## Saturday, June 12, 2021

### 2+2 doesn't always equal 4

[This is a transcript of the video embedded below.]

2 plus 2 makes 5 is the paradigmatic example of an obvious falsehood, a falsehood that everybody knows to be false. Because 2 plus 2 is equal to 1. Right? At the end of this video, you’ll know what I am talking about.

George Orwell famously used two plus two equals five in his novel nineteen eighty-four as an example for an obviously false statement that you can nevertheless make people believe in.

The same example was used already in seventeen eighty-nine by the French priest and writer Emmanuel Sieyès in his essay “what is the third estate”. At this time the third estate – the “bourgeoisie” – made up the big bulk of the population in France, but wasn’t allowed to vote. Sieyes wrote
“[If] it be claimed that under the French constitution two hundred thousand individuals out of twenty-six million citizens constitute two-thirds of the common will, only one comment is possible: it is a claim that two and two make five.” This was briefly before the French revolution.

So you can see there’s a heavy legacy to using two plus two is five as an example of an obvious untruth. And if you claim otherwise that can understandably upset some people. For example, the mathematician Kareem Carr recently got fire on twitter for pointing out that 2+2 isn’t always equal four.

He was accused of being “woke” because he supposedly excused obviously wrong math as okay. Even he was surprised at how upset some people got about it, because his point is of course entirely correct. 2+2 isn’t always equal to four. And I don’t just mean that you could change the symbol “4” with the symbol “5”. You can do that of course, but that’s not the point. The point is that two plus two is a symbolic representation for the properties of elements of a group. And the result depends on what the 2s refer to and how the mathematical operation “+” is defined.

Strictly speaking, without those definitions 2+2 can be pretty much anything. That’s where the joke comes from that you shouldn’t let mathematicians sort out the restaurant bill, because they haven’t yet agreed on how to define addition.

To see why it’s important to know what you are adding and how, let’s back up for a moment to see where the “normal” addition law comes from. If I have two apples and I add two apples, then that makes four apples. Right? Right.

Ok, but how about this. If I have a glass of water with a temperature of twenty degrees and I pour it together with another glass of water at 20 degrees, then together the water will have a temperature of 40 degrees. Erm. No, certainly not.

If both glasses contained the same amount of water, the final temperature will be one half the sum of the temperatures, so that’d still be 20 degrees, which makes much more sense. Temperatures don’t add by the rule two plus two equals four. And why is that?

It’s because temperature is a measure for the average energy of particles and averages don’t add the same way as apples. The average height of women in the United States is 5 ft 4 inches, and that of men 5 ft 9 inches, but that doesn’t mean that the average American has a height of 11 ft 1. You have to know what you’re adding to know how to add it.

Another example. Suppose you switch on a flashlight. The light moves at, well, the speed of light. And as you know the speed of light is the same for all observers. We learned that from Albert Einstein. Yes, that guy again. Now suppose I switch on the flashlight while you come running at me at, say, ten kilometers per hour. At what velocity is the light coming at *you. Well, that’s the speed of light plus ten kilometers per hour. Right? Erm, no. Because that’d be faster than the speed of light. What’s going on?

What’s going on is that velocities don’t add like apples either. They merely approximately do this if all the velocities involved are much smaller than the speed of light. But strictly speaking they have to be added using this formula.

Here u and v are the two velocities that you want to add and w is the result. C is the speed of light. You see immediately if one of the velocities, say u, is also the speed of light, then the resulting velocity stays the speed of light.

So, if you add something to the speed of light, the speed of light doesn’t change. If you come running at me, the light from my flashlight still comes at you with the speed of light.

Indeed, if you add the speed of light to the speed of light because maybe you want to know the velocity at which two light beams approach each other head on, you get c plus c equals c. So, in units of the speed of light, according to Einstein, 1+1 is 1.

That’s some examples from physics for quantities that just have different addition laws. Here is another one from mathematics. Suppose you want to add two numbers that are elements of a finite group, to keep things simple, say one with only three elements. We can give these elements the numbers zero, one, and two.

We can then define an addition rule on this group, which I’ll write as a plus with a circle around it, to make clear it’s not the usual addition. This new addition rule works like this. Take the usual sum of two number, then divide the result by three and take the rest.

So, for example 1+2 = 3, divide by three, the rest is 0. This addition law is defined so that it keeps us within the group. And with this addition law, you have 1 plus 2 equals 0. By the same rule 2 plus 2 equals one.

I know this looks odd, but it’s completely standard mathematics, and it’s not even very advanced mathematics, it just isn’t commonly taught in school. This remainder after division is usually called the modulus. So this addition law can be written as the plus with the circle equals the normal plus mod 3. A set of numbers with this addition law is called a cyclic group.

You can’t only do this with 4, but with any integer number. For example if you take the number 12, that just means if you add numbers to something larger than 12 you start over from zero again. That’s how clocks work, basically, 8+7=3, add another 12 and that gives 3 again. We’re fairly used to this.

Clocks are a nice visual example for how to add numbers in a cyclic group, but time-keeping itself is not an example for cyclic addition. That’s because the “real” physical time of course does not go in a circle. It’s just that on a simple clock we might not have an indicator for the time switching from am to pm or to the next day.

So in summary, if you add numbers you need to know what it is that you are adding and take the right addition law to describe what you are interested in. If you take two integers and use the standard addition law, then, yes, two plus two equals four. But there are many other things those numbers could stand for and many other addition laws, and depending on your definition, two plus two might be two or one or five or really anything at all. That’s not “woke” that’s math.

1. Carr’s case is more complicated. It was a response to 2+2=4 being an example of white supremacy. While one can construct examples where it doesn’t hold, that misses the point that many of the woke actually believe such nonsense. “Remember that just because a woman has a penis, she can still be a lesbian.”. Yes, that is the tenor these days. It seems as if the response to Trump was to be just as stupid, but change the sign. I probably won’t have to wait long before some woke CRT proponent points out that “tenor” is sexist since it prefers male voices. :-|

1. I wonder why so many people think they need to explain this to me.

2. @Dr. Hossenfelder:
I got a lot out of today's video. I wish the maths I learned in high school seemed as tangible as it did when you explain it.

@Phillip Helbig
Biology, gender identity and sexuality are not as simple as many people were told in elementary school. That your understanding is stuck there is a problem you can remedy if you make an effort.
There's a wealth of information out there if you want to look, same as there's this here blog and YouTube channel to explain maths, technology and science to us.
The word 'tenor' has another application in music: it refers to the size of instrument that plays in the tenor range, so the word is gender-neutral in that context. It's pretty funny to see conservatives invent things that progressive people might find offensive like you just did, and get offended by their own imagination.
Being 'woke' is a term used to degrade outspoken, empathetic people, apparently.

3. Because the abbreviated version you give creates a distorted impression for those who don’t know the backstory.

4. Your own comment leaves a distorted impression. The twitter thread that Sabine highlighted had already moved on from the woke wars. Rather, Kareem was trying to educate about actual math and data science and apparently the un-woke weren't having it. Some of the un-woke actually believe the nonsense that 2+2 must always and forever and in all situations equal 4 otherwise they might be giving an inch to the woke masses.

5. I am not sure what you're referring to as an abbreviated version?
I'm reading the back-story now and what Brittany Marshall said seems like a rather poor attempt at saying that much of what is taught and believed is handed down from the perspective of the original white colonisers to the disadvantage of many People of Colour. If I misunderstand I'm happy to be corrected.

Regardless of Brittany Marshall's statement and CRT, transgender sexual orientation and gender identity is not nonsense.

I learned about a whole aspect of mathematics that I didn't know existed until watching today's video, surely you can likewise understand about transgender people.

6. For that matter "a woman has a penis" could be straightforwardly true as long as a person chose a certain definition of "woman". It is not the definition I use but words are defined by usage. Are we really going to pretend, for example, that the term "man" has always, in the past, been synonymous with "adult male"?

Seems to me that if anyone is happy with saying that 2+2 does not necessarily equal 4 because we can make different definitions of "+", then we can hardly complain that someone says that a woman can have a penis because we can make different definitions of "woman".

7. @ C Thompson: Someone once said that no satire can be so extreme that there is not something real which is even worse and/or that someone would mistake such satire for the real thing. Q.E.D.

You don’t know me, hence can’t know if my understanding is stuck, don’t know where I went to elementary school and what I learned there. In any case, even if some concepts have been amended since then, it doesn’t mean that the latest brain fart by the woke person of the week has to be taken as the truth.

Note that “made in Germany” used to be a warning, but evolved into a symbol of quality. Similarly, “woke” was originally a positive self-description, but the absurd antics of many of them are what led to its current pejorative meaning.

8. I'm with Phillip Helbig on this. As Dr. H states, 2+2 can have different answers depending on the definition of the symbols. But there are people out there who claim Maths can depend subjectively on skin colour. More Woke madness.

C Thompson10:56 AM, June 12, 2021
"surely you can likewise understand about transgender people. "

Everybody knows what a transgender person is. There are still only 2 human sexes and genders though. I have told you this several times now, but you won't address it because your whole tower of nonsense will come tumbling down.

There are empirical facts in natural science and definitions and proofs in Maths, and the scientifically illiterate Woke won't be re-writing them.

Understand that.

9. Robin11:38 AM, June 12, 2021

"Seems to me that if anyone is happy with saying that 2+2 does not necessarily equal 4"
Mathematicians are doing what computer scientists call overloading the operator. In some algebra books a different symbol is used for modular addition to make the point, but usually it's left to the context. The point is that the various types of addition are mathematically well-defined though it might not be immediately obvious which addition one is talking about if the context is limited.

