## A DEFENCE OF MATHEMATICAL PLURALISM (2004)

### BibTeX

@MISC{Davies04adefence,

author = {E. B. Davies},

title = { A DEFENCE OF MATHEMATICAL PLURALISM},

year = {2004}

}

### OpenURL

### Abstract

We approach the philosophy of mathematics via a discussion of the differences between classical mathematics and constructive mathematics, arguing that each is a valid activity within its own context.

### Citations

529 |
Methods and Applications of Interval Analysis
- Moore
- 1979
(Show Context)
Citation Context ...levant. Another variation on classical mathematics is Robinson’s non-standard analysis, which has had stochastic applications via the so-called Loeb measure construction (see Cutland [1988], Robinson =-=[1979]-=-). Although a minority interest, it is genuine mathematics which goes against the grain of the classical tradition by reviving the notion of the infinitesimal, which everyone thought had been killed b... |

404 | Constructive Analysis
- Bishop, Bridges
- 1985
(Show Context)
Citation Context ...to two groups, mathematical assumptions and logical rules, but if the subject itself is formal logic the two cannot be distinguished. 2sLet us consider constructive mathematics as developed in Bishop =-=[1967]-=-. In this field one accepts Peano arithmetic, but not the law of the excluded middle. Bishop allowed iterative constructions, for example of the real number system, but required that limiting procedur... |

304 | Introduction to Mathematical Logic - Mendelson - 1987 |

268 | Kernels and Spectral Theory - Davies - 1989 |

136 |
Proofs and refutations: the logic of mathematical discovery
- Lakatos
- 1976
(Show Context)
Citation Context ...des a richer view of the subject than the usual approaches, and even that it demonstrates the irrelevance of some of the existing controversies about the subject. We reiterate the comments of Lakatos =-=[1976]-=- that philosophers’ concentration on foundations does not provide a true picture of the subject. Most mathematicians may not have much more understanding of the philosophy of their subject than crane ... |

114 | One-Parameter Semigroups - Davies - 1980 |

107 | A New Kind of Science (Wolfram - Wolfram - 2002 |

88 | Spectral theory and differential operators - Davies - 1995 |

81 |
Existence and feasibility in arithmetic
- Parikh
- 1971
(Show Context)
Citation Context ...ve opportunism. Nevertheless, the feeling that mathematics is a worthwhile and relevant activity should not completely erase in our minds an honest appreciation of the problems which beset us. (Cohen =-=[1971]-=-) The paradoxical consequences of accepting the ‘truth’ of classical mathematics have become clearer as time has passed. Very recently Friedman has discovered a series of straightforward finite theore... |

55 | Constructive Analysis; Grundlehren der - Bishop, Bridges |

41 | Science without numbers - Field - 1980 |

34 |
Naturalism in mathematics
- Maddy
- 1997
(Show Context)
Citation Context ...s are involved in the same type of human enterprise as the rest of the human race. The historical importance of fruitfulness in the development of mathematics has been explored in some depth by Maddy =-=[1997]-=-. Unfortunately she makes no reference, positively or negatively, to the work of Bishop, Nelson, Parikh or Robinson, and comes to the conclusion that ‘history shows that such an opponent (of the axiom... |

32 |
Predicative Arithmetic
- Nelson
- 1986
(Show Context)
Citation Context ... of logic or mathematics which are of interest, such as the so-called ‘finite arithmetic’ of Parikh [1971], developed by a substantial number of other mathematicians, including Carbone [2000], Nelson =-=[1986]-=- and Sazanov [1995]. This is a more radical departure from the usual axiom systems, but it is of value in certain contexts, for example in relation to computational complexity. In this system exponent... |

28 | Finite functions and the necessary use of large cardinals
- Friedman
- 1998
(Show Context)
Citation Context ...he author himself wrote that his findings raise the specific issue of what constitutes a valid mathematical proof and the general issue of objectivity in mathematics in a down to earth way. (Friedman =-=[1988]-=-) In our language he was questioning the context, not the content, of his results: there is little doubt that his proofs are valid in the limited sense that the steps follow as he claims under the hyp... |

24 |
The Taming of The True
- Tennant
- 1997
(Show Context)
Citation Context ...s are involved in the same type of human enterprise as the rest of the human race. The historical importance of fruitfulness in the development of mathematics has been explored in some depth by Maddy =-=[1997]-=-. Unfortunately she makes no reference, positively or negatively, to the work of Bishop, Nelson, Parikh or Robinson, and comes to the conclusion that ‘history shows that such an opponent (of the axiom... |

24 | Theoretical Mathematics’: Toward a Cultural synthesis of Mathematics and Theoretical - Jaffe, Quinn - 1993 |

