Abstract
I analyze the new emerging role of mathematics for extracting structure from data. This role is different from the traditional role of mathematics as a tool for other sciences. Each such science had provided a theoretical framework in which experiments acquired meaning, with which the quality of data could be assessed and which could be explored with the formal methods of mathematics. The challenge for mathematics now is to handle data without such theoretical guidance from other disciplines. This calls for new principles that are structural, abstract and independent of specific content. Mathematics then is no longer the tool of science, but becomes the science of tools.
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Notes
- 1.
I learned this quote from Peter Schuster.
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The one-body problem, however, was eventually understood as being ill posed. As Leibniz first saw, any physical theory has to be concerned with relations between objects, and an irreducible point mass can only entertain relations with other such objects, but not with itself, because there is no fixed external frame of reference independent of objects. The latter aspect is fundamental in Einstein’s theory of general relativity. (See for instance the exposition by the author in Riemann 2016.)
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The original model was developed only for a particular type of neuron, the giant squid axon, but similarly models have subsequently been developed for other classes of neurons as well.
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In different fields, it may be different what is considered as a larger or a smaller scale. In geography, for instance, larger scale means higher resolution, that is, a smaller reduction factor. In other areas, like physics, a larger scale means the opposite. We shall follow the latter terminology. Thus, at a larger scale, many details from a smaller scale may disappear whereas larger structures might become visible.
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A perhaps somewhat technical point concerning the representation by this diagram: Often, one thinks of a projection as going down, instead of up, and one would then represent X in the top and \(\hat{X}\) in the bottom row. Since, however, we think of \(\hat{X}\) as a higher, more abstract, level, we rather represent that higher level in the top row of our diagram.
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In the terminology of statistical physics, the correlation length of word sequences in texts becomes relatively small after five words. In fact, one would expect that it never becomes exactly zero, that is, there do exist correlations of arbitrary length, whatever small. Thus, in technical terms, when moving from a word to subsequent ones in a text, we have a stochastic process that does not satisfy a Markov property for strings of words of any finite length.
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Acknowledgements
I am very grateful to my partners in the ZIF project, Philippe Blanchard, Martin Carrier, Andreas Dress, Johannes Lenhard and Michael Röckner, for their insightful comments and helpful suggestions which have contributed to shaping and sharpening the ideas presented here. Martin Carrier and Johannes Lenhard also provided very useful comments on an earlier version of my text. Some of my own work discussed here was supported by the ERC Advanced Grant FP7-267087 and the EU Strep “MatheMACS”.
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Jost, J. (2017). Object Oriented Models vs. Data Analysis – Is This the Right Alternative?. In: Lenhard, J., Carrier, M. (eds) Mathematics as a Tool. Boston Studies in the Philosophy and History of Science, vol 327. Springer, Cham. https://doi.org/10.1007/978-3-319-54469-4_14
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