Is mathematics realistic or idealistic?

Stephen Smale

5th Heidelberg Laureate Forum 2017, Heidelberg, Germany, Picture/Credit: Christian Flemming/HLF

The closing talk of the HLF’s main lecture programme (before the young researchers and laureates head off to participate in scientific interaction with SAP representatives to discuss maths and computer science in industry) was given by Fields Medalist Steve Smale.

Speaking without slides, Smale shared with us some of his recent work in the crossover between mathematics and biology, but the central theme of his talk was one which goes to the heart of what mathematics is. Mathematics is embedded in science, and is used to describe and understand many aspects of scientific discovery.

The question was of whether mathematics is realistic or idealistic – do the mathematical models we use to solve problems and understand the universe give a realistic picture of how the world works, or is it all fantasy and we’re ignoring the fine details in order to get a model that works nicely? It’s a constant struggle, and Smale illustrated this with several historical examples.

Heroes of maths and science

Alan Turing’s theory of computation was an inspiring vision of how we can understand and use computers, and has influenced the whole field since. But was it realistic? For some applications, Turing’s approach is the correct model, but for others it fails. Modern study of NP-completeness in computability assumes an infinite amount of input, but obviously this doesn’t model the real world – it’s an idealisation.

Moving on to another giant of maths history, Smale turned to Isaac Newton. Newton’s work on physics, differential equations and mechanics was all an idealisation – to the extent that his calculations didn’t even include friction, which was added to the theory 100 years later.

John Von Neumann, who created the early models of quantum mechanics, introduced the concept of a Hilbert space – again, an idealisation. And even in other fields – Watson and Crick discovered the structure of DNA, but didn’t include the protein core of chromatin, later discovered to be a fundamental part of the more detailed structure.

This idealism is possible because this major work was often done without Newton, Turing or Von Neumann doing lab experiments – they used experimental data from other people’s work, but data which had already been ‘digested’ by the rest of the scientific community. Newton built on the work of Galileo, Copernicus and Kepler before him.

Getting to the heart of the matter

Smale’s current research is on the human body, and in particular the heart. How does it manage to beat in such a coordinated way, with all the myocytes, or muscle cells, acting together to create a regular heartbeat?

The synchronisation was compared to what happens in a crowd applauding for a long time, when the clapping falls into synchronisation almost accidentally. It’s a dynamical system, and can be understood through maths, much like the beating heart.

Smale is working towards a deeper understanding of the heart through mathematics, and in this case standing on the shoulders of Alan Turing. Turing’s final paper on morphogenesis – the biological processes that lead to stripes on a zebra, or the arrangements of seeds in plants – included some differential equations. These were again an idealisation of the reality, as they were based on what happens as the numbers of cells increases to infinity (obviously not the real situation – some biologists disliked Turing’s work for this reason, as they mainly worked with small quantities of cells).

The cells making up the heart are of the same cell type – they all behave in the same way. Smale is looking at how this happens, and outlined how if you consider the set of genes as a graph, with individual genes as the nodes, and directed edges indicating how each transcription factor controls the adjacent gene, elevating protein production, you can build a system of ordinary differential equations to describe how it works.

This system of equations can be seen to reach an equilibrium – a global basin of attraction where the levels of each gene work in exactly the right way to determine how the cell behaves. These basins define the cell types – so a liver cell, or a heart cell, knows exactly what quantities of each protein to produce based on these equations, which are hard-wired into the DNA. You can even consider stable periodic behaviour in the system, to understand the heartbeat’s regular cycle.

Of course, this is again all idealisation – but the workings of the heart are something that have been understood from the biological angle for some time, and now mathematics is providing new ways to model and understand it – which will hopefully lead to powerful medical advances we can implement in reality.

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is a mathematician based in Manchester, who gives talks and workshops on different areas of maths. She finished her PhD in 2011, and since then has talked about maths in schools, at science festivals, on BBC radio, at music festivals, as part of theatre shows and on the internet. Katie writes blog posts and editorials for The Aperiodical, a semi-regular maths news site.

