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cousin_it comments on Supernatural Math - Less Wrong
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Hey, we no longer need to grant mathematical statements special transcendent status now that we have computers and the axiomatic method. Math theorems predict outputs of specified computer programs given specified inputs, period, end of story. Thus it all folds back neatly into experimental science and directly observable facts. And if some proof can't be thus interpreted - can't be single-stepped through an axiomatic checker even in principle - then it's not yet math.
Of course this leaves unanswered the question of why math not only works internally, but also describes our world so well. Maybe we'll learn the answer in time.
Aargh! Every time I come across this argument, I am frustrated that people don't see how this 'problem' is resolved. Even Einstein is said to have remarked on the 'mystery' of why the world is knowable.
We ditch the 'explanations' that don't describe our world well, and keep those that do. That's why our models end up looking like the world. When new data arrives that isn't compatible with those models, eventually we end up discarding them and creating new ones.
There is never any guarantee that some phenomenon we come across won't be beyond our ability to understand. There is never any guarantee that any model we possess is an accurate one, no matter how useful it's been in the past or how well it accounts for known data.
There is no mystery as to how we can know the world. We don't.
Keeping "models that work" won't help in a world of chaos, and it's a useless characterization of intelligence. You fail to properly argue a position on the problem of induction.
This is just silly.
Thanks! I stand 50% corrected. Yes, we keep those models that work. But math seems an unreasonably effective model even after accounting for the selection effect. Why did conic sections turn out useful for describing planetary orbits 2000 years later, and why did Hilbert spaces turn out useful for quantum mechanics 10 years later?
That misses the point. Conic sections are useless for how many things? Likewise for Hilbert spaces. Likewise for all of mathematics. A mathematical construct is useful for the things it is useful for, and useless for everything else.
Mathematics isn't a model. (Well, it is, but not in the sense that you mean it.) It's what we use to build models out of, what makes them possible.
If a branch of mathematics exists, and someone finds a way to use it to describe a set of relationships they find in the world, we call that branch 'useful'. If its behavior doesn't match the relationships we're interested in studying, we ignore it. And if it was needed, but doesn't exist yet, we never realize it.
Do you have a citation for this analysis, or did you make it up? (or non-excluded middle)
Thanks, I stand 100% corrected.