذکر کرده و لذا به نظر می رسد که قضیه فیثاغورث کهن ترین و گسترده ترین توسعه ریاضیاتی بعد از حساب مقدماتی و هندسه باشد. در اسناد تاریخی، در ریاضیات بابلی ها بود که حساب مقدماتی (جمع، تفریق، ضرب و تقسیم) ابتدا پدیدار گشت. بابلی ها همچنین از یک دستگاه مکان-ارزشی بهره می جستند که در آن دستگاه اعداد پایه ۶۰ پیاده سازی شده بود، ازین دستگاه عددی هنوز هم برای اندازه گیری زاویه و زمان استفاده می شود.[۱۱]

با آغاز قرن ششم قبل از میلاد، ریاضیات یونانی ها با فیثاغورسی ها مطالعه ی نظام مندی را در ریاضیات، به هدف شناخت بیشتر خود ریاضیات آغاز نمودند که سرآغاز ریاضیات یونانی ها بود.[۱۲] حدود ۳۰۰ قبل از میلاد، اقلیدس روش اصول موضوعه ای را که هنوز هم در ریاضیات به کار می رود را معرفی کرد که شامل تعاریف، اصول، قضیه و اثبات بود. کتاب مرجع او که به عناصر معروف است به طور گسترده به عنوان موفق ترین و تأثیر گذار ترین کتب مرجع همه زمان ها شناخته می شود.[۱۳] بزرگترین ریاضیدانان باستان را اغلب ارشمیدس (۲۸۷ تا ۲۱۲ قبل از میلاد) اهل سیراکوز می دانند.[۱۴] او فرمول هایی برای محاسبه ی مساحت و حجم اجسام در حال دوران پیدا کرد و از روش افنا برای محاسبه مساحت زیر منحنی سهمی با استفاده از جمع یک سری بی نهایت استفاده کرد به گونه ای که بی شباهت با حساب دیفرانسیل و انتگرال مدرن نمی‌باشد.[۱۵] دیگر دستاورد های قابل توجه در ریاضیات یونان مقاطع مخروطی (آپولونیوس اهل پرگا، قرن سوم قبل از میلاد)،[۱۶] مثلثات (هیپارکوس اهل نیکا (قرن دوم قبل از میلاد))،[۱۷] و آغاز جبر (دیوفانتوس، قرن سوم پس از میلاد) بود.[۱۸]

سیستم عددی هندو-عربی و قواعد استفاده از عملیاتش که امروزه در سراسر جهان استفاده می شود، در طی هزاره ی اول میلادی در هند توسعه یافت و سپس از طریق ریاضیات اسلامی به جهان غرب انتقال یافت. دیگر پیشرفت های مربوط به ریاضیات هندی ها شامل تعریف مدرن سینوس و کسینوس و فرم اولیه سری های بی نهایتی می باشد.

صفحه ای از کتاب جبر خوارزمی.
صفحه ای از کتاب جبر خوارزمی.

در طی عصر طلایی اسلام، که در قرن نهم و دهم شکل گرفت، ریاضیات نوآوری های مهمی را به خود دید که بر اساس ریاضیات یونانی ها پایه ریزی شده بود . مهم ترین دستآورد های ریاضیات اسلامی توسعه ی جبر بود. دیگر دستاورد های مهم ریاضیات دوره ی اسلامی پیشرفت در مثلثات کروی و اضافه شدن اعشار به سیستم عددی عربی بود. بسیاری از ریاضیدانان این دوره فارس زبان بودند مثل الخوارزمی، عمر خیام و شرف الدین توسی.

در طی اوایل عصر مدرن، ریاضیات شروع به توسعه شتابداری در غرب اروپا کرد. توسعه حساب دیفرانسیل و انتگرال توسط نیوتون و لایبنیز در قرن هفدهم ریاضیات را متحول کرد. لئونارد اویلر مهم ترین ریاضیدان قرن هجدهم بود که چندین قضیه و کشفیات را به ریاضیات افزود. شاید مهم ترین ریاضیدانان قرن نوزدهم ریاضیدان آلمانی کارل فردریش گاوس بود که خدمات متعددی به شاخه های مختلف ریاضیات چون جبر، آنالیز، هندسه دیفرانسیل، نظریه ماتریس، نظریه اعداد و آمار کرد. در اوایل قرن بیستم، کورت گودل، ریاضیات را با انتشار قضایای ناتمامیت خویش دچار تغییر کرد. این قضایا نشان دادند که هر سیستم اصول موضوعه سازگاری شامل گزاره های غیر قابل اثبات اند.

ریاضیات از آن زمان به طور گسترده ای توسعه یافته است و کنش و واکنش های ثمربخشی بین ریاضیات و علوم ایجاد شده که به نفع هردو می باشد. کشفیات ریاضیات تا به امروز نیز ادامه دارد. بر اساس نظر میخائیل سوریوک، که در ژانویه ۲۰۰۶ در بولتن انجمن ریاضی امریکا منتشر شد، "تعداد مقالات و کتب پایگاه اطلاعاتی ژورنال Mathematical Review از سال ۱۹۴۰ (اولین سال عملیاتی شدن MR) اکنون به ۱٫۹ میلیون می رسد که سالانه بیش از ۷۵ هزار مورد به این پایگاه افزوده می شود. اکثریت کار های گسترده ای که در این اقیانوس وجود دارد شامل قضایای جدید ریاضیاتی و اثبات هایشان است.

شاخه های ریاضیات[ویرایش]

چرتکه
چرتکه، یک وسیله ساده محاسباتی که از زمان های باستان مورد استفاده قرار می گرفت.

ریاضیات را می توان به طور خیلی کلی به چند قسمت تقسیم کرد: مطالعه کمیت، ساختار، فضا و تغییرات (یعنی حساب، جبر، هندسه و آنالیز). علاوه بر این ها که دغدغه های اصلی ریاضیات هستند، گرایش های دیگری نیز وجوددارند که خود را وقف کاوش ارتباطات بین قلب ریاضیات با دیگر زمینه های ریاضیات کرده اند، مثل ارتباطش با منطق، نظریه مجموعه ها (شالوده های ریاضی)، یا دیگر شاخه های تجربی تر ریاضیات که در علوم مختلف کاربرد دارند (ریاضیات کاربردی)، و اخیراً مطالعه عدم قطعیت. در حالی که برخی از این قلمرو ها ممکن است به ظاهر غیر مرتبط به نظر برسند، برنامه لانگلند ارتباطاتی بین شاخه هایی را یافته است که پیش از این غیر مرتبط تلقی می شدند، مثل گروه های گالوا، رویه های ریمانی و نظریه اعداد.

بنیان ریاضیات و فلسفه[ویرایش]

نظریه مجموعه ها و منطق به منظور تببین بنیان های ریاضیات توسعه یافته اند. منطق ریاضیات شامل مطالعه ی منطق و کاربرد های منطق صوری به شاخه هایی از ریاضیات است؛ نظریه مجموعه ها شاخه ای از ریاضیات است که به مطالعه مجموعه ها یا گردایه ای از اشیاء می پردازد. نظریه رسته ها که به صورت مجرد به مطالعه ساختار های ریاضیاتی و ارتباطشان با هم می پردازد هنوز هم در حال تکوین است. عبارت "بحران بنیان های ریاضیاتی" به دوره ای تاریخی بین ۱۹۰۰ تا ۱۹۳۰ اشاره دارد که در آن دوره جستجویی برای یافتن بنیانی مستحکم برای ریاضیات انجام شد.[۱۹] اختلاف نظر ها در مورد بنیان های ریاضی تا زمان کنونی هم ادامه دارد. این بحران با یک سری بحث ها تحریک شد، از جمله این بحث ها، بحث بر سر نظریه مجموعه های کانتور و جدال هیلبرت-براور بود.

