Author: IMC

一代又一代的科學家將宇宙比作一個巨大而複雜的遊戲，引發了關於誰在玩這個遊戲，以及獲勝意味著什麼的問題。

如果宇宙是場遊戲，那麼誰在玩它？

以下是我們《迷失於時空》電子報的摘錄。每個月，我們都會將鍵盤交給一位物理學家或數學家，向您講述他們宇宙角落中引人入勝的想法。您可以在此免費註冊《迷失於時空》。

宇宙是場遊戲嗎？著名物理學家理查·費曼當然這麼認為：「『世界』就像一場由眾神進行的偉大西洋棋遊戲，而我們是遊戲的觀察者。」當我們觀察時，我們作為科學家的任務是試圖找出遊戲規則。

17 世紀數學家戈特弗里德·威廉·萊布尼茨也將宇宙視為一場遊戲，甚至資助了柏林一所專門研究遊戲的學院的成立：「我非常贊成研究理性遊戲，不是為了遊戲本身，而是因為它們有助於我們完善思考的藝術。」

我們人類喜歡玩遊戲，不僅僅是孩子，成年後也是如此。據信，遊戲是進化發展的重要組成部分——以至於文化理論家約翰·赫伊津哈（Johan Huizinga）提出，我們應該被稱為 Homo ludens，即遊戲物種，而不是 Homo sapiens。有人認為，一旦我們意識到宇宙是由規則控制的，我們就開始開發遊戲，以此來試驗這些規則的後果。

以我們創造的最早的棋盤遊戲之一為例。烏爾王室遊戲可以追溯到西元前 2500 年左右，在美索不達米亞的蘇美爾城市烏爾被發現。四面體形狀的骰子用於競賽，每個玩家的五個棋子沿著共享的 12 個方格的序列前進。對該遊戲的一種解釋是，這 12 個方格代表黃道帶的 12 個星座，它們構成了夜空的固定背景，而五個棋子對應於古代人觀察到的在夜空中移動的五顆可見行星。

但是宇宙本身是否符合遊戲的資格？定義什麼構成遊戲一直是激烈爭論的主題。邏輯學家路德維希·維根斯坦認為，詞語不能被字典定義所束縛，只能通過它們的使用方式來獲得意義，他稱之為「語言遊戲」。他認為，只有通過使用而不是定義才能獲得意義的一個詞的例子是「遊戲」。每次你試圖定義「遊戲」這個詞時，你最終會包括一些不是遊戲的東西，並排除其他你想要包括的東西。

其他哲學家則不那麼悲觀，並試圖確定定義遊戲的品質。包括維根斯坦在內的每個人都同意，所有遊戲的一個共同點是它們由規則定義。這些規則控制著你在遊戲中可以做什麼或不能做什麼。正是因為這個原因，一旦我們理解了宇宙是由約束其演化的規則控制的，宇宙作為遊戲的想法就紮根了。

理論家羅傑·凱約瓦（Roger Caillois）在他的著作《人、遊戲和玩耍》中提出了定義遊戲的其他五個關鍵特徵：不確定性、非生產性、分離性、想像力和自由。那麼，宇宙在多大程度上符合這些其他特徵呢？

不確定性的作用很有趣。我們進入遊戲是因為任何一方都有可能獲勝——如果我們事先知道遊戲將如何結束，它就會失去所有的力量。這就是為什麼確保盡可能長時間的持續不確定性是遊戲設計中的一個關鍵組成部分。

博學家皮埃爾-西蒙·拉普拉斯（Pierre-Simon Laplace）曾著名地宣稱，艾薩克·牛頓（Isaac Newton）對運動定律的識別消除了宇宙遊戲中的所有不確定性：「我們可以將宇宙的現在狀態視為其過去的結果和其未來的起因。一個在某一時刻知道所有推動自然運動的力量，以及組成自然的所有物體的所有位置的智力，如果這個智力也足夠廣闊，可以將這些數據提交給分析，它將在一個單一的公式中包含宇宙中最大物體的運動和最小原子的運動；對於這樣一個智力來說，沒有什麼是不確定的，未來就像過去一樣可以呈現在它的眼前。」

已解決的遊戲遭受同樣的命運。四子棋是一種已解決的遊戲，因為我們現在知道一種演算法，它總是能保證先手玩家獲勝。在完美的遊戲中，沒有不確定性。這就是為什麼純粹的策略遊戲有時會受到影響——如果一個玩家比他們的對手好得多，那麼結果幾乎沒有不確定性。唐納德·川普（Donald Trump）對陣加里·卡斯帕羅夫（Garry Kasparov）的西洋棋比賽將不會是一場有趣的比賽。

然而，20 世紀的啟示重新將不確定性的概念引入了宇宙的規則中。量子物理學斷言，實驗的結果並非由其當前狀態預先決定。遊戲中的棋子可能會根據波函數的崩潰朝多個不同的方向前進。儘管阿爾伯特·愛因斯坦（Albert Einstein）相信什麼，但上帝似乎正在擲骰子玩遊戲。

即使遊戲是確定性的，混沌理論的數學也暗示著玩家和觀察者將無法完全詳細地了解遊戲的當前狀態，並且當前狀態的微小差異可能導致非常不同的結果。

遊戲應該是非生產性的，這是一個有趣的品質。如果我們為了錢或為了教我們一些東西而玩遊戲，凱約瓦認為遊戲已經變成了工作：遊戲是「純粹浪費的場合：浪費時間、精力、才智、技能」。不幸的是，除非你相信某種更高的力量，否則所有證據都指向宇宙的最終無目的性。宇宙不是為了某個原因而存在的。它只是存在。

凱約瓦概述的其他三個品質可能較少適用於宇宙，但將遊戲描述為與宇宙不同的東西，儘管與宇宙平行運行。遊戲是分離的——它在正常的時間和空間之外運行。遊戲有其自己劃定的空間，在設定的時間限制內進行。它有自己的開始和自己的結束。遊戲是我們宇宙的暫停。它是逃往平行宇宙的途徑。

遊戲應該有一個結束也是很有趣的。哲學家詹姆斯·P·卡斯（James P. Carse）在他的著作《有限與無限遊戲》中介紹了無限遊戲的概念。你的目標不是贏得無限遊戲。獲勝會終止遊戲，因此使其成為有限的。相反，無限遊戲的玩家的任務是延續遊戲——確保它永遠不會結束。卡斯以相當隱晦的陳述結束了他的書：「只有一個無限遊戲。」人們意識到他指的是我們都是在我們周圍進行的無限遊戲的玩家，這個無限遊戲就是宇宙。儘管目前的物理學確實提出了一個最終的舉動：宇宙的熱寂意味著這個宇宙可能有一個我們無能為力的殘局。

