What do transitive inference and class inclusion have in common? Categorical (co)products and cognitive development.
What do transitive inference and class inclusion have in common? Categorical (co)products and cognitive development.
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DOI：
10.1371/journal.pcbi.1000599
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发表时间：
2009-12
影响因子：
4.3
通讯作者：
Halford GS
中科院分区：
文献类型：
--
作者：
Phillips S;Wilson WH;Halford GS
传递性推理、类包含和其他各种推理能力有着惊人的相似的发展轮廓，都是在五岁左右获得的。然而，很少有人知道这种对应的原因。范畴论是作为一种在各种数学结构之间建立共性的正式手段而发明的。我们使用范畴理论来证明传递性推理和类包含涉及到双重数学结构，称为产品和余产品。其他具有类似发展特征的推理任务，包括矩阵完成、基数、维度改变的卡片排序、平衡秤（重量-距离整合）和心理理论也涉及这些结构。相比之下，（副）产品不涉及年幼儿童在这些任务中表现出的行为，或在他们能力范围内的简化版本。这些结果指出了儿童时期发展中的一个基本认知原则，即在分类意义上计算（共）产品的能力。儿童在一个非常相似的发展时期获得各种推理技能。然而，这些相似性的原因是一个谜。两个例子是传递性推理和类包含，它们在五岁左右发展。大一点的孩子明白，如果约翰比玛丽高，而玛丽比苏高，那么约翰也比苏高。这种形式的推理被称为传递性推理。大一点的孩子也明白水果比苹果多。这种推理称为类包含。我们用数学的一个分支范畴论来解释为什么这些能力和其他各种能力表现出同样的发展。范畴理论揭示了它们具有相关的潜在结构。因此，尽管这些推理能力表面上有明显的差异，但它们的发展过程是相似的，因为它们涉及相关的过程。
Transitive inference, class inclusion and a variety of other inferential abilities have strikingly similar developmental profiles—all are acquired around the age of five. Yet, little is known about the reasons for this correspondence. Category theory was invented as a formal means of establishing commonalities between various mathematical structures. We use category theory to show that transitive inference and class inclusion involve dual mathematical structures, called product and coproduct. Other inferential tasks with similar developmental profiles, including matrix completion, cardinality, dimensional changed card sorting, balance-scale (weight-distance integration), and Theory of Mind also involve these structures. By contrast, (co)products are not involved in the behaviours exhibited by younger children on these tasks, or simplified versions that are within their ability. These results point to a fundamental cognitive principle under development during childhood that is the capacity to compute (co)products in the categorical sense. Children acquire various reasoning skills during a remarkably similar period of development. Yet, the reasons for these similarities are a mystery. Two examples are Transitive Inference and Class Inclusion, which develop around five years of age. Older children understand that if John is taller than Mary, and Mary is taller than Sue, then John is also taller than Sue. This form of reasoning is called transitive inference. Older children also understand that there are more fruits than apples. This inference is called class inclusion. We explain why these and a variety of other abilities show the same development using a branch of mathematics called category theory. Category theory reveals that they have related underlying structure. So, despite their apparent superficial differences these reasoning abilities have similar profiles of development because they involve related sorts of processes.
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影响因子：
2.6
作者：
HALFORD, GS;WILSON, WH
通讯作者：
WILSON, WH
影响因子：
56.9
作者：
Buschman, Timothy J.;Miller, Earl K.
通讯作者：
Miller, Earl K.
影响因子：
5.7
作者：
Christoff, K;Prabhakaran, V;Gabrieli, JDE
通讯作者：
Gabrieli, JDE
影响因子：
3.7
作者：
Crone, Eveline A.;Wendelken, Carter;van Leijenhorst, Linda;Honomichl, Ryan D.;Christoff, Kalina;Bunge, Silvia A.
通讯作者：
Bunge, Silvia A.
影响因子：
1.3
作者：
EILENBERG, S;MACLANE, S
通讯作者：
MACLANE, S