"we can hardly complain that someone says that a woman can have a penis because we can make different definitions of "woman"."

A woman can have a penis - if she is a transgender man and has one attached in an op. Then he is a woman with a penis. (Where "he" here is a social overload of the pronoun, as you don't want to annoy a woman with a penis pumped full of testosterone ;)

I have been a transgender woman for 2 days now, so I know all about it.

Biologically, there are 2 and only 2 clearly defined human sexes based on the 2 roles in sexual reproduction. Then, according to the dictionary definition of the word "gender", there are only 2 human genders, one associated with each sex. Things like "gender X" are not genders but a collection of human attributes (some possibly subjective) not tied to a biologically defined sex, which is why they don't represent a gender by definition.

10. Here are the initial tweets from Laurie Rubel that apparently started the 2+2=4 discussion:

"the idea that math (or data) is culturally neutral or in any way objective is a MYTH. "

"along with the "of course math is neutral because 2+2=4" trope are the related (and creepy) "math is pure" and "protect math."

reeks of white supremacist patriarchy. "

It's a myth that Maths is objective? So there is a point of view from which there are 2 natural numbers whose ratio when squared equals 2?

And here is Carr:
"It's an objective fact that some groups were more involved in the creation of modern math than others. They may have been *trying* to make it objective but it's not stupid to ask if they actually succeeded!"

So Carr studied a Maths-related subject up to at least PhD level. Did he find any issues in Maths due to the subjectivity of "some groups"?

No. He has found zero issues. So it is stupid to ask.

Once again we are dealing with the insane, people who are off-the-chart mad.

11. Probably nothing could demonstrate the absurdity of much contemporary wokeness more than the fact that Steven Evans and I are in complete agreement about it. :-)

12. White Proofs Matter!

13. More on wokeness. First figure out why Twitter thinks the content is sensitive, then read the article:

14. Hi Philip and Steven,

So are you suggesting that gender has never been even partially a social construct in our society? When Rudyard Kipling says "And what is more you will be a Man" do you think that he only meant "And what is more you will be Biologically Male"?

What do people mean when they say "An unmanly man"? Do they mean a Man who is both biological and not biologically male? Come to that, why do our conservative pundits have fits when a male pop star puts on a frock? Are clothes part of our biology?

Even the patron saint of the antiwoke, Jordan Peterson, regularly uses the words "masculine" and feminine" in ways that have nothing at all to do with biology.

So, what do you say? Do you agree that gendered terms have always been overloaded with social roles and behavioural expectations?

15. @Phillip Helbig
Maybe I missed your mark. I'm aware of Nathan Poe's assertion that done well, a parody of something is indistinguishable from the subject it's satirising (Poe's Law). If that was the case with your comment I apologise. It did look like the ignorant comments I've seen transphobes, homophobes, misogynists, climate-change deniers and other varieties of ignoramuses make.
“...just because a woman has a penis, she can still be a lesbian.” Is a true statement.

@Steven Evans
I hope your journey as a woman continues without the hassles and barriers women often face, especially transgender women.
Scientifically speaking (I will do my best here as I am not as clear and eloquent as I would wish, but I do my best)
Gender and reproductive sex are more like spectra than discrete categories, I assume you know that.
Gender is a social construct, just like mathematics is constructed to interact with and understand the world.
2+2=4 in the case that most people use maths, and 'there are 2 sexes in humans' works on a basic level to explain that our species is sexually dimorphic. There are more chromosomal combinations that people have than XX and XY, and some of these people can sexually procreate, some can't. I.E., there are more than 2 genders. You can look this up if you want scientific confirmation.
We can move on from there to describe how not every case of '2+2' results in '5'. Likewise, not every individual finds themselves fitting into the socially-constructed 'gender binary' and can be ill-served and disadvantaged by the gender-binary-enforcing institutions and amenities we mostly have.
The understanding of the world grows as perceptions changed and biases overturned. Language adjusts accordingly as society progresses.

@Robyn
Hi, we already do have such language to describe people: the prefixes 'cis' and 'trans'. Sorry if you are already aware of this.

16. Robin7:13 PM, June 13, 2021

This is my understanding:
There are 2 and only 2 human sexes which are based on sexual reproduction, and they are clearly defined by biology. A small number of people are born intersex.
By the dictionary definition, gender means the average "traits" associated with a sex. It's less clear what would be included or excluded as a "trait", but it is clear there are only 2 genders in humans because there are only 2 sexes. What people claim are other "genders" besides male and female are human attributes not tied to sex so not a gender, by definition.
Making laws about sex is relatively straightforward as sex is clearly defined biologically. Making laws about gender is problematic because it is fuzzily defined.
Clearly there are people with gender dysmorphia who don't "feel comfortable" as their biological sex, so-called transgender men and women.
But because science can't determine gender exactly, the law should not try. We should deal with it socially. Calling a transgender woman, who is biologically a man, "she" socially is not a burden.
So we can be socially as sympathetic and flexible as possible. But there are instances when the biological maleness of a transgender woman may preclude complete flexibility e.g. transgender women should not participate in women's sports and should not be admitted in to women's prisons if they have a penis.

Because science could not determine gender precisely but the law tried to, in the end they gave up and allowed people to self-declare their gender **legally**. Obviously, this can be abused by liars.
Gendered roles may have been laden with expectations but that's not the case now in Western countries, so there is no need to try to re-write the clear facts of science. In fact, telling lies like this doesn't help the transgender cause, nor does weaponising it politically. I don't really care what clothes people wear.

17. Phillip Helbig wrote:

"Carr’s case is more complicated. It was a response to 2+2=4 being an example of white supremacy. While one can construct examples where it doesn’t hold, that misses the point that many of the woke actually believe such nonsense."

I dug into it a bit. It was Laurie Rubel who wrote that people of the white patriarchy keep saying that "math is neutral because 2+2=4".

So it seems that you are the one missing the point. Her point is that many mathematical algorithms are racist. For example, if a bank uses an algorithm to decide about your loan, there is a good chance that it would discriminate against black people.

It seems to me that your response matches exactly her expectation from the white patriarchy, you attacked her for believing that 2+2 does not equal 4, even though she never claimed it.

18. C Thompson10:37 PM, June 13, 2021

Human sex is based on sexual reproduction. Females produce large gametes and males produce small gametes. That's the definition in biology. Gender are the traits associated with the sex. There are people born intersex with unclear sexual phenotypes, and there are people who possibly due to unexpressed genes don't have "typical traits" for their biological sex. But by definition there are 2 human sexes and therefore 2 genders. Western societies are socially flexible and medical help is available for intersex and transgender people if they need it.
Trying to re-write natural science, the hard sciences, with nonsense from social "sciences" will simply not be accepted by the former communities, though, and doesn't help the transgender and intersex cause.
Maths is an objective subject - the calculations give the same answer in any culture. If you add 2+2 in the natural numbers as defined by the Peano Axioms you always get 4. There are other systems, as Dr. H. pointed out, but they are equally well-defined. The suggestion by Kareem Carr that there may be an issue with subjectivity in Maths due to European and American dominance is nonsense. There are significant non-European contributions in Maths (Arab, Indian, Chinese, Ramanujan) for a start, and Carr has presented not one example of a supposed issue with subjectivity in the whole of Maths. Carr got in to Harvard by getting the Maths right not getting it wrong, and Departments of Education in the US who are now taking up the idea that objectivity in Maths is a myth (nonsense!) will not help people from communities not well-represented in advanced Maths improve the situation. Princeton Maths Dept. won't be letting anyone in for getting the answer wrong.
In society people have opinions. In natural science and Maths we do not; we have empirical evidence and proofs.

19. Udi Fuchs1:35 AM, June 14, 2021

Laurie Rubel wrote this:

"the idea that math (or data) is culturally neutral or in any way objective is a MYTH. "

This is nonsense. It is not a myth that Maths is objective. Maths results do not vary from country to country. Sqrt(2) is irrational everywhere and forever.

Carr wrote this:
"It's an objective fact that some groups were more involved in the creation of modern math than others. They may have been *trying* to make it objective but it's not stupid to ask if they actually succeeded!"

He has presented not one example of this supposed possible issue of group subjectivity skewing Maths results. Again, it is palpable nonsense of the first order.

"if a bank uses an algorithm to decide about your loan, there is a good chance that it would discriminate against black people."

That's a problem with the model that the algorithm is instantiating, not with the whiteness of Euler's and Gauss's skin.

20. 'Transgender women should not...' is pretty damn close to how women and black/Coloured people/homosexuals etc. shouldn't be allowed in certain places or do certain activities.
It's pretty disgusting.
Transgender women aren't the problem, society and our institutions, and the transphobic are.

21. This comment has been removed by the author.

22. C Thompson6:57 AM, June 14, 2021

"'Transgender women should not...' is pretty damn close to how women and black/Coloured people/homosexuals etc. shouldn't be allowed in certain places or do certain activities."

No, it's not. You are incapable of thinking straight and are lost in this ideology. Your thought process consists of thinking what your answer should be based on this week's version of the ideology. We have separate male and female prisons for a reason. A man living as a woman is still a man so goes in the male prison.

You have not shown that there is any biological basis for more than 2 sexes and so have not shown there is any biological basis for more than 2 genders. Natural science will not be bowing to this post-modernist claptrap, even if gullible people like you do.