凱約瓦的想像力品質指的是遊戲是虛構的想法。遊戲包括創造一個與現實生活平行運行的第二現實。這是一個虛構的宇宙，玩家自願召喚它，獨立於我們所屬的物理宇宙的嚴酷現實。

最後，凱約瓦認為遊戲需要自由。任何被迫玩遊戲的人都是在工作而不是在玩耍。因此，遊戲與人類意識的另一個重要方面聯繫在一起：我們的自由意志。

這提出了一個問題：如果宇宙是場遊戲，那麼誰在玩它，獲勝意味著什麼？我們只是這個遊戲中的棋子而不是玩家嗎？有人推測我們的宇宙實際上是一個巨大的模擬。有人編寫了規則，輸入了一些起始數據，並讓模擬運行。這就是為什麼約翰·康威（John Conway）的生命遊戲感覺最接近宇宙可能存在的遊戲類型。在康威的遊戲中，無限網格上的像素根據它們的環境和遊戲規則而誕生、生存和死亡。康威的成功在於創造了一套規則，這些規則產生了如此有趣的複雜性。

如果宇宙是場遊戲，那麼我們也感到幸運，發現自己是遊戲的一部分，這個遊戲在簡單性和複雜性、機遇和策略、戲劇性和危險性之間取得了完美的平衡，使其變得有趣。即使我們發現了遊戲規則，它也承諾在達到殘局的那一刻之前，都將是一場引人入勝的比賽。

如需更多此類見解，請登錄 www.international-maths-challenge.com。

*文章作者：Marcus Du Sautoy*

Des générations de scientifiques ont comparé l’univers à un jeu géant et complexe, soulevant des questions sur qui y joue – et ce que cela signifierait de gagner.

Si l’univers est un jeu, alors qui y joue ?

Ce qui suit est un extrait de notre newsletter Lost in Space-Time. Chaque mois, nous confions le clavier à un physicien ou un mathématicien pour vous parler d’idées fascinantes de leur coin de l’univers. Vous pouvez vous inscrire gratuitement à Lost in Space-Time ici.

L’univers est-il un jeu ? Le célèbre physicien Richard Feynman le pensait certainement : « “Le monde” ressemble à une grande partie d’échecs jouée par les dieux, et nous sommes les observateurs du jeu. » En observant, notre tâche en tant que scientifiques est d’essayer de comprendre les règles du jeu.

Le mathématicien du XVIIe siècle Gottfried Wilhelm Leibniz considérait aussi l’univers comme un jeu et a même financé la fondation d’une académie à Berlin dédiée à l’étude des jeux : « J’approuve fortement l’étude des jeux de raison non pour eux-mêmes mais parce qu’ils nous aident à perfectionner l’art de penser. »

Notre espèce adore jouer, pas seulement en tant qu’enfants mais jusqu’à l’âge adulte. On pense que cela a été une partie importante du développement évolutif – à tel point que le théoricien culturel Johan Huizinga a proposé qu’on nous appelle Homo ludens, l’espèce qui joue, plutôt qu’Homo sapiens. Certains ont suggéré qu’une fois que nous avons réalisé que l’univers est contrôlé par des règles, nous avons commencé à développer des jeux comme moyen d’expérimenter les conséquences de ces règles.

Prenons, par exemple, l’un des tout premiers jeux de plateau que nous avons créés. Le Jeu royal d’Ur remonte à environ 2500 av. J.-C. et a été trouvé dans la cité sumérienne d’Ur, partie de la Mésopotamie. Des dés tétraédriques sont utilisés pour faire courir cinq pièces appartenant à chaque joueur le long d’une séquence partagée de 12 cases. Une interprétation du jeu est que les 12 cases représentent les 12 constellations du zodiaque qui forment un arrière-plan fixe au ciel nocturne et les cinq pièces correspondent aux cinq planètes visibles que les anciens observaient se déplacer dans le ciel nocturne.

Mais l’univers lui-même peut-il être qualifié de jeu ? Définir ce qui constitue réellement un jeu a fait l’objet d’un débat passionné. Le logicien Ludwig Wittgenstein croyait que les mots ne pouvaient pas être définis par une définition de dictionnaire et ne gagnaient leur sens qu’à travers la façon dont ils étaient utilisés, dans un processus qu’il appelait le « jeu de langage ». Un exemple d’un mot qui, selon lui, ne tirait son sens que de l’usage plutôt que de la définition était « jeu ». Chaque fois que vous essayez de définir le mot « jeu », vous finissez par inclure certaines choses qui ne sont pas des jeux et en exclure d’autres que vous vouliez inclure.

D’autres philosophes ont été moins défaitistes et ont essayé d’identifier les qualités qui définissent un jeu. Tout le monde, y compris Wittgenstein, s’accorde à dire qu’une facette commune de tous les jeux est qu’ils sont définis par des règles. Ces règles contrôlent ce que vous pouvez ou ne pouvez pas faire dans le jeu. C’est pour cette raison que dès que nous avons compris que l’univers est contrôlé par des règles qui limitent son évolution, l’idée de l’univers comme jeu s’est imposée.

Dans son livre L’Homme, le Jeu et les Jeux, le théoricien Roger Caillois a proposé cinq autres traits clés qui définissent un jeu : l’incertitude, l’improductivité, la séparation, l’imagination et la liberté. Alors, comment l’univers correspond-il à ces autres caractéristiques ?

Le rôle de l’incertitude est intéressant. Nous entrons dans un jeu parce qu’il y a une chance que l’un ou l’autre camp gagne – si nous savons à l’avance comment le jeu se terminera, il perd tout son pouvoir. C’est pourquoi assurer une incertitude continue le plus longtemps possible est un composant clé dans la conception de jeux.

Le polymathe Pierre-Simon Laplace a déclaré de façon célèbre que l’identification par Isaac Newton des lois du mouvement avait supprimé toute incertitude du jeu de l’univers : « Nous pouvons considérer l’état présent de l’univers comme l’effet de son passé et la cause de son futur. Une intelligence qui à un certain moment connaîtrait toutes les forces qui mettent la nature en mouvement, et toutes les positions de tous les éléments dont la nature est composée, si cette intelligence était aussi vaste pour soumettre ces données à l’analyse, elle embrasserait dans une seule formule les mouvements des plus grands corps de l’univers et ceux du plus petit atome ; pour une telle intelligence rien ne serait incertain et l’avenir tout comme le passé pourrait être présent devant ses yeux. »

Les jeux résolus subissent le même sort. Puissance 4 est un jeu résolu dans le sens où nous connaissons maintenant un algorithme qui garantira toujours la victoire au premier joueur. Avec un jeu parfait, il n’y a pas d’incertitude. C’est pourquoi les jeux de pure stratégie souffrent parfois – si un joueur est bien meilleur que son adversaire, alors il y a peu d’incertitude dans le résultat. Donald Trump contre Garry Kasparov dans une partie d’échecs ne sera pas un jeu intéressant.

