I recently had an interesting conversation with a friend about the methodology of mathematical/scientific inquiry, truth, axioms, and the purported aims of science. I don’t normally think about these sorts of things too hard, but since I’m currently taking classes covering both axiomatic set theory and some topics within the philosophy of science, these ideas have been turning over in my mind as of late.
(I feel like I should probably preface the rest of this post by making it clear that I don’t have any background in mathematics, and that I’ve never taken a philosophy of science class prior to this one. So, yeah, there.)
This conversation started out with me questioning the relationship between intuition and the axioms/theorems of mathematics: When there’s tension between our intuitions and the theorems that follow from our axioms, what do we do – abandon the axiom(s) or admit that our intuitions are mistaken? My friend seemed to favor the latter option, but I don’t see a good reason why the axioms should take precedent over our intuitions when the axioms themselves were chosen by…well…chosen by us, because (and please interpret this charitably) we have intuitions about what makes a good set of axioms.
One such intuition seems to be that good axioms are practical: If we can prove useful theorems from it, or it aids in some important scientific discovery, or something along those lines, then we have good reason to keep the axiom in question.
This emphasis on practicality reminded me of some of the recent discussions we’ve had in my philosophy of science class regarding how we might justify our intuitive preference for simple theories and use of Ockham’s razor. In other words, why should we assume that world behaves according to simple laws? It’s been suggested that simple theories are easier to test, modify, and work with, and thus, we ought to favor them over complex theories that fit the data equally well.
But given this conceptualization of scientific methodology, what can properly be said to be the goal of science? If we’re choosing theories based on their practicality, how can we be sure that they align with reality and truth?
Sorry, I know this was all over the place. (Or it went nowhere at all?) All of this rambling is to motivate a worry I have:
We have intuitions about mathematics and about the way the world works. When we try to justify these intuitions, however, we are forced to resort to practical arguments. But to choose our axioms and theories based on their practical merits is different from choosing them based on their truth, and if we’re not choosing these things based on their truth, then…then what?
Maybe, given these considerations, it’s more accurate to say that math and science are tools to help us conceptualize and understand the workings of our universe in a way that makes sense to us, rather than undertakings with the goal of accurately describing reality.
And I’m completely happy to accept that – I just worry about the normative implications of this conclusion. To elaborate, we tend to think of math and science as “objective”, and therefore use them to back certain normative claims: climate change is a serious issue and we really need to work on reducing our carbon footprint; homosexuality is natural and there’s nothing wrong with being gay; etc. These sorts of ideas are already controversial (even though they absolutely shouldn’t be *eye roll*), and I’m concerned that accepting the proposed understanding of math and science will only exacerbate these issues and give the opposition more reasons to staunchly deny climate change or decry homosexuality. Why care about claims backed by science if science doesn’t even point to objective truths?
These ideas don’t seem incredibly profound or novel to me – in fact I’d be surprised if there didn’t already exist more literature on this than I could devour in a lifetime. But I don’t really have the time (nor the interest, if I’m being completely honest) to look into it; I just enjoy mental masturbation. So I’ll just leave these thoughts here.
Seeing as this is my first real post, I figure I should say a bit about how I plan to organize my blog.
I foresee my posts being categorized in the following way:
- Underdeveloped Philosophical Musings
- Personal Updates
I’ll speak more on the latter two when they become relevant, but seeing as this particular post falls under the first category, some words on that:
I don’t intend for things in this category to be construed as serious philosophical ideas I’m putting forward such that a sophisticated dialectic can ensue (though I of course welcome your responses). Rather, I simply want to record interesting thoughts inspired by my classes or discussions with friends/colleagues. Maybe I’ll come back to them. Maybe not. But I think the practice of documenting them is nonetheless good.