23. I've been spending over a decade with this. I've used the same criteria for evaluating information as I would for my tertiary course assessments.
There used to be reasons for segregating people based on colour and race, we don't do that. It used to be seen as perfectly justified.
It's about treating others fairly, not post-modernism.
At the end of it all every argument for segregating transgender people is based on bigotry.

24. In human sexual reproduction a large gamete fuses with a small gamete. There are 2 types of gamete, so 2 sexes. You haven't identified a 3rd sex in this process, so your tower of nonsense is based on a fiction. A fiction that natural science will not be accepting.

25. Steven Evans wrote about Laurie Rubel's tweet:

"This is nonsense. It is not a myth that Maths is objective. Maths results do not vary from country to country. Sqrt(2) is irrational everywhere and forever."

Laurie never claimed that sqrt(2) is rational or any such nonsense. She did predict that people that belong to the "white supremacist patriarchy" would attack her claims with trivial statements that she must be saying nonsense because "2+2=4". I have no idea what is your skin color, but your behavior exactly matches her predictions.

Steven Evans wrote about discriminatory bank algorithms:

"That's a problem with the model that the algorithm is instantiating, not with the whiteness of Euler's and Gauss's skin."

Again, no one claimed that there is a problem with Euler's or Gauss skin color. The problem is with the mathematical model. You seem to agree about that, so I guess that you are also starting to be "woke".

26. @Udi Fuchs:
Thank you for explaining this.

@Steven Evans:
I mentioned sex chromosomes, not gametes. I'm going off people's lived experiences that they've shared as well as science, that's not nonsense.
I'd expect a transgender woman such as yourself to be less obtuse.

27. Udi Fuchs1:13 AM, June 15, 2021

Laurie Rubel claimed that objectivity in Maths is a myth.
That's nonsense.

Do you accept it is nonsense now, or would you like to present an example of subjectivity for our delight and delectation?

The "white patriarchy" is a meaningless phrase invented by social "scientists" pretending they have an objective domain of study. They don't. I suspect it would be tricky to define "white" scientifically, and suggesting that all men have conspired against all women throughout history is frankly ridiculous. No evidence of such a conspiracy exists.

In the bank loan case, the problem is with the model, not the Maths. Maths is an objective field of study.

Do you understand and accept now that the problem is with the model and not the Maths?

Kareem Carr does indeed suggest that modern maths' domination by "some groups" may have caused subjectivities to creep into Maths based on the nature of people in those "some groups". Fields Medal winners from Japan, Vietnam and Iran got the same answers, though.

Like you and all of your ilk, he provides not one example to support his insane, insane claim.

Do you now accept that? Are you starting to learn some objective facts?

28. C Thompson2:46 AM, June 15, 2021

"Lived experience"?? Good grief. That has nothing to do with natural science.

Now that I have explained the biology to you, do you agree that there are 2 and only 2 human sexes?
Or, are you going to identify a 3rd sex?

29. I suspect it won't do much good but I don't think C Thompson is denying there are two biological sexes (and some rare intersex conditions) just that the relevant criterion shouldn't be sex but gender, which is a choice. I agree on that for most every day proceedings, frankly I couldn't care less if Mary has a penis or not and I totally support people's right to chose their gender and pronouns and whathaveyou. However, for some purposes we use biological sex to make decisions because that's what matters. It's rather idiotic to deny that members of the male sex have on the average a physical advantage over females, and whether or not they identify with a gender that matches their sex is entirely irrelevant. For the same rather obvious reason it'd be utterly idiotic to claim biological males should be given maternity leaves, and since a lot of Americans tend to misunderstand this let me clarify that maternity leave is not the same as parental leave. Maternity leave is the leave briefly before until a few weeks after delivery, it's by design reserved for people with uterusses, no matter what gender they identify with.

The problem with this whole discussion is that just pointing out that biological sex is actually relevant in some cases now counts as transphobic.

30. I agree about social flexibility. Who cares in general? But gender means the average traits of a sex by definition, and there are only 2 sexes so there are only 2 genders. The extra genders aren't genders at all, they are just individual human attributes. By the rules of new gender, I could invent the Steven Evans gender and define it as being exactly me. Which is nonsense.

31. Well, to come back to the point of this post, a definition is just that: a definition. In the end what matters aren't the definitions but the conclusions you draw from them. It's rather moot to fight about the meaning of words in and by themselves. If you use the word gender in a way that the people you communicate with don't, then you'll just talk past each other.

32. I use the word gender as defined in the dictionary, meaning traits associated with a particular sex.

According to the Woke use of gender, I could take my state at every second of my life and declare it a gender, Steven_Evans(t).

So as I write to you my gender is Steven_Evans(1,261,440,000) and my pronouns are He(1,261,440,002)/Him(1,261,440,003).

As you can see, my gender is so fluid, it changes as I type.

33. So? I don't see the problem with that.

34. @Dr Hossenfelder:
You're right, I wasn't denying there are the 2 genders of male and female, and intersex.
Not meaning to talk down to you, but transgender and non-binary people don't choose their gender, as we didn't choose ours. People can find or create descriptions of themselves, and choose those though.

I don't see why Steven couldn't define her own gender if that were to empower her. There's plenty of pronoun sets to choose from already. (I don't think Steven is a woman but hey, I'm using the pronouns I was instructed to.)

You said, 'The problem with this whole discussion is that just pointing out that biological sex is actually relevant in some cases now counts as transphobic.'
Could you say how so?

And for what it's worth, I wrote you in the first place because I'd rather attempt to do or say something I think is important and screw it up than not do or say anything. I was and am mildly terrified to, in this case.

35. C Thompson,

Sorry, I expressed this badly. I meant chose to publicly identify one way or the other. As you know I don't believe in free will so of course no one really has a choice in anything anyway.

As to your question that pointing out biological sex isn't irrelevant counts as transphobic, take the incident with the guy on TicToc some months ago who declared he wouldn't date a transgender woman. Or, for that matter, claiming that it's transphobic to point out that it's a bad idea to allowing biological men at will to transfer to a woman's prison.

36. We already have a term for that though - an individual. Why would we redefine the word "gender" to mean "individual"?

And it's all part of an attempt to deny objectivity in science and maths as illustrated by some of the comments.The Woke Guards are abroad.

37. Steven,

Just because you don't understand why it matters to some people doesn't mean it doesn't matter to them. I'd have thought that as a transgender woman yourself you'd be a little more understanding.

38. C Thompson9:30 AM, June 15, 2021
"I don't see why Steven couldn't define her own gender if that were to empower her. "

I have. It's not "her" anymore, but his(1,261,485,137) at the time of writing.

Anyway, back to the Maths, do you agree:
Maths is objective
Maths is not known to contain any unintended subjectivities due to certain groups being overly represented among modern mathematicians (which isn't wholly true anyway)
Unfair loan approval procedures cannot be caused by the maths, it's the logic in the model

39. Sabine Hossenfelder11:00 AM, June 15, 2021

As I wrote, we should be socially as flexible as possible, but to underpin it in law is tricky. We are already at an impasse in the case of transgender women participating in women's sports.
Laurie Rubel's claim that objectivity in Maths is a myth and Kareem Carr's claim that there may be unintended subjectivities in Maths due to over-representation of certain groups, both claims presented without a single supporting example, are completely ignorant and sinister.

40. Dr Hossenfelder,

In those 2 cases: There's nothing wrong with having a sexual orientation/preference but I'd say the TikTok guy is transphobic because if he wants a particular anatomical configeration to interact with, he can purchase such an item. If he insists on interacting with a person, refusing to date someone with his prefered anatomy who is transgender is definitely transphobia.

As for 'assigned male at birth' women in jails, it isn't transphobic per se to say that the system as it currently stands doesn't work well. The system needs to be overhauled so all inmates' needs are met.

If we wanted to, there are many examples of these issues worth talking about - another whole blog's worth at least. I don't want to keep discussing them here though if you'd rather not. I do think these sorts of conversations are important to have, so thank you for answering my question.

41. C Thompson,

Yeah, I'm not sure this is the right time and place for this discussion but just to provide the context here is the superstraight guy. I think we're talking a little past each other here for what regards his "preferred anatomy" because his problem seems to be exactly that transgender women, who are sexually male, do not have his "preferred anatomy". I was trying to say if someone wants to have sex with people of the sex they are attracted to, that doesn't undermine anyone's rights hence isn't transphobic.

I agree with you on the jails, I think the current arrangement could be improved.

42. Steven Evans wrote:
"Laurie Rubel claimed that objectivity in Maths is a myth.
That's nonsense."

You are misunderstanding what she is talking about. Mathematical proofs are objective and she does not claim otherwise.

Her claim is that when mathematical models are applied to the read world it is not objective. The clerk hits a key and the computer tells him if he should approve the loan or not. The clerk treats this answer as an objective mathematical truth. This is the myth that Laurie is talking about. The clerk does not consider the possibility that the model might be wrong.

Steven Evans wrote:
"Do you accept it is nonsense now, or would you like to present an example of subjectivity for our delight and delectation?"

I will use your binary gender model as an example. I agree with you that it is a beautiful mathematical model. Unfortunately, when you try to apply it to the real world, there are many people for which this binary classification fails. And when this gender model is used to design public restrooms, prisons or sports teams, certain people are harmed by the shortcoming of this model.

You see, a mathematical model can discriminate against certain people. That is why we should not accept math as objective or neutral.

43. @Steven Evans
Again, the problem isn't transgender people participating in sports, it's the systems that divide everyone by gender. It isn't the case in boxing, there's 17 different classes divided by weight. There's no reason not to adjust other competitions accordingly.
The thing about your supposed gender is that it's not something that you're describing as a personal attribute, it's an attempt at mockery. Nice try, bad luck.
Over and out.

