Do you think imaginary numbers, or lateral numbers like Gauss preferred to call them are "real" ? Or are they just tricks to solve some mathematical problems?
Short answer: I think it’s made up.
…
Long answer: I like to assume the universe can “exist” without human perception of it. Therefore I am of the opinion that mathematics, (a tool that we both defined and developed to model the world around us, which I’ll get into later) is not *inherently built into* our universe. It is merely the language that we use to understand it.
For example, just because you can explain the plot of a book in Spanish doesn’t mean it’s been written in Spanish. (It doesn’t mean it’s *not* written in Spanish either.) Most importantly, It doesn’t mean that details written in Spanish that don’t conflict with your explanation of the book are *actually* a part of the book. (It doesn’t mean that they’re not, either.)
So in other words, just because math works on the universe, and certain things like “the square root of -1” work for math won’t mean they inherently “exist” in the universe. Does that make sense to you?
So now here’s my explanation of developing math. If you were to delve into the most low-level form of math, as in “what is math made of”, you would get “axioms” and/or definitions. Axioms are basically your first principles, a fundamental set of assumptions about how things work. They’re a set of “basic truths” and rules that establish the ground-level properties of your mathematical objects. Definitions clarify what those mathematical objects are.
Imagine for a moment that your axioms really are the most “fundamental” rules you’re allowed to make. That is, no two (or more) axioms can be combined to create another axiom. Side note: Anything that is made of two or more combined axioms we can call theorems (or corrolaries, or propositions, etc. depending on how important we feel they are).
Now I want you to logically justify your fundamental rules. Why are they true? Well, if you use an axiom as evidence of itself, you’re assuming the hypothesis is true, which we don’t do in normal logic. And as we just stated before, you can’t prove this axiom with the others. So in essence, you can’t logically justify your own axioms. That means they’re totally arbitrary, even if you use them as your most basic “facts”. Now, some axioms lead to more useful mathematics than others, which is what helps them become popular, and then taught as the mathematics you see in school.
But deep down, all of mathematics rests on arbitrary bullshit decisions. Some of it just happens to be really useful in the long run.
So my opinion is that “i” is totally made up, but that doesn’t detract from the fact that it helps us solve problems, it’s super cool, or that it’s worthwhile to study.
Foreword: sorry, I tried to make this as short as possible but I also wanted to be as clear as possible and those two goals aren’t really as compatible as one would like. I also tried to make this as informal and non-essayish as possible but inevitably fail in some places. All-in-all @the-real-numbers is a great thinker and this topic is one that is tricky to think about, so massive props there. That being said, I’m not sure both sides are being fairly represented here. And so:
Okay, so to be clear - I agree. While I was asleep one night Hartry Field snuck into my bed and whispered sweet nothings of mathematical fictionalism to me. However, deep down I have some latent and repressed Platonic intuitions in me so let me try to present the opposite case.
I certainly think it’s more intuitive to say, as the Platonists do, that numbers exist. But as you’ve already mentioned in a separate post the word “exists” is a little tricky. You write:
““Exists” is kind of a vague word, isn’t it. But what I’m basically saying is mathematical structures are merely our models of whatever makes things happen, they don’t actually *make* things happen. They’re possibly parallel, sure. But I think the distinction is important. I think some people actually believe that math is the *reason* that the world works, for various definitions of “reason” and “world”.“
So before I talk about the axioms which I think is your main point and also the most contentious one, let’s talk a little about the existence of abstract objects more generally.
You made the example of a book being explainable in Spanish and how that doesn’t mean the book was written in Spanish, which I think is an excellent point. Being able to explain a book doesn’t really tell you much about what the book is. I mean, what even is a book? That’s a pretty tough question for the aesthetician, but why? Well, without going too far into the ontological issues, we want to say that a book written in Spanish and the same book written in English are in fact, the same book. Not the same token obviously, but at the very least the same type of thing. So it’s not really dependent on the language used but whatever it is that the language is representing. Let’s call the things that language represents: concepts.
“nieve verde” and “green snow” pick out the same concept without being the same sentence. Moreover, they pick out a concept that I have personally never seen before and yet understand intuitively based on their simple parts. So if I have never empirically experienced what this concept represents, does the concept exist? Well, the language is certainly meaningful - as is evidenced by the possibility of the concept. If you took some snow and dyed it green, it would be unreasonable to assume the laws of reality would collude against the resultant creation without good reason.
That’s just a simple explanation of what we’re getting at when we talk about concepts. You may prefer another method of solving your existence dilemma, it’s not really relevant to the central point.
With that out of the way, we can understand more clearly that when people write “1″ for example, they’re really talking about something that features in reality. For example, one apple and one book are completely different things but they share this property, this ‘oneness’. What’s that about? And how are we supposed to make sense of it? It makes sense to say (as you seem to imply) that it (whatever we represent by “1″) runs parallel to the feature found in reality without being it - but what reason do we have to assume that is true? It looks as though we’re just unnecessarily multiplying entities. The simpler explanation would be that the oneness itself is what we’re representing by “1″.
But maybe simplicity isn’t something we should aim for. After all, we’ve no real reason to assume that simple things are more likely to be true than complex things (pun intended as we are ultimately talking about “i”).
