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I know only a little about set theory and probability, and struggle to infer their implications for many traditional metaphysical concepts and questions.

I was surprised to read the statement, "...infinities are no obstacle to probabilistic calculations" provided they can be assigned a determinate numerical value ("Nihil Unbound," Brassier, p. 68), per Cantor. Brassier then discusses the impossibility of assigning any probability of "an actuality" relative to "all" possible universes. Presumably both are "totalities," so probability is simply 1:1, though I'm still working on it. Seems similar to any refutation of "argument from design."

But aside from that example, could someone list a few rules of thumb that tell you when it is impossible, or makes no sense, to imply a probability. Such as single cases. I suppose the reverse would also answer: what minimal concepts and data are needed to calculate a probability? When I hear even physical cosmologists discussing likelihoods in "the universe" or in "all" possible worlds, it always smells like a simple fallacy.

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The mathematical study of probability does not tell us what probability means. The problem of how we interpret probability, like most philosophical problems, cannot be answered mathematically.

The concept of probability is certainly problematic from a philosophical point of view. This is because there appears to be two distinctly different ways of defining probability; either as a logical concept in terms of frequency, or as a subjective concept as the strength of personal opinions.

The subjective interpretation of probability implies that there are no objective facts about probability, only what people believe. For example, I could say that there is a high probability that the big bang model of creation is true, while you could say that there is a low probability that it is true. Neither of us would be right or wrong. We are simply stating how strongly we believe the statement. This interpretation of probability is not saying that there is no objective fact that determines the truth of the statement, only that there is no objective fact about how probable it is. It is in this context that, as you say, “it makes no sense to assign a probability”.

The logical interpretation of probability rejects this interpretation. It says that the probability that the big bang model is true is objectively true or false according to a specified body of evidence. In other words, a statement’s probability is a measure of the strength of evidence in its favour. In the case of the big bang model, the evidence consists of such objective evidence as the observed uniform expansion of the universe, the cosmic background radiation, etc.. The logical interpretation of probability asserts that a probability can always be assigned based on the available evidence. (This contracts with the subjective interpretation where "it makes no sense to assign a probability".)

The concept of probability is obviously important scientifically since it is at the heart of many of our scientific theories - e.g., quantum theory and genetics. It is also philosophically important because it may shed some light on the problem of inductive inference which lies at the heart of our scientific theories.

(Regarding Cantor’s claims that “infinities are no obstacle to probabilistic calculations”, I believe this would follow from the simple fact that the theory of transfinite numbers ( the ordinals and cardinals of set theory) includes a well-defined transfinite arithmetic. Thus, the same mathematical methods that apply to the finite arithmetic of probability would carry a transfinite interpretation.)


EDIT (Way TL;DR)

I have managed to locate this paper which discusses Brassier’s Nihil Unbounded.

In your comment following my answer, you state that Brassier goes on to say :

while we can assign numerical values to infinities (per Cantor) we cannot "totalize" an infinity

According to the paper I have referenced (above) the term “totalize” means to “count as one”, or “count as a set”. If this is a correct interpretation of “totalize”, then Brassier is plainly incorrect in the context of set theory. It is not clear if Brassier is expressing his own opinion or interpretting that of Badiou. The original naive statement of set theory (as per Cantor), which is based on set as predicate, runs into inconsistencies which imply problems with unrestricted “totalization”. However, these problems are removed by the axiomatic formulation of set theory which followed - i.e. ZF Set Theory. This is reflected in the statement :

In short, Badiou denigrates the power of intuition to totalize its objects.

This is correct, if we read the original naive formulation of set theory (as per Cantor) as being that of our intuition. It is also misleading to suggest that it is Badiou who is identifying these issues, since they were well known and dealt with by mathematicians long before Badiou came along.

A more complete extract, which largely makes sense, is :

This unbinding is made possible under the auspices of post-Cantorian mathematics and the discrete formal object-language of Frege and Russell, which masters the multiple by rendering lexical terms axiomatic. The axioms of set theory are defined compositionally –– rather than conceptually –– since ‘the multiple does not allow its being to prescribed from the standpoint of language alone’ [B&E, p. 40]. Or more precisely, we cannot count-as-one, or count as ‘set’, everything that is subsumable by a property, denying the coherency of any linguistic institution of a universal all-encompassing ontological situation. In short, Badiou denigrates the power of intuition to totalize its objects. Consequently, we can claim that the axiomatic presentation of inconsistent multiplicity annihilates the logical consistency of language and inaugurates the anti-phenomenological reign of the pure multiple (i.e. the void/null-set, the multiple of multiples, or groundless ground of what is presented). This subtractive discipline broadens the discursive range of philosophy, abjuring any previous idealist claims of auto-positional self-sufficiency and deposes the precarious configuration of Oneness. In short, Badiou’s axiomatic decision requires that philosophy be ‘expropriated of its conditions, [and] deprived of the appeal to intuition’ [AR, p.2] which accords him the ability to claim that the One is not, denying the existence of the Whole.

Where this text seems to have problems is the statement “we can claim that the axiomatic presentation of inconsistent multiplicity…”, since it is precisely the axiomatic formulation of set theory that eliminates the inconsistent multiplicities.

While I am not an expert on set theory, it appears clear to me that the author, and by implication Badiou, have some fundamental misconceptions on the status of contemporary set theory. Brassier, on the other hand, seems to be a credible guy.

One assumes that the “actuality” you refer to is the completed “totality” of an infinite collection.

