How can abstract mathematics help describe the world?
Move between structures, spaces and applications without assuming that every useful connection is the same kind of explanation.
Choose an idea to see what it connects to. Each link says how the two are related; the reading takes you further.
How can abstract mathematics help describe the world?
Move between structures, spaces and applications without assuming that every useful connection is the same kind of explanation.
What is preserved when details are removed?
Ask what an abstraction keeps track of. Which relationships remain meaningful when the objects being compared come from different settings?
Connected ideas
← asks about structuresHow can abstract mathematics help describe the world?→offers a way to compareWhich properties survive a change of shape?→← changes the available relationshipsWhat happens when the usual operations change?→can guide a constructionHow does a mathematical idea become a tool?→Read into this question
Which properties survive a change of shape?
Use topology to explore the choice of properties that matter. What changes under a transformation, and what stays the same?
Connected ideas
← asks about spacesHow can abstract mathematics help describe the world?→← offers a way to compareWhat is preserved when details are removed?→can require a different descriptionWhat happens when the usual operations change?→Read into this question
What happens when the usual operations change?
Consider how altering the rules of an algebra changes the structures that can be studied. Follow the definitions before drawing a physical interpretation.
Connected ideas
← asks about mathematical languageHow can abstract mathematics help describe the world?→← can require a different descriptionWhich properties survive a change of shape?→changes the available relationshipsWhat is preserved when details are removed?→Read into this question
How does a mathematical idea become a tool?
Compare an abstract argument with an implemented system. A security application introduces questions about assumptions, implementation and the problem being solved.