15 | Thinking about mathematics: The philosophy of mathematics - Shapiro - 2000 |

14 |
Nonstandard Analysis and its Applications
- Cutland
- 1988
(Show Context)
Citation Context ...he author himself wrote that his findings raise the specific issue of what constitutes a valid mathematical proof and the general issue of objectivity in mathematics in a down to earth way. (Friedman =-=[1988]-=-) In our language he was questioning the context, not the content, of his results: there is little doubt that his proofs are valid in the limited sense that the steps follow as he claims under the hyp... |

12 | D: A constructive proof of Gleason’s theorem - Richman, Bridges - 1999 |

10 |
On finite simple groups and their classification
- Solomon
- 1995
(Show Context)
Citation Context ...tions. They are frequently highly geometrical, but may be purely syntactical, as for example when seeing a pattern in the coefficients of 1 1 − x + x 2 = 1 + x − x3 − x 4 + x 6 + x 7 − ... See Henley =-=[1995]-=-, Lakatos [1976] and Polya [1954] for discussions of this and other examples. Intuitions may also be based upon seeing the possibility of useful analogies, as between the commutator formula for matric... |

8 |
Platonism and Anti-Platonism
- Balaguer
- 1998
(Show Context)
Citation Context ...ly of the human species. If our knowledge is obtained by applying reason, then the application of that reason would lead us to the same conclusions even if they did not exist. 12sWe refer to Balaguer =-=[1998]-=- for a discussion of these issues and references to the literature. This controversy refuses to die. In a recent article Redhead has revived the claims of Gödel, Lucas and Penrose that human beings ha... |

8 | Cycling in proofs and feasibility
- Carbone
- 2000
(Show Context)
Citation Context ...the foundations of logic or mathematics which are of interest, such as the so-called ‘finite arithmetic’ of Parikh [1971], developed by a substantial number of other mathematicians, including Carbone =-=[2000]-=-, Nelson [1986] and Sazanov [1995]. This is a more radical departure from the usual axiom systems, but it is of value in certain contexts, for example in relation to computational complexity. In this ... |

8 | A constructive look at positive linear functionals on L(H - Bridges - 1981 |

7 | M: Computer-Assisted Mathematics at Work The Hahn-Banach Theorem - Bauer, Wenzel |

7 | Gleason’s theorem is not constructively provable - Hellman - 1993 |

7 | Constructive mathematics and quantum mechanics: unbounded operators and the spectral theorem - Hellman |

7 | Trybulec A: Hahn-Banach theorem - Nowak - 1993 |

7 | Intuitionism as generalization - Richman - 1990 |

6 |
K: The Contribution of T. Sunaga to Interval Analysis and Reliable Computing
- Markov, Okumura
- 1999
(Show Context)
Citation Context ...c. This has an ancient history, but is becoming slowly more important in scientific computing, particularly in connection with computer assisted proofs of theorems in analysis; see Markov and Okumura =-=[1999]-=-, Moore [1979] and the Interval Computations Web-Site [2004] for further details and some of the many applications. Its basic entities are rather like real numbers, and may be written in the form x = ... |

6 |
Selected papers of Abraham
- Robinson
- 1979
(Show Context)
Citation Context ...levant. Another variation on classical mathematics is Robinson’s non-standard analysis, which has had stochastic applications via the so-called Loeb measure construction (see Cutland [1988], Robinson =-=[1979]-=-). Although a minority interest, it is genuine mathematics which goes against the grain of the classical tradition by reviving the notion of the infinitesimal, which everyone thought had been killed b... |

5 |
The Bidual of C(X) I
- Kaplan
- 1985
(Show Context)
Citation Context ...parable Banach spaces. In all these cases the description of the dual space is extremely abstract, and involves discussions in set theory which leave most mathematicians cold (but not all; see Kaplan =-=[1985]-=-). In particular the Banach dual space of l∞ (Z) has a description in terms of the very abstract Stone-Čech compactification of the integers. As our second example we consider the intermediate value t... |

5 | Ontology.” Revue Internationale de Philosophie 4: 20-40. Revised and reprinted as Supplement A in Carnap 1956. All page references are to the latter - “Empiricism |

4 | Mathematical constructivism in space-time - Hellman - 1998 |

4 | Unique existence, approximate solutions, and countable choice - Schuster |

3 |
Quantum mechanics does not require the continuity of space
- Davies
- 2003
(Show Context)
Citation Context ...ry useful, even though we no longer regard space and time as separable. A model constructed by the author shows that the predictions of quantum theory are consistent with space being discrete (Davies =-=[2003b]-=-); in this model the apparent, continuous, Euclidean symmetries of space are emergent properties in the large scale limit. The physicist R Newton put it the following way. Conceivably, the resulting p... |

