Can modern society define what a woman is?

Last year I had a debate with someone who insisted that the term "woman" could refer to anything that a person identifying as such wanted it to mean. My objection that if the term could mean anything at all that it became meaningless fell on deaf ears. 

This debate has now gone mainstream. Matt Walsh was invited onto the Dr Phil show and he had this argument with a trans activist (starting from about 7:20):

 

The discussion kicked off when Matt Walsh asked the trans activist (Addison) to define the term "woman". Addison replied: "Womanhood is something that I cannot define because I myself am not [a woman]. Matt Walsh interjected: "But you used the word, so what did you mean when you said transwomen are women if you don't know what it means?" Addison: "So here's the thing, so I do not define what a woman is because I do not identify as a woman. Womanhood is an umbrella term, it includes people who...." Matt Walsh interjecting: "That describes what?" Addison: "People who identify as a woman." Matt Walsh: "Identify as what?" Addison: "As a woman." Matt Walsh: "What is that?" Addison: "To each their own. Each woman, each man, each person is going to have a different relation with their own gender identity and define it differently." Matt Walsh: "You won't even tell me what the word means though".

In my post I suggested that this inability to define a term like "womanhood" could be traced back to the metaphysics of philosophers like Thomas Hobbes who rejected realism in favour of nominalism:

You can see here the logic of nominalism (that there are only individual instances of things) and a certain type of materialism (that we are just matter in motion) in undermining a "teleology" - a view that there are proper ends to human life that are discernible through reason.

  Sohrab Ahmari had a similar take on the Matt Walsh discussion:

And what of the feminists? One made this comment:

This doesn't really help much. Yes, it defines the term "woman" with clarity as "a person with a female body". But it deliberately stops there and refuses to give any meaning to the term "womanhood". Our terf feminist wants "womanhood" to be self-defined in a similar way that the trans activists want "woman" to be self-defined.

Terfs and trans are both running with the same principle, of claiming that our sex has no real content that might influence who we are or what we do. For that reason, championing the terf position prepares the ground for adopting the trans one - it becomes very difficult to hold the line once the general principle is accepted.

Nor is the terf position persuasive even in terms of biology. If it is clear that we are biologically distinct sexes, then it is unreasonable to suggest that these biological distinctions would have no effect on personality. If men evolved more muscular bodies fit for the purposes of hunting large game and for defending the tribe in warfare, and if women's bodies are designed for the bearing and nurture of infants, then how could you possibly claim that this would have no effect at all on who we are in our personhood? 

And why should people even care about manhood and womanhood under the terms suggested by terf feminism? If it is just a matter of different bodies, with no ramifications for the human person, then who would care if they were erased as coherent categories? The categories would be merely accidental to life if what the terfs say is true.

Nor are terf feminists consistent in severing sex and personality. They are not as laid back in claiming that each man can have any kind of personality. Instead, there is a categorising of some kinds of masculine personality as "toxic" and attempts to educate men into adopting more emotional expressions of personality. In other words, there are value judgements when it comes to expressions of manhood, rather than "each man can equally define for himself in any direction he chooses". 

It is better to acknowledge some positive content to manhood and womanhood, rather than making these terms wholly based on subjective, individual preference or practice. This does not have to be overdone, to the point that they are felt to be unnecessarily restrictive or limiting. But look at what happens when a society refuses to define at all and denies any objective meaning to terms like woman or womanhood. The categories are then effectively erased and become meaningless except, as Matt Walsh aptly put it in the video, as "costumes that can be worn".