14 comments Write a comment

  1. I think that of course it’s realistic (analytic; notwithstanding its abstractions) – happening in real time. But thinking (consciousness, mentality “et al.”) – even though also happening in real time – is often kind of idealistic (abstract, immateriell – albeit measurable). Then again this applies to philosophical _and_ mathematical thinking (pondering, calculating) – they are interdependent … smiley …

    • Denken ist immer bilderbezogen oder idealistisch, das Erkenntnissubjekt hantiert mit einem Abzug der Welt, dessen was ist oder ihm sein scheinend, dies ist bereits logisch so festzustellen.
      Die Philosophie, auch gerade das Wort meinend, war am Anfang, später sind dann die Einzeldisziplinen sinnhafterweise aus ihr heraus gelöst worden – vgl. mit ‘interdependent’.
      Die Frage, ob die Mathematik realistisch sei, kann insofern nur so erklärt werden, also ihr Zustandekommen, dass vergessen worden ist, woher die Mathematik kommt.

  2. Mathematik ist ja die “Kunst des Lernens” und insofern idealistisch, also an Bilder und Ideen gebunden, nicht weltlich, auch nicht im Sinne der Realitas (“Sachlichkeit”).
    Sondern eine Methode, die den Weltteilnehmern genügt, auch weil sie physikalisch funktionsfähige und Nutzen bringende Anwendungen unterstützt.
    Das Entstehen der Mathematik ist also den erkennenden und zum sprachlichen Austausch fähigen Subjekten, den Erkenntnissubjekten geschuldet, die wiederum, und hier wird es womöglich interessant, weltlich (“realistisch” sozusagen) gebunden sind.
    In anderen Welten würde, erkennende Subjekte vorausgesetzt, andere Mathematik entstehen.

  3. Bonus-Kommentar hierzu :

    The question was of whether mathematics is realistic or idealistic – do the mathematical models we use to solve problems and understand the universe give a realistic picture of how the world works, or is it all fantasy and we’re ignoring the fine details in order to get a model that works nicely?

    Wenn die Phantasie dazu anleitet ein “realistisches” Bild zu schaffen, meint die zitierte Frage keinen Gegensatz.

    Denn die Sachlichkeit (“Realitas”, “Realismus” (hier steckt auch der Ismus drinnen)) meint die (individuelle und in der Folge gesellschaftlich gepflegte) besondere Phantasie.
    Jedenfalls sofern der Nutzen gemeint bleibt, vgl. mit ‘in order to get a model that works nicely’.

    Der Naturwissenschaftler erfasst ausschnittsartig, näherungsweise und an seine (oder allgemeine) Interessen (!) gebunden und theoretisiert in der Folge ausschnittsartig, näherungsweise und an seine (oder allgemeine) Interessen (!) gebunden.

    Naturwissenschaft in Veranstaltung.

    Naturwissenschaftliche Theorien dürfen gerne empirisch adäquat bleiben.
    (Wobei auch empirisch in-adäquate Theorien einen Wert haben, der bspw. für folgend zu entwickelnde naturwissenschaftliche Theorien, die dann idealerweise wieder empirisch adäquat sind, genutzt werden könnten.)


    Ursich-zynisch formuliert sind Realismus und die Realität, die Sachlichkeit, idealistisch.
    So ist zumindest die konstruktivistische Sicht.

  4. Is mathematics realistic or idealistic?

    Mathematics is as realistic as reality adheres to some form of mathematics. The ultimate nature of reality may be described by a mathematical structure according to Max Tegmark. But – until now – you have to believe that. Furthermore: There is much more mathematics which does not have anything to do with reality as there is mathematics which fits into reality.
    What is the alternative to a mathematical description of a piece of reality? To me, the alternative is the beleive that reality is a funny machine, which must be inspected piece by piece, atom by atom. The human brain project, which tries to simulate a whole brain by simulating all the neurons of the brain, is an example for the believe, that reality, that the brain is a funny machine, which must be inspected neuron by neuron. Even some kind of mathematical proves belong into this category, the proves namely, which enumerate through a big list of cases to. The 4 color theorem seems to be proved in this way. I strongly believe that most of reality is guided by quite general mathematical principles, but that there are also funny machines (and funny creatures). The brain may even be a mixture of funny machine and mathematical guided apparatus.

    • Schwierig.

