Category mistake: Difference between revisions

Latest revision as of 16:25, 2 April 2026

A category mistake can be coarsily described as the "error of assigning to something a quality or action which can only properly be assigned to things of another category".^[1] For instance, a math student saying that "the limit of a function approaches 1" commits a category mistake. The student assigns "approaches 1" which is an attribute of a function to a limit value which is a number.

The purpose of this page is twofold:

Describing the corresponding bottleneck and the related literature.
Collecting category mistakes which hinder student learning in various disciplines.

Description of bottleneck

Students attach properties to the wrong kind of objects.

Collection of category mistakes

Mathematics

Alcock and Simpson^[2] mention four examples for category mistakes in mathematics:
- Saying that every point in a compact set is compact.
- Considering the first two vectors in $\{(1,0),(0,1),(1,1)\}$ as linearly independent and the third as linearly dependent.
- Thinking that a sequence converges in a certain region.
- Viewing $\{4,\{-3,2,-1/7\},\{\{17,5\}\}\}$ as a set of six elements rather than three.
In an Decoding interview on students' difficulties with limits the interviewee mentions the example described at the beginning of this page. See Limits.

References

↑ Magidor, O. (2025). Category Mistakes. In E. N. Zalta & U. Nodelman (Ed.), The Stanford Encyclopedia of Philosophy (Spring 2025). Metaphysics Research Lab, Stanford University.
↑ Alcock, L., & Simpson, A. (2008). Ideas from Mathematics Education—An Introduction for Mathematicians.

[1] Magidor, O. (2025). Category Mistakes. In E. N. Zalta & U. Nodelman (Ed.), The Stanford Encyclopedia of Philosophy (Spring 2025). Metaphysics Research Lab, Stanford University.

[2] Alcock, L., & Simpson, A. (2008). Ideas from Mathematics Education—An Introduction for Mathematicians.

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-{{Hint|text= '''How to use this template'''
+A '''category mistake''' can be coarsily described as the  "error of assigning to something a quality or action which can only properly be assigned to things of another category".<ref>Magidor, O. (2025). [https://plato.stanford.edu/archives/spr2025/entries/category-mistakes/ Category Mistakes]. In E. N. Zalta & U. Nodelman (Ed.), ''The Stanford Encyclopedia of Philosophy'' (Spring 2025). Metaphysics Research Lab, Stanford University. </ref> For instance, a math student saying that "the [[Limits|limit]] of a function approaches 1" commits a category mistake. The student assigns "approaches 1" which is an attribute of a function to a limit value which is a number.
-# Start by describing the bottleneck in the Section '''Description of Bottleneck'''. If you like you can orient yourself on the bottleneck description [[From Derivative to Proportionality]]. If you feel able to turn the bottleneck into a (positive) learning outcome, please do also describe the intended learning outcome.
-# This wiki also serves to connect people interested in certain bottlenecks. If you wish to make yourself known as interested, fill in your name under '''People interested in this Bottleneck'''.
+The purpose of this page is twofold:
-# When done editing, save page.
-# When in reading mode, add suitable tags/categories by pressing the tag symbol on the top of the page. For instance, if the bottleneck is in connection to biology, add "biology" as a tag.
+*Describing the corresponding bottleneck and the related literature.
-# Edit once again and delete the whole Section '''How to use this Template''' and save your changes.
+*Collecting  category mistakes which hinder student learning in various disciplines.
-}}
 ==Description of bottleneck==
-Add description. Please, avoid [[Step_1_-_Identify_a_Bottleneck_to_Learning#Describing bottlenecks|vague descriptions]].
+Students attach properties to the wrong kind of objects.
-==Related scholarly work on this bottleneck==
+==Collection of category mistakes==
-==People interested in this bottleneck==
+===Mathematics===
+#Alcock and Simpson<ref>Alcock, L., & Simpson, A. (2008). ''[https://s3.eu-west-2.amazonaws.com/assets.creode.advancehe-document-manager/documents/hea/private/alcocksimpsonbook_1568036775.pdf Ideas from Mathematics Education—An Introduction for Mathematicians]''. </ref> mention four examples for category mistakes in mathematics:
+#*Saying that every point in a compact set is compact.
+#*Considering the first two vectors in <math>\{(1,0),(0,1),(1,1)\}</math> as linearly independent and the third as linearly dependent.
+#*Thinking that a sequence converges in a certain region.
+#*Viewing <math>\{4, \{-3,2,-1/7\},\{\{17,5\}\}\}</math> as a set of six elements rather than three.
+#In an Decoding interview on students' difficulties with limits the interviewee mentions the example described at the beginning of this page. See [[Limits]].
 ==References==
 [[Category:Bottleneck]]
+[[Category:Mathematics]]
+[[Category:Philosophy]]
+<references />