chriswarbo-net: 342e26426c5efe4f04ca51ab16af058285218183

     1: ---
     2: title: Intuitionism in physics
     3: ---
     4: 
     5: Sabine Hossenfelder
     6: [published a video recently](https://www.youtube.com/watch?v=oEWm3yPUosg)
     7: discussing whether questions in Physics could benefit from an intuitionistic
     8: approach to mathematics. It's an interesting idea, but it's rather subtle; and
     9: tripping over those nuanced details can lead us astray (or give us the wrong
    10: intuition, so to speak).
    11: 
    12: In particular, Sabine focused on the idea of
    13: [real numbers](https://en.wikipedia.org/wiki/Real_number) and their "decimal
    14: places", which is a useful model for introducing these ideas; but taking it too
    15: far can result in some confusion. For example, something as simple as the number
    16: ⅓ may appear "infinitely complicated" when viewed as a never-ending decimal
    17: 0.333… On the other hand, using a different
    18: [numerical base](https://en.wikipedia.org/wiki/Radix) can make it simple again
    19: (e.g. in [dozenal](https://en.wikipedia.org/wiki/Duodecimal) it's just 0.4).
    20: 
    21: There *is* an important question lurking in the decimal places of real numbers,
    22: but to understand it we can't "just" calculate or measure more digits.
    23: 
    24: ### Intuitionism and constructivism ###
    25: 
    26: Full disclaimer, I'm not a trained mathematician, but I have done work in formal
    27: systems, proof assistants and type theory; so I want to get across a high-level
    28: picture, and try to clarify some of the points Sabine made.
    29: 
    30: I'll also note that the "intuitionist" maths that Sabine describes (where
    31: mathematical objects 'exist in the mind', and are only 'real' once thought of)
    32: is closely related to the idea of "constructive" maths (where mathematical
    33: proofs must be direct, not double-negatives). I'll actually focus on the latter
    34: (since I'm more familiar, and inclined to agree, with that perspective), which
    35: is a bit of a philosophical bait-and-switch; but the practical results are
    36: essentially the same (e.g. axiomatic systems which don't allow a general law of
    37: excluded middle).
    38: 
    39: <!--
    40: Since Constructive theories can have firstly they can have
    41: richer structure than their classical counterparts (since the latter permit
    42: extra axioms, leading to more equalities, which collapse those structures).
    43: Secondly, since constructive theories don't force everything into a true/false
    44: dichotomy, they let us naturally consider aspects like (un)provability,
    45: (un)computability, etc.
    46: -->
    47: 
    48: ### Information, not decimal places ###
    49: 
    50: Sabine rightly points out a problem when considering a quantity's "number of
    51: decimal places": that it can change dramatically. She gives logarithms and
    52: exponentials as an example.
    53: 
    54: We can clarify the situation by taking a step back, and recalling what a
    55: "decimal place" *actually is*: a way to narrow-down the possible range of a
    56: number, into one of ten possible divisions (0... or 1... or 2... etc. up to
    57: 9...). Each decimal place we state narrows-down the range by another factor of
    58: ten (although [the edges do overlap](/blog/2024-05-14-spigot.html)).
    59: 
    60: Grouping numbers into finite decimals and infinite decimals doesn't take us far
    61: enough. We need to split up those infinite decimals further, into those which
    62: are rational (whose digits repeat) and those which are irrational; then we split
    63: those irrationals into the algebraic   A phrase like "infinite decimal" can encompass many things, from the
    64: otherwise-simple ⅓ to something as inscrutable as
    65: [Chaitin's constant](https://en.wikipedia.org/wiki/Chaitin%27s_constant); similar
    66: loose, rational or "irrational"
    67: https://en.wikipedia.org/wiki/Algorithmic_information_theory
    68: 
    69: 
    70: ### Algorithmic information theory ###
    71: 
    72: The finite/infinite distinction we should be making is not directly at the level
    73: of decimal places, but more fundamentally at the amount of *information* present
    74: in a number (or any other mathematical construct). Information is a *relative*
    75: quantity, depending on how much it constrains an existing (or "prior")
    76: probability distribution.
    77: 
    78: For computing the digits of a number, the relevant probability distribution is
    79: over computations which give rise to those digits.