دغدغه ی منطق ریاضیاتی، ایجاد چارچوب مستحکم اصول موضوعه ای برای ریاضیات است. منطق ریاضی الزامات چنین چارچوبی را مطالعه می کند. مثلاً قضایای عدم کمال گودل به طور ضمنی می گویند که هر نظام صوری اگر معنا دار باشد (یعنی تمام قضیه هایی که می توان آن ها را اثبات کرد درست باشند)، الزاماً ناکامل اند (یعنی قضیای درستی هستند که نمی توان آن ها را در این سیستم اثبات کرد). گودل نشان داد که هر گردایه متناهی از اصول موضوعه های نظریه اعداد را به عنوان اصول موضوعه در نظر بگیریم، می توان یک جمله صوری ساخت که از نظر حقایق نظریه اعداد صحیح باشد ولی از این اصول موضوعه بدست نیایند. لذا در نظریه اعداد هیچ نظام صوری که از نظر اصول موضوعه ای کامل باشد وجود ندارد. منطق نوین به چند بخش تقسیم می شود: نظریه بازگشت، نظریه مدل و نظریه اثبات و ارتباط نزدیکی با علوم کامپیوتر و نظریه رسته ها دارد. در زمینه ی نظریه بازگشت، عدم امکان وجود سیستم اصول موضوعه ای کامل را می توان به صورت صوری از طریق پیامد های قضیه MRDP نشان داد.

علوم کامپیوتر شامل نظریه محاسبه پذیری، نظریه پیچیدگی محاسباتی و نظریه اطلاعات است. نظریه ی محاسبه پذیری محدودیت های مدل های مختلف نظری رایانه ها را بررسی می کند که شامل بسیاری از مدل های شناخته شده چون ماشین تورینگ می شود. نظریه پیچیدگی به مطالعه ی رام پذیری حل مسائل در رایانه می پردازد. برخی مسائل وجود دارند که با وجود این که از لحاظ نظری توسط رایانه قابل حل هستند، اما در عمل هزینه حل کردنشان از نظر زمان یا فضا زیاد است و عملاً با وجود پیشرفت های سریع سخت افزاری در دنیای کامپیوتر حل آن ها به نظر نامعقول می آید. یک مسئله مشهور در این وادی مسئله ی "P=NP"؟ است که برای حل آن جایزه ی مسائل هزاره تعیین شده است[۲۰]. در نهایت، نظریه اطلاعات با حجمی از داده ها سر و کار دارد که بتوان آن ها را بر روی یک وسیله خاص ذخیره کرد، پس این علم با مفاهیمی چون فشرده سازی و انتروپی سروکار دارد.

Venn A intersect B.svg Commutative diagram for morphism.svg DFAexample.svg
منطق ریاضیاتی نظریه مجموعه ها نظریه رسته ها نظریه محاسبات

ریاضی محض[ویرایش]

کمیت[ویرایش]

مقاله اصلی: حساب

مطالعه ی کمیت با اعداد آغاز می گردد، ابتدا مطالعه ی اعداد طبیعی و اعداد صحیح و عملیات حسابی روی آن ها که در شاخه حساب انجام می گردد. خواص عمیق تر اعداد در نظریه اعداد صورتی می پذیرد، که قضایای معروفی چون آخرین قضیه فرما از آن بیرون می آید. اعداد اول دوقلو و حدس گلدباخ دو تا از مسائل لاینحل نظریه اعدادند.

با پیشرفت دستگاه اعداد، اعداد صحیح به عنوان زیر مجموعه ای از اعداد گویا ("کسر ها") شناخته شدند. خود اعداد گویا زیر مجموعه ی اعداد حقیقی می باشند که از آن ها برای نمایش مفهوم کمیت های پیوسته استفاده شده است. خود اعداد حقیقی زیر مجموعه ی اعداد مختلط اند. این ها اولین قدم ها در سلسله مراتب اعداد می باشد که شامل چهارگان ها و هشتگان ها می باشد. با در نظر گرفتن اعداد طبیعی، می توان به اعداد ترامتناهی رسید که مفهوم "بی نهایت" بودن را صوری می کنند. بر اساس قضیه بنیادی جبر، تمام جواب های چند جمله ای های تک متغیره با ضرایب مختلط، صرف نظر از درجه‌شان مختلط هستند. یکی دیگر از قلمرو های مطالعاتی مربوط به اندازه مجموعه ها می شود، که در اعداد کاردینال توصیف گشته اند. مثل اعداد الف که امکان مقایسه ی مجموعه های نامتناهی را با هم می دهند.

اعداد طبیعی اعداد صحیح اعداد گویا اعداد حقیقی اعداد مختلط

ساختار[ویرایش]

مقاله اصلی: جبر

بسیاری از اشیاء ریاضیاتی، مثل مجموعه اعداد و توابع، ساختار داخلی از خود بروز می دهند که می تواند پیامد عملیات یا روابطی باشند که بر روی یک مجموعه اعمال می شود. سپس ریاضیات به مطالعه خواص آن مجموعه هایی می پردازد که می توان آن ها را بر اساس آن ساختار مورد نظر بیان کرد؛ به عنوان مثال نظریه اعداد به مطالعه خواص مجموعه اعداد صحیح می پردازد که می توان آن ها را با عملیات حساب بدست آورد. به علاوه، معمولاً اتفاقی که می افتد این است که چنین مجموعه های ساخت یافته (ساختار ها) خواص مشابهی از خود بروز می دهند که امکان انجام یک مرحله تجرید دیگر بر روی آن ها را داده و لذا در چنین شرایطی می توان اصول موضعه هایی برای آن دسته خاص از مجموعه ها ارائه داد، و سپس به مطالعهٔ همه آن ها به صورت یکجا پرداخت (همه آن مجموعه هایی که در آن اصول موضوعه صدق می کنند). ازین رو، می توان گروه ها، حلقه ها، میدان ها و دیگر نظام های مجرد را مطالعه کرد؛ چنین مطالعاتی (برای ساختار های تعریف شده با عملیات جبری) تشکیل یک قلمرو از ریاضیات به نام جبر مجرد را می دهند.

جبر مجرد را می توان در حالت کلی آن به مسائل به ظاهر غیر مرتبط اعمال کرد؛ به عنوان مثال، تعدادی از مسائل مربوط به ساخت به کمک خط کش و پرگار در نهایت با کمک نظریه گالوا حل شدند، که در آن از نظریه میدان و گروه ها استفاده شد. یکی دیگر از مثال های مرتبط با نظریه جبری، جبر خطیست، که عناصر آن بردار ها می باشند. بردار ها هم اندازه دارند و هم جهت و می توان از آن ها برای مدل سازی روابط بین نقاط درون فضا استفاده کرد. این مثالی از پدیده ای است که پیشتر اشاره شد، یعنی ارتباط قلمروهای به ظاهر غیر مرتبط مثل هندسه و جبر، به گونه ای که مشخص می شود این قلمروهای به ظاهر غیر مرتبط ارتباطاتی بس عمیق تر با یک دیگر در ریاضیات مدرن دارند. ترکیبیات به مطالعه راه های شمارش تعدادی اشیاء می پردازد که آن اشیاء در ساختار داده شده ای صدق می کنند.