Les révélations du XXe siècle, cependant, ont réintroduit l’idée d’incertitude dans les règles de l’univers. La physique quantique affirme que le résultat d’une expérience n’est pas prédéterminé par son état actuel. Les pièces du jeu pourraient se diriger dans plusieurs directions différentes selon l’effondrement de la fonction d’onde. Contrairement à ce qu’Albert Einstein croyait, il semble que Dieu joue un jeu avec des dés.

Même si le jeu était déterministe, les mathématiques de la théorie du chaos impliquent aussi que les joueurs et les observateurs ne pourront pas connaître l’état présent du jeu dans tous ses détails et que de petites différences dans l’état actuel peuvent résulter en des résultats très différents.

Qu’un jeu doive être improductif est une qualité intéressante. Si nous jouons à un jeu pour de l’argent ou pour nous apprendre quelque chose, Caillois croyait que le jeu était devenu du travail : un jeu est « une occasion de pure perte : perte de temps, d’énergie, d’ingéniosité, d’habileté ». Malheureusement, à moins que vous ne croyiez en une puissance supérieure, toutes les preuves pointent vers l’absence de but ultime de l’univers. L’univers n’est pas là pour une raison. Il est, tout simplement.

Les trois autres qualités que Caillois décrit s’appliquent peut-être moins à l’univers mais décrivent un jeu comme quelque chose de distinct de l’univers, bien qu’évoluant parallèlement à lui. Un jeu est séparé – il fonctionne en dehors du temps et de l’espace normaux. Un jeu a son propre espace délimité dans lequel il se joue dans une limite de temps définie. Il a son propre commencement et sa propre fin. Un jeu est un temps mort de notre univers. C’est une évasion vers un univers parallèle.

Le fait qu’un jeu doive avoir une fin est aussi intéressant. Il y a le concept d’un jeu infini que le philosophe James P. Carse a introduit dans son livre Jeux finis et infinis. Vous ne cherchez pas à gagner un jeu infini. Gagner termine le jeu et le rend donc fini. Au lieu de cela, le joueur du jeu infini a pour tâche de perpétuer le jeu – s’assurer qu’il ne finisse jamais. Carse conclut son livre avec la déclaration plutôt cryptique : « Il n’y a qu’un seul jeu infini. » On réalise qu’il fait référence au fait que nous sommes tous des joueurs dans le jeu infini qui se déroule autour de nous, le jeu infini qu’est l’univers. Bien que la physique actuelle postule un coup final : la mort thermique de l’univers signifie que cet univers pourrait avoir une fin de partie que nous ne pouvons rien faire pour éviter.

La qualité d’imagination de Caillois fait référence à l’idée que les jeux sont de la fiction. Un jeu consiste à créer une seconde réalité qui fonctionne en parallèle avec la vraie vie. C’est un univers fictif que les joueurs invoquent volontairement indépendamment de la réalité sévère de l’univers physique dont nous faisons partie.

Enfin, Caillois croit qu’un jeu exige la liberté. Quiconque est forcé de jouer à un jeu travaille plutôt qu’il ne joue. Un jeu, par conséquent, se connecte avec un autre aspect important de la conscience humaine : notre libre arbitre.

Cela soulève une question : si l’univers est un jeu, qui y joue et que signifiera gagner ? Ne sommes-nous que des pions dans ce jeu plutôt que des joueurs ? Certains ont spéculé que notre univers est en fait une énorme simulation. Quelqu’un a programmé les règles, saisi des données de départ et a laissé la simulation tourner. C’est pourquoi le Jeu de la vie de John Conway ressemble le plus au type de jeu que l’univers pourrait être. Dans le jeu de Conway, des pixels sur une grille infinie naissent, vivent et meurent selon leur environnement et les règles du jeu. Le succès de Conway a été de créer un ensemble de règles qui a donné naissance à une complexité si intéressante.

Si l’univers est un jeu, alors il semble que nous aussi avons eu de la chance de nous retrouver partie d’un jeu qui a l’équilibre parfait de simplicité et de complexité, de hasard et de stratégie, de drame et de péril pour le rendre intéressant. Même quand nous découvrons les règles du jeu, il promet d’être un match fascinant jusqu’au moment où il atteint sa fin de partie.

Pour plus de telles perspectives, connectez-vous sur www.international-maths-challenge.com.

*Crédit pour l’article donné à Marcus Du Sautoy*

Generationen von Wissenschaftlern haben das Universum mit einem riesigen, komplexen Spiel verglichen und dabei Fragen aufgeworfen, wer spielt – und was es bedeuten würde, zu gewinnen.

Wenn das Universum ein Spiel ist, wer spielt es dann?

Das Folgende ist ein Auszug aus unserem Newsletter „Lost in Space-Time“. Jeden Monat überlassen wir einem Physiker oder Mathematiker die Tastatur, um Ihnen faszinierende Ideen aus ihrem Bereich des Universums zu erzählen. Sie können sich hier kostenlos für „Lost in Space-Time“ anmelden.

Ist das Universum ein Spiel? Der berühmte Physiker Richard Feynman dachte sicherlich so: „‚Die Welt‘ ist so etwas wie ein großes Schachspiel, das von den Göttern gespielt wird, und wir sind Beobachter des Spiels.“ Während wir beobachten, ist es unsere Aufgabe als Wissenschaftler, die Regeln des Spiels herauszufinden.

Der Mathematiker Gottfried Wilhelm Leibniz aus dem 17. Jahrhundert betrachtete das Universum ebenfalls als ein Spiel und finanzierte sogar die Gründung einer Akademie in Berlin, die sich der Erforschung von Spielen widmete: „Ich befürworte nachdrücklich das Studium von Denkspielen, nicht um ihrer selbst willen, sondern weil sie uns helfen, die Kunst des Denkens zu perfektionieren.“

Unsere Spezies liebt es, Spiele zu spielen, nicht nur als Kinder, sondern bis ins Erwachsenenalter. Es wird angenommen, dass dies ein wichtiger Teil der evolutionären Entwicklung war – so sehr, dass der Kulturtheoretiker Johan Huizinga vorschlug, wir sollten Homo ludens, die spielende Spezies, genannt werden, anstatt Homo sapiens. Einige haben vorgeschlagen, dass wir, sobald wir erkannten, dass das Universum von Regeln kontrolliert wird, begannen, Spiele zu entwickeln, um mit den Konsequenzen dieser Regeln zu experimentieren.

