3 ms·
> Is there any (uncontroversial) theory that rigorously defines what a 50 % probability for heads and tails means? Yeah, Kolmogorov's axioms: https://en.wikip
by jordigh 10y ago
> Is there any (uncontroversial) theory that rigorously defines what a 50 % probability for heads and tails means?
Yeah, Kolmogorov's axioms:
https://en.wikipedia.org/wiki/Probability_axioms https://en.wikipedia.org/wiki/Probability_axioms
To interpret these axioms for 50% probability means that the measure underneath the density function corresponding to the event "heads" is one-half.
But "rigourous" doesn't have anything to do with the natural world. You can't make physics "rigourous", for example, but you can make the mathematics inspired by physics rigourous. Kolmogorov's axioms just give a mathematical description of probability in a formal sense i.e. only discussing its form, not its meaning. Formalism is about saying, "whatever this means, this is how it should behave". It's a very 20th century notion of mathematics.
- danbruc 10y agoI know the Kolmogorov's axioms but I am really more interested in that part they avoid - what is the meaning of a probability of 0.5? It is surly nice that we can operate with probabilities in a (hopefully) self-consistent way, but it bugs me quite a bit that I don't really precisely unterstand what the result of a calculation implies for the real world.
- jordigh 10y agoThe modern approach to mathematics is that there is no "meaning", just like "2" or "derivative" has no meaning. We just say how it behaves, or define it in terms of other things, which eventually bottoms out with undefined terms, such as sets and set membership. This is formalism. How you apply mathematics to the world is not the business of formal mathematics. Whatever you want to do with it is "mere" philosophy. ;-)
- danbruc 10y agoThat is what I meant - I am more asking for a solid philosophical interpretation than a mathematical theory, after all frequentism, Bayesianism and all the other interpretations seem to carry quite a bit of philosophy. On the other hand in another comment it just boiled down to the question whether there is a measure that gives 1 for the set of all infinite binary sequences with 50/50 zeros and ones and 0 for the set of all the other sequences. So it is not pure philosophy what I am interested in.