Elliptic curve simple.png Group diagram d6.svg
جبر مجرد نظریه اعداد نظریه گروه‌ها
Torus.jpg MorphismComposition-01.png Lattice of the divisibility of 60.svg
توپولوژی نظریه مدول‌ها نظریه ترتیب

فضا[ویرایش]

مقاله اصلی: هندسه

مطالعه فضا از هندسه آغاز شد، بخصوص هندسه اقلیدسی که فضا و اعداد را با هم ترکیب کرده و قضیه معروف فیثاغورس را بوجود آورد. مثلثات شاخه ای از ریاضیات است که درگیر ارتباطات بین اضلاع و زاویه های مثلث و توابع مثلثاتی می باشد. در مطالعات مدرن فضا، این ایده ها تعمیم یافته تا به هندسه هایی با ابعاد بالاتر، فضاهای غیر-اقلیدسی (که نقش بنیادینی در نسبیت عام دارند) و توپولوژی برسد. کمیت و فضا هردو نقش بنیادینی در هندسه تحلیلی، هندسه دیفرانسیل و هندسه جبری دارند. هندسه محدب و گسسته برای حل مسائلی در نظریه اعداد و آنالیز تابعی توسعه یافتند، اما اکنون به نیت کاربرد هایشان در بهینه سازی و علوم کامپیوتر دنبال می شوند. در هندسه دیفرانسیل مفاهیم کلاف های تاری و حساب دیفرانسیل و انتگرال بر روی منیفلد ها، بخصوص بردار ها و حساب تانسوری وجود دارد. در هندسه جبری توصیف اشیاء هندسی مربوط به مجموعه جواب چند جمله ای ها بحث می شود که مفاهیم کمیت و فضا را با هم ترکیب می کند. همچنین در مطالعه بر روی گروه های توپولوژی نیز به دنبال ترکیب ساختار و فضاییم. گروه های لی در مطالعه فضا، ساختار و تغییرات استفاده می شود. توپولوژی در تمام شاخه های متعدد خویش را می توان به عنوان بزرگترین رشد در ریاضیات قرن بیستم تلقی کرد. شاخه های توپولوژی شامل توپولوژی نقطه ای، توپولوژی نظریه مجموعه ای، توپولوژی جبری و توپولوژی دیفرانسیل است. به عنوان مثال توپولوژی عصر جدید شامل قضیه ی مترپذیری، نظریه اصول موضوعه ای مجموعه ها، نظریه هوموتوپی و نظریه مورس می باشد. توپولوژی همچنین اکنون شامل حدس اثبات شده ی پوانکاره بوده و هنوز قلمروهای لاینحلی چون حدس هاج را در بر دارد. دیگر نتایج هندسه و توپولوژی شامل قضیه چهار رنگ و حدس کپلر می باشد که به کمک رایانه ها اثبات شده اند.

Torus.jpg Pythagorean.svg Taylorsine.svg Osculating circle.svg Koch curve.svg
توپولوژی هندسه مثلثات هندسه دیفرانسیل هندسه فراکتال ها

تغییر[ویرایش]

مقاله اصلی: حساب دیفرانسیل و انتگرال

فهم و توصیف تغییر تم اصلی علوم طبیعی بوده و حساب دیفرانسیل و انتگرال به عنوان ابزاری برای تحقیق در این ارتباط ساخته شد. توابع در اینجا به عنوان مفهوم مرکزی توصیف کننده یک کمیت متغیر ظهور پیدا کردند. مطالعه مستحکم اعداد حقیقی و توابع تک متغیره ی حقیقی را آنالیز حقیقی گویند، آنالیز مختلط هم فیلد مشابهی است که بر روی میدان اعداد مختلط کار می کند. آنالیز تابعی بر روی فضاهای (اغلب بی نهایت بعدی) توابع متمرکز است. یکی از کاربرد های متعدد آنالیز تابعی در مکانیک کوانتومی است. بسیاری از مسائل به طور طبیعی به رابطه ی بین یک کمیت و نرخ تغییراتش منجر می شوند. بسیاری از پدیده ها در طبیعت را می توان به وسیله سیستم های دینامیکی توصیف کرد؛ نظریه آشوب به طور دقیق بررسی می کند که چگونه یک سیستم می تواند پیش بینی ناپذیر باشد و در حالی که همزمان رفتار قطعی خود را نیز حفظ می کند.

Integral as region under curve.png Vectorfield jaredwf.png
حساب حسابان حساب برداری آنالیز ریاضی
Limitcycle.jpg LorenzAttractor.png
معادلات دیفرانسیل سیستم‌های دینامیکی نظریه آشوب

ریاضیات کاربردی[ویرایش]

مقاله اصلی: ریاضیات کاربردی

ریاضیات کاربردی به دنبال روش های ریاضیاتی است که اغلب در علوم، مهندسی، بازرگانی و صنعت به کار برده می شوند. لذا "ریاضیات کاربردی" یک علم ریاضیاتی است با دانش تخصصی. همچنین عبارت ریاضیات کاربردی تخصصی حرفه ای را توصیف می کند که بر روی مسائل عملی تمرکز کرده است، ریاضیات کاربردی بر روی "فرمول بندی، مطالعه و استفاده از مدل های ریاضیاتی" در علوم، مهندسی و دیگر حوزه هایی که ریاضیات به کار می رود تمرکز می کند.

در عمل، کاربرد های عملی منجر به توسعه قضایای ریاضیاتی شده، که این قضایا خود، موضوع مطالعه در ریاضیات محض شده اند، که در آن ریاضیات به هدف توسعه خود ریاضیات مطالعه می شود. ازین رو، فعالیت ریاضیات کاربردی به طور حیاتی به تحقیقات در ریاضیات محض گره خورده است.

Arbitrary-gametree-solved.svg BernoullisLawDerivationDiagram.svg Composite trapezoidal rule illustration small.svg Maximum boxed.png Two red dice 01.svg Oldfaithful3.png Caesar3.svg
نظریه بازی ها دینامیک سیالات آنالیز عددی بهینه سازی نظریه احتمال آمار رمز نگاری
Market Data Index NYA on 20050726 202628 UTC.png Gravitation space source.svg CH4-structure.svg Signal transduction pathways.svg GDP PPP Per Capita IMF 2008.svg Simple feedback control loop2.svg
ریاضیات مالی ریاضی فیزیک ریاضیات شیمی ریاضیات زیستی ریاضیات اقتصاد نظریه کنترل

جوایز ریاضیاتی[ویرایش]

می توان مدعی شد که مهم ترین جایزه ریاضیاتی جایزه ی فیلدز است،[۲۱][۲۲] که در سال ۱۹۳۶ تأسیس شد و در این سال ها، هر چهار سال یک بار (به جز حدود جنگ جهانی دوم) به حداکثر ۴ ریاضیدان تعلق گرفته است. مدال فیلدز اغلب به عنوان معادلی برای نوبل در ریاضیات در نظر گرفته شده.