Nehmen wir zum Beispiel eines der allerersten Brettspiele, die wir geschaffen haben. Das Königsspiel von Ur stammt aus der Zeit um 2500 v. Chr. und wurde in der sumerischen Stadt Ur, einem Teil Mesopotamiens, gefunden. Tetraederförmige Würfel werden verwendet, um fünf Figuren jedes Spielers über eine gemeinsame Abfolge von 12 Feldern zu bewegen. Eine Interpretation des Spiels ist, dass die 12 Felder die 12 Sternbilder des Tierkreises darstellen, die einen festen Hintergrund des Nachthimmels bilden, und die fünf Figuren den fünf sichtbaren Planeten entsprechen, die die Alten beim Bewegen durch den Nachthimmel beobachteten.

Aber qualifiziert sich das Universum selbst als Spiel? Die Definition dessen, was ein Spiel tatsächlich ausmacht, war Gegenstand hitziger Debatten. Der Logiker Ludwig Wittgenstein glaubte, dass Wörter nicht durch eine Wörterbuchdefinition festgelegt werden könnten und ihre Bedeutung nur durch die Art und Weise erhielten, wie sie verwendet wurden, in einem Prozess, den er das „Sprachspiel“ nannte. Ein Beispiel für ein Wort, von dem er glaubte, dass es seine Bedeutung nur durch Gebrauch und nicht durch Definition erhielt, war „Spiel“. Jedes Mal, wenn man versucht, das Wort „Spiel“ zu definieren, schließt man am Ende einige Dinge ein, die keine Spiele sind, und schließt andere aus, die man einschließen wollte.

Andere Philosophen waren weniger defätistisch und haben versucht, die Eigenschaften zu identifizieren, die ein Spiel definieren. Alle, einschließlich Wittgenstein, sind sich einig, dass ein gemeinsamer Aspekt aller Spiele darin besteht, dass sie durch Regeln definiert sind. Diese Regeln bestimmen, was man im Spiel tun oder lassen kann. Aus diesem Grund setzte sich die Idee des Universums als Spiel durch, sobald wir verstanden, dass das Universum von Regeln kontrolliert wird, die seine Entwicklung bestimmen.

In seinem Buch Man, Play and Games schlug der Theoretiker Roger Caillois fünf weitere Schlüsselmerkmale vor, die ein Spiel definieren: Ungewissheit, Unproduktivität, Abgrenzung, Vorstellungskraft und Freiheit. Wie passt das Universum also zu diesen anderen Merkmalen?

Die Rolle der Ungewissheit ist interessant. Wir treten in ein Spiel ein, weil die Chance besteht, dass eine der beiden Seiten gewinnt – wenn wir im Voraus wissen, wie das Spiel enden wird, verliert es seine ganze Kraft. Deshalb ist die Gewährleistung einer anhaltenden Ungewissheit so lange wie möglich ein Schlüsselbestandteil des Spieldesigns.

Der Universalgelehrte Pierre-Simon Laplace erklärte bekanntlich, dass Isaac Newtons Identifizierung der Bewegungsgesetze alle Ungewissheit aus dem Spiel des Universums entfernt habe: „Wir können den gegenwärtigen Zustand des Universums als die Wirkung seiner Vergangenheit und die Ursache seiner Zukunft betrachten. Ein Intellekt, der zu einem bestimmten Zeitpunkt alle Kräfte kennen würde, die die Natur in Bewegung setzen, und alle Positionen aller Gegenstände, aus denen die Natur besteht, wenn dieser Intellekt auch groß genug wäre, um diese Daten der Analyse zu unterziehen, würde er in einer einzigen Formel die Bewegungen der größten Körper des Universums und die des kleinsten Atoms umfassen; für einen solchen Intellekt wäre nichts ungewiss und die Zukunft könnte ebenso wie die Vergangenheit vor seinen Augen liegen.“

Gelöste Spiele erleiden dasselbe Schicksal. Vier gewinnt ist ein gelöstes Spiel in dem Sinne, dass wir jetzt einen Algorithmus kennen, der dem ersten Spieler immer einen Gewinn garantiert. Bei perfektem Spiel gibt es keine Ungewissheit. Deshalb leiden reine Strategiespiele manchmal – wenn ein Spieler viel besser ist als sein Gegner, gibt es wenig Ungewissheit über den Ausgang. Donald Trump gegen Garry Kasparow in einer Schachpartie wird kein interessantes Spiel sein.

Die Offenbarungen des 20. Jahrhunderts haben jedoch die Idee der Ungewissheit wieder in die Regeln des Universums eingeführt. Die Quantenphysik behauptet, dass das Ergebnis eines Experiments nicht durch seinen aktuellen Zustand vorbestimmt ist. Die Figuren im Spiel könnten sich je nach dem Kollaps der Wellenfunktion in mehrere verschiedene Richtungen bewegen. Entgegen dem, was Albert Einstein glaubte, scheint es, dass Gott ein Spiel mit Würfeln spielt.

Selbst wenn das Spiel deterministisch wäre, impliziert die Mathematik der Chaostheorie auch, dass Spieler und Beobachter den gegenwärtigen Zustand des Spiels nicht vollständig kennen können und kleine Unterschiede im aktuellen Zustand zu sehr unterschiedlichen Ergebnissen führen können.

Dass ein Spiel unproduktiv sein sollte, ist eine interessante Eigenschaft. Wenn wir ein Spiel um Geld spielen oder um uns etwas beizubringen, glaubte Caillois, dass das Spiel zu Arbeit geworden sei: Ein Spiel ist „ein Anlass reiner Verschwendung: Verschwendung von Zeit, Energie, Einfallsreichtum, Geschicklichkeit“. Leider deutet, sofern man nicht an eine höhere Macht glaubt, alles darauf hin, dass das Universum letztlich zwecklos ist. Das Universum ist nicht aus einem bestimmten Grund da. Es ist einfach.

Die anderen drei Eigenschaften, die Caillois skizziert, treffen vielleicht weniger auf das Universum zu, beschreiben aber ein Spiel als etwas vom Universum Getrenntes, das jedoch parallel dazu verläuft. Ein Spiel ist getrennt – es funktioniert außerhalb der normalen Zeit und des Raums. Ein Spiel hat seinen eigenen abgegrenzten Raum, in dem es innerhalb einer festgelegten Zeitbegrenzung gespielt wird. Es hat seinen eigenen Anfang und sein eigenes Ende. Ein Spiel ist eine Auszeit von unserem Universum. Es ist eine Flucht in ein Paralleluniversum.

