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5:47 AM
@Zanna I was pleased to hear you say that, in that it gave me hope other British people will say "math" instead of "maths" as well. :)
I had meant to ask you what math you were referring to. I imagine it will eventually become tiresome, if it hasn't already, for me to disagree with you every time you suggest you don't know something. After, you know what you do and don't know better than I do! However, my understanding is that math is a subject in which you have considerable knowledge and experience.
@EliahKagan (that was me attempting to search the chat for math, failing to press Ctrl, and not noticing until it was too late to delete the message)
6:04 AM
@EliahKagan :D sometimes I adopt something from another dialect because it seems better
That sigh was expressing sadness that I'm missing out on the fun to be had with radians
Not expressing dread of the learning to be done
I think what I would actually like would be to take some kind of course in high school level math so I could fill in all the gaps I have
Which topics?
Well it's a case of barely knowing what I don't know. I usually help kids who are either younger than 12 or struggling with simple stuff (I'm not a math teacher) but sometimes I get asked to help with something I've never seen. I had never seen surds until one of my students showed me. But in general math is something I find hard. There are algebra things where I'm like hmm am I allowed to do this operation, will this be the same if I do that
Really I want to learn everything again and more
Wish I had continued with math after the compulsory bit, even though I would have found it really tough
We only get to choose 4 topics at most. I did literature, philosophy, psychology and photography. (Books are my favourite thing, but I quit literature after a few months)
This is for high school in the UK?
The four topics, I mean.
6:24 AM
@EliahKagan that's what we call sixth form... from age 16-18. But these days there is different stuff in the 14-16 curriculum and i want to do all that
I had one student who I helped with most of the subjects he did from when he was 13 to when he was 18 because he was very dyslexic. That was awesome for me because I got to do so many different things. When we were doing GCSE math papers it was great fun for me because there were lots of types of problems I had never seen or totally forgotten about
GCSEs are compulsory exams at 16
Many times I learned something just well enough to explain it to the student but since then I've forgotten it
I don't have that problem much with things that can be explained narratively
I remember science things well
Can't math be explained narratively?
I don't know, sometimes
I feel that things in physics can be explained either narratively or quantitatively. These are the labels I myself have given to different ways of thinking through the same thing that I try to bring together
6:40 AM
I think that makes sense. But can you give an example?
Of narrative and quantitative explanations of something in physics, I mean.
I am also wondering if some abstract interpretations of mathematical operations are narrative, or almost so.
For example, is the combinatoric interpretation of arithmetic merely quantitative?
(By this I mean, for example, the interpretation of a × b as the number of ways to choose from a things and then from b things.)
6:58 AM
If I think for a while, I'll hopefully come up with some helpful examples. Commonly there is a question in the exam about a skydiver, what happens when the parachute opens? Why do they slow down?
The student will have drawn this free body force diagram with weight and air resistance or even a series of them but those diagrams don't show the current state of motion. The kids don't understand why things reach terminal velocity. They don't understand why the parachute will decrease the terminal velocity.
I have them talking through the situation over and over. I will get them to put some arbitrary numbers into the situation. Then we will draw graphs which are awesome because they tell a story with numbers
@EliahKagan seems like a story to me
When I'm explaining how to do math things I am always telling some story, putting in concrete stuff, drawing pictures, because those are the kinds of explanations that seem to work for me, I mean people understand them
There are concepts like momentum that we can describe through lots of scenarios but I usually tell students that the description they should remember for momentum is that it's what you get when you do mass x velocity.
There is another common question in exams about how seatbelts improve safety and what they have to write is sort of counter intuitive. They have to say it increases the time taken for the change in momentum
I find that kids can't get their heads around this until I point out that the time taken for the change in momentum is the denominator in that equation for the force that's required for the change in momentum to occur. This is the force acting on the thing accelerating (that is, the person in the car). What happens when we increase the denominator? Ohhh
7:15 AM
This reasoning strikes me as quantitative as well. What would a quantitative, rather than narrative, explanation sound like?
Well that was the quantitative explanation of the situation
Oh. What's the narrative explanation?
Was that less of a story than the example I gave about choosing from a things and then from b things?
I wonder if one of the reasons the exam question about seatbelts is counterintuitive to students is that it is a factually incomplete account of how seatbelts improve safety.
@EliahKagan I don't know :)
@EliahKagan yeah :(
Some exam papers are better written than others. In June I marked two physics papers. One was very well written imho, the other not so much
Is the goal of the seatbelt question to test students' knowledge that force is the rate of change of momentum?
