3:24 PM
Do you feel like talking about what had not made sense initially but now does? (And what still does not, if anything?) I fully understand if you are not up for, or do not feel like, talking about that now.
@Zanna Btw I think I can say why types in a type theory are called types without becoming too caught up in type theory.
4:53 PM
you started talking about definite descriptions at the point where function symbols had run into trouble when trying to talk about things that didn't exist
and since definite descriptions didn't solve that problem, I hadn't really seen what it was that was preferable
but it seems that it at least requires us to not skim over the dangerous possibility of nonexistent things
@Zanna Definite descriptions use a notation that looks like, and is often said to be a form of, quantification. A sentence (atomic or compound) that contains one or more definite descriptions can be translated into a compound sentence that does not contain any definite descriptions, and resulting sentence will use quantification in a predictable way.
5:08 PM
There are two approaches to the problem of what happens when an n-ary function symbol combines with n terms to produce a term that fails to refer to anything. (I'm folding the case where it seems like it would refer to multiple things into this. For example, if you have more than one paintbrush, then you have no unique paintbrush, so if "
zanna" is a name for you and "paintbrush" is a function symbol that means "the paintbrush of," then paintbrush(zanna) fails to refer to anything.)
I agree with that sentiment, but it's worth mentioning that there are useful applications of function symbols even with this approach. For example, if our universe of discourse is the nonnegative integers and we have a function symbol
s that means "the successor of," then that's no problem. We can write the term sx (where x is a variable of quantification) and, no matter how that is instantiated, it will refer to a unique object.
Similarly, although it would not be a good idea for these to be primitives (i.e., for them to appear in the signature of a theory) since they can be, and are, defined in terms of "∈" and quantification, it's worth mentioning that a number of symbols commonly used in set theory can be regarded as either special notations for particular definite descriptions or as function symbols...
...and they can even be regarded as function symbols even if we operate with the the restriction that a term produced by binding n terms to an n-ary function symbol must always succeed at referring to a unique object.
If we have only set, being a set and existing are the same thing, and every object has a power set. Furthermore, the power set is always unique. Every set has exactly one power set.
The cardinality of a set
S, denoted |S| is informally the number of elements in S. I of course intend to present this more formally later. The important thing about cardinality is that if you have sets A and B and you can pair up every element of A with an element of B so that separate elements of A are paired with separate elements of B and nothing in B is left over, then A and B have the same cardinality.
Except for the cardinality 0, there are multiple sets of each cardinality. But every set has exactly one cardinality.
(To define cardinality formally, not only is it necessary to say what I said about pairing up elements more formally, but also it is necessary to define each cardinality as a particular set. This includes 0, 1, 2, and so on, but also infinite cardinalities, such as ℵ₀, the cardinality of the set of integers. There are many infinite cardinalities.)
Given any two sets
S and T, there is exactly one set S ∩ T that contains all and only those members that are in both: S ∩ T := {x | x ∈ S ∧ x ∈ T}. That is, the intersection of two sets is unique. Similarly, there is exactly one set S ∪ T that contains all and only those members that are in either: S ∪ T := {x | x ∈ S ∨ x ∈ T}.
(I know you are familiar with "∩" and "∪" already; I'm bringing them up as examples of symbols that could be function symbols even if one takes the approach that no term made by binding arguments to a function symbol may ever fail to refer to an object.)
Although sets are unordered, it is possible to create ordered structures out of them. In particular, there are set-theoretic constructions of ordered pairs. Typically the Kuratowski construction is used:
The important property of ordered pairs is that pairs are equal iff their corresponding entries are equal. That is:
The reason that construction is the one that is usually used, and the ways it may be extended to n-tuples for any finite n, is something we should go into in detail, but the above is sufficient for giving ordered pairs as an example of a binary operation that is defined for all combinations of objects in the whose universe of discourse of sets.
Given sets
S and T, the Cartesian product of S and T, denoted S × T and sometimes pronounced "S cross T" (but not to be confused with the cross product), is the set of ordered pairs whose first entry is taken from S and whose second entry is taken from T. That is:
5:58 PM
But many operations are not so nice. This includes operations that are "low level" in set theory, but the most universally recognizable example is that, in most number systems, one cannot divide by zero. So "
1 / 0" is a term (because what qualifies as a term is entirely a matter of syntax), but it does not name an object.
The thing is, when function symbols are introduced, they are often appealing, and even argued for, based on the idea that they facilitate formalizing mathematics in a way that preserves the syntax we actually use to do math. They are also often presented as being permitted only when you can't form any terms from them that fail to refer to a unique object.
But that restriction does not apply, and cannot be taken to apply, to operations in mathematics, especially when mathematics is built on top of set theory (because set theory gives a huge ontology of sets, and mathematical operations like
+ and / will simply not be defined for argument outside a tiny sliver of that universe of discourse).