44. Dr Hossenfelder,

Got that, thanks for the link. I'm listening to music and your version of 'Talk to me' was just playing, lol.

45. Udi Fuchs12:29 AM, June 16, 2021

"Her claim is that when mathematical models are applied to the read world it is not objective."

That's not what she wrote. She wrote this:

1:04 AM · Aug 4, 2020· Laurie Rubel wrote
"the idea that math (or data) is culturally neutral or in any way objective is a MYTH."

And apparently she is a "mathematics education professor"!

There are no examples of cultural or any other subjectivity being discovered in Maths. This is insane nonsense.

"The clerk treats this answer as an objective mathematical truth. "
No, the clerk treats it as the policy of the bank, which it is. Can these policies be unfair? I'm sure they can. But that ain't the fault of Maths.

Your example of the definition of gender is not a problem with the Maths, since 1 + 1 = 2 when we're counting types of cells, it's a problem with your definition of gender versus that of the dictionary. Of course, if services based on gender are a problem for some groups of people, we can extend the services. We do not need to redefine the meaning of words which are based on biological facts.

46. Steven Evans wrote:
"That's not what she wrote. She wrote this:"

1:04 AM · Aug 4, 2020· Laurie Rubel wrote
"the idea that math (or data) is culturally neutral or in any way objective is a MYTH."

It is a one-line tweet. It is a bit ambiguous, but she explains herself in the next tweet:

It seems that you misunderstood her first tweet and therefore assumed it was total nonsense. Maybe the next time someone says something that sounds like nonsense, you should give them the benefit of the doubt and try to understand what they are talking about. This way you won't make a fool out of yourself.

Steven Evans wrote:
"it's a problem with your definition of gender versus that of the dictionary."

Lets see what the Oxford dictionary has to say about gender:
"the fact of being male or female, especially when considered with reference to social and cultural differences, not differences in biology; members of a particular gender as a group"

"The term gender is also used more broadly to mean a range of identities that do not necessarily fit in with the usual division between male and female."

So the main definition is binary, but it can also be a range of identities. In any case you are wrong when you claim that gender is defined based on sexual reproduction. Gender is defined by social and cultural differences.

47. Udi Fuchs1:46 AM, June 17, 2021

It's not ambiguous. She states very strongly that objectivity in Maths is a "MYTH" (all caps - she's shouting it)

Laurie Rubel wrote
"the idea that math (or data) is culturally neutral or in any way objective is a MYTH."

This is laughable nonsense. It's terrifying that a so-called "mathematics education professor" would write this. This is someone who doesn't know the first thing about Maths and yet is involved with Maths education. Parents in the US should make sure she has no input into DoE policy.

Mathwashing is separate to the question of objectivity in Maths. Of course, Maths can be used in algorithms that are unfair in some way, and then disingenuous people might claim the algorithms aren't unfair because they use Maths. But that doesn't mean that the Maths isn't objectively correct - it is. Obviously.

Don't chop off definitions. That's naughty. This is the relevant OED defintion:
"The state of being male or female as expressed by social or cultural distinctions and differences, rather than biological ones; the collective attributes or traits associated with a particular sex, or determined as a result of one's sex. "

As I wrote.

So not part of the definition. Just a comment that some people don't know the precise meaning of words. Who knew?

If you want to show that Maths is not objective, just provide an example.
If you want to show there is a 3rd gender based on a 3rd sex, then provide the natural scientific evidence.

48. Steven Evans wrote:
"It's not ambiguous. She states very strongly that objectivity in Maths is a "MYTH" (all caps - she's shouting it)"

The ambiguity is in the word "math". You are thinking that she is referring to pure math, but she is talking about applied math. It is obvious from her other tweets. I couldn't find a single tweet in which she claims that pure math is not objective.

I guess that I was ambiguous in my statement about her tweet being ambiguous. This is the nature human language. It is very hard to write well defined rigorous sentence. Pure mathematicians sometimes try to be rigorous. The result is exhausting and mostly incomprehensible.

Steven Evans wrote:
"Don't chop off definitions. That's naughty."

I didn't. It is the full definition from the freely available Oxford Learners Dictionaries. In any case I don't see any meaningful difference between the two. It still states that gender is defined by social distinctions, not biological.

Steven Evans wrote:
"If you want to show there is a 3rd gender based on a 3rd sex, then provide the natural scientific evidence."

You failed to show that there are exactly 2 genders, since some people do not fit into this binary classification. But it doesn't matter. Gender is a social construct and as such it is not rigorously defined.

49. Udi Fuchs12:07 AM, June 18, 2021

The maths used in an applied maths model is still maths.There is no example of maths itself being subjective.

Apologies. From the Oxford Learners Dictionaries; "the fact of being male or female, especially when considered with reference to social and cultural differences, not differences in biology"

So two genders (which are based originally on the two biological sexes)

50. Steven Evans wrote:
"The maths used in an applied maths model is still maths.There is no example of maths itself being subjective."

You have a talent of making true but meaningless statements. Saying that the math part of applied math is objective may be technically correct, but it is meaningless. The whole point of applied math is that it applied.

Pure math is objective, but applied math can be subjective.

Steven Evans wrote:
"So two genders (which are based originally on the two biological sexes)"

Your definition of gender is very rigid, but not rigorous. The math is correct and objective, 1+1=2. But the model is subjective. Specifically, it only considers humans that can reproduce.

51. @Udi:
I've been thinking that gender is a splendid demonstration of 1+1≠2, perhaps.

52. Udi Fuchs1:19 AM, June 19, 2021

There are no known subjectivities in Math itself whether it is applied or not. Laurie Rubel stated that "the idea that math (or data) is culturally neutral or in any way objective is a MYTH."

It is a MYTH that Maths is objective in any way, she claims.

This is an incredibly strong claim and completely wrong. It is completely the opposite of the truth, that there are no known subjectivities in Maths.

She has confused Maths with people's favourite things.

If she meant that models or algorithms using maths can be poor models of reality or unfair to certain groups, then that's what she should have written. That's completely different in meaning though. Words have meanings. Otherwise we can't understand each other.

Maths is considered to be objective, that is not a "MYTH".

"Your definition of gender is very rigid"

It's not my definition. It's the dictionary definition. Gender means the traits associated with a particular sex. So whatever one might consider those traits to be, by definition there can only be two genders because there are only two sexes. Note this definition does not rule out individual attributes, or concepts like cisgender and transgender.

"it only considers humans that can reproduce."
Presumably because the biological concept of sex is based on the rather important process of sexual reproduction. And the concept of gender is based on the biological concept of sex.

If you are claiming any collection of human attributes can constitute a gender, then it is not clear what you mean by gender, other than a random collection of human attributes.

Either present a subjectivity in Maths, or admit that Laurie wrote nonsense. Arguing that she didn't write nonsense because she didn't mean what the sentence she wrote means is just insanity.

Either admit that gender means the traits associated with a particular sex of which there are two, or provide a new definition of gender.

53. C Thompson4:46 AM, June 19, 2021

"I've been thinking that gender is a splendid demonstration of 1+1≠2, perhaps."

Brilliant. Fortunately, people who "think" like this will never get a job in a university STEM department or become a government minister. Human society is safe from your "thoughts".

54. It was a joke. Holy heck.
You should try that thinking business.

55. Steven Evans wrote:
"If she meant that models or algorithms using maths can be poor models of reality or unfair to certain groups, then that's what she should have written."

That is what she wrote, using her own words. You don't have a monopoly over freedom of expression. People are free to express their message however they see fit.

Steven Evans wrote:
"That's completely different in meaning though. Words have meanings. Otherwise we can't understand each other."

I understood what she meant, which proves that it is possible to understand her. Moreover, her message is more focused than your version. It expresses the fact that people mistakenly trust such models because they are based on math and therefore are assumed to be objective. You see, in one tweet she summarized a fairly elaborated idea on which she expanded on in followup tweets.

Steven Evans wrote:
"Either admit that gender means the traits associated with a particular sex of which there are two, or provide a new definition of gender."

And what traits are those? Facial hair, pitch of voice, long hair, wearing dresses. These traits are not binary, instead they define a multidimensional spectrum. Therefore I don't need a new definition, because the dictionary definition is generic enough.

2. This reminds me of the loaves and fishes story in the New Testament. Religious conservatives in America regularly use this as a example of why science is wrong, justifying the conspiracy theory nonsense that is destroying the country.

3. That only means that you can change the truth value of a proposition by changing the meaning of any of the terms in it. That goes for anything at all, not just mathematics. The Earth is the largest planet in the Solar System, just as long as you change the meaning of "Earth" to refer to the fifth planet from the Sun.

1. Sure, but in mathematics context is very important.

We regularly assume that maths we encounter without any specific context is "regular" maths in base 10.

Cyclic group mathematics is _very_ common, it underpins public key cryptography (well, the most popular kind anymay) which in turn is what almost all secure communications that happens online is using.

Likewise, when using matrices you have at least three different kinds of multiplication and the most common one does _not_ work like you naively expect it to.

2. Context is important. If I say "2+2" to my butcher, she has no trouble grasping that I am not talking about cyclic groups or the combining of two fluids or whatever.

My butcher and I both understand that, the way I am using "2" and "+" that 2+2 always equals 4.

It is not that we are unaware that there are different kinds of addition, just that we can judge which meaning is being used

And, let's face it, all of those other kinds of addition would have no meaning if it were not the case that 2+2 (in the sense that most people mean) is always equal to 4.