@EliahKagan well we try to think through the situation, but in this case I think to explain without thinking in terms of equations you have to keep breaking it down to a point where it doesn't flow as a story hahaha
@EliahKagan yeah
There are various reasons why we have such silly questions
7:31 AM
@Zanna It seems to me that this may illustrate an additional benefit of the narrative approach.
I mean... is it that it doesn't flow as a story, or is it that the story is compelling because it does not explain the actual mechanism of injury, which relates to deformation of the human body?
The exam problem seems to be telling a story about how injury is explained by impulse and momentum (and their derivatives).
But that strikes me as misleading or inaccurate. I think the injury is explained by work and energy (and their derivatives).
@EliahKagan * or that the story is not compelling
7:48 AM
This is a very inexact question, because how severely someone is injured is not really just a number... but if I've been shot out of a large slingshot (like they have at circuses and call "cannons") and something goes wrong, and I hit a wall at speed 2v, does that hurt me only twice as much as if I hit the will at speed v?
@EliahKagan yes I agree with you
they sometimes have to say the same thing about crumple zones etc
but it's obvious that the zone is crumpling instead of you
that's what I find counterintuitive about it
It seems to me that, in math classes, math is often taught as a story about the manipulation of symbols.
I think this still a story, but not a very illuminating story, and I think it causes a lot of problems.
This is based on my knowledge of math classes (at the primary and secondary level) in the United States, though.
I cannot as strongly support my belief that math is taught this way in other societies (including the UK) as I think I can for the US.
I feel that math literacy (or numeracy) is really poor in the UK
Why is this?
@EliahKagan As a sort of shallow example, there's something weird going on with how students are taught to remember "FOIL".
There's nothing special about the case where each factor is a sum of two terms.
8:05 AM
I really don't know what goes wrong (except that a very great number of children have completely disengaged from school and learning emotionally by the age of about 14 or 15 due to school being a horrible environment for them for various variable reasons) but it must be the way of teaching. I don't know how math should be taught. When I started learning algebra, I was totally confused because the teacher was giving us some tricks to do it.
Then in the next year we got streamed and I had the head of maths teaching me for the rest of secondary school. He was my favourite teacher and he would say if you can't understand anything you've got no hope of remembering it. That way of thinking helped me with everything.
@Zanna I definitely think your parenthetical point is very important and should not be ignored.
@Zanna Or at least of applying it to new situations.
@EliahKagan yes
The reason I call my objection to "FOIL" shallow is that it's a procedure; it's not intended to say what a formula to which it applies means.
@EliahKagan do you mean, that doing that in some consistent order is only a way of making sure that all the terms get multiplied by all the other terms and that we end up with something written in the conventional style, and the students won't realise that because it's presented in a way that makes it seem critical?
I mean I think it's less useful than just applying the distributive property itself.
And also that, for example, (x + y)(a + b + c) is not conceptually harder than (x + y)(a + b), but "FOIL" specifically applies only when the form is (x + y)(a + b).
8:15 AM
one thing I find better about the USian / American way of talking mathematically is that y'all say "negative 2" instead of "minus 2" when you mean the number -2. Because in British English we say "three minus two equals one" and also, "the temperature is minus two" and I really think that this is the reason why kids find it so hard to deal with any operation at all that involves negative numbers
I hear (and have heard) "minus 2" quite a lot, but yes, for some reason, people in the US seem to think "negative" is more correct or more formal than "minus" for forming negative numbers.
@EliahKagan these little tricks can end up being very limiting because you get thrown off as soon as you meet something that it won't apply to
a lot of kids get taught algebra the way the first teacher I had for it taught it
@Zanna Yes, they can. Though my problem with "FOIL" isn't that it limits other cases but that it doesn't really help the one case is for.
@Zanna Can you say more about that approach that was bad (or at least that was bad for you)? Was it entirely based on tricks?
@EliahKagan If things were bad enough, then only shallow objections would be possible, because nobody would ever say or think anything about what anything means. Though my deep objections are more severe, my ability to come up with them shows things are not as bad as they could be.
the trick is, "if you change the side, change the sign"
so if we have on the left side -4, we can instead put +4 on the right side
Yes, or as I (and I think you) would prefer to put it, we can add four to both sides without affecting the truth of the equation or inequality. :)
"if you change the side, change the sign" is a case where a specific syntactic transformation is being taught. I think the effect of the way math is taught is that many, many students come away from it with the belief that math is about the meaningless manipulation of symbols.