Die Tatsache, dass ein Spiel ein Ende haben sollte, ist ebenfalls interessant. Es gibt das Konzept eines unendlichen Spiels, das der Philosoph James P. Carse in seinem Buch Finite and Infinite Games einführte. Man zielt nicht darauf ab, ein unendliches Spiel zu gewinnen. Gewinnen beendet das Spiel und macht es somit endlich. Stattdessen ist der Spieler des unendlichen Spiels damit beauftragt, das Spiel fortzusetzen – sicherzustellen, dass es niemals endet. Carse schließt sein Buch mit der eher kryptischen Aussage: „Es gibt nur ein unendliches Spiel.“ Man erkennt, dass er sich auf die Tatsache bezieht, dass wir alle Spieler in dem unendlichen Spiel sind, das sich um uns herum abspielt, dem unendlichen Spiel, das das Universum ist. Obwohl die aktuelle Physik einen letzten Zug postuliert: Der Hitzetod des Universums bedeutet, dass dieses Universum ein Endspiel haben könnte, das wir nicht vermeiden können.

Caillois’ Eigenschaft der Vorstellungskraft bezieht sich auf die Idee, dass Spiele ein Trugbild sind. Ein Spiel besteht darin, eine zweite Realität zu schaffen, die parallel zum realen Leben verläuft. Es ist ein fiktives Universum, das die Spieler freiwillig heraufbeschwören, unabhängig von der strengen Realität des physikalischen Universums, dessen Teil wir sind.

Schließlich glaubt Caillois, dass ein Spiel Freiheit erfordert. Jeder, der gezwungen wird, ein Spiel zu spielen, arbeitet eher, als dass er spielt. Ein Spiel verbindet sich daher mit einem weiteren wichtigen Aspekt des menschlichen Bewusstseins: unserem freien Willen.

Dies wirft eine Frage auf: Wenn das Universum ein Spiel ist, wer spielt es dann und was bedeutet es, zu gewinnen? Sind wir nur Bauern in diesem Spiel statt Spieler? Einige haben spekuliert, dass unser Universum tatsächlich eine riesige Simulation ist. Jemand hat die Regeln programmiert, Startdaten eingegeben und die Simulation laufen lassen. Deshalb fühlt sich John Conways Spiel des Lebens am ehesten wie die Art von Spiel an, die das Universum sein könnte. In Conways Spiel werden Pixel auf einem unendlichen Gitter geboren, leben und sterben gemäß ihrer Umgebung und den Regeln des Spiels. Conways Erfolg bestand darin, eine Reihe von Regeln zu schaffen, die eine so interessante Komplexität hervorbrachten.

Wenn das Universum ein Spiel ist, dann haben wir anscheinend Glück gehabt, Teil eines Spiels zu sein, das die perfekte Balance aus Einfachheit und Komplexität, Zufall und Strategie, Drama und Gefahr aufweist, um es interessant zu machen. Selbst wenn wir die Regeln des Spiels entdecken, verspricht es ein faszinierendes Match zu werden, bis zu dem Moment, in dem es sein Endspiel erreicht.

Für weitere solcher Einblicke melden Sie sich unter www.international-maths-challenge.com an.

*Artikelgutschrift an Marcus Du Sautoy*

How can you figure out which points lie on a certain curve? And how many possible curves do you count by a given number of points? These are the kinds of questions Pim Spelier of the Mathematical Institute studied during his Ph.D. research. Spelier received his doctorate with distinction on June 12.

What does counting curves mean on an average day? “A lot of sitting and gazing,” Spelier replies. “When I’m asked what exactly do, can’t always answer that easily. Usually give the example about the particle traveling through time.”

All possible curves

Imagine a particle moving through space and you follow the path the particle makes through time. That path is a curve, a geometric object. How many possible paths can the particle follow, if we assume certain properties? For example, a straight line can only pass through two points in one way. But how many paths are possible for the particle if we look at more difficult curves? And how do you study that?

By looking at all possible curves at the same time. For example, all possible directions from a given point form with each other a circle, and that is called a modulspace. And that circle is itself a geometric object.

The mathematical magic can happen because this set of all curves itself has geometrical properties, Spelier says, to which you can apply geometrical tricks. Next, you can make that far more complicated with even more complex curves and spaces. So not counting in three but, for example, in eleven dimensions.

Spelier tries to find patterns that always apply to the curves he studies. His approach? Breaking up complicated spaces into small, easy spaces. You can also break curves into partial curves. That way, the spaces in which you’re counting are easier. But the curves sometimes get complicated properties, because you have to be able to glue them back together.

Spelier says, “The goal is to find enough principles to determine the number of curves exactly.”

In addition to curves, Spelier also counted points on curves. He studied the question: how many solutions does a given mathematical equation have?

These are equations that are a bit more complicated than the a² + b² = c² of the Pythagorean theorem. That equation is about the lengths of the sides of a right triangle. If you replace the squares with higher powers, it is more difficult to investigate solutions. Spelier studied solutions in whole numbers, for example, 3² + 4² = 5².

Meanwhile, there is a method to find those solutions. Professor of Mathematics Bas Edixhoven, who died in 2022, and his Ph.D. student Guido Lido developed an alternative approach to the same problem. But to what extent the two methods match and differ was still unclear. During his Ph.D. research, Spelier developed an algorithm to investigate this.

The first person with an answer

Developing that algorithm is necessary to implement the method. If you want to do it by hand, you get pages and pages of equations. Edixhoven’s method uses algebraic geometry. Through clever geometric tricks, you can calculate exactly the whole number points of a given curve. Spelier proved that the Edixhoven-Lido method is better than the old one.

David Holmes, professor of Pure Mathematics and supervisor of Spelier, praises the proof provided. “When you’re the first person to answer a question that everyone in our community wants an answer to, that’s very impressive. Pim proves that these two methods for finding rational points are similar, an issue that really kept mathematicians busy.”

Doing math together

The best part of his Ph.D.? The meetings with his supervisor. After the first year, it was more collaboration than supervision, both for Spelier and Holmes. Spelier says, “Doing math together is still more fun than doing it alone.”

Spelier starts in September as a postdoc in Utrecht and is apparently not yet done with counting. After counting points and curves, he will soon start counting surfaces.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to Leiden University.

Whether you are sharing a cake or a coastline, maths can help make sure everyone is happy with their cut, says Katie Steckles.

One big challenge in life is dividing things fairly. From sharing a tasty snack to allocating resources between nations, having a strategy to divvy things up equitably will make everyone a little happier.

But it gets complicated when the thing you are dividing isn’t an indistinguishable substance: maybe the cake you are sharing has a cherry on top, and the piece with the cherry (or the area of coastline with good fish stocks) is more desirable. Luckily, maths – specifically game theory, which deals with strategy and decision-making when people interact – has some ideas.

When splitting between two parties, you might know a simple rule, proven to be mathematically optimal: I cut, you choose. One person divides the cake (or whatever it is) and the other gets to pick which piece they prefer.

