Thinking like a Mathematician

Warning! Math-related post! Don’t let that scare you off! Math can actually be fun, and will help your reasoning abilities.

As I was wandering around the blogosphere, I stumbled upon an interesting article over at Michellalianna that has nothing to do with what I am going to write about. Well, nothing but the image that was used for the post. The image is in fact the featured image for this post. This picture bothers me. Not as an androsexual transwoman, but instead as a mathematician—in training. As a math-loving person, the notation bothers me. I think that the usage of the equals sign doesn’t make sense. So, I am going to argue that the proper subset notation should be used.

Some preliminary stuff first: I don’t disagree with the message of the picture; I simply think the notation is off. Clearly this post will have some math in it, but I will do my best to keep it simple, easy to understand, and informal.

Before I dive into why I think the proper subset notation should be used, let me give you some mathematical details on the two symbols. I am thinking in terms of set theory. A set is simply a collection of elements—like an array in programming or simply an unordered list. Now, a set A is a subset of the set B if and only if every element in A is an element of B, i.e. A is contained in B. In set theory we would denote this by , and read it as A is a subset of B. It’s important to note here that every set is subset of itself because every set contains itself. Now, two sets are equal if and only if , i.e. they both contain each other thus A=B. Lastly, A is a proper subset of B if and only if A is contained in B, but A != B. This is denoted in set theory as , and read as A is a proper subset of B. Easy enough, right?

Well, I am going to state my case mathematically, then I will use words to describe what I am talking about. Let P be the set of all human beings. Let G be the set of all sets of people that share a common trait, and let GP be in G and be equal to P. Let Ra denote a unary relation that produces the set of all civil rights related specifically to that set—how this looks, I have no idea.

Now, lets  —this constrains X to only being groups in G—then  .  We want X to be equal to GP because then RX would contain everyone’s civil rights. In the case of the picture, X is the set of all trans people—as broadly speaking as possible—so, —while X is in G, it is not equal to GP which implies that RX isn’t equal to RGP. So, in order for the equals sign to make sense, the set on the left must be the human species, i.e.  . Now, RGP contains everyone’s civil rights, thus —for every group g in G that is not equal to GP, g is a proper subset of GP which implies that Rg is a proper subset of RGP.

Did you skip that? It might make more sense after I give some explanation. In order for X rights—the rights of the set we are talking about—to be equal to civil rights, it needs to contain everyone. In the picture, X doesn’t contain everyone; it only contains trans people. Thus the equals sign should be a proper subset sign.

Realistically speaking, this isn’t important at all. The message of the picture is clear. It’s really about the perspective you are using. If you look at the image mathematically, then it’s invalid. If you instead look at it sociopolitically, then everything is in order. The math perspective is useful for skeptics because math improves one’s reasoning abilities. Skepticism and Math both require attention to detail, and biasless—okay, minimized bias—approaches to finding the truth.

Gratefully,
Belle

P.S. While wordpress is supposed to support LaTeX out of the box, it’s not parsing it. If you would like a copy of the article, email me.

1. First off, I dig this post. More math is a happy place for me.

Secondly, I want to consider this: People tend to consider = to mean “the same in value” rather than meaning “containing the same set.” While value has a mathematical meaning, the word ALSO has a social meaning, which has more to do with quality than quantity.

Also, this got me thinking about rights issues as being a set, similar to numbers. I think a graphic including various rights issues, similar to the graphics many math textbooks use to show sets of numbers (something like this: http://www.mathsisfun.com/sets/images/number-sets.gif) could be an interesting project. It would probably need to be a wiki, so that it could be constantly updated and improved but I think that thinking about human rights issues as sets could be a fun thought experiment.

• Thanks!

Well, mathematically, = stands for an equivalence relation (http://en.wikipedia.org/wiki/Equivalence_relation); I decided against putting this into the post for reasons I find obvious. I do recognize that = has a social or more loose definition socially, and that’s fine; I was simply thinking mathematically.

That actually is a pretty neat idea. The only problem is the rights of a given group aren’t quantifiable, and would vary from perspective to perspective.

2. Another interesting argument that occured to me is through the use of first order logic with equality. If we look at the second axiom of equality, we see:

2) x = y -> f(…,x,…) = f(…,y,…)

The value of any function with x as an argument must be equal to the value of the same function with y replaced for the corresponding argument. If we allow R to be a placeholder for civil rights, and T to be a placeholder trans rights, then we may examine the predicate p(r, x), where the value of p is true if the right r is a property of the rights placeholder x. To show that R =/= T, we just have to find a right r for which p(r, R) =/= p(r,T). As a selection, we could consider E, “the freedom from employment discrimination because of race”. Then p(E,R) is true, this is a property of civil rights, while p(E,T) is false, as this is not a property of trans rights explicitly. The relation of axiom 2 does not hold, and therefore civil right do not have (mathematical) equality with trans rights.

Without further extending our calculus with an inclusion or membership operator (and thus making it equivalent to a basic set theory), we can’t express the proper subset concept directly. We can, however posit that p(r,T) iff p(r,R) and get a similar relation.

Just thought you might be interested in a different approach which leads to the same result ^_^

• Sorry, should be p(r,T) -> p(r,R), ~p(r,T) -> ~p(r,R), not iff

• I like this approach more than the set theory approach. Unfortunately, I think most readers would become confused more easily than they would by the set theory argument.

My goal here, i.e. in this article and on Queereka, is to bring a mathematical perspective to queerness, skepticism, science, and feminism; accordingly, I need to keep things as simple as possible without losing any or much rigor.

I really enjoyed your comment! This is one of the type of comments I wanted–the other being non-math people asking questions–and you did not disappoint.