8:23 AM
yes. I find that students who are taught to "change the side and change the sign" will soon be getting everything horribly wrong because they don't know why they are doing it
@EliahKagan yessss. I find the thing I most often repeat, to the point that it's a joke, when trying to help people with algebra is "is that equals sign still true?"
science teachers give another algebra trick, which is the formula triange
I don't think I know that one.
because all the algebra we have to do in science up to GCSE is very simple
but still, the kids haven't learned to do the most basic algebra
so the science teachers give this trick
I'll show you, let me draw it...
@Zanna What I find interesting is that, while this sort of confusion can occur anywhere, in practice it seems to occur specifically in mathematics, and it does not seem to go away with greater proficiency.
that is interesting
this is what I call a formula triangle
what we are taught to do is to cover the quantity we want to find
then we can see which operation we need to do
like, say we want to find F, we have m | a so we have to do m x a
That embeds information about the relationship between multiplication and division, and the way it does so is readily discoverable and invariant with respect to the terms written in the different placeholders, which makes me wonder if it may actually be conceptually useful.
8:35 AM
this works ok for some but very quickly becomes confusing
Yeah, I don't much like the idea of using it...
like, they learn to do it in the lesson, but later when they get an equation and they try to make a triangle out of it, they do it wrong
this method is supposed to help students who don't know how to do algebra, but my experience is that students who know how to do algebra can use this method (but they don't need to), and those students who don't know how to do algebra can't use this method
I mark exams so I see thousands of these formula triangles drawn and what happens next
@Zanna This seems like writing all the factors related to a particular physical quantity along with itself in a triangle. No really helpful, is it? Especially because you have a chance of mixing up ól operators and constants
I added the example for KE because this is an example of how even the students who can do algebra are misled by the triangle method
I have put v^2 in one box because that's the only way it can work (meaning the triangle hasn't told us we have to take the square root of KE/m to get v, if that's what we want)
but students almost always put m | v | 2
Hey isn't KE ½mv²? Or do you omit constants while writing the basic formula?
8:41 AM
yes you're totally right
it's 1/2 mv^2
the 1/2 should be in the box with the m
and that also gets written separately
I hate formula triangles
they are evil
and math teachers are always telling science teachers to stop using them
@BaiduryaMathaddict Yeah, I don't think it has any advantages compared to writing it as a fraction and saying that fraction is equal to 1 -- at least so long as the denominator does not come out as zero.
@Zanna Do the science teachers keep using them because of a poor opinion of the math teachers' judgment?
I can understand how science teachers might think math teachers don't know what they're doing... even when that belief would sometimes be misguided. :)
@EliahKagan I think they keep using them because the students find it easier for the duration of the lesson to do it that way
it's easy to learn a trick, but then they forget some detail of it and they haven't understood anything so the trick is worse than useless to them
@Zanna it occurs to me that I could produce a piece of research on this that would convince science teachers to stop using formula triangles
You should do that research! :)
but I am not sure I would be allowed to do that because the material is confidential
Oh, perhaps you should not do that then.
Confidential how?
8:48 AM
I could do the research and then ask the exam board how to go about correctly sharing it
I was reminded of something about numeracy that I had wanted to share, because it is broadly relevant, especially to what we were talking about before: ams.org/notices/200502/fea-kenschaft.pdf
@EliahKagan exam scripts, I mean the things students have written, are not to be shown to anyone outside the marking process
@Zanna I think the meaning of = is tied up very closely with the distinction between things and their names, and I have observed confusions about this in people's thinking about mathematics to be common and very conceptually deep, including among mathematicians. In contrast, conceptually deep confusion about the difference between things and their names does not appear common outside of mathematics.
For example, if we discover that your friend Jill is the same person as my friend Nancy--that is, that "Jill" and "Nancy", as we are using them, are different names for the same person--then we are not going to have any problem realizing that a raincoat that belongs to Jill is a raincoat that belongs to Nancy.
Nor are we going to have problems in the other direction: we aren't going to be confused about how it could possibly be that Jill is Nancy, even though Jill is four letters long and Nancy is five letters long.
> in my first visit in 1986 to a K-6 elementary school, I discovered that not a single teacher knew how to find the area of a rectangle
this reminds me that I was working with this boy last year who was generally super smart and awesome at maths but could never get how to find the area of a rectangle. I just couldn't find a way to explain it to him that he would find helpful :(
9:04 AM
I fully understand there is likely a limit to how much detail you can give, due to the importance of protecting privacy. Within the limits of that, however, I am interested in what didn't make sense to him about each approach (and thus, about which approaches you tried).
@EliahKagan Right, me too. This is interesting...!
