Saved from the cellar. Gerhard Gentzen’s shorthand notes on logic and foundations of mathematics.

*(English)*Zbl 1414.03002
Sources and Studies in the History of Mathematics and Physical Sciences. Cham: Springer (ISBN 978-3-319-42119-3/hbk; 978-3-319-42120-9/ebook). x, 315 p. (2017).

The book contains translations of shorthand notes which survived in the Nachlass of the mathematical logician Gerhard Gentzen.

Gentzen invented the formal calculi of natural deduction and sequent calculus, proved the consistency of (formalized) arithmetic in terms of transfinite induction up to \(\varepsilon_0\). The book shows carefully taken notes of Gentzen which shed light on the history of his accomplishments. As the editor writes (p. 16): “Gentzen’s scientific results begin with his paper on what are known as Hertz systems …. The last work published during his life was the 1943 work on transfinite induction. These are the two fixed points of his achievement, and there do not seem to be any great results hidden in the stenographic manuscripts. There is, instead, one result Gentzen left unpublished … namely the proof of normalization for intuitionistic natural deduction …. For the rest, Gentzen’s central achievements can be read from this publications, and the manuscript sources serve mainly to enrich the understanding of the published papers.” In addition, there are some loose ends (p. 50): “Gentzen’s unfinished work, as it appears from the stenographic notes was on four topics. … first, his attempt at a semantical decision method for intuitionistic propositional logic. … The second unfinished piece of work is the proof theory of intuitionistic arithmetic, and the third the great problem of the consistency of analysis. Finally, he planned to write a popular book on the foundational research in mathematics.”

Part I of the book contains an extensive discussion of Gentzen’s life and work by the editor. Part II gives background information on the shorthand notes (including copies of some sample pages) and on the edition principles. Part III contains the English translations of the shorthand notes; which are complemented by the (survived) correspondence of Gentzen with Arend Heyting and Paul Bernays.

The book is valuable source for the history of modern logic; the editor did an excellent work in getting the shorthand notes, first transcribed in normal German text, and then translating it to English. For the translation (p. 8): “the guiding principle has been to always give precedence to thought over word.” What the editor accomplished we will recognize in full when also the German and stenographic originals are published.

Gentzen invented the formal calculi of natural deduction and sequent calculus, proved the consistency of (formalized) arithmetic in terms of transfinite induction up to \(\varepsilon_0\). The book shows carefully taken notes of Gentzen which shed light on the history of his accomplishments. As the editor writes (p. 16): “Gentzen’s scientific results begin with his paper on what are known as Hertz systems …. The last work published during his life was the 1943 work on transfinite induction. These are the two fixed points of his achievement, and there do not seem to be any great results hidden in the stenographic manuscripts. There is, instead, one result Gentzen left unpublished … namely the proof of normalization for intuitionistic natural deduction …. For the rest, Gentzen’s central achievements can be read from this publications, and the manuscript sources serve mainly to enrich the understanding of the published papers.” In addition, there are some loose ends (p. 50): “Gentzen’s unfinished work, as it appears from the stenographic notes was on four topics. … first, his attempt at a semantical decision method for intuitionistic propositional logic. … The second unfinished piece of work is the proof theory of intuitionistic arithmetic, and the third the great problem of the consistency of analysis. Finally, he planned to write a popular book on the foundational research in mathematics.”

Part I of the book contains an extensive discussion of Gentzen’s life and work by the editor. Part II gives background information on the shorthand notes (including copies of some sample pages) and on the edition principles. Part III contains the English translations of the shorthand notes; which are complemented by the (survived) correspondence of Gentzen with Arend Heyting and Paul Bernays.

The book is valuable source for the history of modern logic; the editor did an excellent work in getting the shorthand notes, first transcribed in normal German text, and then translating it to English. For the translation (p. 8): “the guiding principle has been to always give precedence to thought over word.” What the editor accomplished we will recognize in full when also the German and stenographic originals are published.

Reviewer: Reinhard Kahle (Lisboa)

##### MSC:

03-03 | History of mathematical logic and foundations |

01A60 | History of mathematics in the 20th century |

03F03 | Proof theory, general (including proof-theoretic semantics) |

03F05 | Cut-elimination and normal-form theorems |

03F25 | Relative consistency and interpretations |

03F40 | Gödel numberings and issues of incompleteness |

03F50 | Metamathematics of constructive systems |

03F55 | Intuitionistic mathematics |

03A05 | Philosophical and critical aspects of logic and foundations |

01A70 | Biographies, obituaries, personalia, bibliographies |