The Irrationality of Mathematics?

When I write, I often choose my words very carefully in order to remove any opportunity the reader might have to make a quick judgment about the content of what I am saying.  I’m hoping they will keep thinking about what I am saying.  The unexpected pairing of words often accomplishes this, and in this spirit, I remember telling my husband (a particle physicist) that I thought that mathematics was probably the most irrational of the sciences.  I don’t remember how he responded, so I may not have accomplished very much.  But I thought about it again today when I looked into the recently published book Thinking, Fast and Slow, by Daniel Kahneman.  The first chapter of the book appeared on this past June.  Kahneman has written what he calls “a psychodrama with two characters.”  The characters are mental processes he identifies as System 1 and System 2 which Kahneman uses to describe the relationship between two ways that the brain works.

• System 1 operates automatically and quickly, with little or no effort and no sense of voluntary control.
• System 2 allocates attention to the effortful mental activities that demand it, including complex computations. The operations of System 2 are often associated with the subjective experience of agency, choice, and concentration.

The narrative begins like this:

When we think of ourselves, we identify with System 2, the conscious, reasoning self that has beliefs, makes choices, and decides what to think about and what to do. Although System 2 believes itself to be where the action is, the automatic System 1 is the hero of the book. I describe System 1 as effortlessly originating impressions and feelings that are the main sources of the explicit beliefs and deliberate choices of System 2. The automatic operations of System 1 generate surprisingly complex patterns of ideas, but only the slower System 2 can construct thoughts in an orderly series of steps. (emphasis my own)

It was this last sentence that got my attention.  The automatic activities that are attributed to System 1 are things as fundamental as detecting that one object is more distant than another, or orienting the source of a sudden sound, as well as the uniquely human actions developed by prolonged practice like, reading words on large billboards, driving a car or even finding a strong move in chess.  But the mental actions of System 1 are involuntary and include innate skills that we share with other animals. It cannot be turned off at will. The involuntary action of System 1 will mobilize the voluntary attention of System 2, described as deliberate, effortful, and orderly. The control of attention is shared by the two systems. System 1 and System 2 are not actually a pair of little agents in our head.  They are what Kahneman calls “useful fictions” that help explain how the mind is working.  I find it interesting that the extent to which an individual’s pupils are dilated indicates the extent to which System 2 is in use.

Jim Holt reviewed the book for the New York Times in November, 2011. He summarized the roles of System 1 and System 2 in this way:

More generally, System 1 uses association and metaphor to produce a quick and dirty draft of reality, which System 2 draws on to arrive at explicit beliefs and reasoned choices. System 1 proposes, System 2 disposes. So System 2 would seem to be the boss, right? In principle, yes. But System 2, in addition to being more deliberate and rational, is also lazy. And it tires easily. (The vogue term for this is “ego depletion.”) Too often, instead of slowing things down and analyzing them, System 2 is content to accept the easy but unreliable story about the world that System 1 feeds to it. “Although System 2 believes itself to be where the action is,” Kahneman writes, “the automatic System 1 is the hero of this book.” System 2 is especially quiescent, it seems, when your mood is a happy one.

The fast reads of System 1 create many of our mistakes of judgment, biases and illusions that are not easily overcome.  And while the more willful deliberations of System 2 can override these judgments, it can’t take over for System 1.  Holt’s review raises an interesting question. He points out that Kahneman “never grapples philosophically with the nature of rationality,” or what rationalilty is there to accomplish.  It can’t undo the biases and illusions created by System 1’s fast action, and it is impractical for us to reflect on every impression System 1 creates.  But System 2 does draw on the action of System 1 to formalize beliefs and make choices.  It should be clear, however, that Systems 1 and 2 are not isolatable systems with interacting aspects or parts.  There is no one part of the brain where they live.

This System 1/System 2 scheme caused me to think about mathematics on multiple levels. I can imagine System 1 and System 2 paralleling the relationship between an intuition and a proof in mathematics.  It does seem that much of the action in mathematics comes from hunches and perceived possibilities, which are then formally explored.  It may be that the quick actions of System 1, the ones that make associations and metaphors actually drive the creation of new ideas or perceived possibilities in mathematics, while the laborious rigor of proof gives us the way to talk about it, or to make our fast read of the situation useful. The interesting thing about this possibility is that it puts the perception of mathematical possibilities outside what we usually think of as the rational side of our nature.   I can imagine that our more immediate experiences of magnitudes related to space, time, and quantity, after prolonged practice, were transformed (by associations and metaphors) into math ideas that got the attention of our ‘lazier’ System 2.  Studies in cognitive science already suggest that mathematics may be built on the brain circuitry that encodes these perceived magnitudes.  And as Kahneman says, “The automatic operations of System 1 generate surprisingly complex patterns of ideas, but only the slower System 2 can construct thoughts in an orderly series of steps.

This ‘orderly series of steps’ may be related to the inductive reasoning of logic and mathematics, what we normally think of as the content of mathematics.  But if the ideas in mathematics are generated by actions more like the actions of System 1, then the source of mathematics (as Poincare pointed out) would remain obscure and, as I have wanted to suggest, not fully rational.





4 comments to The Irrationality of Mathematics?

  • Bill

    What you have describe is the creative process.

    Study a topic
    Incubation – Rest
    Insight acquired – usually incomplete and the order in which it comes out is a little jumbled – Level 1 – Unconscious mind.
    Polishing up – More detail provided. Sequence of thought arrange in a more linear fashion – Level 2 – Conscious mind.

    Anyone who engages in the creative process will attest to this is just the way it is – conversing with The Muses.

    The classic example is Kekule’s description of how the structure of benzene came to him in a dream of a snake biting its tail.

    It is a conundrum to contemplate how the advancement of the rational can come about from “irrational” sources. However, each, is a matter of perspective. To the irrational, rational must seem rather odd.

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  • Joselle

    The chaotic attractor idea is an interesting one. And I think there is an attempt here to reduce the bias toward System 2.

  • happyseaurchin

    nice 🙂

    what came to me was
    the interaction between 1st/2nd as a chaotic attractor between the two

    didn’t particularly like the negative connotation for 1st system
    clearly in a medium that is biased towards the 2nd system…
    does he talk about this contextualisation of his writing and our reading?

    maps very closely to the left-right hemisphere discussion
    instead of grounding it in “hardware” biology
    this is the “software” side of mental activity

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