I think one thing that can be confusing about lengths are areas is that they are not conceptually numbers, but their presentations in math classes sometimes gloss over this.
To be more precise, there are questions we can ask about numbers and get meaningful answers, that have no meaningful answers when asked about dimensionful quantities (in the sense of dimensional analysis) like lengths, areas, mass, and so forth. We can take any particular (strictly positive) length and call it the unit length, and the most we've done is change our coordinate system.
@EliahKagan Similarly, {0, 1} = {1, 0} (due to the axiom of extensionality, i.e., sets are equal just when they have the same elements), but you might be surprised at the prevalence of the view that this shows = doesn't really mean "is the same as", because the second symbol (or, if you prefer, first-listed element) in {0, 1} is a 0 but the second symbol (or, if you prefer, first listed element) in {1, 0} is a 1.
Also 1 + 1 = 2, yet (depending how you count them) there are at least three symbols in 1 + 1 = 2 but only one in 2; I have also heard the claim made that this means = doesn't mean "is the same as."
If I wanted to make my prose as formally correct as possible, then I should write things like "there are at least three symbols in '1 + 1 = 2'" rather than "there are at least three symbols in 1 + 1 = 2". Note, however, that even sloppy quoters don't have any trouble understanding that it is the names, and not the thing they name, that are being talked about when one says, "Jill is four letters long and Nancy is five letters long" (or "My name is Eliah.").
These are claims I have heard from professional mathematicians with doctorates in mathematics!
It is with slight trepidation that I recount this, because any specific account runs the risk of being confused with the others: I've had multiple conversations with mathematicians about these topics, and they have not all been equally unimpressive. Most of what I am drawing on here is not from conversations I have personally had with mathematicians in public on the Internet.
A mathematician I knew IRL, who has (and had) a doctorate in mathematics and whose career was (and I presume still is) teaching mathematics to college students and doing research in mathematics, once made that specific claim to me about how things that are equal are nonetheless not always the same thing because {0, 1}, {1, 0} are equal.
9:32 AM
well I was helping his older brother at the same time so probably I just didn't give it enough attention. He would say he got it (but I wouldn't be convinced) and do it correctly and then a few weeks later be worrying about it and asking for help with it again.

What I would do is draw some rectangle and then divide it into squares of side 1. The first rectangle would be a square of side 1. How many squares? 1. So if this length 1 is 1m we can say the area is 1m squared. Then a rectangle made of two such squares side by side. How many squares? 2... Then we'll make one with sides 2x2... How
@EliahKagan oh ho, this set theory stuff is actually the number 1 topic I want to learn
I never saw anything of it in school
now I see kids in year 8 and 9 with all these beautiful symbols in their books and I'm like "what is that?"
@EliahKagan oooh
@EliahKagan My impression, which is of course unscientific as otherwise I wouldn't refer to it as an "impression," is that these sorts of mistakes are extremely uncommon among mathematicians who work on foundations (as one would expect) or topology, and otherwise fairly widespread among mathematicians. My degree of belief in that impression is somewhat weak.
However, I feel more confident in what I'd call my strong guess that the confusion between objects and their names is part of the culture of mathematics at every level, rather than something kids go to school with presystematic confusions about that they must then unlearn.
@EliahKagan haha it seems unlikely indeed that kids go to school with that problem
I think this might relate to the source of humor when you say "is that equal sign still true?" In non-mathematical contexts, the same principles seem obvious to everyone.
although one of our guests on Saturday evening teaches reception class (age 4-5) and she has a child in her class who doesn't know his own name :(
@EliahKagan haha yes
@Zanna Did he understand the reason behind the more basic case of a 1-by-1 square?
Also, did you try with units (for example: meters and meters^2), and if so, did that help at all?
9:43 AM
maybe I don't understand the reason myself
@EliahKagan yes I would say the units and write them. I don't know whether that helped
his older brother was doing compound areas
I don't think I know the term "compound area," though it rings a bell.
that one was comfortable finding the area of a rectangle by doing length x height (or width or whatever)
@Zanna Well, the sort of thing I mean is, if one adopts the convention that the area of the unit square is 1 (or, better, 1 unit^2), that satisfies the hypothesis that the area of a rectangle is the product of its length and width, and any other rectangle you can make from those square satisfies it too. I basically just mean, did he understand that a unit of area was different from a unit of length, and what it meant to be a unit of area.
