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So if we could swap the chromosomes, would the sex of a person change?

Sex is not just chromosomes.
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Why is the chromosomal definition not binary? Presence of Y chromosome => male, everything else => female. To me that is binary. >

I guess it depends on who you're talking to about it. Please note, I'm not an expert in these fields, just someone who has some interest.
If you are interested in "binary sex" from a scientific perspective, Y chromosome certainly is a very important thing to understand, but ultimately it doesn't correspond 100% to binary sex, so you should understand the definition around gamete size (eg it's possible to have an XX with small gametes). There is very useful science to be studied around chromosomes. However, modern science relies on the gamete size definition. Honestly, as a lay-person, this makes sense since "sex" is meant to describe an aspect of sexual reproduction, and gametes are the key to sexual reproduction.
From a medical perspective, binary sex is only useful to a degree. Biological definitions around gamete size (or presence of Y chromosome) don't correspond well enough to the various phenotypic traits you describe, and if we try to apply a binary sex model to individuals in a medical setting, a decent percentage of people will have worse medical outcomes. Eg. it's possible for an XX to have more "male" phenotypical traits. So a nonbinary model is very important here.
If you're talking about it socially, the social idea of "sex" is very much nonbinary. Y chromosome (and also gamete size) doesn't correlate very well with the sexual characteristics that people often think of when discussing sex, with or without medical intervention.  So a nonbinary model is useful here too.
Conflating social, medical, and scientific definitions often leads to arguments that are unproductive as people talk past one another. So attempting to raise binary sex into social or medical conversations may end up being frustrating or counterproductive in some cases. >

!delta yeah I love this the social definition, and the most common one definitely isn’t binary. >

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!delta  

If you edit your comment, the bot will pick it up. >

Might i just say that biology is never binary. everybody, literaly every "body" is unique. the production of hormones that detemine your primary and secondary sexual features, the way you think and how you identify are all regulated by physical processes. for example the amount and ratio of hormones you produce, the development of specific areas of the brain, and mutations on the genes that turn off, limit, promote or turn on certain aspects of the complex chains of steps that your body takes to develop into what ever it developes into. there are so many factors that determen your biological sex, and variations there of, that it becomes difficult in some cases to round someone up or down to the nearest sex i.a male or female. sometimes a body can truely be between or beyond these two options. and how the sense of self is altered by your enviourment, genes, and culture is so personal that gender can sometimes not align with someones biological sex. 
in a black and white world there are only 2 sexes for humans. in biology there are no real sexes, there is just life and we classify it and put it in boxes. 2 boxes is the easiest to work with. but there are spiecies that have 1 sex, 2 sexes, 3 sexes, 5 sexes. there is no limit to the variation. and what definition we use to determine a sex is also made up. nature didn't write it in stone or in the skys.
as long as you're happy with how you are and can accept someone esle for how they see themselfs there is no reason for being argy about people not fitting in our self made boxes, or wanting to be places in the other box. >


Male => has Y chromosome, female => does not have Y chromosome. This definition is inclusive toward those with chromosomal differences such as those with Kleinfelters, Turners, etc.

This isn't strictly speaking entirely correct because XY gonadal dysgenesis (with Y chromosome, but dysfunctional SRY group) exists, which presents externally as female. 46,XX/46,XY chimerism also exists, which can result in either indeterminate sex characteristics or just entirely male or female presentation. >

My understanding is that if you move the binary to functional SRY gene, you have a near binary with the possible exception of androgen insensitivity. So is sex binary with the claim: a person is male if they have a functioning SRY and are androgen sensitive, else they are female?
There's probably some genetic abnormality aflicting 0.001% of the population I'm not familiar with, but this seems near enough to binary to me. >

Okay but surely it's just easier to say that it's bimodal then, and then you've accurately represented all the genetic situations that you are and aren't familiar with >

If 1/100 people or even 1/1000 people couldn't be neatly classified as male or female, I'd agree, but if you start to get into  1/1,000,000 cases, bimodal starts to seem disingenuous to me. In the natural world nothing is ever perfect, there are always exceptions or genetic defects. For all I (or it sounds like you) know, the presupposition I just put forth may be accurate, I doubt it, as there are always exceptions, but at what level of rarity do these exceptions become inconsequential to all but a few experts? Suppose 5 people in the world currently do not fit my preposed binary, would you still call that bimodal?
For what it's worth, I think bimodal is an appropriate way to define gender, but it seems like a stretch for sex. >



Sex is binary. Male => has Y chromosome, female => does not have Y chromosome. This definition is inclusive toward those with chromosomal differences such as those with Kleinfelters, Turners, etc.


It might be inclusive. It's also useless in practice and doesn't conform to the common perception.
No one is going to call an XY-female who has an Y-choromsome but either suffers from androgyn insensitivity or a damaged SRY-gene a “male”. This person will most likely only find out about this when 15 years old or never at all in some cases when they experience some manner of female puberty. No one is going to call an XX-male with a translocated SRY-gene to one of the other chromosomes a “female” either and this person will most often never find out about this.
One can make up all sorts of “definitions” for words. One could also simply say that anyone shorter than 175 is female, and anyone taller male, but these definitions simply don't align with how people use these terms, and thus will never catch on.

Sexual traits are strongly bimodal

Actually no. This is a common myth: so common in fact that human height is often used in statistics classes to explain bimodal distribution as an example, because people expect it to be bimodal, but it's actually unimodal. Many people for cultural reasons expect secondary sex characteristics to be bimodal, but they're actually usually unimodal.
Height, mass, physical strength, voice pitch, wast-to-hip-ratio, breast-size, shoulder-widfth; they've all been found to be unimodal in most countries, and certainly across the entire human population of adults.
Primary sex characteristics are distributed heavily unimodally, of course, but secondary sex characteristics rarely are. >

> but these definitions simply don't align with how people use these terms, and thus will never catch on.
Great point
> Primary sex characteristics are distributed heavily unimodally, of course, but secondary sex characteristics rarely are.
Could you clarify? why do I see females with boobs and males with penises then? >


Primary sex characteristics are distributed heavily unimodally, of course, but secondary sex characteristics rarely are. >

I both agree and disagree. Biologically speaking, people are normally born male/female and that's about all there is to it. However, sex is also a social construct, just like everything else that we give names to and is subject to change, but by your definition as according to the chromosomes it is binary. 
In my personal opinion, it really doesn't matter what people call themselves and what you call them. To many people care far to much about what others define them as over actually caring what they define themselves as, just believe whatever you want as long as your not causing harm to others, and stop caring what people say about themselves or you. >

Sex is not social construct. Societies accepted the fact who's male and female not arbitraly deciding about it.

In my personal opinion, it really doesn't matter what people call themselves and what you call them.

It matters very much since se exlusive zones and laws exist. >

Sex is a social construct. Society has defined sex by chromosomes for forever so it does not feel that way, but if tomorrow everyone swapped one sex with the other, it would be so. My point is not arguing it is arbitrary, simply speaking to the ways words are defined.
You are right on the ways they matter, my apologies I was not clear I meant on a personal level, day-to-day kind of mattering. >


. Society has defined sex by chromosomes

nature did that. 

simply speaking to the ways words are defined.

It's not about mere words. It's about fundamental ideas which societies and cultures are built upon. >

So if we could swap the chromosomes, would the sex of a person change?
And my point was the nature of words. Not in the accuracy of how something is defined behind said words.