4. "Divide by three and take the rest" would not have been understandable to me from what I learned in pre-college school in the USA. What you called the rest, we called the remainder. Perhaps in some other English speaking countries they call it the rest. Also maybe a visual example was given which wasn't in the transcript. Anyway, it is a small point, but it confused me for a second or so. I decided to mention it in case it is useful in some other presentation.

I gather there is now "woke" and "unwoke", and some of the "unwoke" are "woke" about the "woke"--or something like that. I prefer to analyze each example as best I can, without using broad categories. It seems a little less likely to make mistakes that way. Although we will never stop making mistakes.

1. Hi JimV,

Yes, sorry about that. I put a note on this in the info below the video.

5. The theory of rings and fields is not usually taught until sophomore college mathematics. It is nifty and gives on a different sense of things. The modular group or cyclic group of n letters ℤ_n = ℤ/nℤ, which leads us into the theory of quotients of groups. I love to see the looks on people's faces when I tell them about dividing an algebra by another algebra and getting a space as an answer. This is still a bit away from p-adic number theory.

I am not exactly sure how 2 + 2 = 5. This would be some sort of non-Archimedian field. The Archimedean field obeys |x + y| ≤ |x| + |y|, and a violation of this would have |x + y| > |x| + |y|, so we could maybe get some version of 2 + 2 = 5. These things do occur with quirky number fields, such as Robinson numbers.

1. 2 + 2 = 5 mod 1

2. 2 + 2 = 5 is not a result in a cyclic group. The Z_5 group can count from 0 to 4, containing 5 elements, and then it starts over at 0. Here 2 + 2 = 4. For Z_3 there are the element 0, 1, 2, 3 so that 2 + 2 = 0.

3. little erratum: I wrote about dividing an algebra by an algebra. I meant dividing a group by a group.

4. One could write [2] + [2] = [5] in Z/Z isomorphic to the zero ring, meaning 2 + 2 ≡ 5 (modulo 1)
And then we drop the brackets of the equivalence classes, a not unknown convention, to get:
2 + 2 = 5

It's almost arguable. And meets the Oregon Department of Education's requirement that no answers are wrong.

5. If you want to call it that go ahead. I think most people would call it 1 or one.

6. Sabine wrote: ...velocities don’t add like apples... according to Einstein, 1+1 is 1.

I think it's not good to tell people that "Einstein says 1+1=1". Special relativity doesn't say u+v = (u+v)/(1+uv/c^2). That would clearly be an abuse of notation, because the symbols "u+v" appear on both sides of the equation but must obviously mean different things. The expression u+v on the right side is ordinary addition of speeds. In contrast, the expression "u+v" on the left side - if anyone actually wrote it that way - would have to signify not the arithmetical sum of u and v (as it does in the numerator on the right side), but rather the composition of two speed transformations, involving the evaluation of speeds in terms of two different systems of inertial coordinates.

Instead of bewildering students and undermining their confidence in basic rationality by telling them that "addition doesn't work for speeds in special relativity" (which is obviously untrue), we should just explain what "composition of speeds" means, and then explain that although the composition of speeds was given by addition in Galilean relativity, it is not given by addition in special relativity. This explanation may not be as much fun, but I think it's more clear and accurate.

Admittedly, many relativity texts - including Einstein's - have often used the word "addition" to mean composition, and for mathematically inclined people this isn't too confusing, because the phrase "addition theorem" is familiar from trigonometry, and of course the linear velocity composition formula is just the addition theorem for the hyperbolic tangent. But for most students I think it isn't helpful to conflate addition of two quantities with formulas relating some function of the sum of two quantities in terms of that function of the individual quantities.

To give an example of how the word "addition" is used in various ways, I think it would be better to leave Einstein and special relativity out of it, and just point to trigonometric "addition" formulas, such as tanh(x+y) = [tanh(x)+tanh(y)]/[1+tanh(x)tanh(y)]. Someone might mischievously refer to the left side of this equation as "the addition of tan(x) and tan(y)", but it really isn't, it is tan(x+y). It's best to avoid conflating those two things, so we don't give anyone the impression that ordinary arithmetic is invalid in trigonometry (or special relativity).

7. little erratum: I wrote about dividing an algebra by an algebra. I meant dividing a group by a group.

1. Its possible to divide and multiply and much wider variety of algebras than just groups, rings and modules. It's a theorem from universal algebra that any variety, which is a set with a number of operations and satisfying a certain set of axioms, can be multiplied and divided. Not all algebraic objects are varieties, for example a field is not. The problematic axiom here is that division is always possible by any number other than zero. In other words, it is the lack of universal quantification over division thats the problem.

I'm sure you know this, but multiplication and division not also applies to algebraic entities but also to spatial entities. We can multiply and divide topological and smooth spaces. In fact, we can add them too. And so we get a ring like structure or rather a rig - a ring without negatives and hence the missing 'n' in rig. Its a mathematical joke, even though the structure is no joke.

2. An algebra generates a group. For a group g and an algebra A the group is in cryptic form g = e^A or as we physicists put it g = e^{iA}. This means products or quotients of groups implies summands of algebras. It gets gets kind of complicated with the Baker-Campbell-Haussdorf theorem on doing this.

8. Well Sabine, you blew me out of the water!
I knew in advance that your video was going to be about 2+2≠4, so I did some advance research and had a prepared comment for you that was, in my mind, brilliant!
Watching your video with my fingers poised on the ctrl+v keys, I realize that I had underestimated you. I thought that you were going to pull some kind of math trick on us. I even ‘woke’ up early so that I could be the first to comment on it.
Disappointed that I wasn’t going to impress you with my brilliance, I removed my hands from the keyboard and went back to bed. But I want to thank you for the math lesson. I should have known better than to think that you would have stooped to trickery in your videos.
{jcrc}

p.s. Can someone please explain to me what the difference is between 'math' and 'maths.'

1. Glad you liked it :)

"Can someone please explain to me what the difference is between 'math' and 'maths.'"

They're both abbreviations for mathematics, "math" is the American English version, "maths" is the British English version. I'm afraid my English is rather inconsistent and tends to flip between AE and BE.

2. Your accents, both verbal and written, are part of you.
I wouldn't change a thing.

9. We did finite groups and modulus arithmetic in 7th grade, but I was in a special math program.

1. This comment has been removed by the author.

10. At first your statement that 2 plus 2 = 5 confused me then I realised maybe 2+2 does equal 5 by using ceil(2.1)+2=5 using programming code, thereby getting around issue as presented by the falsehood of the logic statement.

But then you touched on the modulus operator. Something I am familiar with in programming. Yes it is very useful as a cyclic operator.

Say we have a 5x4 grid. One way we can present this is through nested loops. In this case the first loop goes to 5, the second loop goes to 4. While/for/do it does not matter.

Another way is through the modulus operator. Both require data stored as a list or array - the loop method ideally reading from a 2D array and the modulus from a 1D array.

The modulus however does save processing power as it requires fewer lines of code per iteration.. however it is not flawless and there are times when one does need to use the nested loop logic to id where the data is coming or going from memory to the screen.

11. "Does 2 + 2 = 4? No! Because two cats plus two sausages is what? Two cats. Two drops of water plus two drops of water? One drop of water." -- Vasiliy Georgievich Bogin explaining the philosophy of Georgy Shchedrovitsky (https://www.nytimes.com/2011/09/18/magazine/my-familys-experiment-in-extreme-schooling.html)

12. Well its obvious. In the expression 2+2=5 there are 5 symbols. QED as they say.

1. That is so. Also, an example of apophenia. :)

13. - I think you may have confused your audience, to be honest I was a bit dismayed with the 20C-water example, not according to usual your science-standards I believe. Perhaps a more interesting discussion would be : some seem to believe math becomes an absolute truth-reference frame when thinking about or describing nature. Is that so ? Or are for instance string-theorists doomed forever ? There is one good point on the 20c-mixing example. Most of our math comes from observations, hence later a strict formalism was created. How does that limit theoretical physics in the end ?

1. What's your problem? You thought 20C+20C=40C and now you're dismayed I told you it's wrong? I find it stunning how many men who know nothing about social media think they can tell me how do my job.

2. I apologize I didn't mean to be offensive, I found the example a bit 'weak' as stated, I find the deeper-discussion I referred too , more interesting.

3. Sorry, apology accepted. I thought it's an example that most people will immediately understand. Much fewer people have heard of the special relativistic addition for velocities.

4. It was a great example. I can have my seven year old kids watch this and that example will help them understand what you mean. Thank you.

5. I agree about the relativistic additions, of course math=math , physical phenomena should not 'disturb' it, but as I say there is an implicit deeper connection, perhaps a topic for another day!

6. I thought the temperature example interesting because it only involves scalar quantities.

Actually, there is an algebraic gadget that handles the relativistic velocity addition rule in its full glory: Ungers gyrocommmutative gyrogroups. The gyro in the name arises from the gyration that occurs when we add two skew velocities.

14. This comment has been removed by the author.

1. "For the mod-3-example we can write 1=4"

4 isn't in the group, this equation doesn't even make sense.

15. This comment has been removed by the author.

16. High Sabine,

I love the addition of frequencies like energies, and the addition of wavelengths like the inverse of the addition of the inverse of wavelengths. That is much more fun than apples. Best of all the addition of volumes of phase: h^3 + h^3 = ?

Best,
J.