جایزه ی وولف در ریاضیات، در ۱۹۷۸ تأسیس شد و به هدف قدردانی از دستاورد هایی است که یک ریاضیدان در عمر خویش بدان ها نایل گشته. جایزه ی آبل در ۲۰۰۳ تأسیس شد. مدال چرن در ۲۰۱۰ معرفی شد برای قدردانی از دستاورد های یک عمر. این جوایز برای اهمیت دادن به برخی کار های نوآورانه، یا برای پیدا کردن راه حل برای مسائل مهم در یک شاخه خاص در نظر گرفته شده اند.

لیستی از ۲۳ مسئله باز که به آن ها "مسائل هیلبرت" می گویند در سال ۱۹۰۰ توسط ریاضیدان آلمانی دیوید هیلبرت معرفی شد. این لیست به معروفیت زیادی بین ریاضیدانان دست یافت. حداقل نه تا از این مسائل اکنون حل شده اند. لیست جدیدی از هفت مسئله مهم به نام "مسائل جایزه هزاره" نیز در سال ۲۰۰۰ منتشر شد. تنها یکی از آن ها با لیست مسائل هیلبرت اشتراک دارد. جایزه حل هر مسئله در لیست جایزه هزاره ۱ میلیون دلار می باشد.

جستارهای وابسته[ویرایش]

منابع[ویرایش]

  1. ۱٫۰ ۱٫۱ "mathematics, n.". Oxford English Dictionary. Oxford University Press. 2012. Retrieved June 16, 2012. The science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis.
  2. Kneebone, G.T. (1963). Mathematical Logic and the Foundations of Mathematics: An Introductory Survey. Dover. p. 4. ISBN 978-0-486-41712-7. Mathematics ... is simply the study of abstract structures, or formal patterns of connectedness.
  3. LaTorre, Donald R.; Kenelly, John W.; Biggers, Sherry S.; Carpenter, Laurel R.; Reed, Iris B.; Harris, Cynthia R. (2011). Calculus Concepts: An Informal Approach to the Mathematics of Change. Cengage Learning. p. 2. ISBN 978-1-4390-4957-0. Calculus is the study of change—how things change, and how quickly they change.
  4. Ramana (2007). Applied Mathematics. Tata McGraw–Hill Education. p. 2.10. ISBN 978-0-07-066753-2. The mathematical study of change, motion, growth or decay is calculus.
  5. Ziegler, Günter M. (2011). "What Is Mathematics?". An Invitation to Mathematics: From Competitions to Research. Springer. p. vii. ISBN 978-3-642-19532-7.
  6. Eves, p. 306
  7. Peterson, p. 12
  8. Dehaene, Stanislas; Dehaene-Lambertz, Ghislaine; Cohen, Laurent (Aug 1998). "Abstract representations of numbers in the animal and human brain". Trends in Neurosciences. 21 (8): 355–61. doi:10.1016/S0166-2236(98)01263-6. PMID 9720604.
  9. See, for example, Raymond L. Wilder, Evolution of Mathematical Concepts; an Elementary Study, passim
  10. Kline 1990, Chapter 1.
  11. Boyer 1991, "Mesopotamia" p. 24–27.
  12. Heath, Thomas Little (1981) [originally published 1921]. A History of Greek Mathematics: From Thales to Euclid. New York: Dover Publications. ISBN 978-0-486-24073-2.
  13. Boyer 1991, "Euclid of Alexandria" p. 119.
  14. Boyer 1991, "Archimedes of Syracuse" p. 120.
  15. Boyer 1991, "Archimedes of Syracuse" p. 130.
  16. Boyer 1991, "Apollonius of Perga" p. 145.
  17. Boyer 1991, "Greek Trigonometry and Mensuration" p. 162.
  18. Boyer 1991, "Revival and Decline of Greek Mathematics" p. 180.
  19. Luke Howard Hodgkin & Luke Hodgkin, A History of Mathematics, Oxford University Press, 2005.
  20. Clay Mathematics Institute, P=NP, claymath.org
  21. Monastyrsky 2001, p. 1: "The Fields Medal is now indisputably the best known and most influential award in mathematics."
  22. Riehm 2002, pp. ۷۷۸–۸۲.

کتاب‌شناسی[ویرایش]

  • Kline, M. , Mathematical Thought from Ancient to Modern Times (1973);

پیوند به بیرون[ویرایش]

  • فرهنگ جامع ریاضیات
  • اطلس ریاضیات
  • اریک ویستن، دنیای ریاضیات، http://www.mathworld.com دانشنامهٔ برخط ریاضیات.
  • سیارهٔ ریاضی (به انگلیسی:Planet Math) دانشنامهٔ بر خط ریاضیات که هنوز در دست ساخت است. به دلیل استفاده از اجازهٔ GFDL امکان تبادل مقالات با ویکی‌پدیا وجود دارد. این دانشنامه از روش نشان‌گذاری TeX استفاده می‌کند.
  • Metamath یک وبگاه و یک زبان که به شرح و بسط ریاضیات از پایه می‌پردازد.

Euclid (holding calipers), Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens (1509-1511)[a]

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") includes the study of such topics as quantity (number theory),[1] structure (algebra),[2] space (geometry),[1] and change (mathematical analysis).[3][4][5] It has no generally accepted definition.[6][7]

Mathematicians seek and use patterns[8][9] to formulate new conjectures; they resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, mathematical reasoning can be used to provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements.[10] Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.[11]

Mathematics is essential in many fields, including natural science, engineering, medicine, finance, and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics (mathematics for its own sake) without having any application in mind, but practical applications for what began as pure mathematics are often discovered later.[12][13]

History

The history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, which is shared by many animals,[14] was probably that of numbers: the realization that a collection of two apples and a collection of two oranges (for example) have something in common, namely quantity of their members.

As evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have also recognized how to count abstract quantities, like time – days, seasons, years.[15]

The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.

Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy.[16] The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC.[17] Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.[18] It is in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication and division) first appear in the archaeological record. The Babylonians also possessed a place-value system, and used a sexagesimal numeral system [18]which is still in use today for measuring angles and time.[19]

Archimedes used the method of exhaustion to approximate the value of pi.

Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics.[20] Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom, theorem, and proof. His textbook Elements is widely considered the most successful and influential textbook of all time.[21] The greatest mathematician of antiquity is often held to be Archimedes (c. 287–212 BC) of Syracuse.[22] He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.[23] Other notable achievements of Greek mathematics are conic sections (Apollonius of Perga, 3rd century BC),[24] trigonometry (Hipparchus of Nicaea (2nd century BC),[25] and the beginnings of algebra (Diophantus, 3rd century AD).[26]

The numerals used in the Bakhshali manuscript, dated between the 2nd century BCE and the 2nd century CE.

The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics.[27] Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine[27], and an early form of infinite series.

A page from al-Khwārizmī's Algebra

During the Golden Age of Islam, especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system.[28] Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī.

During the early modern period, mathematics began to develop at an accelerating pace in Western Europe. The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics.[29] Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries.[30] Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss[31], who made numerous contributions to fields such as algebra, analysis, differential geometry, matrix theory, number theory, and statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show in part that any consistent axiomatic system — if powerful enough to describe arithmetic — will contain true propositions that cannot be proved.[32]

Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs."[33]

Etymology

The word mathematics comes from Ancient Greek μάθημα (máthēma), meaning "that which is learnt",[34] "what one gets to know", hence also "study" and "science". The word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times.[35] Its adjective is μαθηματικός (mathēmatikós), meaning "related to learning" or "studious", which likewise further came to mean "mathematical". In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), Latin: ars mathematica, meant "the mathematical art".

Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense.[36]

In Latin, and in English until around 1700, the term mathematics more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This has resulted in several mistranslations. For example, Saint Augustine's warning that Christians should beware of mathematici, meaning astrologers, is sometimes mistranslated as a condemnation of mathematicians.[37]

The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural τὰ μαθηματικά (ta mathēmatiká), used by Aristotle (384–322 BC), and meaning roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from Greek.[38] In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math.[39]

Definitions of mathematics

Leonardo Fibonacci, the Italian mathematician who introduced the Hindu–Arabic numeral system invented between the 1st and 4th centuries by Indian mathematicians, to the Western World.

Mathematics has no generally accepted definition.[6][7] Aristotle defined mathematics as "the science of quantity" and this definition prevailed until the 18th century.[40] In the 19th century, when the study of mathematics increased in rigor and began to address abstract topics such as group theory and projective geometry, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions.[41] Three leading types of definition of mathematics today are called logicist, intuitionist, and formalist, each reflecting a different philosophical school of thought.[42] All have severe flaws, none has widespread acceptance, and no reconciliation seems possible.[42]

An early definition of mathematics in terms of logic was Benjamin Peirce's "the science that draws necessary conclusions" (1870).[43] In the Principia Mathematica, Bertrand Russell and Alfred North Whitehead advanced the philosophical program known as logicism, and attempted to prove that all mathematical concepts, statements, and principles can be defined and proved entirely in terms of symbolic logic. A logicist definition of mathematics is Russell's "All Mathematics is Symbolic Logic" (1903).[44]

Intuitionist definitions, developing from the philosophy of mathematician L. E. J. Brouwer, identify mathematics with certain mental phenomena. An example of an intuitionist definition is "Mathematics is the mental activity which consists in carrying out constructs one after the other."[42] A peculiarity of intuitionism is that it rejects some mathematical ideas considered valid according to other definitions. In particular, while other philosophies of mathematics allow objects that can be proved to exist even though they cannot be constructed, intuitionism allows only mathematical objects that one can actually construct. Intuitionists also reject the law of excluded middle — a stance which forces them to reject proof by contradiction as a viable proof method as well. [45]

Formalist definitions identify mathematics with its symbols and the rules for operating on them. Haskell Curry defined mathematics simply as "the science of formal systems".[46] A formal system is a set of symbols, or tokens, and some rules on how the tokens are to be combined into formulas. In formal systems, the word axiom has a special meaning different from the ordinary meaning of "a self-evident truth", and is used to refer to a combination of tokens that is included in a given formal system without needing to be derived using the rules of the system.

A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable.[6] There is not even consensus on whether mathematics is an art or a science.[7] Some just say, "Mathematics is what mathematicians do."[6]

Mathematics as science

Carl Friedrich Gauss, known as the prince of mathematicians

The German mathematician Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences".[47] More recently, Marcus du Sautoy has called mathematics "the Queen of Science ... the main driving force behind scientific discovery".[48] In the original Latin Regina Scientiarum, as well as in German Königin der Wissenschaften, the word corresponding to science means a "field of knowledge", and this was the original meaning of "science" in English, also; mathematics is in this sense a field of knowledge. The specialization restricting the meaning of "science" to natural science follows the rise of Baconian science, which contrasted "natural science" to scholasticism, the Aristotelean method of inquiring from first principles. The role of empirical experimentation and observation from the external world is arguably negligible in mathematics[49], especially when compared to natural sciences such as biology, chemistry, or physics. Albert Einstein stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."[50]

Some modern philosophers consider that mathematics is not a science.[51] The philosopher Karl Popper observed that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently."[52]

Mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences. Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics.

The opinions of mathematicians on this matter are varied. Many mathematicians[53] feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts; others feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics.[54] One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is created (as in art) or discovered (as in science). It is common to see universities divided into sections that include a division of Science and Mathematics, indicating that the fields are seen as being allied but that they do not coincide. In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the philosophy of mathematics.[55]

Inspiration, pure and applied mathematics, and aesthetics

Isaac Newton
Gottfried Wilhelm von Leibniz
Isaac Newton (left) and Gottfried Wilhelm Leibniz developed infinitesimal calculus.

Mathematics arises from many different kinds of problems. At first these were found in commerce, land measurement, architecture and later astronomy; today, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. For example, the physicist Richard Feynman invented the path integral formulation of quantum mechanics using a combination of mathematical reasoning and physical insight, and today's string theory, a still-developing scientific theory which attempts to unify the four fundamental forces of nature, continues to inspire new mathematics.[56]

Some mathematics is relevant only in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. A distinction is often made between pure mathematics and applied mathematics. However pure mathematics topics often turn out to have applications, e.g. number theory in cryptography. This remarkable fact, that even the "purest" mathematics often turns out to have practical applications, is what Eugene Wigner has called "the unreasonable effectiveness of mathematics".[13] As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: there are now hundreds of specialized areas in mathematics and the latest Mathematics Subject Classification runs to 46 pages.[57] Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and computer science.

For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty in a simple and elegant proof, such as Euclid's proof that there are infinitely many prime numbers, and in an elegant numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in A Mathematician's Apology expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic.[58] Mathematical research often seeks critical features of a mathematical object. A theorem expressed as a characterization of the object by these features is the prize. Examples of particularly succinct and revelatory mathematical arguments has been published in Proofs from THE BOOK.

The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions. And at the other social extreme, philosophers continue to find problems in philosophy of mathematics, such as the nature of mathematical proof.[59]

Notation, language, and rigor

Leonhard Euler created and popularized much of the mathematical notation used today.

Most of the mathematical notation in use today was not invented until the 16th century.[60] Before that, mathematics was written out in words, limiting mathematical discovery.[61] Euler (1707–1783) was responsible for many of the notations in use today. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. According to Barbara Oakley, this can be attributed to the fact that mathematical ideas are both more abstract and more encrypted than those of natural language.[62] Unlike natural language, where people can often equate a word (such as cow) with the physical object it corresponds to, mathematical symbols are abstract, lacking any physical analog.[63] Mathematical symbols are also more highly encrypted than regular words, meaning a single symbol can encode a number of different operations or ideas.[64]

Mathematical language can be difficult to understand for beginners because even common terms, such as or and only, have a more precise meaning than they have in everyday speech, and other terms such as open and field refer to specific mathematical ideas, not covered by their laymen's meanings. Mathematical language also includes many technical terms such as homeomorphism and integrable that have no meaning outside of mathematics. Additionally, shorthand phrases such as iff for "if and only if" belong to mathematical jargon. There is a reason for special notation and technical vocabulary: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".

Mathematical proof is fundamentally a matter of rigor. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "theorems", based on fallible intuitions, of which many instances have occurred in the history of the subject.[b] The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of Isaac Newton the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics. Today, mathematicians continue to argue among themselves about computer-assisted proofs. Since large computations are hard to verify, such proofs may be erroneous if the used computer program is erroneous.[c][65] On the other hand, proof assistants allow verifying all details that cannot be given in a hand-written proof, and provide certainty of the correctness of long proofs such as that of Feit–Thompson theorem.[d]

Axioms in traditional thought were "self-evident truths", but that conception is problematic.[66] At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable formulas; and so a final axiomatization of mathematics is impossible. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.[67]

Fields of mathematics

The abacus is a simple calculating tool used since ancient times.

Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry, and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty. While some areas might seem unrelated, the Langlands program has found connections between areas previously thought unconnected, such as Galois groups, Riemann surfaces and number theory.

Discrete mathematics conventionally groups together the fields of mathematics which study mathematical structures that are fundamentally discrete rather than continuous.

Foundations and philosophy

In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic includes the mathematical study of logic and the applications of formal logic to other areas of mathematics; set theory is the branch of mathematics that studies sets or collections of objects. Category theory, which deals in an abstract way with mathematical structures and relationships between them, is still in development. The phrase "crisis of foundations" describes the search for a rigorous foundation for mathematics that took place from approximately 1900 to 1930.[68] Some disagreement about the foundations of mathematics continues to the present day. The crisis of foundations was stimulated by a number of controversies at the time, including the controversy over Cantor's set theory and the Brouwer–Hilbert controversy.

Mathematical logic is concerned with setting mathematics within a rigorous axiomatic framework, and studying the implications of such a framework. As such, it is home to Gödel's incompleteness theorems which (informally) imply that any effective formal system that contains basic arithmetic, if sound (meaning that all theorems that can be proved are true), is necessarily incomplete (meaning that there are true theorems which cannot be proved in that system). Whatever finite collection of number-theoretical axioms is taken as a foundation, Gödel showed how to construct a formal statement that is a true number-theoretical fact, but which does not follow from those axioms. Therefore, no formal system is a complete axiomatization of full number theory. Modern logic is divided into recursion theory, model theory, and proof theory, and is closely linked to theoretical computer science,[citation needed] as well as to category theory. In the context of recursion theory, the impossibility of a full axiomatization of number theory can also be formally demonstrated as a consequence of the MRDP theorem.

Theoretical computer science includes computability theory, computational complexity theory, and information theory. Computability theory examines the limitations of various theoretical models of the computer, including the most well-known model – the Turing machine. Complexity theory is the study of tractability by computer; some problems, although theoretically solvable by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with the rapid advancement of computer hardware. A famous problem is the "P = NP?" problem, one of the Millennium Prize Problems.[69] Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence deals with concepts such as compression and entropy.

Venn A intersect B.svg Commutative diagram for morphism.svg DFAexample.svg
Mathematical logic Set theory Category theory Theory of computation

Pure mathematics

Quantity

The study of quantity starts with numbers, first the familiar natural numbers and integers ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory, from which come such popular results as Fermat's Last Theorem. The twin prime conjecture and Goldbach's conjecture are two unsolved problems in number theory.

As the number system is further developed, the integers are recognized as a subset of the rational numbers ("fractions"). These, in turn, are contained within the real numbers, which are used to represent continuous quantities. Real numbers are generalized to complex numbers. These are the first steps of a hierarchy of numbers that goes on to include quaternions and octonions. Consideration of the natural numbers also leads to the transfinite numbers, which formalize the concept of "infinity". According to the fundamental theorem of algebra all solutions of equations in one unknown with complex coefficients are complex numbers, regardless of degree. Another area of study is the size of sets, which is described with the cardinal numbers. These include the aleph numbers, which allow meaningful comparison of the size of infinitely large sets.

Natural numbers Integers Rational numbers Real numbers Complex numbers Infinite cardinals

Structure

Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of operations or relations that are defined on the set. Mathematics then studies properties of those sets that can be expressed in terms of that structure; for instance number theory studies properties of the set of integers that can be expressed in terms of arithmetic operations. Moreover, it frequently happens that different such structured sets (or structures) exhibit similar properties, which makes it possible, by a further step of abstraction, to state axioms for a class of structures, and then study at once the whole class of structures satisfying these axioms. Thus one can study groups, rings, fields and other abstract systems; together such studies (for structures defined by algebraic operations) constitute the domain of abstract algebra.

By its great generality, abstract algebra can often be applied to seemingly unrelated problems; for instance a number of ancient problems concerning compass and straightedge constructions were finally solved using Galois theory, which involves field theory and group theory. Another example of an algebraic theory is linear algebra, which is the general study of vector spaces, whose elements called vectors have both quantity and direction, and can be used to model (relations between) points in space. This is one example of the phenomenon that the originally unrelated areas of geometry and algebra have very strong interactions in modern mathematics. Combinatorics studies ways of enumerating the number of objects that fit a given structure.

Elliptic curve simple.svg Rubik's cube.svg Group diagdram D6.svg Lattice of the divisibility of 60.svg Braid-modular-group-cover.svg
Combinatorics Number theory Group theory Graph theory Order theory Algebra

Space

The study of space originates with geometry – in particular, Euclidean geometry, which combines space and numbers, and encompasses the well-known Pythagorean theorem. Trigonometry is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Convex and discrete geometry were developed to solve problems in number theory and functional analysis but now are pursued with an eye on applications in optimization and computer science. Within differential geometry are the concepts of fiber bundles and calculus on manifolds, in particular, vector and tensor calculus. Within algebraic geometry is the description of geometric objects as solution sets of polynomial equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may have been the greatest growth area in 20th-century mathematics; it includes point-set topology, set-theoretic topology, algebraic topology and differential topology. In particular, instances of modern-day topology are metrizability theory, axiomatic set theory, homotopy theory, and Morse theory. Topology also includes the now solved Poincaré conjecture, and the still unsolved areas of the Hodge conjecture. Other results in geometry and topology, including the four color theorem and Kepler conjecture, have been proved only with the help of computers.

Illustration to Euclid's proof of the Pythagorean theorem.svg Sinusvåg 400px.png Hyperbolic triangle.svg Torus.svg Mandel zoom 07 satellite.jpg Measure illustration (Vector).svg
Geometry Trigonometry Differential geometry Topology Fractal geometry Measure theory

Change

Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a tool to investigate it. Functions arise here, as a central concept describing a changing quantity. The rigorous study of real numbers and functions of a real variable is known as real analysis, with complex analysis the equivalent field for the complex numbers. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equations. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior.

Integral as region under curve.svg Vector field.svg Navier Stokes Laminar.svg Limitcycle.svg Lorenz attractor.svg Conformal grid after Möbius transformation.svg
Calculus Vector calculus Differential equations Dynamical systems Chaos theory Complex analysis

Applied mathematics

Applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge. The term applied mathematics also describes the professional specialty in which mathematicians work on practical problems; as a profession focused on practical problems, applied mathematics focuses on the "formulation, study, and use of mathematical models" in science, engineering, and other areas of mathematical practice.

In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. Thus, the activity of applied mathematics is vitally connected with research in pure mathematics.

Statistics and other decision sciences

Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially with probability theory. Statisticians (working as part of a research project) "create data that makes sense" with random sampling and with randomized experiments;[70] the design of a statistical sample or experiment specifies the analysis of the data (before the data be available). When reconsidering data from experiments and samples or when analyzing data from observational studies, statisticians "make sense of the data" using the art of modelling and the theory of inference – with model selection and estimation; the estimated models and consequential predictions should be tested on new data.[e]

Statistical theory studies decision problems such as minimizing the risk (expected loss) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best. In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost, under specific constraints: For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence.[71] Because of its use of optimization, the mathematical theory of statistics shares concerns with other decision sciences, such as operations research, control theory, and mathematical economics.[72]

Computational mathematics

Computational mathematics proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis includes the study of approximation and discretisation broadly with special concern for rounding errors. Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic matrix and graph theory. Other areas of computational mathematics include computer algebra and symbolic computation.

