Ordinary talk is full of ambiguity and vagueness.
Mathematics tends towards being much more precise and rigorous, with many terms explicitly defined. Sometimes these terms are made up for mathematical use, such as “homomorphism.” Other times, they are terms that already have some meaning (or several meanings) in everyday talk, but in mathematics they are co-opted for a specific, explicit purpose, that may be only loosely inspired by the everyday meaning. Some examples of this are words like “lattice” and “compact.”
In everyday talk a “lattice” might be a fence pattern like this. In mathematics it has more specific meanings.
However, although mathematics is definitely more precise and explicit than ordinary, everyday talk, it still ends up being messy. This might not be surprising, given how many different people contribute to mathematics over so many years.
One kind of messiness is that the same words can be used in different parts of mathematics to mean different things. For example, the word “lattice” means one thing if you’re talking about mathematical orders and/or algebras, and something different if you’re talking about mathematical groups. The word “compact” means one thing if you’re talking about topological spaces, and something different if you’re talking about logical theories. And so on. (If you don’t know what these different branches of mathematics are about, that’s fine. I hope you can understand the general point without knowing that.)
Another kind of messiness is that sometimes mathematicians define a word in different ways, even when — in some sense — they intend it to have a single meaning. I said that one way to use “lattice” is to talk about mathematical orders and/or algebras. Even for this one intended use, there are different definitions offered.
The details aren’t important, but if you’re curious, here is one definition and here is another.
This can sometimes be harmless, if the two definitions are after all equivalent. But sometimes it happens that that the mathematicians start out thinking that some definitions are equivalent, but then later it turns out that they’re not. Or it might turn out that if we make some assumptions, they are equivalent, but on other assumptions they’re not. And those assumptions might be controversial: accepted by some theories but rejected by others.
When it comes to definitions, philosophy often tends towards the mathematical model. Sometimes it’s just as explicit and precise as mathematics, sometimes it’s just loosely headed in that direction. This can often be philosophically useful.
It’s important to realize though that the kinds of messiness described in mathematics all show up in philosophy too, even more often and making more of a mess.
Sometimes one and the same word will be used in different parts of philosophy with different meanings. Prominent example of this are words like “internal” and “external”. Another example is the word “intentional.”
And even when we focus on what’s intended (in some sense) to be a single meaning, sometimes different philosophers (or even the same philosopher in different places) will define the word in different ways. Sometimes these differences are minor subtleties, that might matter for some purposes but not others. Other times these differences can be pretty substantial. Usually the philosophers offering the definitions expect that they are equivalent, and choose one of them rather than the other because it’s easier to work with in the discussion they’re engaged in. But often these expectations of equivalence are controversial. Often it’ll be the case that if we make some assumptions, the definitions are equivalent, but on other assumptions they’re not. And those assumptions might be accepted by some theories but disputed by others.
Sorry about all this. It does makes philosophy harder to learn. But I think it’s better to acknowledge it, and help you see where it’s happening, than to paper it over and pretend it’s not going on.
The alternative would be for your teachers to choose one of the definitions, either the one they like best, or that’s used in the textbook they’re using, and just declare that this is the official definition of a term like “substance”, or “physicalism”, or “reduction,” or what have you. But then sooner or later you’re going to come across some other philosopher working with a different “official” definition. And it can then be confusing what’s going on. Until you get enough experience to figure out what I’ve told you above.
In class I gave an example where we start out associating the word/concept “mother” with two kinds of properties. (We’re simplifying the details about motherhood a lot here.)
It might start out that everyone we consider a mother has both of these properties. Then we start noticing that there can be cases where these properties come apart. There can be cases where one person, Alison, gave birth to you, and another person, Beatrice, is your female caretaker. Who should we say is your mother, in such cases?
You can imagine people arguing about this. One camp might say that the first property is the most important or dominant one, so it’s only Alison who counts as your mother. (Of course, you might still say to Beatrice, “You’re much more important to me than my mother is.”) A second camp might say that the second property is the most important or dominant one, so it’s only Beatrice who counts as your mother. (Then you’d say to Beatrice, “You’re the only mother I have.”) A third camp might say that the word/concept “mother” is ambiguous, and to be more explicit, we should distinguish whether we mean mother in sense 1 (biological mothers) or mother in sense 2. A fourth camp might say that someone only counts as your mother if they have both properties, so in our example neither Alison nor Beatrice is your mother.
For different concepts it might be more reasonable to understand them in one of these ways, and for other concepts to understand them in another way. We said that for “mother,” to us it seems like the third camp is most plausible. But you can imagine people arguing about this, and you can imagine what if they hadn’t yet settled what’s the best way to understand “mother,” in relation to the possibility that condition 1 and condition 2 might come apart.
I said that we’re going to introduce some technical concepts over the coming classes.
As I said in class, it’s going to be relatively easy and straightforward for me to explain notion 3 and the views mentioned in 4. But explaining the notions in 1 and 2 is challenging.
Part of what makes it hard is that with the notions in 1 and 2, the situation is like I described with “mother” a moment ago. And philosophers still haven’t agreed and settled upon whether they want to understand these notions in the way the first camp does, or the second, or the third…
This is one of the places where philosophy is messy and harder to learn. Maybe I should paper it over and just pretend like the messiness isn’t there. But I’ve chosen to be upfront about it.
I hope that despite the messiness, we can still build some understanding of what philosophers are getting at with the notions in 1 and 2. That’s what we’ll be trying to do over the next classes.