## Nelson’s Work on Logic and Foundations and Other Reflections on Foundations of Mathematics (2006)

Citations: | 2 - 0 self |

### BibTeX

@MISC{Buss06nelson’swork,

author = {Samuel R. Buss},

title = {Nelson’s Work on Logic and Foundations and Other Reflections on Foundations of Mathematics},

year = {2006}

}

### OpenURL

### Abstract

This paper starts by discussing Nelson’s philosophy of mathematics, which is a blend of mathematical formalism and a radical constructivism. As such, it makes strong assertions about the foundations of mathematic and the reality of mathematical objects. We then offer our own suggestions for the definition of mathematics and the nature of mathematical reality. We suggest a second characterization of mathematical reasoning in terms of common sense reasoning and argue its relevance for mathematics education. Nelson’s philosophy is the foundation of his definition of predicative arithmetic. There are close connections between predicative arithmetic and the common theories of bounded arithmetic. We prove that polynomial space (PSPACE) predicates and exponential time (EXPTIME) predicates are predicative. We discuss Nelson’s formalist philosophies and his unpublished work in automatic theorem checking. This paper was begun with the plan of discussing Nelson’s work in logic and foundations and his philosophy on mathematics. In particular, it is based on our talk at the Nelson meeting in Vancouver in June 2004. The main topics of this talk were Nelson’s predicative arithmetic and his unpublished work on automatic theorem proving. However, it proved impossible to stay within this plan. In writing the paper, we were prompted to think carefully about the nature of mathematics and more fully formulate our own philosophy of mathematics. We present this below, along with some discussion about mathematics education. Much of the paper focuses on Nelson’s philosophy of mathematics, on how his philosophy motivates his development of predicative arithmetic, and on his unpublished work on computer assisted theorem proving. We also discuss the

### Citations

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Citation Context ...hematics education, saying that the goal of mathematics education should be to inculcate “an ability to work with words to which precise meanings are assigned.” He draws on research of Stanovich-West =-=[30]-=- and others that concludes that individuals have two different styles of thinking. The first style, called “System I” thinking, is intuitive, associative, heuristic, automatic, fast, and compatible wi... |

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Citation Context ...the superexponential function is not predicative. It is also possible to give a predicative development of parts of real analysis, at least up through standard results on integration. Ko and Friedman =-=[16, 15]-=- showed that polynomial space computability is sufficient for the definition of integration (more precisely, they showed that the counting class #P is sufficient). However, they only considered comput... |

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Citation Context ...ivist philosophy underlies Nelson’s predicative arithmetic; predicative arithmetic is a weak formal theory of the integers that is mathematically very similar to the theories of bounded arithmetic of =-=[24, 31, 2]-=-. The original definition of bounded arithmetic, I∆0, by Parikh [24] was motivated in part by constructivism and in part by feasible computability (see the survey [3]); however, most researchers in bo... |

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Citation Context ...the superexponential function is not predicative. It is also possible to give a predicative development of parts of real analysis, at least up through standard results on integration. Ko and Friedman =-=[16, 15]-=- showed that polynomial space computability is sufficient for the definition of integration (more precisely, they showed that the counting class #P is sufficient). However, they only considered comput... |

45 |
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Citation Context ...ivist philosophy underlies Nelson’s predicative arithmetic; predicative arithmetic is a weak formal theory of the integers that is mathematically very similar to the theories of bounded arithmetic of =-=[24, 31, 2]-=-. The original definition of bounded arithmetic, I∆0, by Parikh [24] was motivated in part by constructivism and in part by feasible computability (see the survey [3]); however, most researchers in bo... |

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Citation Context ...imilar to [22], but now theorems are stated and proved with sufficient formality to be computer-checked. The proof system qed is a deduction proof system (similar to a deduction proof system of Fitch =-=[7]-=-, but using very different notations). To illustrate the system qed, consider using the axiom to prove the equality ∀x∀y(x + Sy = S(x + y)) (3) x = 0 + x → Sx = 0 + Sx. (4) As a precursor to the actua... |

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Citation Context ...e computability (see the survey [3]); however, most researchers in bounded arithmetic adopt only the mathematical trappings of constructivity and very few subscribe to radical constructivism. Sazonov =-=[26, 27, 28]-=-, however, advocates a form of radical 3sconstructivism. Along with doubting the existence of the integers, and thereby doubting the existence of a fixed semantics for reasoning about infinite objects... |

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Citation Context ...Hook in his 1983 Ph.D. thesis [11] under Nelson developed a predicative version of real analysis with the additional assumption that exponentiation is not total. More recently, Fernandes and Ferreira =-=[6]-=- have given a predicative treatment of parts of real analysis within the framework of reverse mathematics for bounded arithmetic. They show explicitly that their theory of real analysis is interpretab... |

5 | Mathematics: A Very Short Introduction - Gowers - 2002 |

2 |
Nelson’s work on logic and foundations: Formalism and radical constructivism. Talk at a Workshop on Analysis
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(Show Context)
Citation Context ...ix gives a new, weaker base theory Q − for predicative arithmetic. The second appendix proves that exponential time computability is predicative. 1 Platonism, constructivism and formalism In our talk =-=[1]-=-, we described Nelson’s philosophy of mathematics as being “radical constructivism.” Afterward, Nelson suggested that he thinks of himself as a “formalist” rather than a “radical constructivist.” In f... |