Arbitrary-gametree-solved.svg BernoullisLawDerivationDiagram.svg Composite trapezoidal rule illustration small.svg Maximum boxed.png Two red dice 01.svg Oldfaithful3.png Caesar3.svg
Game theory Fluid dynamics Numerical analysis Optimization Probability theory Statistics Cryptography
Market Data Index NYA on 20050726 202628 UTC.png Gravitation space source.svg CH4-structure.svg Signal transduction pathways.svg GDP PPP Per Capita IMF 2008.svg Simple feedback control loop2.svg
Mathematical finance Mathematical physics Mathematical chemistry Mathematical biology Mathematical economics Control theory

Mathematical awards

Arguably the most prestigious award in mathematics is the Fields Medal,[73][74] established in 1936 and awarded every four years (except around World War II) to as many as four individuals. The Fields Medal is often considered a mathematical equivalent to the Nobel Prize.

The Wolf Prize in Mathematics, instituted in 1978, recognizes lifetime achievement, and another major international award, the Abel Prize, was instituted in 2003. The Chern Medal was introduced in 2010 to recognize lifetime achievement. These accolades are awarded in recognition of a particular body of work, which may be innovational, or provide a solution to an outstanding problem in an established field.

A famous list of 23 open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert. This list achieved great celebrity among mathematicians, and at least nine of the problems have now been solved. A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. Only one of them, the Riemann hypothesis, duplicates one of Hilbert's problems. A solution to any of these problems carries a 1 million dollar reward.

See also

Notes

  1. ^ No likeness or description of Euclid's physical appearance made during his lifetime survived antiquity. Therefore, Euclid's depiction in works of art depends on the artist's imagination (see Euclid).
  2. ^ See false proof for simple examples of what can go wrong in a formal proof.
  3. ^ For considering as reliable a large computation occurring in a proof, one generally requires two computations using independent software
  4. ^ The book containing the complete proof has more than 1,000 pages.
  5. ^ Like other mathematical sciences such as physics and computer science, statistics is an autonomous discipline rather than a branch of applied mathematics. Like research physicists and computer scientists, research statisticians are mathematical scientists. Many statisticians have a degree in mathematics, and some statisticians are also mathematicians.

