Foundations as truths which organize mathematics.

*(English)*Zbl 1271.18001This article is a detailed and analytical answer to Feferman’s claims about categorical foundations of mathematics. Summarising, Feferman objects that categorical foundations are not a foundation for mathematics since (1) it is based on earlier set theories, and (2) membership is not really derived or constructed but just reduced by means of a trick to an arrow-oriented representation.

The author argues against these objections: in the first place, the development of category theory, and of its foundations, started from set-theoretic ideas, but it was the categorical view that emerged as the guiding principle to developments; in the second place, categorical set theory was developed not as a foundation to derive all the mathematical truths, even if this is possible in principle, but, on the contrary, as a way to organise the mathematical body of knowledge.

In fact, the author favours a view of foundations which develops as a recognised body of truths adequate to organise definitions and proofs, at least in their presentation. He claims that finding concise principles to organise mathematical knowledge has been a huge achievement by mathematician and logicians.

In the spirit of the journal it is published in, the article is conversational rather than rigidly technical, and quite easy to read even for the non-expert, although previous knowledge of Feferman’s objections is required to have a correct picture of the discussed questions.

The author argues against these objections: in the first place, the development of category theory, and of its foundations, started from set-theoretic ideas, but it was the categorical view that emerged as the guiding principle to developments; in the second place, categorical set theory was developed not as a foundation to derive all the mathematical truths, even if this is possible in principle, but, on the contrary, as a way to organise the mathematical body of knowledge.

In fact, the author favours a view of foundations which develops as a recognised body of truths adequate to organise definitions and proofs, at least in their presentation. He claims that finding concise principles to organise mathematical knowledge has been a huge achievement by mathematician and logicians.

In the spirit of the journal it is published in, the article is conversational rather than rigidly technical, and quite easy to read even for the non-expert, although previous knowledge of Feferman’s objections is required to have a correct picture of the discussed questions.

Reviewer: Marco Benini (Buccinasco)

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##### References:

[1] | DOI: 10.1093/bjps/axl030 · Zbl 1122.01017 |

[2] | DOI: 10.1093/acprof:oso/9780199296453.003.0015 |

[3] | DOI: 10.1093/philmat/2.1.36 · Zbl 0798.18001 |

[4] | Abelian Categories: An Introduction to the Theory of Functors. Harper and Row. Reprinted with author commentary in: Reprints in Theory and Applications of Categories pp 23– (1964) |

[5] | Reports of the Midwest Category Seminar III pp 201– (1969) · Zbl 0184.03504 |

[6] | DOI: 10.2307/2215789 · Zbl 1366.03098 |

[7] | DOI: 10.1007/BF01049179 · Zbl 0780.03001 |

[8] | Logic, Foundations of Mathematics, and Comput- ability Theory pp 149– (1977) |

[9] | DOI: 10.2307/2275291 · Zbl 0764.18002 |

[10] | Commutative Algebra (1995) |

[11] | DOI: 10.1093/bjps/41.3.351 · Zbl 0709.18002 |

[12] | Gesammelte mathematische Werke (1932) |

[13] | Journal of Philosophical Logic 17 pp 75– (1988) |

[14] | Gesammelte Abhandlungen mathematischen und philosophischen Inhalts (1932) · JFM 58.0043.01 |

[15] | From a Geometrical Point of View: The Categorical Perspective on Mathematics and its Foundations (2009) |

[16] | Mathematische Annalen 46 pp 481– (1897) |

[17] | Categories for the Working Mathematician (1998) · Zbl 0906.18001 |

[18] | Mathematics: Form and Function (1986) · Zbl 0675.00001 |

[19] | Categories for the Working Mathematician (1971) · Zbl 0232.18001 |

[20] | Reports of the Midwest Category Seminar. III pp 192– (1969) · Zbl 0184.03504 |

[21] | Category Theory, Homology Theory and their Applications, II (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Two) pp 146– (1969) |

[22] | Axiomatic Set Theory (1964) |

[23] | Theorie der Transformationsgruppen (1893) |

[24] | Manifolds and Differential Geometry (2009) |

[25] | DOI: 10.1111/j.1746-8361.1969.tb01194.x · Zbl 0341.18002 |

[26] | An Elementary Theory of the Category of Sets pp 1– (1965) |

[27] | DOI: 10.1073/pnas.52.6.1506 · Zbl 0141.00603 |

[28] | Algebra (2005) · Zbl 1374.13001 |

[29] | Set Theory: An Introduction to Independence Proofs (1983) · Zbl 0534.03026 |

[30] | Strict Finitism and the Logic of Mathematical Applications (2011) · Zbl 1279.03006 |

[31] | Tool and Object: A History and Philosophy of Category Theory (2007) · Zbl 1114.18001 |

[32] | DOI: 10.1093/philmat/nkp014 · Zbl 1279.00013 |

[33] | Axiomatic Set Theory pp 189– (1971) |

[34] | Principia Mathematica 1 (1910) · JFM 41.0083.02 |

[35] | Indagationes Mathematicae 14 pp 334– (1952) |

[36] | Lehrbuch der Algebra 2 (1896) |

[37] | General Topology (1955) · Zbl 0066.16604 |

[38] | Lehrbuch der Algebra 1 (1895) |

[39] | The Higher Infinite (1994) |

[40] | A Comprehensive Introduction to Differential Geometry (1999) |

[41] | DOI: 10.1090/S0002-9947-1958-0131451-0 |

[42] | Topology (2000) · Zbl 0951.54001 |

[43] | Set Theory (2006) |

[44] | Introduction to Mathematical Logic (1964) |

[45] | Topological Dynamics (1955) · Zbl 0067.15204 |

[46] | DOI: 10.1093/acprof:oso/9780199296453.003.0014 |

[47] | Topologie Algébrique et Théorie des Faisceaux (1958) |

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