Anyway, this all has little to do with your original question concerning probabilities. I am guessing that the “possible worlds” referred to are those of the universe of set theory, in which case it is the transfinite arithmetic I have referenced my original answer would be the key (or so I believe).

The only reference to probability in the linked paper (above) is in relation to Chaitin’s constant Ω, which “measures” the probability of an arbitrary (or random) program halting. This doesn’t have anything to do with set theory, at least not directly.

I’m not sure if any of this will be of any help to you, but hopefully it will help clarify some of the terms.

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  • Thanks, not to confuse, it was Brassier not Cantor who made the statement. He goes on to say that while we can assign numerical values to infinities (per Cantor) we cannot "totalize" an infinity. So, I think, when we speak of "all" possible universes we are verbally and illegitimately totalizing, as if we could then speak of the "improbability" is this actual universe. Seems like an obvious fallacy, yet you hear some cosmologists talking this way. Commented Sep 19, 2015 at 21:09
  • Also, my understanding was that Bayesiansim convincingly reduced the subjectivity problem, as long as you can define a mutual, countable "totality" of possibilities, though I understand these are all philosophically controversial interpretations. Perhaps my confusion over the statement hinges more on "infinity" than probability. "Counting" infinities does not mean "totaling," and without some "total" no probability of either type. Commented Sep 19, 2015 at 21:24
  • @NelsonAlexander At my sophomore level, I struggle to see how Bayes' Theorem can be interpreted as a measure of degrees of belief, primarily because it is formulated in terms of existing probabilities - P(A|B)P(A)/P(B). I understand that many do interpret the theorem as rebutting the subjective interpretation, but I (currently) read it as dealing with probability exclusively in terms of statistical frequency. Perhaps when my view are more sophisticated I will better understand the subtlety of this view. I'm still thinking about Brassier's view and hope to come up with a comment (soon?).
    – nwr
    Commented Sep 19, 2015 at 21:58
  • @NelsonAlexander In addition to the above comment, I have posted an unnervingly long edit to my original post with regards to Brassier's thoughts.
    – nwr
    Commented Sep 20, 2015 at 0:00
  • If you are in fact a "sophomore" you have more formal training than I do, so don't take what I say too seriously. My understanding is that the Bayesian model, while "frequentist," can start with "any" set of convictions, provided they are self-consistent. Hence it could begin with "any" belief or subjective stance. Of course, the devil is in the details of application. I'll have to think about your qualification of "existing probabilities." Not sure what that means. Commented Sep 20, 2015 at 0:23
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This question is made rigorous by the mathematical field of measure theory. In measure theory, a probability measure is one in which the measure of the full set is 1. Most of the obvious examples of measures are called translation invariant - just a fancy way of saying that all events are equally likely. However, it's possible to construct non-translation invariant measures on infinite sets. A specific example of this can be seen in the Poisson distribution (examples: arrival times, number of children in a family, number of passengers on a train) - although each integer itself has non-zero measure, the sum of the probabilities of all numbers together is still 1.

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  • Thanks, applying this is a bit over my head. But I see there are many web intros to "measure theory," so I will investigate and try to work it out. Commented Sep 21, 2015 at 17:42
  • Mathematicians are famous for inventing words to scare people away. Fortunately, the concepts are pretty intuitive once you peel back the fancy words. Commented Sep 21, 2015 at 18:02
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It depends on which kind of mathematical infinity; generally the two infinities to take into account here are the first two: the cardinality of all the natural numbers, and of all the real numbers.

It's this all that requires Set Theory; in the sense we can formally manipulate sets according to precise rules in a way we can't by concepts; though of course the concept of a totality - the all - is available.

The higher cardinalities don't generally make an appearance in probability theory or in physics.

Probability first made an appearance in physics with random walks; and then of course and famously QM; there it is its alignment with experiment that is crucial - the empirical method.

Further away from this it becomes increasingly speculative - which aligns with your intuition it becomes closer to a fallacy; and it is - if it's asserted; but generally the speculative character is understood; even if some would like to take it as certainty - as some do, and through the media, many more do. ie speculative multiverse theories

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  • Thanks, though still mulling the second part. In Brassier book he says infinities can be assigned numerical value, but cannot be "totalized." As you say, "all" becomes the problem, as would "one." So I take it a "total of" something is needed for any probability, but this is only defined and countable by fiat. So fallacies pop up when we confuse two ways of using "totality:" in set theory as opposed to metaphysical or common parlance. Commented Sep 19, 2015 at 19:36
  • Well, you can assign numbers to anything - including infinities; but they wouldn't be probabilities without some sense; Brassier is in the continental tradition - so it may be that he is using this concept in a non-specific way, or in a way specific to him; the same for totalising - ie when I used the concept totality or all, I was drawing on how Aristotle used this sense, but didn't bother mentioning it. Commented Sep 19, 2015 at 19:42
  • For all I know, not having read Brassier - he might be using total in the sense of totalitarianism - joke, or maybe not; given his interest in Badiou, and Badious interest in politics. Commented Sep 19, 2015 at 19:44
  • Actually, Brassier is kind of anti-Continental Continental. Pretty committed to science, math, anti-Heideggerian, though with radical implications. Like Badiou, he is keen on set theory, so definitely not using the terms loosely. Not defending him, but it is often a mistake, even with people like Hegel, Bergson, or Heidegger to assume "loose lips" or unfamiliarity with the math, science, and logic of their day. Admittedly, Badiou is an odd duck but his set theory work is supposedly quite respectable. Commented Sep 19, 2015 at 20:06

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