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In this short contribution:

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  • On mathematical foundations and mathematical practice

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  • Thinking along with CPM

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  • An additional concept and an additional argument in support of CPM

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  • Three step argumentation

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1. What philosophy of mathematical practice(s) has already shown:

 

  • Heterogeneity

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  • No single notion of proof

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  • No single notion of explanation

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  • No single notion of thought experiment(s)

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  • No single notion of beauty

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2. The primary purpose of foundations

 

  • Making mathematics homogeneous

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  • Preferably one single mathematical theory

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  • Hence, striving for the absolute

 

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Includes a promise:

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All could be rewritten if asked (but nobody seems to ask)

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<a, b> = {{a}, {a, b}}

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3. However, heterogeneity is not eliminated:

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  • The ‘Three Magi’ have failed
  • Too many alternatives within a single foundational theory
  • Too many competitors

 

Hence:

  • Foundations are just as heterogeneous as mathematics itself
  • Important because Heterogeneity is a strong indicator
  • Of the Indefinite and
  • Of the social and the ethical