So I think we’re left with the question: is the relationship between mathematics and reality like the relationship between a language and a book (accidental (just one embedding of concepts out of many)) or is it more like the relationship between (no article here) language and a book (purposeful (there is no possible explanation of a book without the use of language. indeed, a languageless book is almost an incoherent concept))?
To begin to answer that question we may want to ask ourselves: just how general is mathematics? But I fear we’re heading astray here. Just some food for thought. Now, on to your main point.
Your claim is that: “… deep down, all of mathematics rests on arbitrary bullshit decisions.” But that’s not quite true. Because the decisions as to which axioms we use aren’t entirely arbitrary. We generally think that there are two kinds of justification for the axioms we use, extrinsic justifications and intrinsic justifications.
Starting with extrinsic justifications, let’s draw a parallel with the game of chess. I think you want to say something like: we decide upon the rules of chess and some sets of rules lead to interesting games and others lead to non-interesting games. An extrinsic justification for the rules we choose is “interestingness” whatever that is.
But I feel like that just pushes the question back one step rather than actually answering it. After all, why should this thing be interesting? Well, it’s interesting because it does the same kinds of things that we see in reality (or behaves in a way we think is possible (don’t want to go too heavy into modal realities here but the kinds of mathematical inquiry that don’t approximate reality can be explained by their approximation to a possible reality)). So are we attempting to approximate the rules of reality or a facsimile of these rules? Is that what interestingness is? The approximation of rules similar to reality. Does that mean the end goal is to describe some fundamental part of the world?
Maybe, maybe not. But it is interesting to note that every interesting system of axioms is co-extensive to ZFC. So there is obviously a level of power that is required before a system becomes interesting and a level of power where a system becomes un-interesting (EFQ proves everything trivially).
So if some rules produce games of chess that are objectively more interesting then the rules have some tie to objectivity. They can’t be entirely arbitrary.
To put it another, more condensed/ simpler, way: any fictionalist account for mathematical entities is going to run against the Quine-Putnam indispensability argument. Which goes roughly as follows:
1) You can’t do physics without maths 2) Physics is describing reality (as it really is) 3) Therefore we have good reasons to believe that mathematical sentences are true 4) The simplest and most likely explanation for their being ‘true’ is their primary entities existing
Tbh, I just tagged 4 on at the end in favour of Platonism, but it doesn’t look out of place here eh?
Now maybe you’ve got a case against 1, maybe, but nobody has been able to convincingly nominalize a scientific theory to accord with reality absent mathematics. Arguably Field manages to nominalize Newtonian gravity by extending metric Hilbert spaces - but that’s one theory and one that we know isn’t even correct. The actual project you’d have before you is monumental if you tried to nominalize a future unified theory of physics.
Furthermore extrinsic justifications only make up one half of the kind of things we use to justify our axioms. There are also intrinsic justifications that we need to discuss before we can make the claim of absolute arbitrarity.
You make the claim that the most basic parts of maths are the axioms or definitions. And well… Maybe. But we know that no formal system can completely or consistently represent the natural numbers. So there has to be something more going on beneath the surface. Maybe something like absolutely unsolvable diophantine problems. But if there are absolutely unsolvable problems this suggests that mathematics isn’t a creation of the human mind (according to godel here, maybe this isn’t implied but it’s certainly an intuitive response).
Mathematicians seem to have an idea of what they’re doing. And for the most part, that’s an attempt at understanding the various interactions between particular complex features of reality. Not playing games and deciding what rules to play by. The V-hierarchy is a model of how we think the universe of sets should work. And it’s a model based entirely upon reason.
We know that not all of ZFC is extrinsically motivated at the very least, because (and I may be wrong here, this part is based on a convo I had a few months ago) there exist axioms such as the axiom of infinity that accord with our reasoning about the V-hierarchy but are not (yet) found in any physical theory about the universe.
But you may want to say something like: “well, the extrinsic motivation, in this case, is how well our axioms produce mathematical statements proven in non-axiomatic mathematical fields (i.e. calc)”. And yeah, maybe that’s the case, I mean, from a historical perspective the kinds of conversations we have about axioms and sets come at a far later date than the kinds of conversations we have about infinitesimals and limits.
So I’m willing to accept that axioms what’s important (as opposed to numbers) - but it’s not like that’s not in harmony with the Platonic world-view either. In fact, it’s kind of more in-line with Platonism than Fictionalism. If numbers are constructed from arbitrary rules (easily done, up to the real numbers from sets) then it’s the rules that are our abstract objects, not the numbers. That’s stranger than it appears at first so I’ll give you some time for it to sink in. While it does, consider the Fictionalist account of the same phenomenon. Science without numbers is maybe (super contentious maybe) MAYBE possible. But science without axioms?
I think your view needs a lot more justification than you’ve presented here. It’s a more extraordinary claim than the claim ‘imaginary numbers are real, just like the real and natural numbers’.
But I realize I’ve droned on for way too long now. And I’m eager to hear your response. Sorry if this is a little disjointed I had to edit down because I don’t think anybody has the willpower to make it through this long and probably wrong (b/c this topic is hard and we’re all stupid and have a lot to learn) post as it is, before cutting it was too long to even be comprehensible.