Since the person cutting the cake doesn’t choose which piece they get, they are incentivised to cut the cake fairly. Then when the other person chooses, everyone is satisfied – the cutter would be equally happy with either piece, and the chooser gets their favourite of the two options.

This results in what is called an envy-free allocation – neither participant can claim they would rather have the other person’s share. This also takes care of the problem of non-homogeneous objects: if some parts of the cake are more desirable, the cutter can position their cut so the two pieces are equal in value to them.

What if there are more people? It is more complicated, but still possible, to produce an envy-free allocation with several so-called fair-sharing algorithms.

Let’s say Ali, Blake and Chris are sharing a cake three ways. Ali cuts the cake into three pieces, equal in value to her. Then Blake judges if there are at least two pieces he would be happy with. If Blake says yes, Chris chooses a piece (happily, since he gets free choice); Blake chooses next, pleased to get one of the two pieces he liked, followed by Ali, who would be satisfied with any of the pieces. If Blake doesn’t think Ali’s split was equitable, Chris looks to see if there are two pieces he would take. If yes, Blake picks first, then Chris, then Ali.

If both Blake and Chris reject Ali’s initial chop, then there must be at least one piece they both thought was no good. This piece goes to Ali – who is still happy, because she thought the pieces were all fine – and the remaining two pieces get smooshed back together (that is a mathematical term) to create one piece of cake for Blake and Chris to perform “I cut, you choose” on.

While this seems long-winded, it ensures mathematically optimal sharing – and while it does get even more complicated, it can be extended to larger groups. So whether you are sharing a treat or a divorce settlement, maths can help prevent arguments.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Katie Steckles*

“We wanted to investigate how second pillar pension funds react to financial crises and how to protect them from the crises,” says Kaunas University of Technology (KTU) professor Dr. Audrius Kabašinskas, who, together with his team, discovered a way to achieve this goal. The discovery in question is the development of stress tests for pension funds. Lithuanian researchers were the first in the world to come up with such an adaptation of the stress tests.

Stress tests are usually carried out on banks or other financial institutions to allow market regulators to determine and assess their ability to withstand adverse economic conditions.

According to the professor at KTU Faculty of Mathematics and Natural Sciences, this innovative pension fund stress testing approach will benefit both regulators and pension fund managers.

“Making sure your pension fund is resilient to harsh financial market conditions will help you sleep better, save more, and have increased trust in your funds and the pension system itself,” Kabašinskas adds.

Results based on two major crises

First, the study needed to collect data from previous periods. “Two major events that shocked the whole world—COVID-19 and the first year of Russian invasion of Ukraine—just happened to occur during the project. This allowed us to gather a lot of relevant information and data on changes in the performance of pension funds,” says Kabašinskas.

The Hidden Markov Model (HMM), which, according to a professor at KTU Department of Mathematical Modelling, is quite simple in its principle of operation, helped to forecast future market conditions in this study.

The paper is published in the journal Annals of Operations Research.

“The observation of air temperature could be an analogy for it. All year round, without looking at the calendar, we observe the temperature outside and, based on the temperature level, we decide what time of the year it is. Of course, 15 degrees can occur in winter and sometimes it snows in May but these are random events. The state of the next day depends only on today,” he explains vividly.

According to the KTU researcher, this describes the idea of the Hidden Markov Model: by observing the changes in value, one can judge the state of global markets and try to forecast the future.

“In our study, we observed two well-known investment funds from 2019 to 2022. Collected information helped us identify that global markets at any given moment are in one of four states: no shock regime, a state of shock in stock markets, a state of shock in bond markets, and a state of global financial shock—a global crisis,” says Kabašinskas.

Using certain methods, the research team led by a professor Miloš Kopa representing KTU and Charles University in Prague found that these periods were aligned with the global events in question. Once the transition probabilities between the states were identified, it was possible to link the data of pension funds to these periods and simulate the future evolution of the pension funds’ value.

That’s where the innovation of stress testing came in. The purpose of this test is to determine whether a particular pension fund can deliver positive growth in the future when faced with a shock in the financial markets.

“In our study, we applied several scenarios, extending financial crises and modeling the evolution of fund values over the next 5 years,” says a KTU researcher.

This methodology can be applied not only to pension funds but also to other investments.

Example of Lithuanian pension funds

The research and the new stress tests were carried out on Lithuanian pension funds.

Kabašinskas says that the study revealed several interesting things. Firstly, on average, Lithuanian second pillar pension funds can withstand crises that are twice as long.

“However, the results show that some Lithuanian funds struggle to cope with inflation, while others, the most conservative funds for citizens who are likely to retire within next few years or who have already retired, are very slow in recovering after negative shocks,” adds the KTU expert.

This can be explained by regulatory aspects and the related investment strategy, as stock markets recover several times faster than bond markets, and the above-mentioned funds invest more than 90% in bonds and other less risky instruments.

A complementary study has also been carried out to show how pension funds should change their investment strategy to avoid the drastic negative consequences of various financial crises and shocks.

“Funds that invest heavily in stocks and other risky instruments should increase the number of risk-free instruments slightly, up to 10%, before or after the financial crisis hits. Meanwhile, funds investing mainly in bonds should increase the number of stocks in their holdings. In both cases, the end of the crisis should be followed by a slow return to the typical strategy,” advises a mathematician.

Although the survey did not aim to increase people’s confidence in pension funds, the results showed that Lithuania’s second pillar pension funds are resilient to crisis and are worth trust. Historically they have delivered long-term growth, some have even outperformed inflation and price increases.

“Although short-term changes can be drastic, long-term growth is clearly visible,” says KTU professor Dr. Kabašinskas. “Lithuania, by the way, has a better system than many European countries,” he adds.

For more insights like this, visit our website at www.international-maths-challenge.com.

Finding bats is hard. They are small, fast and they primarily fly at night.  But our new research could improve the way conservationists find bat roosts. We’ve developed a new algorithm that significantly reduces the area that needs to be searched, which could save time and cut labour cost.

Of course, you may wonder why we would want to find bats in the first place. But these flying mammals are natural pest controllers and pollinators, and they help disperse seeds. So they are extremely useful in contributing to the health of our environment.

Despite their importance though, bat habitats are threatened by human activities such as increased lighting, noise and land use. To ensure that we can study and enhance the health of our bat population, we need to locate their roosts. But finding bat roosts is a bit like finding a needle in a haystack.

Our previous work measured and modelled the motion of greater horseshoe bats in flight. Having such a model means we can predict where bats will be, depending on their roost position. But the position of the roost is something we often don’t know.

Our new research combines our previous mathematical model of bat motion with data gathered from acoustic recorders known as “bat detectors”. These bat detectors are placed around the environment and left there for several nights.