@EliahKagan oh simply this is what we call finding areas of shapes that are irregular but can be broken up into two or more shapes that are easy to find the area of, rectangles or triangles. Such as an L shaped patio or a trapezium or a rectangular pond with a square island in the middle
9:49 AM
he had no problem finding the area of a rectangle, but he couldn't do these tasks
I wonder if, for some students, volume might be more intuitive.
He would count up the lengths of all the sides on the figure and multiply them all together
Of a plane figure?
@Zanna Set theory is very fun and widely used. I recommend it.
9:53 AM
that student had confidence issues. He would pretend to know how to do things that he did not know how to do. He would insist on doing them himself and very secretly look up how to do them on the internet. I would tell him that looking up how to do things on the internet is perfectly fine and he doesn't have to do it secretly. That not knowing how to do things is universal and I myself look up how to do things I don't know how to do on the internet on a daily basis.
@EliahKagan I don't know at what level of depth you wish to study it.
But because of the way they're taught kids think that they have to know things already. Looking things up on the internet is cheating, just like helping someone else is cheating
That's messed up.
@EliahKagan any level at all would be good. I don't know the first thing about it :)
Unfortunately, I don't think I have good recommendations to make about sources on it. I do have one idea...
10:00 AM
oh yes?
I was thinking plato.stanford.edu/entries/set-theory/basic-set-theory.html might be a good recomendation, but I have not recently read it (though I am looking at it now).
@EliahKagan maybe not, maybe that was the problem.
@EliahKagan It is notationally heavy though. This is good for its intended purpose, which I believe is to prepare readers to read other encyclopedia articles about set theory and topics related to it, but one need not be introduced to quite so much notation before getting onto the interesting parts of set theory. (The notation exists for a reason, but it is not all needed to understand things like functions and Russell's paradox.)
Yeah, I don't think that article would be all that accessible if one doesn't already know some set theory.
I do not say that to discourage you from reading it though.
either I am quite good at remembering symbols, or people are overly generous in praising my ability to remember symbols in the the contexts that has happened
10:16 AM
> An hour! With eight-year-olds! Totally focussed.
this reminds me of the week being a trainee TA in primary school I did as part of my (secondary) teaching qualification
I witnessed something I will never forget, that I think about every time someone complains about attention spans, especially of kids
What was that?
I was in a year 2 class. That's 6-7. The teacher was so great. The class loved her
One day she told me we were going to do writing. They had already done some preparation for the writing so she would only take a few minutes to start them off
The kids came in and she told them to get out the preparation stuff, words they had collected or whatever it was
they were a lively and enthusiastic class, but when she said ok I'm lighting the silent writing candle they all sat and wrote silently for an entire hour
I don't think I would believe that actually happened if I hadn't seen it
@Zanna Have you done formal logic?
That was a very definitively stated answer. :)
10:28 AM
haha :)
However, I don't know which of several possible attitudes or emotions the ! expresses. :)
I am guessing that it is not a factorial symbol in this context.
it means, this is yet another field in which I am amazingly ignorant
@EliahKagan that article is wonderful
I agree.
I should clarify that "set theory" is not a specific theory, but a kind of theory. There are various set theories. Also, this is with the meaning of "theory" that applies in formal logic and mathematics.
I know of two major approaches to learning set theory: the more formal approach and the less formal approach.
The less formal approach is usually followed in whatever undergraduate mathematics course introduces foundations of mathematics and teaches proof techniques.
In order to get qualified teacher status here at the time I trained, we had to pass tests in English and math... I can't remember for sure what they were called now, maybe "basic skills". The math test was quite hard for me, but it was entirely mental arithmetic. Stuff like, "there are 173 children going on a school trip and the proportion of adults to children required on the trip is 1:8. How many coaches with a capacity of 60 people will be needed for everyone going on the trip?"
@EliahKagan Traditionally, at least in the US, this was done in linear algebra, but nowadays it is often covered in another course (and linear algebra is often taught in a way that dos not require it). That may be a specific course just for that, or it may be another course, such as real analysis. There are a number of textbooks for such courses.
@Zanna For a bit I was confused by the question until I understood which meaning of "coaches" applies.
10:40 AM
oh haha I had to think for a minute what the other meaning of "coaches" was
The more formal approach is the one taken when set theory is introduced late in a formal logic class, or when set theory is covered for its own sake and knowledge of formal logic is presupposed.
@EliahKagan but I am not at all good at understanding explanations of what things mean or how to do them unless I get to practise doing them a lot
In what I'm calling the "less formal approach," formal logic is still introduced and used, but formal logic and set theory are taught together; formal logic is not extensively covered before set theory is introduced. Set theory may not be fully presented--by this I mean, the axioms of a formal set theory may or may not be explicated fully--and formal logic is almost invariably not fully presented. I am not criticizing this approach, btw.
