Philosophical Perspectives on the Essential Structure of Complex Numbers
By
FillMaths
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Summary
This philosophical mathematics article examines different perspectives on the essential structure of complex numbers, exploring whether they should be understood primarily as an algebraic field, with topological structure, or with distinguished subfields like the real numbers. The author discusses how these different views engage with structuralism in the philosophy of mathematics, examining foundational questions about what constitutes the 'true' nature of mathematical objects.
Key quotes
· 5 pulledHow are we to think of the complex numbers? What I mean is, with what fundamental structure at bottom do we take the complex numbers to be endowed?
They form the complex field, of course, with the corresponding algebraic structure, but do we think of the complex numbers necessarily also with their smooth topological structure?
Is the real field necessarily distinguished as a fixed particular subfield of the complex numbers?
Do we understand the complex numbers necessarily to come with their rigid coordinate structure?
I discuss several commonly held perspectives on the complex numbers and explore how their differences engage with several aspects of structuralism in the philosophy of mathematics.
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