Seeing with sound

Bats use echolocation, which allows them to “see with sound” when they’re flying. If these ultrasonic calls are made within ten to 15 metres of a bat detector, the device is triggered to make a recording, providing an accurate record of where and when a bat was present.

The sound recordings also provide clues about the identity of the species. Greater horseshoe bats make a very distinctive “warbling” call at almost exactly 82kHz in frequency, so we can easily tell whether the species is present or not.

Assuming that a bat detector’s batteries last for a few nights, its memory card is not full, and the units are not stolen or vandalised, then we can use the bat call data to generate a map that shows the proportion of bat calls at each detector location.

Our model can also be used to predict the proportion of bat calls based on a given roost location. So, we split the environment up into a grid and simulate bats flying from each grid square. The grid square, or squares, whose simulations best reproduce the bat detector data will then be the most likely locations of the roost.

This simple algorithm can then be applied to whole terrains, meaning that we can create a map of likely roost locations. Cutting out the regions that are least likely to contain the roost can mean we shrink the search space to less than 1% of the initially surveyed area. Simplifying the process of finding bat roosts allows more of an ecologist’s time to be spent on conservation projects, rather than laborious searching.

In 2022, we developed an app that uses publicly available data to predict bat flight lines. At the moment the app can help ecologists, developers or local authority planners, know how the environment is used by bats. However, it needs a roost location to be specified first, and this information is not always known. Our new research removes this barrier, making the app easier to use.

Our work offers a way of identifying likely roost locations. These estimates can then be verified either by directly observing particular features, or by capturing bats at a nearby location and following them back home, using radiotracking.

Over the past two decades, bat detectors have gone from simple hand-held machines to high-performance devices that can collect data for days at a time. Yet they are usually deployed only to identify bat species. We have shown they can be used to identify the areas most likely to contain bat roosts, uncovering critical information about these most secretive of animals.

We hope that this will provide further tools for ecologists to optimise the initial microphone detector locations, thereby providing a holistic way of detecting bat roosts.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Thomas Woolley and Fiona Mathews, The Conversation

By rapidly analysing large amounts of data and making accurate predictions, artificial intelligence (AI) tools could help to answer many long-standing research questions. For instance, they could help to identify new materials to fabricate electronics or the patterns in brain activity associated with specific human behaviours.

One area in which AI has so far been rarely applied is number theory, a branch of mathematics focusing on the study of integers and arithmetic functions. Most research questions in this field are solved by human mathematicians, often years or decades after their initial introduction.

Researchers at the Israel Institute of Technology (Technion) recently set out to explore the possibility of tackling long-standing problems in number theory using state-of-the-art computational models.

In a recent paper, published in the Proceedings of the National Academy of Sciences, they demonstrated that such a computational approach can support the work of mathematicians, helping them to make new exciting discoveries.

“Computer algorithms are increasingly dominant in scientific research, a practice now broadly called ‘AI for Science,'” Rotem Elimelech and Ido Kaminer, authors of the paper, told Phys.org.

“However, in fields like number theory, advances are often attributed to creativity or human intuition. In these fields, questions can remain unresolved for hundreds of years, and while finding an answer can be as simple as discovering the correct formula, there is no clear path for doing so.”

Elimelech, Kaminer and their colleagues have been exploring the possibility that computer algorithms could automate or augment mathematical intuition. This inspired them to establish the Ramanujan Machine research group, a new collaborative effort aimed at developing algorithms to accelerate mathematical research.

Their research group for this study also included Ofir David, Carlos de la Cruz Mengual, Rotem Kalisch, Wolfram Berndt, Michael Shalyt, Mark Silberstein, and Yaron Hadad.

“On a philosophical level, our work explores the interplay between algorithms and mathematicians,” Elimelech and Kaminer explained. “Our new paper indeed shows that algorithms can provide the necessary data to inspire creative insights, leading to discoveries of new formulas and new connections between mathematical constants.”

The first objective of the recent study by Elimelech, Kaminer and their colleagues was to make new discoveries about mathematical constants. While working toward this goal, they also set out to test and promote alternative approaches for conducting research in pure mathematics.

“The ‘conservative matrix field’ is a structure analogous to the conservative vector field that every math or physics student learns about in first year of undergrad,” Elimelech and Kaminer explained. “In a conservative vector field, such as the electric field created by a charged particle, we can calculate the change in potential using line integrals.

“Similarly, in conservative matrix fields, we define a potential over a discrete space and calculate it through matrix multiplications rather than using line integrals. Traveling between two points is equivalent to calculating the change in the potential and it involves a series of matrix multiplications.”

In contrast with the conservative vector field, the so-called conservative matrix field is a new discovery. An important advantage of this structure is that it can generalize the formulas of each mathematical constant, generating infinitely many new formulas of the same kind.

“The way by which the conservative matrix field creates a formula is by traveling between two points (or actually, traveling from one point all the way to infinity inside its discrete space),” Elimelech and Kaminer said. “Finding non-trivial matrix fields that are also conservative is challenging.”

As part of their study, Elimelech, Kaminer and their colleagues used large-scale distributed computing, which entails the use of multiple interconnected nodes working together to solve complex problems. This approach allowed them to discover new rational sequences that converge to fundamental constants (i.e., formulas for these constants).

“Each sequence represents a path hidden in the conservative matrix field,” Elimelech and Kaminer explained. “From the variety of such paths, we reverse-engineered the conservative matrix field. Our algorithms were distributed using BOINC, an infrastructure for volunteer computing. We are grateful to the contribution by hundreds of users worldwide who donated computation time over the past two and a half years, making this discovery possible.”

The recent work by the research team at the Technion demonstrates that mathematicians can benefit more broadly from the use of computational tools and algorithms to provide them with a “virtual lab.” Such labs provide an opportunity to try ideas experimentally in a computer, resembling the real experiments available in physics and in other fields of science. Specifically, algorithms can carry out mathematical experiments providing formulas that can be used to formulate new mathematical hypotheses.

“Such hypotheses, or conjectures, are what drives mathematical research forward,” Elimelech and Kaminer said. “The more examples supporting a hypothesis, the stronger it becomes, increasing the likelihood to be correct. Algorithms can also discover anomalies, pointing to phenomena that are the building-blocks for new hypotheses. Such discoveries would not be possible without large-scale mathematical experiments that use distributed computing.”

Another interesting aspect of this recent study is that it demonstrates the advantages of building communities to tackle problems. In fact, the researchers published their code online from their project’s early days and relied on contributions by a large network of volunteers.