@Zanna I think that's universal or nearly so.
So, the articles in the Stanford Encyclopedia of Philosophy, including that article I linked to take the more formal approach. But the more formal approach can be taken more accessibly than that article--for example, parts of that article assume knowledge of forma logic rather than developing it.
Since set theory is the number 1 topic you want to learn, you might want to start with something like the less formal approach, since that doesn't painstakingly present formal logic first.
@Zanna I'm not sure what does convey well what sets mean. I think that, usually, in teaching technical subjects, the history of the subject is emphasized too much. But in set theory, I think it might not be emphasized enough. Sets are very intuitive to many people... so intuitive, vital information about why are they are used is often not conveyed.
Do you know what a set is?
(I ask this because you said you don't know anything about set theory.)
11:00 AM
@EliahKagan no, I don't, but I think it's a group of things
Yes, a set is an unordered collection of things.
A sets knows its members and doesn't know anything else.
That's an immensely informal characterization. What I mean is that one can ask, given any thing and any set, if that thing is a member of the set.
I'll use "object" to mean "thing." Sets contain objects. Sets also are objects. One can have a set whose members are sets. In the most popular set theories, everything is a set. There are no objects that are are not set. In those theories, when one says, "for all x, ..." that is the same as saying "for all sets x, ..." (except that, in those theories, there is not typically any formal way to express that something is a set, since there is no need for that, since everything is a set).
ok, this seems fine
@EliahKagan In formal logic is there a notion of a domain of discourse. When one say "there is some x such that" or all x are such that" one means this within that domain of discourse. Even set theorists who do set theory as part of a wider philosophical project do not--as far as I am aware--usually intend to make a metaphysical claim by choosing a set theory where everything is a set.)
I don't want to overstate my claim: there are set theories in use that have things that are not sets.
@Zanna Cool! :)
Because sets are defined by their elements -- by what objects are their members -- two sets with the same members are the same set; they are equal.
The more obvious claim that two sets that are equal have all the same members is also true.
Set theory can be developed on top of an underlying logic that has a notion of = or one that does not.
If your underlying logic supplies identity, then "equal sets have the same members" has to hold.
This is very closely related to what you were talking about, about how the equals sign is still true. :)
That is, if S = T and x ∈ S, then x ∈ T. S and T are the same thing, so everything true of S is true of T.
"∈" is pronounced "is a member of" or "is in".
11:17 AM
that's clear :)
However, you could imagine ordered collections, or collections that bear grudges against specific people, or whatever, that have more information that just what objects are members of them.
yes :)
So one's underlying logic's notion of = is not sufficient to tell us that sets with exactly the same members are the same set. You need an axiom for that. This is called an axiom of extensionality.
One of the appeals of the set theory is that the machinery of sets can get a lot done, including, it seems, most of mathematics. It can even get = done. That is, you can do set theory on topic of a logic not equipped with identity, and it works.
When this is done, one does not have an axiom of extensionality (not in the above sense). Instead, extensionality becomes the definition of =.
That is, if our underlying logic doesn't equip us with =, we say S = T means "for all x, x ∈ S if and only if x ∈ T"
Although the = is sometimes used in bizarre ways, including in mathematics, including in ways that are highly useful, that's not what we're doing here. That is, even when your underlying logic doesn't provide identity, you can still use = with the expected meaning and with the usual benefits so long as your system can never distinguish between things that are equal.
We don't have that yet, though, because saying that S = T means S and T have all the same elements isn't enough to let us disprove all conjectures that say something is true of S but not of T.
When your underlying logic has identity, this is taken care of already. Nothing can be true of S but not T when S = T. (Basically, what it means for a logic to be equipped with identity is that it provides a = predicate and an axiomatic basis to prove everything of that form.)
When your underlying logic does not have identity, and you define ideneity as meaning "have all the same members," it is not taken care of, and one needs an axiom for it.
Fortunately, since our only predicate is "∈", there's only one thing left to pin down: sets that are equal are members of all the same sets. Or to put it without explicitly talking about identity: sets that have all the same members are members of all the same sets.
what does it mean that our only predicate is "∈"?
11:32 AM
@EliahKagan a more formal way to write "S and T are members of all the same sets" is "for all x, S ∈ x if and only if T ∈ x"
@EliahKagan oh hahaha but that sounds completely different
@Zanna The two things in that same message or two things in different messages?
oh no my bad
I meant that "for all x, S ∈ x if and only if T ∈ x" seems different from "for all x, x ∈ S if and only if x ∈ T"
@Zanna Yes, that's because they're totally different.
