Comment by zkmon
3 hours ago
We are entering into a probabilistic era where things are not strictly black and white. Things are not binary. There is no absolute fake.
A mathematical proof is an assertion that a given statement belongs to the world defined by a set of axioms and existing proofs. This world need not have strict boundaries. Proofs can have probabilities. Maybe Reimann's hypothesis has a probability of 0.999 of belonging to that mathematical box. New proofs that would have their own probability which is a product of probabilities of the proofs they depend on. We should attach a probability and move on. Just like how we assert that some number is probably prime.
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.
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.
> 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.