Because definite descriptions are extremely flexible, and because their syntax reflects the use of quantification that they employ, this confusion does not arise.
You could have
/ as a binary infix function symbol. But not with a rule that says x / y must always succeed at referring to a unique object. If you let x / y fail (when y = 0), then you can do it. Then it works like a special case of a definite description. But the thing about definite descriptions that people don't like is that they can fail. In many, probably most applications, useful function symbols can also fail.
@EliahKagan oh yes I definitely recommend that book. I think it's helpful in a lot of ways. For example if the result seems astonishing and hard to make sense of and people are saying that the electorate are stupid, then we can consider the indictment of the "folk theory of democracy" it briefly mentions
Rather than gathering information about the various parties and thoughtfully weighing up all the factors before deciding who to vote for, most people have very little political knowledge and no interest in acquiring more (in the UK, there is nothing at all in compulsory education about politics and "politics" "political" are negative words), and even those who do acquire information tend to use it to reinforce their convictions rather than challenging them.
So, most people vote based on who they see themselves to be (as a member of a particular community or social group) or on some issue about which they or their community or social group has some particular view.
He also points out that money spent on election campaigns translates into votes ("an analysis by US political scientists found an almost perfectly linear relationship, across 32 years, between the amount of money available to Congressional candidates and their share of the vote")
(also it's really laughably easy to design a fair political funding system, which makes it really obvious that the people who have already bought power through the really obviously unfair system don't want the system to be fair and that Something Is Wrong)
But above all that it starts from the premise that humans are mostly naturally altruistic and compassionate and cooperative and connection-making. We aren't really these ultracompetitive individualistic beings that neoliberalism says we are or should be. (And think about it, aren't most of the people you meet actually pretty kind and friendly and just totally not the kind of person you can imagine deliberately voting for that awful party/leader/policy etc?)
Many aspects of our society and political system have caused us to become alienated and frustrated and to feel powerless etc etc but we should know that the chances are that nearly everyone else is also feeling the same kinds of things
and so we can regenerate community and participatory democracy (and here are various examples of these sorts of desirable things) and reform the political system using the Big Organising techniques that almost worked for Bernie Sanders (but they started too late), because talking to a person who believes in what they're saying will always be more effective than anything else...
but of course without doing that it's not going to happen. Like Audre Lorde said without community there is no liberation.
And now people are saying Labour lost because they were too leftist and we should learn from New Labour, but like so many other traditionally social-democratic parties in the neoliberal era, New Labour did what the book said - they adopted the ideology of their opponents and implemented the kind of policies their opponents would have implemented, and this is not a victory.
6:15 PM
I should have perhaps written that, that, and this with universal quantification, rather then as open sentences.
And two sets that are subset of each other (i.e., two sets where one is both a subset and a superset of the other) are equal:
@EliahKagan There is another reason function symbols do not directly capture the meaning of most operations in mathematics, though. Definite descriptions do not directly capture them either, but they don't appear to--even though definite descriptions are more versatile than function symbols. The set-theoretic operations I said we could use function symbols for are extremely general, and there are limitations that make it difficult to treat them as objects.
But most operations in mathematics that have meaning for particular mathematical structures (such as integers modulo 11, integers, rationals, constructible numbers, algebraic numbers, computible numbers, reals, complex numbers, quaternions, 7-dimensional vector spaces over the reals, and so on) don't apply to the vast majority of the universe if discourse, and we need to be able to treat them as objects.
6:30 PM
@EliahKagan The reason we need to be able to treat them as objects -- that is, why in at least many cases it is important that we are able to regard "
+" in 2 + 3 as naming a particular object in the same sense that "2" and "3" name particular objects -- is so they can be part of mathematical structures.
For example, consider the mathematical function that describes the position of some physical (classical) particle. You put time in and get position out. The first derivative of that function is another function. With this other function, you put time in and get velocity out. Taking the derivative is an operation on a function. To study the derivative, one must be able to treat functions as things that we can talk about and do operations on.
There are two broad ways to do this. One is to use a second-order logic, and represent mathematical functions as function symbols (though the term "function symbol" is no longer quite so appropriate, since a second-order logic can talk about them as names of things; they are no longer purely syntactic). Or if we must also study the differential operator where you put a function in a get its first derivative out--which we must!--then a third-order logic.
In this approach, to do all of mathematics, one needs a higher-order logic, like a type theory. This is the sort of approach that Whitehead and Russel developed in Principia Mathematica.
The other broad approach, though, is to regard mathematical functions as objects. The most popular way to do this is with set theory--functions like f: ℝ ⟶ ℝ where f(x) = x² are constructed as specific sets. Then we can talk about them, and the differential operator is just a function that maps from some such functions to other functions.