They do not claim the same thing at all.
yes, I realise that
11:34 AM
Cool! :)
@Zanna I have not forgotten that you have asked this, I'll reply shortly.
I misread the first part of that message in a way that made me think they were supposed to be the same
@EliahKagan I'm trying to resist asking questions since I don't want you to feel obliged to explain set theory to me!
@Zanna You should most definitely not resist asking questions!!!
hahaha :)
So, when your underlying logic gives you identity, you assert an axiom of extensionality that says sets with all the same members are the same set. Your underlying logic's axioms of identity are what let you know that a set is a member of all the same sets as itself.
But when your underlying logic doesn't give you identity, you define identity with the meaning that two sets are identical just when they have all the same members. Then you assert what I have often heard called an "axiom of intentionality" that says sets are the same (which here means, they have all the same members) if and only if they are members of all the same sets.
@Zanna All the things you can say about anything come down to what's a member of what.
In a set theory, one doesn't define the meaning of "∈". Instead, it is a primitive (much as "point" is a primitive in traditional presentations of Eucludean geometry) and how we interpret it is elucidated by the axioms that contain it.
However, there's also a more down-to-Earth question there that I should answer. The meaning of predicate in logic is motivated by and similar to, but not the same as, that of a predicate in a natural language.
In formal logic, when one formalizes a sentence like "Smith is away", one will usually decide to us a predicate for "away"; one might write that sentence like As or Away(smith)
There, the predicate for "away", spelled A in one dialect and Away in another, is unary -- it takes a single argument.
You bind one argument to it and you get a sentence.
But the predicate A by itself and the name s by itself (or: the predicate Away by itself and the name smith by itself) are not sentenes.
One can have predicates of higher arities.
For example, I might use F to mean "found" and express "Smith found the towel" by Fst or Found(smith, towel). That's a binary predicate.
11:49 AM
the niceness of your examples always makes me smile
Thanks. :)
@EliahKagan the order of things matters in this F
@EliahKagan Though probably not the same, some of these examples--as well as the approach to devising examples--are motivated by Methods of Logic by Quine and, to a lesser extent, Language Proof and Logic by Barwise and Etchemendy.
@Zanna Correct.
I can think of two things you may mean by that. They are both correct.
One is that the order is syntactically meaningful, which is true.
The other is that changing the order may change the truth value of the sentence that is formed. That is true with some predicates and not others, and this F is one such predicate.
So, the sentences formed by binding n arguments to an n-ary predicate are called atomic sentences.
So As is an atomic sentence. So is Fst.
An example of a sentence that is not atomic is: As and Fst
That is more commonly written with a symbol for "and," often "∧". So that sentence would commonly be written as As ∧ Fst. In a more verbose dialect, it might be written Away(smith) ∧ Found(smith, towel).
There are various phrases used for "∧" and the like. I (and a number of people) call them truth-functional sentential connectives.
They connect sentences together to form new sentences, so they are sentential connectives. And the truth values of the resulting sentences are determined entirely by the truth value of the sentences you started with. That is, As ∧ Fst is true only when As is true and Fst.
I didn't need to know what the A and F predicates meant to know about this relationship between their truth and that of the compound sentence built from them.
@EliahKagan I was thinking of both of the things you mentioned
The other most commonly used truth functional sentential connectives (but note, notation varies) besides "∧" (and), are "∨" (or), "→" (only if), "↔" (if and only if), and "¬" (not).
For example, ¬As expresses the denial of "Smith is away".
All these symbols vary, but especially the symbol for "not".
~ is common. A horizontal bar above a sentence is also sometimes used to express its logical negation.
12:05 PM
my keyboard has that ¬
Nice. Mine doesn't. :)
Note that the syntactic function of these truth-functional sentential connectives work different syntactically than predicates.
Although I am partial to calling them expressions, the most common word for the sort of thing that can be an argument to an atomic sentence is a term, so I'll go with that.
An n-ary predicate combines with n terms to produce a sentence. (That sentence is an atomic sentence.)
In contrast, an n-ary sentential connective combines with n sentences to produce a sentence.
All the connectives shown above are binary, except "¬" which is unary.
So, besides F, you know some other binary predicates. Because we've used two of them, as they are formally used.
Predicates are often written in prefix position as shown with those examples. But "=" and "∈" are also predicates. They are binary infix predicates.
what's this infix?