17. First, if the question is whether 2+2=4, I must point out this delightfully relevant and well-done DUST sci-fi short (15 minutes) story, “The Secret Number.” Search for: dust the secret number. (The kid in the first confusing minute is a flashback memory by the adult psychiatrist.)

Mainly because our brains do it so well, we usually have no clue as to both the incredible levels of temporal stability and the massive cognitive resources required to make sense of the notation 2+2=4. A minor part of that complexity is realizing that the notation style itself took centuries if not millennia to mature into its current form of positional Arabic numerals, implicit use of zero, and infix (recall reverse Polish notation?) operators. A more recent twist is the concept of overloading, the idea that the same operator can have vastly different operational meanings depending on the nature of the operands, that is, the items added.

Shall we also delve into the issue of how, exactly, to represent those numbers in a fully formal fashion? By that, I mean a representation of numbers that strictly forbids biological shortcuts, fast-thinking, and general sloppiness? You get that by requiring numbers to be represented only by finite state machines that lack such ambiguities. These are more commonly known as computers.

[The idea that programmers are the only true formalists in human history while traditional mathematicians tend to be dangerously sloppy has non-trivial consequences for physics. One delightfully pathological example of mathematical sloppiness that has wasted human-centuries of precious intellectual resources is Kruskal-Szekeres (K-S) coordinates. General relativity theorists like K-S because it creates a plane for analyzing event horizons that is “well-behaved everywhere outside the physical singularity.” No, it does not! The plane is “well-behaved” only if you completely ignore that both point density and numeric precision explode to infinity as you get near the singularity. The original infinities are still there, unchanged. This disregard for the computational and energetic implications of formal equations has caused generations of general relativity theorists to think they can “solve” event horizons paradoxes by using K-S. It is just a Jedi mind trick that distracts its users from the painful reality that the actual infinity problem remains unchanged. Unfortunately, that is just one minor example of how mathematical precision sloppiness has led physicists astray over the past century. The most brutal examples are in particle physics and, especially, quantum mechanics.]

Um, where was I? Oh yes! I just wanted to point out the immense “cognitive load” implied by a statement as simple as 2+2=4. Here’s the crux of the problem: “2” of what? As the deeply insightful modern philosopher Ron Green points [1][2], no two non-quantum, classical objects are ever exactly identical, not even in particle physics (except in Bose-Einstein condensates). So how do you decide two entities are sufficiently similar to be addable?

That’s where even 2+2=4 gets weird. Adding requires addable entities. Any fully formal definition of addable entities, in turn, requires an intelligent entity with sufficient perceptive and modeling complexity to define approximately which items are enough alike to be addable. You cannot fully and formally define addition without including that entity. You can do it sloppily by ignoring the beautifully ironic (formality requires approximation?) role that humans or computers play. But without including approximation, that is all such definitions can be: sloppy. Set theory is worse: It assumes set cognition as its most fundamental operator!

Folks often lament the need for observers in quantum mechanics. Hah! You can’t even get rid of them when saying 2+2=4!

-----
[1] Green, R. Nothing Matters: a book about nothing, John Hunt Publishing, 2011.
[2] Green, R. Time to Tell: a look at how we tick. John Hunt Publishing, 2018.

1. Terry,
How are computers getting on checking the truth of Fermat's Last Theorem? Still made 0% progress? ;)
In practice which is more believable - several thousand Mathematicians agreeing that Wiles' several hundred steps of logic prove FLT, or that there have been no software or hardware errors when running a server farm for a year to check quintillions of cases, or exactly 0%, of FLT?

Do you have an example of where continuous Maths has led Physicists to an actual mistake about Physics?
Do we know that space-time isn't continuous?

I agree counting depends on perception and the evolved brain, but surely neuroscientists will be able to or can explain this in terms of brain structures and perception.

Why is set cognition Set Theory's most fundamental operator? Isn't ZFC a formal first order theory which defines the concept of a (iterative) set regardless of our intuition?

2. [1 of 4]

Hi Steven, I hope you are doing well.

Thanks for the excellent (but at times a bit puzzling) questions:

----------
>… In practice, which is more believable — several thousand mathematicians agreeing that Wiles’ several hundred steps of logic prove FLT or that there have been no software or hardware errors when running a server farm for a year to check quintillions of cases, or exactly 0%, of FLT?

Wile’s beautiful proof, of course. It’s beautiful and well done. But what has that got to do with obfuscation of infinities by sloppy, non-axiomatic use of real numbers?

----------
>… Do you have an example of where continuous maths has led physicists to an actual mistake about physics?

Wow, where to start? You did read what I said about the utter disaster of K-S coordinates, which destabilized the GR theory community and led to mutually contradictory interpretations of the event horizon that over decades have flopped around as wildly as an unanchored firehose? Firewalls, anyone? When theory flops around that wildly, there’s a good chance there’s a flaw in its axiom set. In this case, the flaw is pretty easy to identify: It’s the inadvertent use of what amount to distraction tricks like K-S to gloss over infinities inherent in the real problem, specifically by leaving number density out of the axiom set for representing the physics accurately. That same dumb indifference to number density causes the occasionally disastrous engineering effect called gimbal lock. The omission of number density from the axiom set, especially for mass-and-energy extrema in theories like GR and quantum field theory, is dumb because it is inherently non-physical. The fact that entire careers and entire university departments arisen from such sloppy thinking doesn’t make that kind of axiomatic sloppiness more acceptable. If anything, it just makes the situation worse.

If that’s not enough, one especially mind-boggling example of how number density sloppiness “has led physicists to an actual mistake about physics” is Everett’s “ultra-density catastrophe,” the matter-wave analog (superset, really) of the ultraviolet catastrophe that first got quantum mechanics moving in the early 1900s. Einstein knew very well that both electromagnetic waves and matter waves are quantized. After all, he got a Nobel Prize for realizing the former and effectively booted out of the quantum community for pointing out the latter at a Solvey meeting.

So decades later and two years after Einstein dies, Everett gets excited about the idea that the harmonics of matter waves might make them grow infinitely detailed over time, just as earlier classical physicists worried wondered if harmonics in electromagnetic waves might end up incinerating the universe. Everett appears never to have realized that all he was doing was recreating the same “energy is free” perspective captured in the ultraviolet catastrophe thought problem.

In any case, even more so than K-S coordinates in general relativity, Everett’s ideas depend on axiomatic indifference to both number density and number precision since the waves to support all of those universes must quickly grow infinitely information-rich to support the infinitely detailed waveforms. The peak damage from many worlds was decades ago, but the fact that some excellent physicists still take Everett’s ideas seriously is an indicator of just how much damage this kind of sloppy-math thinking has done to theoretical physics.

3. [2 of 4]

Finally, while the availability of renormalization “fixes” makes it more acceptable, it’s hard not to point out that the original sin of infinities in calculating the mass of point-like charged particles such as the electron never really disappeared. That, too, is a more subtle consequence of assuming that it’s perfectly to model space as having more precision than the mass of the electron locally supports. Models that made space itself uncertain, versus the now more common view that space is infinitely detailed and particles are “smeared out” across it, were explored as late as the 1970s. Such ideas led to S-matrix and later (somewhat ironically) string theory but never became a self-contained theory. The idea that each observer and particle might have its version and precision of spacetime is, however, in many ways just a quantum generalization of the frames of special relativity.

----------
>… Do we know that spacetime isn’t continuous?

All available experimental evidence points powerfully towards space being smooth and continuous down to any limit we can reach experimentally. The most impressive example of the enduring smoothness of spacetime was the 2020 HAWC Consortium study of gamma rays above 100 TeV [1]. This outstanding effort showed that simpler versions of both string theory quantum gravity are too lumpy to accommodate good ol’ special relativity. I seriously doubt any experiment will ever encounter a limit in the smoothness of spacetime.

Attempts to make spacetime discrete via any variant of precise matrix-like or lattice-like topologies, such as Gerard ’t Hooft’s 1993 holographic assertion that “One Boolean variable [bit] per Planckian surface element should suffice” [2], fall prey to the same axiomatic precision indifference (time for an abbreviation: API) misuse of real numbers that makes K-S and multi-worlds non-physical. The brilliant ’t Hooft — one of my favorite physicists — was on the right track of holography but got tripped up by the community-wide API mindset.

However, a holographic model in which bits are sparse and directly proportional to total mass-energy is also possible [3], and that type of holographic model is both more straightforward and fully compatible with Planck uncertainty. The trick to making bits compatible with known physics, as opposed to inserting them deus ex machina, is to acknowledge a point that is well-known to anyone who has tried to use holography for digital storage: bits emerge from holography, not vice-versa. In other words, bits in holography are emergent, non-fundamental entities created by internal convolution of the holographic medium. It then becomes the total quantity of holographic medium plus the desired level of certainty for each bit together that determines the maximum number of bits possible.

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[1] Albert, A. et al. Constraints on Lorentz invariance violation from HAWC observations of gamma rays above 100 TeV. Physical Review Letters, APS, 2020, 124, 131101.

[2] ’t Hooft, G., Dimensional Reduction in Quantum Gravity. arXiv preprint gr-qc/9310026, 1993.

