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Trick or Truth? pp 5–20Cite as

Children of the Cosmos

Presenting a Toy Model of Science with a Supporting Cast of Infinitesimals

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Part of the book series: The Frontiers Collection ((FRONTCOLL))

Abstract

Mathematics may seem unreasonably effective in the natural sciences, in particular in physics. In this essay, I argue that this judgment can be attributed, at least in part, to selection effects. In support of this central claim, I offer four elements. The first element is that we are creatures that evolved within this Universe, and that our pattern finding abilities are selected by this very environment. The second element is that our mathematics—although not fully constrained by the natural world—is strongly inspired by our perception of it. Related to this, the third element finds fault with the usual assessment of the efficiency of mathematics: our focus on the rare successes leaves us blind to the ubiquitous failures (selection bias). The fourth element is that the act of applying mathematics provides many more degrees of freedom than those internal to mathematics. This final element will be illustrated by the usage of ‘infinitesimals’ in the context of mathematics and that of physics. In 1960, Wigner wrote an article on this topic [4] and many (but not all) later authors have echoed his assessment that the success of mathematics in physics is a mystery. At the end of this essay, I will revisit Wigner and three earlier replies that harmonize with my own view. I will also explore some of Einstein’s ideas that are connected to this. But first, I briefly expose my views of science and mathematics, since these form the canvass of my central claim.

[A]ll our science, measured against reality, is primitive and childlike – and yet it is the most precious thing we have.

Albert Einstein [1, p. 404]

[...] I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.

Isaac Newton [2, p. 54]

The above quote is attributed to Isaac Newton shortly before his death (so in 1727 our shortly before), from an anecdote in turn attributed to [Andrew Michael] Ramsey by J. Spence [2]. See also footnote 31 in [3].

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Notes

  1. 1.

    Making these connections involves developing narratives. Ultimately, science is about storytelling. “The anthropologists got it wrong when they named our species Homo sapiens (‘wise man’). In any case it’s an arrogant and bigheaded thing to say, wisdom being one of our least evident features. In reality, we are Pan narrans, the storytelling chimpanzee.”—Ian Stewart, Jack Cohen, and Terry Pratchett (2002) [6, p. 32].

  2. 2.

    This is my translation of the German quote [7, p. 206]: “Wie Schiffer sind wir, die ihr Schiff auf offener See umbauen müssen, ohne es jemals in einem Dock zerlegen und aus besten Bestandteilen neu errichten zu können.”

  3. 3.

    In the current context, I differentiate little between ‘models’ and ‘theories’. For a more detailed account of scientific models, see [5].

  4. 4.

    I will have more to say on the ancient Greek view on mathematics and science in section “A Speculative Question Concerning the Unthinkable”.

  5. 5.

    Or ‘spatiotemporal’, if you like to talk like a physicist.

  6. 6.

    The same trade-off occurs, for instance, in medical testing and law cases.

  7. 7.

    In this example, considering the negation of the planar assumption—rather than any of the other background assumptions—is prompted by troubles in physics.

  8. 8.

    This is fine, of course, since this is not the goal of mathematics.

  9. 9.

    Here, I recommend humming a Shania Twain song: “So, you’re a rocket scientist. That don’t impress me much.” If you are too young to know this song, consult your inner teenager for the appropriate dose of underwhelmedness.

  10. 10.

    Here, I mean by infinitesimals numbers larger than zero, yet smaller than 1/n for any natural number n.

  11. 11.

    It has been shown that physical problems can be rephrased in terms of NSA [17], both in the context of classical physics (Lagrangian mechanics [18]) and of quantum mechanics (quantum field theory [19], spin models [20], relativistic quantum mechanics [21], and scattering [18]). Apart from formal aspects (mathematical rigour), such a translation also offers more substantial advantages, such as easier (shorter) proofs.

  12. 12.

    In case this remark made you wonder: the T-shirt was invented about a century ago.

  13. 13.

    My view of mathematics might raise the question: “Why, then, should we expect that anything as human and abstract as mathematics applies to concrete reality?” I think this question is based on a false assumption, due to prolonged exposure to Platonism—remnants of which are abundant in our culture.

  14. 14.

    For the influence of Moritz Schlick on Einstein’s ideas, see [39].

  15. 15.

    Original quote by J.L. Austin, 1979 [41]: “It’s not things, it’s philosophers that are simple.”

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Acknowledgments

Science is a multiplayer game. Therefore, I am grateful to Danny E.P. Vanpoucke for his feedback on an earlier version of this essay and to all participants in the discussion on the FQXi forum [42]. However, we do play with real money. This work was financially supported by a Veni-grant from the Dutch Research Organization (NWO project “Inexactness in the exact sciences” 639.031.244). I am grateful to FQXi for organizing the 2015 essay contest “Trick or Truth: the Mysterious Connection Between Physics and Mathematics”, thereby giving me an incentive to write this piece. And, of course, I am very thankful that they awarded me the first prize for it.

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Authors and Affiliations

  1. KU Leuven, Centre for Logic and Analytic Philosophy, Institute of Philosophy, Kardinaal Mercierplein 2—Bus 3200, 3000, Leuven, Belgium

    Sylvia Wenmackers

  2. University of Groningen, Faculty of Philosophy, Oude Boteringestraat 52, 9712 GL, Groningen, The Netherlands

    Sylvia Wenmackers

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Correspondence to Sylvia Wenmackers .

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Editors and Affiliations

  1. Dept of Physics, UC Santa Cruz, Santa Cruz, California, USA

    Anthony Aguirre

  2. Foundational Questions Institute, Decatur, Georgia, USA

    Brendan Foster

  3. Foundational Questions Institute, Decatur, Georgia, USA

    Zeeya Merali

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Wenmackers, S. (2016). Children of the Cosmos. In: Aguirre, A., Foster, B., Merali, Z. (eds) Trick or Truth?. The Frontiers Collection. Springer, Cham. https://doi.org/10.1007/978-3-319-27495-9_2

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