“How can I stay in any field and continue to prove theorems if the fundamental notions I’m using are problematic?” asks Peter Koellner, a professor of philosophy at Harvard University who specializes in mathematical logic.

yeah, i hear you. i didn't even make it that far before i gave up. i mean, i switched to math because physics was clearly bullshit, only to find out that math was also bullshit.

worse, nobody seemed to care. "bullshit or not, i still get paid, right?". fuck this, give me my guitar....

i think they need to take a step back with the axiom of choice, actually. it needs to be weaker: an axiom of countable choice. once they do that, a lot of this debate gets sort of meaningless.

and as far the "continuum hypothesis"? well, note the name. hypothesis. hypotheses are to be determined using empirical methods, not converted into axioms. the undecidability of the statement seems to somehow verify this, in a strange kind of way.

so, we should be constructivist about it. if we can build a set with cardinality between that of the reals and naturals, then one exists. if we cannot demonstrate the truth of the negation, we should remain agnostic. the addition of new axioms may be verified for other reasons, and may have an effect on what can be deduced, but should not be affected by one's intuitive/arbitrary views on the truth of the continuum hypothesis.

and to stick with that step backwards, we should be constructivist about our choice functions. these abstract choice functions strike me as entirely incoherent nonsense. and they lead to ridiculous conclusions that are observably false.

if you're following this, you're going to think i'm some kind of completely dour wet blanket that seeks to rob mathematics of it's beauty and enjoyment, or something. not intentionally. i just don't think mathematics should be operating in the realm of fantasy.

if i'm going to operate in the realm of fantasy, i'd rather do it through sound.

i've alluded to this before, but i'll come out flatly and state it. i think that before we can get anywhere further with axiomatic/deductive thought on an abstract level we need to take some very complicated measurements of space as it exists outside of the earth's atmosphere. once we've done that, we need to empirically model the space we actually inhabit. we then need to construct our axioms based on an empirical understanding of the world around us. it is only then that we can come to any kind of fruitful discussion about how mathematics relates to the world that we live in.

in the meantime, there's two approaches that i think are worthwhile.

1) we need to restrict ourselves to very limited, finite, constructivist assumptions if we wish to apply anything to reality.

2) we should undertake a systematic evaluation of all possible positions, and catalog them as possible models. we should then explore the possible consequences of *all* of them. we can philosophize abstractly about how things might be, but until the data comes in we can't know how things are.

mathematics is not as out of touch as philosophy is, but it's not much further evolved. we have to understand the limits of this type of reasoning and get ourselves out of the nineteenth century.

we also have to understand that the necessary modelling is going to be fraught with difficulties in the attempt to apply as few assumptions as possible.

another thing that empirical study can help us understand, besides the shape of space, is whether or not space is quantized. if the answer is 'yes', mathematics need to go back to the drawing board.

(translation: it would answer the question about the continuum hypothesis (yes - our model of space in it's construction) in a way that would explode the whole field.)