[3] T. Bollinger, The Dual Denysuk Holographic Universe, Parts 1, 2, 3.
Comments in "Backreaction: Is the universe REALLY a hologram?":
https://backreaction.blogspot.com/2021/03/is-universe-really-hologram.html
2021-03-31.17:36: ?showComment=1617226571494#c4195291181306303905
2021-04-02.12:37: ?showComment=1617381420045#c1570206509549898588
2021-04-02.13:17: ?showComment=1617383833975#c373334779288985000

4. [3 of 4]

Saying that a holographic universe does not begin with bits is equivalent to saying it does not begin with classical time or classical physics, both of which require the specificity of “information” to exist. Instead, the holographic medium (call it mass-energy) grows bits (entropy) as it becomes more convoluted and self-determined. The maximum density of bits possible in such a “low resolution” universe is enormous by human standards but infinitesimal in comparison to any of the deus ex machina, API-enabled “Planck foam” models of bits (or strings, loops, or other even more bit-intensive objects). The maximum bits per kilogram in a low-resolution holographic universe is rather mundane, being essentially the same as how much useful information one kilogram of matter can encode.

I call this low-resolution, emergent-bits (and thus also emergent-classical-physics) holographic approach the Dual Denisyuk or DD model of the universe. It’s dual because it simultaneously encodes the xyz and momentum space images of the universe. It’s Denisyuk because Yuri Denisyuk discovered 3D emulsion holography. However, unlike the ’t Hooft holographic model, no distant or hidden 2D surface encodes the 3D space of a DD universe. Instead, the DD image encodes and evolves itself; that is, the convolutions of the medium are identical to the image it projects. In combination with the absolute conservation of quantum numbers, this rather severe dynamical constraint creates much of what we think of as classical physics, including the prevalence of so many phenomena with smooth, differentiable behavior.

Thus in DD, you are your own holographic image. Entanglement with the rest of the universe (from both of them, really) keeps your image stable. We see and interpret such entanglements and their preserving actions in terms of concepts such as “space” and “time,” but these are just different ways of expressing the strict constraint rules imposed by DD self-imaging in combination with absolute quantum number conservation. Notably, there is no concept of “particles” in DD, just mutually bound bundles of quantum numbers whose precision (apparent size to observers) varies primarily with the level of detail of the local holographic image.

Interestingly, the bundle concept helps analyze some recent Wigner’s friend [4] experiments that look for whether two observers can “see” two divergent realities. Different quantum numbers in the same bundle, such as spin and mass for electrons, can have different levels of holographic resolution simultaneously. Proietti et al. beautifully tested and documented this very point for the case of spin (polarization) and mass-energy (location) in “Experimental test of local observer-independence” [5]. One analysis [6] of the Proietti paper asserts that “… Wigner’s friend measures [and stores] the polarization of a photon [while] Wigner … performs an interference measurement… [The] experiment [shows] both realities can coexist even though they produce irreconcilable outcomes.”

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[4] Wikipedia, Wigner’s friend. 2021-06-15.
https://en.wikipedia.org/wiki/Wigner%27s_friend

[5] Proietti, M., et al. Experimental test of local observer-independence. Science advances, American Association for the Advancement of Science, 2019, 5, eaaw9832.

[6] Kentuck FC. A quantum experiment suggests there’s no such thing as objective reality. MIT Technology Review: Emerging Technology from the arXiv, 2019.
https://www.technologyreview.com/2019/03/12/136684/a-quantum-experiment-suggests-theres-no-such-thing-as-objective-reality/

5. [4 of 4]

Well, no, not really. The “two realities” in question are nothing more than two independent conserved quantum quantities of a single particle, those being spin (polarization) and location (interference). I was excited when I first heard about this paper as a possible new insight into quantum mechanics. Alas, on closer examination, it’s just another example of how some particle bundles can be temporarily teased apart and made to show independent quantum behaviors. You cannot get a reality paradox out of that any more than you can by Wigner examining the front end of an elephant and his poor friend getting rear-end duty.

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>… I agree that counting depends on perception and the evolved brain, but surely neuroscientists can explain this in terms of brain structures and perception?

Well, yes. That was my point: Such abstractions exist only as material systems that are both complex and non-intuitive. That is true not just for biology but also for silicon.

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>… Why is set-cognition set theory’s most fundamental operator? Isn’t ZFC a formal first-order theory that defines the concept of an (iterative) set regardless of our intuition?

ZFC Axiom 1, the axiom of extensionality:
Two sets are equal (are the same set) if they have the same elements.

Five apples sit on a desk. You and a friend look at them, and you both decide they are a set. You compare your sets and decide they are the same set.

Would you agree that the two sets are “real”? Otherwise, you would have no way to talk about them and compare them for equality.

Question: Where in the physical world do these two “real” sets reside, and of what, exactly, are they composed?

ZFC Axiom 7, the axiom of infinity
With quite a bit of paraphrasing on my part: Start with a root node. Expand it into a binary tree so that the right branch always looks exactly like the immediately previous version of the tree. Call the initial node zero, the first tree 1, the second tree 2, and on. These become the numbers N. For completeness, pretend you managed somehow to expand the tree an infinite number of times. Reach out into the infinitely distant future, bring back the final tree, and label that one “infinity.”

Well, you’ve got me there. There’s nothing in that definition that remotely reeks of computer programming, arbitrary decisions (Left or right branch? Up or down branch? Red or blue branch? All of the above?), or sloppy non-physical assumptions such as time travel to and from the infinitely far future. :)

But surely I’m exaggerating the importance of at least the time travel part? As in, no physicist would fall for including that step in an actual model, right? Alas, whenever someone uses K-S coordinates to “cross” an event horizon, that is exactly what they are doing: Assuming that the part of the infalling object that an outside observer sees lingering forever on the event horizon nonetheless also manages to pass through the horizon and somehow, more-or-less, return to “now” on the inside. After all, if you cannot cross the boundary outside-observer finite time, you cannot create a singularity before the end of the universe, and where’s the fun in that?

In any case, give me a break. ZFC is nothing more than programming, and not even a particularly persuasive example of it. Binary trees make inefficient and needlessly inelegant numbers.

There is at least one critical insight in all this ZFC stuff, one that does get close to fundamental physics. Here’s my way of phrasing it:

The simplest step forward in causal time is a binary choice.

Using the DD model, I can even tell you why: The smallest possible unit of causal time occurs when the holographic image of object gains sufficient resolution to choose, irreversibly in terms of hologram elaboration, between two possible configurations, such as spin-up or spin-down for an electron.

6. Terry,
Thanks for the response. I think I got a flavour of what you mean. I thought that your "The idea that programmers are the only true formalists in human history while traditional mathematicians tend to be dangerously sloppy has.." meant you didn't trust traditional pure maths proofs, but see that is not the case. I have heard before of computer scientists claiming that even the most rigorous Weirstrassian maths isn't rigorous by CS standards.

I get the impression that this is about information-theoretic aspects being ignored in models, which you have written about before and mentioned in your comment to the embedded space post. I have copied and pasted your comment into a text file and will try to get a handle on this information-theoretic pov. This statement appears to be an info-theoretic check on the validity of the model: "The maximum bits per kilogram in a low-resolution holographic universe is rather mundane, being essentially the same as how much useful information one kilogram of matter can encode."

One point I simplistically cling to, though, is that one would expect any grand new theory of the physical universe to be able to tell us something the old theories couldn't, something that can be checked by observation. But I take your point that theoretical cul-de-sacs could have been avoided.

On ZFC, it seems that any model of ZFC will have an underlying set in some meta-universe so I guess this is equivalent to saying set cognition is fundamental. Though I suppose set theorists are happy for that to be the case, your problem seems to be with its physical consequences if it sneeks into a physical model.

Anyway, I shall have a read and a think...

7. [1 of 2]

Steve,

Thank you for your thoughtful response. I appreciate questions such as what you have asked because they are a great way to force me to examine my ideas more skeptically. Who could be better to demolish a flawed idea than its creator?

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>… I have heard of computer scientists claiming that even the most rigorous Weierstrass maths aren’t rigorous by CS standards.

We have no shortage of idiots, possibly even a glut of them. But if they demean the clear-headed foundational work of someone like Karl Weierstrass, they are basket cases living in a digital fantasy, folks who likely think analog is a dialect of Prolog.

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>… “The maximum bits per kilogram in a low-resolution holographic universe is rather mundane, being essentially the same as how much useful information one kilogram of matter can encode” … [is] an info-theoretic check on the validity of the [DD] model

Yes. Alas, it’s not as helpful as it might seem. To the best of my knowledge, no one has ever proposed a way to access the kinds of data storage proposed in, for example, ’t Hooft’s Planck-scale bits. Saying that mundane storage is the limit thus does not immediately suggest any lab-scale direct comparison tests.

On the other hand, such a low bit density should have observable and decidedly non-standard consequences for cosmology where it would make space itself into a finite resource.

Think of it this way: What if general relativity is perfectly correct, but the assumption that the spacetime fabric on which it operates is finite? What if, for example, stretching out the universe beyond the scale of galaxies and galaxy clusters cause space to become so bit-poor that even though GR still works the same way numerically, it longer has sufficient precision to give the same uniformly smooth results as a ’t Hooft Planck-bit-density projective holographic universe?

What I think (nothing more at this point) that would result in is baryonic matter acting a bit like gluten in a pizza, making it tend to stick together so that gravity can “do its thing,” unhindered by bit-sparse space. The bit-starved regions would no longer fully implement gravity and would lose out in the competition for keeping baryonic matter.

That’s just unquantified speculation at this point, but I can’t see any reason why a precision-aware variant of GR could not convert such speculation into a precise predictive model. The baryonic and bosonic mass of the universe would define a maximum volume for smooth space with a uniform bit density. Until it reaches that volume, an expanding universe should be capable of maintaining high cosmic-scale smoothness in the distribution of matter. However, once the universe surpasses that volume, it would, for lack of a better phrase, begin to rip apart at the space-fabric level — not gravity, but space itself. Gravity would continue to operate normally, but only across regions that maintain a high baryon density.