• I’m glad you enjoyed it. I’m one of those weird people for whom formal languages are fun, so I jump to thinking like this. I don’t really know how to formally define the predicate without going to regular languages though.

I agree that set membership is the easier concept of the two; it does encapsulate that “is a part of” relationship very nicely. And, you get to use a lot more funny symbols, which makes it feel way more “mathy” 😀

3. Belle, this was beautifully written! I barely understood it, but interestingly enough, I had the same reaction in a more sociopolitical and less math based approach. Well done though! Thanks for linking to my post too. 🙂

Michelle

• I really appreciate that! I don’t have any confidence in my writing abilities.

I would love to see you expand on your sociopolitical thoughts on the topic.

^^

4. Just consider the equality as an equivalence relation on the quotient group defined by some superset which includes all rights mod the set of rights. Then, we get the more formal statement:

trans rights = civil rights (mod rights)

5. I would argue that for any sufficiently large Gp group defined by a non-trans trait (Since there is going to be a lot of overlap of Gp groups) will contain some transgender people. That is the group of blacks will contain some black transgender individuals, the group of women will contain trans woman, etc. and so on.

This means that most of your large Gp sets will contain trans people. So if you’re trans and are also a member of a non-trans minority, it’s not going to matter if your civil rights are being violated because you’re trans or a member of the other minority.

Thus:
Trans rights = civil rights (for all sufficiently large Gp minorities) as otherwise the trans members of those minority groups they belong to, will be lacking in civil rights.

Also, as others have pointed out, the statement trans rights are civil rights, is probably what the poster was supposed to mean, and in common English, = is often substituted for “are”.
-Jeremy

• There are some problems with your argument. First, there is only one G_p, and it’s equal to P. Second, the relation defined outputs Civil rights specific to that group; so, if X is an element of G such that X is equal to the set of women, then X would contain all transwomen. RX would contain the rights of women, but RX wouldn’t contain RT, where T is an element of G such that T is equal to the set of all transgender people, because those rights are specific to transgender people–also, X doesn’t contain any transmen by definition.

Even with elements of G with large cardinalities, my argument still stands. Let’s take the largest set in G, G_p. RT is a proper subset of RG_p, i.e. RT is contained in RG_p but isn’t equal to RG_p–in order for sets to be equal, they must contain each other.

I recognize that the poster wasn’t making a mathematical statement, and I stated that in the post. I was using the image as an opportunity to teach some mathematics because I think the perspective is interesting, it’s fun, and useful to skeptics.

In general, you need to reread the way I defined everything. Also, Zoe’s comment is a great first order logic explanation.

Anyways, thanks for you comment! I hope I don’t come across as harsh; I’m used to talking with other mathematicians, and the nature of mathematics tends to color our communication when it comes to logic and math.

• Sorry it’s been awhile since I’ve studied set theory and read your argument wrong. Therefore, I will try and avoid embarrassing myself and simply make my argument in plain English.

We can interpret the sign several ways.

One: the set containing all civil rights is equivalent the set of trans rights. I think this interpretation, which you appear to have adopted, is a straw man.

Two: Equals means are, so yes, trans rights are civil rights. Not much else to see here.

Three: They’re using the equals sign as a logical bi-conditional. That is, trans rights iff civil rights for all other groups.

Let’s break this down.
Let A = the set of all trans rights specific to trans people.
B = the set of all other civil rights not specific to trans people.
B can also be broken down into other subsets specific to other non-trans groups, denoted Bx
Thus civil rights are the union of A and B.

Let us assume the following:
For all sufficiently large populations X, there will be at least one trans person.

Thus, all groups worth considering will contain a subset of the trans population and the trans population will contain a subset of every other group.

Assume A civil rights are granted but not B.
Every group will have at least one member who has not obtained full civil rights.

Assume B civil rights are granted but not A.
Therefore, there will be trans people who have not obtained full civil rights.
Furthermore, assume any subset, Bx, is not granted, then there will still exist at least one trans individual who has not obtained full civil rights.

Thus trans rights are a necessary condition for full civil rights to take place. Likewise, civil rights for all other groups are necessary for trans people to obtain full civil rights.

Conclude
Trans rights = civil rights

• I don’t quite understand what you mean by I am using a straw man. I stated that my perspective was going to be mathematical; therefore, I am going to use the equals sign as defined by mathematics, ie. an equivalence relation. I even gave the definition for set equality. The only way I can possible way I can conceive of that being a straw man is if I were commenting on the social aspects of the poster. I stated in the post that that wasn’t what I was talking about. I dont’ see a problem with the poster from a sociopolitical perspective. It’s obvious that the poster is using the equals sign to say that Trans rights are civil rights. There’s no questions there, but if you translate the statement into mathematics, i.e. be extremely literal, then the statement breaks down.

Now, concerning the biconditional. This is an interesting argument, but it breaks down for one specific reason: you seem to be mixing a sociopolitical perspective and a mathematical one. You are adding qualitative weights to the sets that neither of us have defined. The defined relation only outputs rights that are specific to the trait that defines the set. It has no concern for whether or not those rights are being granted or not, because that’s not what I’m talking about. It also doesn’t matter if the relation returns enough rights to be just. It only outputs rights specific to that group. Now, we could define a relation that outputs a set of rights that is ‘fair,’ but I would think that would be equal to RG_p, i.e. it’s not necessary and somewhat meaningless. It would, however, be a context in which you are right, but it doesn’t refute my math.

• Tangentially, I really appreciate that you’re taking the time to discuss this with me. I honestly thought that this post was going to be largely ignored because of the math content. I didn’t expect someone to dissect my argument. It actually brings me a fair amount of joy. I plan on posting more math related content, and I hope to see more feedback from you in the future. ^^

Anyway, back to the debate ;D