      There is much more mathematics which does not have anything to do with reality as there is mathematics which fits into reality.
      What is the alternative to a mathematical description of a piece of reality? To me, the alternative is the beleive that reality is a funny machine, which must be inspected piece by piece, atom by atom.

      Realität ist Mathematik, die Kunst des Lernens, oder andersherum, aus bestimmter Sicht, die gehoben die Sachbearbeitung im Grundsatz der Mathematik anvertraut ist.
      Nichts gegen schlechtes Englisch btw.
      Die Sprache darf bestmöglich verhunzt werden, sie ist an sich wertlos.

  5. For all the challenging and entertaining value of these explorations into the various niches of thinking, there is something bugging me about the notion of ‘rationality’ as the file drawer into which the discussion has been categorized. From the point of view of design and planning that I have tried to explore issues — such as the problem of prediction of how a proposed plan will perform over time, — there is one aspect that explains why such prediction is so ‘hard’, — especially for social issues — and why the very effort to do it with greater certainty (as an argument for acceptance of the plan) may actually be counterproductive. That aspect may be crudely described as the desire of people to ‘make a difference’ in their lives. At least once they go beyond the basic problem of survival, of being accepted in their social context.
    ‘Making a difference’ can then be tracked through many of the stages described in the article (I haven’t read the book): getting better at recognizing, understanding, then using the existing rules; but then going on to breaking some of them (breaking the rules is meaningless unless one has recognized the current rules) and making one’s own rules. The new rules are meaningless if they are just ‘new’ but can’t be assessed against some principles, criteria, standards (also current or ‘new’) of plausibility, functionality, ethics, logic, aesthetics: are they a p p e a l i n g enough to those others from whom you wish to be recognized, appreciated as ‘different’?

  6. There is a simple reason why life, the brain and most other things must follow simple mathematical rules: Limited information. The human DNA e.g. consists of 3 billion base pairs, an information which fits onto a memory stick but enables a brain which outperforms all supercomputers for certain tasks. Mathematics enables generality and without generality nothing makes sense.

    • Mathematics enables generality and without generality nothing makes sense.

      Mathematics is the gate and key of the sciences. [Roger Bacon]

      The human DNA e.g. consists of 3 billion base pairs, an information which fits onto a memory stick but enables a brain which outperforms all supercomputers for certain tasks.

      Nicht die hier genannte Datenhaltung “DNA” ist ausschlaggebend für ihre spätere Nutzung, auch ihre Größe meinend nicht, sondern die (biologisch feststellbare) Umgebung, die sie eben zu nutzen weiß.
      Diese ist in ihrem Wesen, als Betriebssystem sozusagen, noch weitgehend unerforscht, wird wohl auch so, “wegen Komplexität”, so bleiben.

  7. Dr. Webbaer,
    “Denken ist immer idealistisch”, das gilt nur so lange, so lange man nicht in Zahlen denkt.
    Mathematik ist eine formales System und unserer gesprochenen Sprache ebenbürtig, im Bereich der Physik und Informatik sogar überlegen.
    Was die Mathematik als formales System betrifft, so kann sie ein idealisiertes (vereinfachtes) Bild der Wirklichkeit bieten. Sie kann der Wirklichkeit aber sehr nahe kommen (also realistisch sein), wie es die Atombombe ja schrecklicherweise beweist.
    Dass man Gefühle nicht mathematisch darstellen kann, zeigt die Grenzen der Mathematik auf. Dafür sind dann die Musik und die Künste zuständig.

  8. “Denken ist immer idealistisch”

    -> Der war gut.

    Aber vor allem ist die Mathematik idealistisch, weil sie formale Systeme erstellt und eine solche Erstellung ist ja nun immer irgendwie “am Ideal” (oder auch Original) orientiert. Also idealistisch.

    Das macht sie nur eben nicht irgendwie schlechter.

  9. Oder eigendlich ist sie dann aber auch realistisch, wenn sie sich am Gegenstand, am Ideal orientiert.

    Wieso muß man sich da so eine Frage stellen?

    • Oder eigendlich ist sie dann aber auch realistisch, wenn sie sich am Gegenstand, am Ideal orientiert.

      Wieso muß man sich da so eine Frage stellen?

      Weil der Gegenstand unzureichend idealisiert sein könnte.

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