Comment by teiferer
14 days ago
Definitely not.
"Probability" does not mean "maybe yes, maybe not, let me assign some gut feeling value measuring how much I believe something to be the case." The mathematical field of probability theory has very precise notions of what a probability is, based in a measurable probability space. None of that applies to what you are suggesting.
The Riemann Hypothesis is a conjecture that's either true or not. More precisely, either it's provable within common axioms like ZFC or its negation is. (A third alternative is that it's unprovable within ZFC but that's not commonly regarded as a realistic outcome.)
This is black and white, no probability attached. We just don't know the color at this point.
>> "Probability" does not mean "maybe yes, maybe not, let me assign some gut feeling value measuring how much I believe something to be the case."
That's exactly what Baeysian probabilities are: gut feelings. Speaking of values attached to random variables, a good Bayesian basically pulls their probabilities out their ass. Probabilities, in that context, are nothing but arbitrary degrees of belief based on other probabilities. That's the difference with the frequentist paradigm which attempts to set the values of probabilities by observing the frequency of events. Frequentists ... believe that observing frequencies is somehow more accurate than pulling degrees of belief out one's ass, but that's just a belief itself.
You can put a theoretical sheen on things by speaking of sets or probability spaces etc, but all that follows from the basic fact that either you choose to believe, or you choose to believe because data. In either case, reasoning under uncertainty is all about accepting the fact that there is always uncertainty and there is never complete certainty under any probabilistic paradigm.
Baffling to see such a take on HN.
If I give you a die and ask about the probabiliy for a 6, then it's exactly 1/6. Being able to quantify this exactly is the great success story of probability theory. You can have a different "gut feeling", and indeed many people do (lotteries are popular), but you would be wrong. If you run this experiment a large number of times, then about 1/6 of the outcomes will be a 6, proving the 1/6 right and the deviating "gut feeling" wrong. That number is not "pulled out of somebody's ass" or some frequentist approach. It's what probability means.
Yes, that's the frequentist approach. Surely, even on HN, there is an understanding that there are two interpretations of probability?
8 replies →
It's time that mathematics need to choose it's place. Physical world is grainy and probabilistic at quantum scale and smooth amd deterministic at larger scale. Computing world is grainy and deterministic at its "quantum" scale (bits and pixels) and smooth and probabilistic at larger scale (AI). Human perception is smooth and probabilistic. Which world does mathematics model or represent? It has to strongly connect to either physical world or computing world. For being useful to humans, it needs to be smooth and probabilistic, just like how computing has become.
Please elaborate what “quantum scale” means if possible.
> Physical world is grainy and probabilistic at quantum scale and smooth amd deterministic at larger scale.
This is almost entirely backwards. Quantum Mechanics is not only fully deterministic, but even linear (in the sense of linear differential equations) - so there isn't even the problem of chaos in QM systems. QFT maintains this fundamental property. It's only the measurement, the interaction of particles with large scale objects, that is probabilistic.
And there is no dilemma - mathematics is a framework in which any of the things you mentioned can be modeled. We have mathematics that can model both deterministic and nondeterministic worlds. But the mathematical reasoning itself is always deterministic.