• Noah Te Stroete
    1.3k
    Can’t a real number go on and on for infinity? Or is that an irrational number...?
  • Noah Te Stroete
    1.3k
    "
    I can't speak to non-pointed set-theoretic probability theory. I know about ETCS (elementary theory of the category of sets) so I understand that sets don't require points. But as to probability, I can't say. However if you take finite line segments as sets, you seem to lose intersections. Are these closed or open segments? You have a reference for this interpretation of probability theory?
    — fishfry

    Let's look at Kolmogorov axioms here: (http://mathworld.wolfram.com/KolmogorovsAxioms.html)

    Everything that is needed is a set W
    W
    , some Qi
    Q
    i
    , that can be "anything", a function Qi
    Q
    i
    from the Qi
    Q
    i
    to real numbers, and a function "complement" on the Qi
    Q
    i
    .

    Let's consider as our probability space the segment [0, 1].

    I can take for Qi
    Q
    i
    the closed sets included in [0, 1] made of countable number of non overlapping segments with non zero length, and for W
    W
    the set of all these sets. The complement of a Qi
    Q
    i
    will be the closure of the remaining part of [a, b] when I remove the Qi
    Q
    i
    . There are no Qi
    Q
    i
    of zero measure (and this is very reasonable for a probability theory: every event that can happen must have a non zero probability to happen).

    The complement of a Qi
    Q
    i
    overlaps with Qi
    Q
    i
    only on the end points, and that is compatible with the axioms: the sum of measures adds up to 1.

    The elements of W
    W
    are simply ordered pairs of real numbers instead of single real numbers, but everything works at the same way: from the point of view of set theory two segments are equal if and only if the extremes are equal: no mention of overlapping segments at all.

    The definition of overlapping segments is the usual one: the higher number of the first pair is bigger than the lower number than the second pair.

    There is no need to consider infinite sets of points, and for probability theory there is no need to speak about points at all: probability theory does not need zero-measure events, and no physical possible event has zero probability.

    P.S. This is only a very simple example to show that it's not contradictory to define probability without points. Pointless topology is much more general than this and makes use of the concept of "locales" (https://en.wikipedia.org/wiki/Pointless_topology)
    Mephist

    This to me seems to be what @Dfpolis was saying that mathematics must be instantiated in nature first otherwise it is pointless to talk about numbers existing in a Platonic ideal realm. That’s what I got from what you both were saying. Pardon the intrusion.
  • Mephist
    178
    Well, I wrote about physical events because probability is a concept that belongs both to physics and mathematics, but from the point of view of mathematics only (Kolmogorov axioms), if axioms do not require zero to be a valid probability measure, you can avoid to consider events with zero probability.
    But of course in this case the axioms are chosen with the intent to be the minimal constraints that have to be satisfied by any definition of probability that makes sense in physics.
    This is very common: the mathematical definition is derived from the physical intuition, but is used as an axiomatic theory completely disconnected from physics. That's why it often happens that after several years somebody finds a "model" of the axiomatic theory that does not correspond to the physical intuition at all but verifies all the axioms, so from the point of view of mathematics it is a valid model.
    The point is that axiomatizations are needed if you want to use formal logic (and you want to use formal logic because physical intuition is not enough to avoid paradoxes), but the axiomatization very often does not capture exactly the physical concept that you have in mind. And that happens even for the most elementary physical concepts, such as the concept of natural numbers.
  • fishfry
    699
    ↪fishfry Can’t a real number go on and on for infinity? Or is that an irrational number...?Noah Te Stroete

    You mean the decimal representation of a real number? Yes, they're infinitely long. For example 1/3 = .333... and sqrt(2) = 1.4142... and so forth. Even terminating decimals like 1/2 = .5 = .4999... have infinitely long representations.

    But the representation isn't the real number. The real number is the abstract thing pointed to by the representation, just like 2 and 1 + 1 are two different representations of the same abstract number.
  • fishfry
    699
    I guess I got confused because you guys were talking about numbers as points, where I was thinking about zero-dimension points on a two-dimensional line. This stuff to me right now is very esoteric, as I don’t remember the terminology for the different kinds of numbers. I was always better at calculations like an engineer than I was so much interested in or ever had any exposure to theory.Noah Te Stroete

    A line is only one dimensional. A plane is two dimensional. Points are zero-dimensional. How a bunch of zero dimensional points make a line is a bit of an ancient mystery. Newton thought of a point as tracing out a curve as the point moves through space. But that only pushes the mystery to space itself.

    There's really only one standard real number line that most people care about, the one from high school math used by physicists and engineers. The other stuff is esoteric. Constructivism is the idea that every mathematical object must be able to be explicitly constructed. That leads to a different kind of notion of the real line. But I really can't speak for constructivist philosophy, since I don't know much about it. You can see the difficulties @Mephist is having in explaining it to me!
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