One way "⊆" (on sets) differs from "≤" on numbers is that ⊆ is a partial order but ≤ is a total order.
Also, ⊆ is not a partial order on a set (that's why I didn't linkify "partial order" above, but see this WP section.
To clarify, this does not reflect an asymmetry between partial and total orders, but merely an asymmetry on what articles I was able to find quickly.
⊆ is a partial order but not a total order because there exist sets
S and T for which S ⊈ T ∧ T ⊈ S. In contrast, we don't have numbers x and y (in number systems for which ≤ is defined) for which x ≰ y ∧ y ≰ x.
Also, I should have been more nebulous and said we need to treat them a "things." In a first-order theory, all terms that succeed at referring to anything refer to objects. All names refer to objects, and all variables of quantification reign over objects. In second and higher order logics including type theories, this is not so, and what we call objects in first-order logic are called individuals.
In higher order logics, mathematical functions and relations don't need to be individuals and typically are not developed/constructed as individuals, because one has higher-order terms and higher-order quantification. It's specifically in first-order logic that, to do mathematics reasonably, one must have mathematical functions and relations as objects.
(So "partial" is not being used in the sense of "non-total" as it is sometimes used in ordinary language.)
Partial orders that are not not total orders are very common. For example, suppose you have several tasks, some of which depend on the others. You may know you must do A before B, and A before C, and B before D, and C before D.
I don't know how (or if I can) specify that in the code. But selecting it doesn't change the URL that is produced.
This graph adds the edge from
A to D, so that the presence of an edge from x to y means "x must be done before y" rather than "x was specified as a prerequisite for y" or "x is a direct dependency of y". Selecting "circo" instead of "dot" for the engine makes it display more readably.
In this situation, there will also of course be no loops, i.e., we will not have
x < x, since if we did, we would also have x < x. :)
So for satisfiable dependencies, "is a (direct or indirect) dependency of" is a strict partial order, which is why it makes sense to write "<". If we instead had a non-strict partial order, where
x ≤ x and where (x ≤ y ∧ y ≤ x) → x = y, it would make more sense to write "≤".
S is said to be a proper subset of T, and we write S ⊊ T, when S is a subset of T but is not equal to T. In this situation, we also say T is a proper superset of S and write T ⊋ S. That is, T ⊋ S means T is a superset of S but is not equal to S.
⊆ ("is a subset of") is a non-strict partial order. But ⊊ ("is a proper subset of") is a strict partial order.
@EliahKagan Likewise, ⊇ ("is a superset of") is a non-strict partial order, but ⊋ ("is a proper superset of") is a strict partial order.
In this graph, which is the same as that one except the loops (self-edges) have been removed, an edge from x to y means
x ⊊ y.
As far as symbols go, we have "⊆" (is a subset of), "⊊" (is a proper subset of), "⊈" (is not a subset of), "⊇" (is a superset of), "⊋" (is a proper superset of), and "⊉" (is not a superset of). Unambiguous single symbols for "is not a proper subset of" and "is not a proper superset of" are rare; I don't know how to write them in either TeX or Unicode. Of the symbols just mentioned, "⊆" is overwhelmingly the most commonly used.
But you may also see the symbols "⊂" and "⊃", unfortunately. By "⊂" some authors mean "⊆" ("is a subset of"), while others mean "⊊" ("is a proper subset of"). Likewise, by "⊃" some authors mean "⊇" ("is a superset of"), while others mean "⊋" ("is a proper superset of"). Furthermore, "⊃" is separate also sometimes used in logic to mean "→" (i.e., the conditional, "only if"). So it is best to avoid using "⊂" and "⊃" in any meaning related to subsets and supersets.
It is tempting to use "⊂" and "⊃" to mean "⊆" and "⊇", respectively, because they're faster and easier to write and we speak of subsets far more often than of proper subsets (and special of supersets far more often than of proper supersets). And it is tempting to use "⊂" and "⊃" to mean "⊊" and "⊋", respectively, by analogy to "<" and ">". It's not worth it, though.
This is perhaps a good time to mention... I've been using Unicode rather than TeX out of the belief that it would be more convenient for you, but I do not recall ever actually having asked you about your preference. There are user scripts that render TeX in chat. I have noticed that you don't type symbols not on keyboard as much as I do. TeX might make this easier for you, even if you don't yet know any TeX.
OTOH, the reason MathJax (the most common way to render TeX on websites) is not enabled in SE chat, without a user script, is performance. If your computer is slow, I anticipate that rendering TeX in chat may make things slower. I am also not sure if you'd be able to get it to render when you're using chat from a mobile device.
If you do want to try using TeX in this room, I suggest ChatJax++. (A small modification to the code would be needed to enable it for this room, since the site this room is associated with does not have MathJax enabled on it, and ChatJax++ checks for that. But that should not be a problem.) It's up to you.
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