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
- Academia

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.

Hi Brielle! I was wondering if you had any examples of when “thereโs tension between our intuitions and the theorems that follow from our axioms”. Upon reading your post I thought of Galileo throwing weights off a tower and proving our intuition about falling objects incorrect, but I wasn’t sure if that was what you were thinking of.

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Hi! ๐ The specific example I had in mind when writing that was the Axiom of Choice and the Banach-Tarski theorem. (Axioms are the given starting points of mathematics and theorems are those which are directly entailed, or proven by, the axioms.)

The Axiom of Choice (informally) states that if you have a collection of (non-empty) sets, there is a way to select one element from each set. Intuitively, this seems correct: if I have 3 jars (sets) each with 5 marbles (elements), then I can take 1 marble from each of the 3 jars. Similarly this can be done for any other collection of non-empty sets.

The Banach-Tarski theorem, put simply, says that you can cut up a sphere into a finite number of pieces which, when rearranged and reassembled (i.e., moving the pieces around and rotating them without changing their shape or size), give you two spheres which are both the same size of the original sphere. This theorem is proven using the axiom of choice.

It seems completely counter-intuitive to say that you can double the volume of something simply by rearranging its pieces, yet this result directly follows from the axiom of choice.

So it seems we have two options: A) Deny the Axiom of Choice, despite its intuitive correctness, to avoid the counter-intuitive result that is the Banach-Tarski theorem, or B) Accept the Axiom of Choice, seeing as it appears intuitively correct, and simply accept the counter-intuitive Banach-Tarski theorem as a consequence. (Accepting the axiom yet simultaneously denying the theorem is not an option, since the theorem is directly entailed by the axiom.)

So, when it seems to be a battle of intuition vs. intuition, how do we decide which option to go with? As you can probably guess from my post’s focus on practicality, one way to decide whether or not to keep the Axiom of Choice is to look at how practical it is, i.e., does it seem to be so important a foundation of mathematics that we’d want to keep it around? And this seems to be the general response of the majority of the mathematical community: they justify their continued use of the Axiom of Choice (despite the paradoxical theorem that follows from it) by pointing out how many useful theorems rely on it, and by how many fundamental ideas we would have to get rid of without it.

To make my overarching point explicit with respect to this, my worry is whether or not the Axiom of Choice is actually true – or are we just keeping it around because it’s useful? We tend to think of mathematics as this body of objective knowledge that’s not up for debate (e.g. that 1+1=2 is not something that is up for questioning in the same way “capitalism is an exploitative system” is), but if we’re just choosing the axioms (upon with all of mathematics is based) based on what’s useful, then how can we really be sure that our math is right or correct or true?

Thanks for your question – hope this made sense! ๐

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Thanks for the reply ๐ I was reading more about the Banach-Tarski theorem and correct me if I’m wrong, but the pieces of the sphere(s) it describes are not sphere pieces as we would intuitively define them but rather “infinite scatterings of points”. It seems like our intuition about the Banach-Tarski theorem only feels paradoxical when we think that the sphere(s) described are broken down into pieces the way we’d imagine a regular solid ball would be. Once it’s understood that the “pieces” the theorem speaks of are not actually what we intuitively think of as pieces, our intuition about the Banach-Tarski theorem doesn’t seem so at odds with our intuition about the Axiom of Choice.

But I say this with a layman’s intuition. Maybe if mathematical intuition were sharper, I would struggle to reconcile the two statements. Or maybe I’ve missed what you’re saying completely! If you have time, please let me know what you think.

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Well to me it still seems counterintuitive, but you can take any sort of mathematical paradox or counterintuitive result and my worries would still be able to move forward. There are plenty of other instances to choose from – the BT theorem was simply the example that I was using. Thanks for reading! ๐

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