In the same way that prefix notation puts something before things that attach to it, and posfix (or suffix) notation puts something after the things that attach to it, infix notation puts something between the things that attach to it.
I have not yet described a logical foundation for terms other than simple names (like "Smith" or s or Smith, and like "the towel" or t or towel). But an example from mathematics is, I think, helpful here, in illustrating fixity.
In x + y, the operator + is infix.
In f(x), the function f is prefix.
In the factorial notation n!, the ! is postfix (or suffix).
Bringing it back to logic, suppose = were prefix. Then instead of writing x = y, we would write =xy in a terser dialect or =(x, y) in a more verbose dialect.
That is, if = were a prefix predicate, then we would use it like F or Found.
12:21 PM
ok, I should have realised that :)
So, in a set theory where the universe of discourse is sets (which is not so for all set theories, but is common), your primitive predicates are either "=" and "∈" or just "∈". Depending on whether or not your underlying logic supplies identity.
By "primitive" I mean that we don't define it in terms of something else.
ok :)
A more modern way to say that a predicate is primitive is to say that it's in the signature of the system, but I think talking about what is primitive is adequate for now.
So, if your underlying logic doesn't supply "=" then your set theory (whose universe of discourse is just sets) has "∈" as its only predicate. That is, even if you have a formal way of introducing new non-primitive predicates, the meaning of any sentence that uses them can be expressed fully in a sentence that uses "∈" as its only predicate; furthermore, sentences with predicates you introduce for convenience can be mechanically translated to those that do not use them and use only "∈".
One philosophical approach to such a system is to say that it doesn't even have any other predicates and that you're just writing shorthand or that you're using a different language that represents more complex things in its language (though there are some problems with this).
So, what I am about to say should be taken with a vast asterisk.
Which I'll spill the tea on shortly.
In a system whose only predicate is "∈", everything you can say about a sets is reducible to a sentence all of whose constituent atomic sentences have the form s ∈ x or x ∈ s for some term x.
Also, because the fonts are weird, I should clarify that is how SE chat formats ∈ as code.
If we define "=" by saying s = t means "for all x: x ∈ s ↔ x ∈ t" then we can never distinguish sets by their members. That definition of = effectively expresses that. It is an extensional definition of "=".
(Recall that "↔" means "if and only if".)
This is to say that, by definition of =, it can never be true that s = t but, for some x, it's true that x ∈ s but false that x ∈ t.
But there remains a possible way to distinguish s and t even when s = t. What if, for some x, it were true that s ∈ x but false that t ∈ x?
If = were provided by our underlying logic, then our axioms of identity would let us refute that; since we would know s = t, our underlying logic would let us replace one with the other. We could start with s ∈ x and replace s with t to yield t ∈ s.
But when our underlying logic doesn't furnish us with identity, = means only what we have taken care to ensure it means.
So we are in need of an axiom that says, "for all s, for all t, for all x: s ∈ x ↔ t ∈ x"
12:42 PM
which is "intentionality", right?
I have often heard it called that but I am not sure if that word is really correct. I will try to remember to look into it. But yes.
@EliahKagan So, that is obviously an error and not what I meant to write, though. (Because not all sets are members of the same set. Some sets are different!)
What I should have said was:
So we are in need of an axiom that says, "for all s, for all t: (s = t ↔ for all x: (s ∈ x ↔ t ∈ x))"
That is, no matter what sets s and t you pick, they're equal (meaning: they have all the same members) if and only if they're members of all the same sets.
More formally but still in prose: No matter what sets s and t you pick, they're equal (meaning: for all x, x is in s if and only if x is in t) if and only if, for all x, s is in x if and only if t is in x.
And that axiom (of "intensionality") can be written this way if one doesn't wish to use one's non-primitive = symbol:
for all s, for all t: ( (for all x: (x ∈ s ↔ x ∈ t)) ↔ (for all x: (s ∈ x ↔ t ∈ x)) )
If you find that confusion, note that the xes on either side of the central "↔" have nothing to do with each other. I could use a different variable on one side than on the other. That is, I could write this:
for all s, for all t: ( (for all x: (x ∈ s ↔ x ∈ t)) ↔ (for all y: (s ∈ y ↔ t ∈ y)) )
@EliahKagan yes I see that
@EliahKagan so, this is the definition of = that we have made only out of ∈ and the things it talks about
12:57 PM
Yes. And it doesn't depend on any particular thing being talked about. Really we have defined "=" in terms of "∈".
When building a set theory on a logic that doesn't provide identity, such an axiom is what one uses instead of an axiom of extensionality (and that phrase is universally used).
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