“Our study shows that scientific research can be conducted without exclusive access to supercomputers, taking a substantial step toward the democratization of scientific research,” Elimelech and Kaminer said. “We regularly post unproven hypotheses generated by our algorithms, challenging other math enthusiasts to try proving these hypotheses, which when validated are posted on our project website. This happened on several occasions so far. One of the community contributors, Wolfgang Berndt, got so involved that he is now part of our core team and a co-author on the paper.”

The collaborative and open nature of this study allowed Elimelech, Kaminer and the rest of the team to establish new collaborations with other mathematicians worldwide. In addition, their work attracted the interest of some children and young people, showing them how algorithms and mathematics can be combined in fascinating ways.

In their next studies, the researchers plan to further develop the theory of conservative matrix fields. These matrix fields are a highly powerful tool for generating irrationality proofs for fundamental constants, which Elimelech, Kaminer and the team plan to continue experimenting with.

“Our current aim is to address questions regarding the irrationality of famous constants whose irrationality is unknown, sometimes remaining an open question for over a hundred years, like in the case of the Catalan constant,” Elimelech and Kaminer said.

“Another example is the Riemann zeta function, central in number theory, with its zeros at the heart of the Riemann hypothesis, which is perhaps the most important unsolved problem in pure mathematics. There are many open questions about the values of this function, including the irrationality of its values. Specifically, whether ζ(5) is irrational is an open question that attracts the efforts of great mathematicians.”

The ultimate goal of this team of researchers is to successfully use their experimental mathematics approach to prove the irrationality of one of these constants. In the future, they also hope to systematically apply their approach to a broader range of problems in mathematics and physics. Their physics-inspired hands-on research style arises from the interdisciplinary nature of the team, which combines people specialized in CS, EE, math, and physics.

“Our Ramanujan Machine group can help other researchers create search algorithms for their important problems and then use distributed computing to search over large spaces that cannot be attempted otherwise,” Elimelech and Kaminer added. “Each such algorithm, if successful, will help point to new phenomena and eventually new hypotheses in mathematics, helping to choose promising research directions. We are now considering pushing forward this strategy by setting up a virtual user facility for experimental mathematics,” inspired by the long history and impact of user facilities for experimental physics.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to Ingrid Fadelli , Phys.org

Direct reciprocity facilitates cooperation in repeated social interactions. Traditional models suggest that individuals learn to adopt conditionally cooperative strategies if they have multiple encounters with their partner. However, most existing models make rather strong assumptions about how individuals decide to keep or change their strategies. They assume individuals make these decisions based on a strategy’s average performance. This in turn suggests that individuals would remember their exact payoffs against everyone else.

In a recent study, researchers from the Max Planck Institute for Evolutionary Biology, the School of Data Science and Society, and the Department of Mathematics at the University of North Carolina at Chapel Hill examine the effects of realistic memory constraints. They find that cooperation can evolve even with minimal memory capacities. The research is published in the journal Proceedings of the Royal Society B: Biological Sciences.

Direct reciprocity is based on repeated interactions between two individuals. This concept, often described as “you scratch my back, I’ll scratch yours,” has proven to be a pivotal mechanism in maintaining cooperation within groups or societies.

While models of direct reciprocity have deepened our understanding of cooperation, they frequently make strong assumptions about individuals’ memory and decision-making processes. For example, when strategies are updated through social learning, it is commonly assumed that individuals compare their average payoffs.

This would require them to compute (or remember) their payoffs against everyone else in the population. To understand how more realistic constraints influence direct reciprocity, the current study considers the evolution of conditional behaviours when individuals learn based on more recent experiences.

Two extreme scenarios

This study first compares the classical modeling approach with another extreme approach. In the classical approach, individuals update their strategies based on their expected payoffs, considering every single interaction with each member of the population (perfect memory). Conversely, the opposite extreme is considering only the very last interaction (limited memory).

Comparing these two scenarios shows that individuals with limited payoff memory tend to adopt less generous strategies. They are less forgiving when someone defects against them. Yet, moderate levels of cooperation can still evolve.

Intermediate cases

The study also considers intermediate cases, where individuals consider their last two or three or four recent experiences. The results show that cooperation rates quickly approach the levels observed under perfect payoff memory.

Overall, this study contributes to a wider literature that explores which kinds of cognitive capacities are required for reciprocal altruism to be feasible. While more memory is always favourable, reciprocal cooperation can already be sustained if individuals have a record of two or three past outcomes.

This work’s results have been derived entirely within a theoretical model. The authors feel that such studies are crucial for making model-informed deductions about reciprocity in natural systems.

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Credit of the article given to Michael Hesse, Max Planck Society

The purpose of education is to ensure that students acquire the skills necessary for succeeding in a world that is constantly changing. Self-assessment, or teaching students how to examine and evaluate their own learning and cognitive processes, has proven to be an effective method, and this competence is partly based on metacognitive knowledge.

A new study conducted at the University of Eastern Finland shows that metacognitive knowledge, i.e., awareness of one’s cognitive processes, is also a key factor in the learning of mathematics. The work is published in the journal Cogent Education.

The study explored thinking skills and possible grade-level differences in children attending comprehensive school in Finland. The researchers investigated 6th, 7th and 9th graders’ metacognitive knowledge in the context of mathematics.

“The study showed that ninth graders excelled at explaining their use of learning strategies, while 7th graders demonstrated proficiency in understanding when and why certain strategies should be used. No other differences between grade levels were observed, which highlights the need for continuous support throughout the learning path,” says Susanna Toikka of the University of Eastern Finland, the first author of the article.

The findings emphasize the need to incorporate elements that support metacognitive knowledge into mathematics learning materials, as well as into teachers’ pedagogical practices.

Self-assessment and understanding of one’s own learning help to face new challenges

Metacognitive knowledge helps students not only to learn mathematics, but also more broadly in self-assessment and lifelong learning. Students who can assess their own learning and understanding are better equipped to face new challenges and adapt to changing environments. Such skills are crucial for lifelong learning, as they enable continuous development and learning throughout life.

“Metacognitive knowledge is a key factor in learning mathematics and problem-solving, but its significance also extends to self-assessment and lifelong learning,” says Toikka.

In schools, metacognitive knowledge can be effectively developed as part of education. Based on earlier studies, Toikka and colleagues have developed a combination of frameworks for metacognitive knowledge, which helps to identify students’ needs for development regarding metacognitive knowledge by offering an alternative perspective to that of traditional developmental psychology.

“This also supports teachers in promoting students’ metacognitive knowledge. Teachers can use the combination of frameworks to design and implement targeted interventions that support students’ skills in lifelong learning.”

According to Toikka, the combination of frameworks enhances understanding of metacognitive knowledge and helps to identify areas where individual support is needed: “This type of understanding is crucial for the development of metacognitive knowledge among diverse learners.”

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Credit of the article given to University of Eastern Finland