References

  1. ^ a b "mathematics, n.". Oxford English Dictionary. Oxford University Press. 2012. Retrieved June 16, 2012. The science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis.
  2. ^ Kneebone, G.T. (1963). Mathematical Logic and the Foundations of Mathematics: An Introductory Survey. Dover. p. 4. ISBN 978-0-486-41712-7. Mathematics ... is simply the study of abstract structures, or formal patterns of connectedness.
  3. ^ LaTorre, Donald R.; Kenelly, John W.; Biggers, Sherry S.; Carpenter, Laurel R.; Reed, Iris B.; Harris, Cynthia R. (2011). Calculus Concepts: An Informal Approach to the Mathematics of Change. Cengage Learning. p. 2. ISBN 978-1-4390-4957-0. Calculus is the study of change—how things change, and how quickly they change.
  4. ^ Ramana (2007). Applied Mathematics. Tata McGraw–Hill Education. p. 2.10. ISBN 978-0-07-066753-2. The mathematical study of change, motion, growth or decay is calculus.
  5. ^ Ziegler, Günter M. (2011). "What Is Mathematics?". An Invitation to Mathematics: From Competitions to Research. Springer. p. vii. ISBN 978-3-642-19532-7.
  6. ^ a b c d Mura, Roberta (December 1993). "Images of Mathematics Held by University Teachers of Mathematical Sciences". Educational Studies in Mathematics. 25 (4): 375–385. doi:10.1007/BF01273907. JSTOR 3482762.
  7. ^ a b c Tobies, Renate & Helmut Neunzert (2012). Iris Runge: A Life at the Crossroads of Mathematics, Science, and Industry. Springer. p. 9. ISBN 978-3-0348-0229-1. [I]t is first necessary to ask what is meant by mathematics in general. Illustrious scholars have debated this matter until they were blue in the face, and yet no consensus has been reached about whether mathematics is a natural science, a branch of the humanities, or an art form.
  8. ^ Steen, L.A. (April 29, 1988). The Science of Patterns Science, 240: 611–16. And summarized at Association for Supervision and Curriculum Development Archived October 28, 2010, at the Wayback Machine, www.ascd.org.
  9. ^ Devlin, Keith, Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe (Scientific American Paperback Library) 1996, ISBN 978-0-7167-5047-5
  10. ^ Wise, David. "Eudoxus' Influence on Euclid's Elements with a close look at The Method of Exhaustion". jwilson.coe.uga.edu. Retrieved October 26, 2019.
  11. ^ Eves, p. 306
  12. ^ Peterson, p. 12
  13. ^ a b Wigner, Eugene (1960). "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". Communications on Pure and Applied Mathematics. 13 (1): 1–14. Bibcode:1960CPAM...13....1W. doi:10.1002/cpa.3160130102. Archived from the original on February 28, 2011.
  14. ^ Dehaene, Stanislas; Dehaene-Lambertz, Ghislaine; Cohen, Laurent (August 1998). "Abstract representations of numbers in the animal and human brain". Trends in Neurosciences. 21 (8): 355–61. doi:10.1016/S0166-2236(98)01263-6. PMID 9720604.
  15. ^ See, for example, Raymond L. Wilder, Evolution of Mathematical Concepts; an Elementary Study, passim
  16. ^ Kline 1990, Chapter 1.
  17. ^ "Egyptian Mathematics - The Story of Mathematics". www.storyofmathematics.com. Retrieved October 27, 2019.
  18. ^ a b "Sumerian/Babylonian Mathematics - The Story of Mathematics". www.storyofmathematics.com. Retrieved October 27, 2019.
  19. ^ Boyer 1991, "Mesopotamia" p. 24–27.
  20. ^ Heath, Thomas Little (1981) [originally published 1921]. A History of Greek Mathematics: From Thales to Euclid. New York: Dover Publications. ISBN 978-0-486-24073-2.
  21. ^ Boyer 1991, "Euclid of Alexandria" p. 119.
  22. ^ Boyer 1991, "Archimedes of Syracuse" p. 120.
  23. ^ Boyer 1991, "Archimedes of Syracuse" p. 130.
  24. ^ Boyer 1991, "Apollonius of Perga" p. 145.
  25. ^ Boyer 1991, "Greek Trigonometry and Mensuration" p. 162.
  26. ^ Boyer 1991, "Revival and Decline of Greek Mathematics" p. 180.
  27. ^ a b "Indian Mathematics - The Story of Mathematics". www.storyofmathematics.com. Retrieved October 27, 2019.
  28. ^ "Islamic Mathematics - The Story of Mathematics". www.storyofmathematics.com. Retrieved October 27, 2019.
  29. ^ "17th Century Mathematics - The Story of Mathematics". www.storyofmathematics.com. Retrieved October 27, 2019.
  30. ^ "Euler - 18th Century Mathematics - The Story of Mathematics". www.storyofmathematics.com. Retrieved October 27, 2019.
  31. ^ "Gauss - 19th Century Mathematics - The Story of Mathematics". www.storyofmathematics.com. Retrieved October 27, 2019.
  32. ^ "20th Century Mathematics - Gödel". The Story of Mathematics. Retrieved October 27, 2019.
  33. ^ Sevryuk 2006, pp. 101–09.
  34. ^ "mathematic". Online Etymology Dictionary. Archived from the original on March 7, 2013.
  35. ^ Both meanings can be found in Plato, the narrower in Republic 510c, but Plato did not use a math- word; Aristotle did, commenting on it. μαθηματική. Liddell, Henry George; Scott, Robert; A Greek–English Lexicon at the Perseus Project. OED Online, "Mathematics".
  36. ^ "Pythagoras - Greek Mathematics - The Story of Mathematics". www.storyofmathematics.com. Retrieved October 27, 2019.
  37. ^ Boas, Ralph (1995) [1991]. "What Augustine Didn't Say About Mathematicians". Lion Hunting and Other Mathematical Pursuits: A Collection of Mathematics, Verse, and Stories by the Late Ralph P. Boas, Jr. Cambridge University Press. p. 257.
  38. ^ The Oxford Dictionary of English Etymology, Oxford English Dictionary, sub "mathematics", "mathematic", "mathematics"
  39. ^ "maths, n." and "math, n.3". Oxford English Dictionary, on-line version (2012).
  40. ^ Franklin, James (July 8, 2009). Philosophy of Mathematics. p. 104. ISBN 978-0-08-093058-9.
  41. ^ Cajori, Florian (1893). A History of Mathematics. American Mathematical Society (1991 reprint). pp. 285–86. ISBN 978-0-8218-2102-2.
  42. ^ a b c Snapper, Ernst (September 1979). "The Three Crises in Mathematics: Logicism, Intuitionism, and Formalism". Mathematics Magazine. 52 (4): 207–16. Bibcode:1975MathM..48...12G. doi:10.2307/2689412. JSTOR 2689412.
  43. ^ Peirce, Benjamin (1882). Linear Associative Algebra. p. 1. Archived from the original on September 6, 2015.
  44. ^ Russell, Bertrand (1903). The Principles of Mathematics. p. 5.
  45. ^ "The Definitive Glossary of Higher Mathematical Jargon — Proof by Contradiction". Math Vault. August 1, 2019. Retrieved October 27, 2019.
  46. ^ Curry, Haskell (1951). Outlines of a Formalist Philosophy of Mathematics. Elsevier. p. 56. ISBN 978-0-444-53368-5.
  47. ^ Waltershausen, p. 79
  48. ^ du Sautoy, Marcus (June 25, 2010). "Nicolas Bourbaki". A Brief History of Mathematics. Event occurs at min. 12:50. BBC Radio 4. Archived from the original on December 16, 2016. Retrieved October 26, 2017.
  49. ^ Markie, Peter (2017), "Rationalism vs. Empiricism", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Fall 2017 ed.), Metaphysics Research Lab, Stanford University, retrieved October 27, 2019
  50. ^ Einstein, p. 28. The quote is Einstein's answer to the question: "How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?" This question was inspired by Eugene Wigner's paper "The Unreasonable Effectiveness of Mathematics in the Natural Sciences".
  51. ^ Shasha, Dennis Elliot; Lazere, Cathy A. (1998). Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists. Springer. p. 228.
  52. ^ Popper 1995, p. 56
  53. ^ See, for example Bertrand Russell's statement "Mathematics, rightly viewed, possesses not only truth, but supreme beauty ..." in his History of Western Philosophy
  54. ^ "The science checklist applied: Mathematics". undsci.berkeley.edu. Retrieved October 27, 2019.
  55. ^ Borel, Armand (March 2017). "Mathematics: Art and Science". EMS Newsletter. 3 (103): 37–45. doi:10.4171/news/103/8. ISSN 1027-488X.
  56. ^ Meinhard E. Mayer (2001). "The Feynman Integral and Feynman's Operational Calculus". Physics Today. 54 (8): 48. Bibcode:2001PhT....54h..48J. doi:10.1063/1.1404851.
  57. ^ "Mathematics Subject Classification 2010" (PDF). Archived (PDF) from the original on May 14, 2011. Retrieved November 9, 2010.
  58. ^ Hardy, G. H. (1940). A Mathematician's Apology. Cambridge University Press. ISBN 978-0-521-42706-7.
  59. ^ Gold, Bonnie; Simons, Rogers A. (2008). Proof and Other Dilemmas: Mathematics and Philosophy. MAA.
  60. ^ "Earliest Uses of Various Mathematical Symbols". Archived from the original on February 20, 2016. Retrieved September 14, 2014.
  61. ^ Kline, p. 140, on Diophantus; p. 261, on Vieta.
  62. ^ Oakley 2014, p. 16: "Focused problem solving in math and science is often more effortful than focused-mode thinking involving language and people. This may be because humans haven't evolved over the millennia to manipulate mathematical ideas, which are frequently more abstractly encrypted than those of conventional language."
  63. ^ Oakley 2014, p. 16: "What do I mean by abstractness? You can point to a real live cow chewing its cud in a pasture and equate it with the letters c–o–w on the page. But you can't point to a real live plus sign that the symbol '+' is modeled after – the idea underlying the plus sign is more abstract."
  64. ^ Oakley 2014, p. 16: "By encryptedness, I mean that one symbol can stand for a number of different operations or ideas, just as the multiplication sign symbolizes repeated addition."
  65. ^ Ivars Peterson, The Mathematical Tourist, Freeman, 1988, ISBN 0-7167-1953-3. p. 4 "A few complain that the computer program can't be verified properly", (in reference to the Haken–Apple proof of the Four Color Theorem).
  66. ^ "The method of 'postulating' what we want has many advantages; they are the same as the advantages of theft over honest toil." Bertrand Russell (1919), Introduction to Mathematical Philosophy, New York and London, p. 71. Archived June 20, 2015, at the Wayback Machine
  67. ^ Patrick Suppes, Axiomatic Set Theory, Dover, 1972, ISBN 0-486-61630-4. p. 1, "Among the many branches of modern mathematics set theory occupies a unique place: with a few rare exceptions the entities which are studied and analyzed in mathematics may be regarded as certain particular sets or classes of objects."
  68. ^ Luke Howard Hodgkin & Luke Hodgkin, A History of Mathematics, Oxford University Press, 2005.
  69. ^ Clay Mathematics Institute, P=NP, claymath.org
  70. ^ Rao, C.R. (1997) Statistics and Truth: Putting Chance to Work, World Scientific. ISBN 981-02-3111-3
  71. ^ Rao, C.R. (1981). "Foreword". In Arthanari, T.S.; Dodge, Yadolah (eds.). Mathematical programming in statistics. Wiley Series in Probability and Mathematical Statistics. New York: Wiley. pp. vii–viii. ISBN 978-0-471-08073-2. MR 0607328.
  72. ^ Whittle (1994, pp. 10–11, 14–18): Whittle, Peter (1994). "Almost home". In Kelly, F.P. (ed.). Probability, statistics and optimisation: A Tribute to Peter Whittle (previously "A realised path: The Cambridge Statistical Laboratory upto 1993 (revised 2002)" ed.). Chichester: John Wiley. pp. 1–28. ISBN 978-0-471-94829-2. Archived from the original on December 19, 2013.
  73. ^ Monastyrsky 2001, p. 1: "The Fields Medal is now indisputably the best known and most influential award in mathematics."
  74. ^ Riehm 2002, pp. 778–82.

Bibliography

Further reading

Wikiversity
At Wikiversity, you can learn
more and teach others about Mathematics at the School of Mathematics.