3 |
The Truth of Science
- Newton
(Show Context)
Citation Context ...s are involved in the same type of human enterprise as the rest of the human race. The historical importance of fruitfulness in the development of mathematics has been explored in some depth by Maddy =-=[1997]-=-. Unfortunately she makes no reference, positively or negatively, to the work of Bishop, Nelson, Parikh or Robinson, and comes to the conclusion that ‘history shows that such an opponent (of the axiom... |

3 | et al.: Responses to ‘Theoretical mathematics: towards a cultural synthesis of mathematics and theoretical - Atiyah - 1994 |

3 | Never say ’Never’! On the communication problem between intuitionism and classicism - Hellman - 1989 |

3 | Gleason’s theorem has a constructive proof - Richman |

3 | M: Countable choice as a questionable uniformity principle - Schuster |

2 |
Did Bishop have a philosophy of mathematics
- Billinge
- 2003
(Show Context)
Citation Context ...luded middle imposed unacceptable penalties. However, Bishop’s philosophy was not well worked out, and one does not need to buy into it totally in order to benefit from what he achieved (see Billinge =-=[2003]-=-). We can think of the difference between the two systems as depending upon different interpretations of the quantifier ∃. Its interpretation is classified under the heading of intuition, and should b... |

2 |
Comments on the foundations of set theory. p 9-15
- Cohen
- 1971
(Show Context)
Citation Context ...ve opportunism. Nevertheless, the feeling that mathematics is a worthwhile and relevant activity should not completely erase in our minds an honest appreciation of the problems which beset us. (Cohen =-=[1971]-=-) The paradoxical consequences of accepting the ‘truth’ of classical mathematics have become clearer as time has passed. Very recently Friedman has discovered a series of straightforward finite theore... |

2 |
B: Empiricism in Arithmetic and Analysis
- Davies
- 2003
(Show Context)
Citation Context ...ld count from 1 up to 10 100 if one were sufficiently determined. It is known that the law of induction may be proved within D100 in a few pages, using only routine logic and the law of substitution (=-=[2003a]-=-). Peano arithmetic is obtained by abstracting, or idealising, the properties of this system, but by itself D100 goes far beyond the needs of scientists, who cannot measure anything with greater than ... |

2 |
Home Page for the Axiom of Choice’. http://www.math.vanderbilt.edu/∼schectex/ccc/choice.html
- Schechter
(Show Context)
Citation Context ...ics. Many different mathematical universes are possible. When we accept or reject the Axiom of Choice, we are specifying which universe we shall work in. Both possibilities are feasible... (Schechter =-=[2004]-=-) The first two groups of mathematicians may be identified as Platonists and constructivists respectively. One still has to ask whether the pluralist view is philosophically coherent. One might regard... |

2 | Constructive mathematics and unbounded operators – a reply to - Bridges - 1995 |

2 | Quantum Mechanical Unbounded Operators and Constructive Mathematics a Rejoinder to Bridges - Hellman - 1997 |

2 | On Errett Bishop’s Constructivism – Some Expositions, Extensions and Critiques - Ye - 1998 |

1 |
Set Theory and the Continuum
- Cohen
- 1966
(Show Context)
Citation Context ...a justification would later appear. Now we know that there can be no proof of the axiom, but it seems that many mathematicians regard it as being so obvious that it needs no serious discussion. Cohen =-=[1966]-=-, the person who finally proved that the continuum hypothesis was independent of ZFC, disagreed. Historically, mathematics does not seem to enjoy tolerating undecidable propositions. It may elevate su... |

1 |
S: Syntax-directed discovery in mathematics. Erkenntnis 43
- Henley
- 1995
(Show Context)
Citation Context ...tions. They are frequently highly geometrical, but may be purely syntactical, as for example when seeing a pattern in the coefficients of 1 1 − x + x 2 = 1 + x − x3 − x 4 + x 6 + x 7 − ... See Henley =-=[1995]-=-, Lakatos [1976] and Polya [1954] for discussions of this and other examples. Intuitions may also be based upon seeing the possibility of useful analogies, as between the commutator formula for matric... |

1 |
I: ‘A Course in Mathematical Logic
- Yu
- 1977
(Show Context)
Citation Context ...use the goal in one field is to prove theorems and in the other to beat one’s opponent. As World 3 objects, the fact that theorems are human constructs does not rob them of objective validity (Popper =-=[1977]-=-). Every chess player agrees about what is a checkmate for one player, just as every mathematician agrees about what is a valid proof. These are objective facts within the appropriate contexts. Mathem... |