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Conjectures and Refutations
Open accessAbstract
To the outside observer, mathematics may appear as the definite system of truth. From this vantage point conjectures may seem a fringe phenomenon to be shunned; the poor man's theorem and an admittance of one's unsuccessful attempts of proving it. I will argue that aforementioned conjecture is false. I will argue for Karl Popper's view that "there is no more rational procedure than the method of trial and error--of conjecture and refutation: of boldly proposing theories; of trying our best to show that these are erroneous; and of accepting them tentatively if our critical efforts are unsuccessful." We will see that conjectures are an integral and essential part of research in the mathematical sciences. This will be illustrated with some surprising examples of conjectures and their refutations concerning the complexity of solving linear systems, the dimensions of algebraic varieties, and the equality of symmetric and nonsymmetric rank of symmetric tensors.
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Cites
ALEXANDER, P. (1963) · Philosophical Books · doi:10.1111/j.1468-0149.1963.tb00798.x · Cited by 2,854
Gaussian elimination is not optimal
Strassen, V. (1969) · Numerische Mathematik · doi:10.1007/bf02165411 · Cited by 2,509
Tensors: Geometry and Applications
Landsberg, J. (2011) · Graduate Studies in Mathematics · doi:10.1090/gsm/128 · Cited by 623
Symmetric Tensors and Symmetric Tensor Rank
Comon, P., Golub, G., Lim, L.-H., & Mourrain, B. (2008) · SIAM Journal on Matrix Analysis and Applications · doi:10.1137/060661569 · Cited by 544
Cited by
(1994) · Cited by 13,126
Falsification and the Methodology of Scientific Research Programmes
Lakatos, I. (1976) · Can Theories be Refuted? · doi:10.1007/978-94-010-1863-0_14 · Cited by 3,915
The Nature of Theory in Information Systems1
Gregor, S. (2006) · MIS Quarterly · doi:10.2307/25148742 · Cited by 3,306
Hofer, B. K., & Pintrich, P. R. (1997) · Review of Educational Research · doi:10.3102/00346543067001088 · Cited by 3,181
Developing a framework for responsible innovation
Stilgoe, J., Owen, R., & Macnaghten, P. (2013) · Research Policy · doi:10.1016/j.respol.2013.05.008 · Cited by 3,184
Metadata
- Venue
- Radical Treatment
- Publisher
- MQUP
- Published
- 2020-02-27
- Pages
- 54-77
- OpenAlex
- W2073984231
- Cited by
- 3,373
- References
- 4
- FWCI
- 275.00
- Type
- book chapter
- Language
- en
- Open access
- green
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