2 |
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Citation Context ... of bounded arithmetic of [24, 31, 2]. The original definition of bounded arithmetic, I∆0, by Parikh [24] was motivated in part by constructivism and in part by feasible computability (see the survey =-=[3]-=-); however, most researchers in bounded arithmetic adopt only the mathematical trappings of constructivity and very few subscribe to radical constructivism. Sazonov [26, 27, 28], however, advocates a ... |

2 |
A Many-Sorted Approach to Predicative Mathematics
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(Show Context)
Citation Context ...owed that the counting class #P is sufficient). However, they only considered computability, not provability, so this did not say anything per se about predicatively. J. Hook in his 1983 Ph.D. thesis =-=[11]-=- under Nelson developed a predicative version of real analysis with the additional assumption that exponentiation is not total. More recently, Fernandes and Ferreira [6] have given a predicative treat... |

2 |
Advanced mathematical thinking in computerized environment. Topic Study Group 13
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Citation Context ...mastered, the same problems become very simple and intuitively obvious. Once a mathematical concept is fully mastered, it may even be difficult to apply System II reasoning to the concept. (See Khait =-=[12, 13]-=- for similar discussions on how mathematical thinking integrates both systems of thinking.) In addition, the emphasis on a dichotomy between System I and System II is potentially very harmful to the p... |

2 |
as a tool to develop intuition. Topic Study Group 19
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(Show Context)
Citation Context ...mastered, the same problems become very simple and intuitively obvious. Once a mathematical concept is fully mastered, it may even be difficult to apply System II reasoning to the concept. (See Khait =-=[12, 13]-=- for similar discussions on how mathematical thinking integrates both systems of thinking.) In addition, the emphasis on a dichotomy between System I and System II is potentially very harmful to the p... |

1 |
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Citation Context ...r proposal for a definition of mathematics: Mathematics is the study of objects and constructions, or of aspects of objects and constructions, which are capable of being fully and completely defined. =-=[4]-=- Our original statement of this definition was on the foundations of mathematics (FOM) mailing list, and the reader might refer to that for some related discussion. At the time it was met by a modest ... |

1 |
Re: Interpretability in Q. FOM mailing list posting
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Citation Context ...tive cut. However, their conjunction is equivalent to the totality of exponentiation and is not interpretable in Q. (Another disproof of the compatibility problem was recently outlined by H. Friedman =-=[8]-=-.) 4 What is mathematics? — Two definitions In this section, we will set aside Nelson’s philosophy and present some of our own ideas on the nature of mathematical reality. This will turn out to be a s... |

1 |
What is Mathematics
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(Show Context)
Citation Context ...rs there has been discussion about ‘post-modern’ ideas about the nature of mathematical reality, e.g., that it is a social activity and mathematics does not have any independent existence (cf. Hersch =-=[10]-=-). An extreme form of postmodernism might assert that mathematical truth depends on the culture or bias of the mathematician. This post-modern idea is completely silly if it is taken as saying that pa... |

1 |
Confessions of an apostate mathematician. Presented at the
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- 1995
(Show Context)
Citation Context ..., this means he maintains that there are no platonic mathematical objects which mathematics is about, rather that mathematics consists purely of formal manipulation of first-order formulas. The paper =-=[17]-=- contains Nelson’s most emphatic declaration of the formalist philosophy. A more recent discussion is in [20]. The pure formalist philosophy is usually coupled with a rejection of any non-formal intui... |

1 | Presented at Jubilee for Men and Women from the World of Learning, The Vatican, May 2000. Available at http://www.math.princeton.edu/∼nelson/papers.html - Mathematics, faith |

1 |
and the mind. Presented at Toward a Science of Consciousness — Fundamental Approaches
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(Show Context)
Citation Context ...Nelson’s expects for computer-based mathematics research, namely that for the foreseeable future (centuries) computers will not attain all the capabilities of humans. Nelson discusses this further in =-=[19]-=-. Mathematicians no more discover truths than the sculptor discovers the sculpture inside the stone. (Surely you are joking, Mr. Buonarroti!) But unlike sculpting, our work is tightly constrained, bot... |

1 |
and semantics. Presented at Foundations and the Ontological Quest
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(Show Context)
Citation Context ...that mathematics consists purely of formal manipulation of first-order formulas. The paper [17] contains Nelson’s most emphatic declaration of the formalist philosophy. A more recent discussion is in =-=[20]-=-. The pure formalist philosophy is usually coupled with a rejection of any non-formal intuition or reasoning. When I was a graduate student, Nelson told me on more than one occasion that his approach ... |

1 |
induction. Typeset manuscript, fragmentary chapter 1 of uncompleted book
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- 1979
(Show Context)
Citation Context ...is in the process of being defined. Section 3 will discuss how Nelson formulates a definition of the integers in a way that avoids this second objection. First, however, we shall try to further 2 See =-=[21]-=- for another account of Nelson’s doubt about the integers. 3 The set theoretic method of defining finite in terms of not being equinumerous with a proper subset is too technical and is not convincing ... |

1 |
manuscript. Contains four chapters
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(Show Context)
Citation Context ... understanding and computer verification. In an untitled, unpublished, and unfinished manuscript dated 1993, Nelson revisited the development of predicative arithmetic with an automated proof checker =-=[23]-=-. For this, he wrote an automated proof checker, qed, which 20sworks directly from text in his TeX files. 13 The system allows theorems to be stated and proved in a formal system with all details auto... |