If you play that out, I’m pretty sure you end up with a universe with lots of large, exceptionally empty voids, some sheets, and (depending on what stage you are at) and lots of string-like features. The strings would survive longest since the odds of new rip-voids disrupting them is less with just one dimension of stretching. Eventually, however, even the string-dominated web structure fails as new voids break the strings and gravity contracts their pieces into smaller, more spherical clusters. In the end, it’s nothing but isolated roughly spherical islands of galaxies and galaxy clusters, separated by exceedingly empty “bit ripped” space where gravity is no longer welcome.

Is there a MOND opportunity in this scenario? I don’t know, but the idea of space ripping would seem to present an opportunity for MOND-style mathematics. MOND would apply at the lower and more fundamental space-ripping level rather than by altering gravity directly.

8. [2 of 2]

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>… I … would expect any … new theory … to tell us something … that can be checked by observation.

See above (part [1 of 2]).

There are also many predictions that things like wormholes cannot exist because the fabric of space fails before gravity can get there. However, since the existence or non-existence of such exotica does not appear to be testable, I’m not sure those predictions matter. Cosmology, however, is an increasingly data-rich option for exploring and testing alternative theories.

A very different domain of possible validation for low-precision direct holographic models of the universe is this: It may prove that folks like Karl Weierstrass were doing physics all along and didn’t realize it. That is, in a dual holographic universe the rules of closeness and smoothness that enable calculus to work emerge from bits being created rather than fundamental. If you begin by assuming an infinite number of bits are available in the fabric of space, then there is no good reason why spacetime shouldn’t be as complicated as the Mandelbrot set. If you instead assume that bits are emergent, smoothness becomes the default and fractals the expensive exception. A Dual Denisyuk universe is profoundly friendly towards the founders and elaborators of the calculus. It’s precision indifference that has caused problems, not the calculus itself.

Another possible future test point is that a quantum wave collapse can’t be genuinely random in DD. The collapse is just a new reconfiguration or convolution of the hologram, predicated by its previous state. The trick is that this “previous state” may, through entanglement, include state information from the other side of the universe or even from its antiverse. But still, with more specific models, it should eventually be possible to show that collapses have dependencies on experimentally accessible variables.

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>… in some meta-verse

Sorry, not going there. Not a parsable phrase.

>… if it sneaks into a physical model

No. My point is simple: There is no example anywhere in the universe of a “set” that does not consist, even in its simplest possible form, of at least a few bits within some information processing unit.

No one can call “set theory” fundamental without first giving at least one example of what a set is. You cannot create a foundation for all of mathematics by first refusing to accept that the actual form of its most fundamental unit “doesn’t count.”

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>… Anyway, I shall have a read and a think

Hey, thanks. I appreciate your willingness to take a look!

9. Terry, Thanks for the further details. I'm interested in understanding properly this information-theoretic angle you always bring up, in particular the ZFC comments.

">… in some meta-verse
Sorry, not going there. Not a parsable phrase."

I thought I had it in my head what it meant having had a shufty at some ZFC postings on Math Overflow, etc., but deep down I knew I wouldn't get away with it as I was writing it ;)

18. I'm going to speak up for the ordinary man in the street who believes that 2+2=4. Given that generally he's been taught this all the way from primary school to high school, is shouldn't be surprising that he's astonished when told that this need not be the case. And in a way, he's right, because when he imagines addition he imagines it for a particular kind of addition which of course doesn't obtain everywhere.

What is even more astonishing is non-classical logics where the law of the excluded middle doesn't hold. This seems like an unhinged possibility which has only been raised since imagining non-classical anything became all the rage after non-classical geometry became all the rage. The question is, whether it has anything solid to say.

The seed of such a logic was hinted at in Aristotles time when he pointed out that the usual logic doesn't apply for statements about the future. Hegel also suggested that logic was involved in motion, in that the reason why anything moves, is that for an instantbit is both here and there.

Whilst classical logic is, as everyone knows today, can be algebracised through Boolean algebras it tirns out a particular flavour of non-classical logic, intuitionistic logic, can be algebraicised through Heyting algebras. This makes this notion much more concrete.

Moreover, calculus can be developed in a synthetic way tgrough intuitionistic logic because it allows for infinitesimals which are defined to be nonzero numbers d such that d squared, that is d^2, is zero. Classically, the only such numbers is zero. Intuitionistically, we have the infinitesimals.

As the calculus is implicated in the science of motion, this makes concrete Hegels suggestion that a particle moves because it is both here and there.

Of course, its natural to wonder whether intuitionistic logic can help understand the magic of the quantum. This to my mind is an underexplored idea. Personally, I like the idea that logic is not classical because of the futurity of time as it implicates time in an ireversible way. After all, the most pervasive observational fact of our world is that time moves in a single direction.

1. Mozibur, your observations are nicely precise and intriguing. I'd never heard of the intuitionist Heyting algebras before. I'll be sure to read up on them.

Although I came to my views by way of cognitive science, they fall under what I now know is "intuitionism," the idea that math is a construct of human minds. I do wish they had come up with a less wishy-washy sounding name for what I perceive as being brutally, relentlessly precise in describing how numbers really work. It's some non-intuitionists who get sloppy when they start pretending their own cognition does not exist.

Regarding "whether intuitionistic logic can help understand the magic of the quantum," the law of the excluded middle asserts that every situation is a bit, a deterministic either-or choice. But if bits themselves are an emergent phenomena, such as when they are generated by holographic data storage media, then classical logic becomes an overly limited model that scoops the deterministic "bit cream" off such a system while ignoring the emphatically non-binary medium from which those bits emerged. Low-resolution quantum holographic models of the universe do much the same thing by interpreting classical reality as the excluded-middle image projected by a much broader, blurrier, and profoundly entangled quantum reality.

19. Vectors are the objects that come to mind when I think about non-classical addition. It's not often pointed out that they also have a non-classical multiplication. Well, it is pointed out with the dot and cross product. But the dot product only returns a scalar whilst the cross product only works in 3d and 7d. The question arises is there a natural product that works in all dimensions and returns a vector.

The obvious product was pointed by Grassmann (actually by his father), he suggested that the product of two vectors is the parallelogram spanned by the two vectors. This is more or less the tensor product. An interesting question arises when the two vectors are parallel, then the parallelogram is degenerate in that it doesn't sweep out some area. This suggests an additional axiom: u x u = 0. That is the square of any vector vanishes. When we apply this to the sum of two vectors, u+v, we obtain: (u+v)^2 = 0 and then multiplying out and cancelling the squares we find: u x v = - v x u! This is the paradigmatic rule for Grassmans exterior algebra and which when lifted into the manifold context, gives us differential forms.

20. For those of you who insist on talking about penises, here is an example of how 11+1=1.
My doctor told me that I was lucky because if mine was an inch longer it'd be a foot.

21. Not sure if this is on topic, but here is an example where 2+2=5 makes perfect sense. Apparently this is something children come up with independently. A kid holds one hand with three fingers up and says "two", then the other hand with three fingers up and says "plus two", then both with the six fingers up and says "equals five". It makes sense because it is meant to represent the number of gaps between the fingers. With three up there are two gaps, and with six up, when you hold the hands next to each other, there are five gaps.

22. Based upon my credentials in Public Administration if I were tasked with the design of a program to determine whether a convicted transgender person should go to a prison for men or women it would be something like this:

A clinical evaluation would be made to determine the status of the relevant individual's mental and physical gender identity along with other relevant factors. Then a determination would be made based on consideration of the likely hood of physical violence up to and including rape in light of the state's legal obligation to house the individual in a reasonably safe manner.

Along with the individual evaluation, available data regarding incidences of violence in the housing of transgender individuals must be considered. Then the individual can be placed where there is the least possibility of adverse outcomes for the individual and other inmates. It does not automatically matter whether the transgender person has or has not a penis. There is not likely to be one solution that fits all cases.

Frankly there has been enough discussion of this issue. I hope the above puts an end to it.

1. Hi Steve,
I think that's potentially a good idea for all inmates.
I would hope that others will stop thinking of transgender people, especially women, as causing harm to others, invading spaces, erasing women's rights and all that guff, then there won't be the need to argue that.

23. Determining which stuff happening in the world matches with which mathematical ideas was always an empirical task, if hidden in plain sight.

24. I fully accept that non-normal addition is part of mathematics and physics. However, I'm also in favour of the person in the street for whom 2+2=4 is always the standard sensible result. To break this one has to wriggle and introduce other stuff such as units (e.g. velocity) or redefine addition (e.g. cyclic groups). It's all a bit like lawyers twisting facts in a court case.
I notice that in the law for adding velocities, the numerator is the addition of the two velocities. And how do we add velocities? We have to use the law for adding velocities, and we have a nice infinite regress. Of course, we simply wriggle out of this one and redefine something.
Back in primary school, when I first learnt about addition, we were taught "one and one makes two"; we were not taught "one plus one equals two". So doing sums with "and" & "makes" was in a different place from maths with "plus" & "equals".
My favourite example of non-normal addition is "one plus one makes four", as in father "and" mother have two children "makes" four.

25. 2 + 2 always equals 4. All the comments to the contrary, including Sabine's, is pettifoggery...and literally insane.

1. I recommend that you consider that you don't know what you are talking about. Though, going by your comment, you are almost certainly incapable of such self-reflection.

2. Behold the UnWoke!

26. I don't want to lower the tone around the points made here but there is a mundane problem in presenting statistical tables of percentages due to rounding errors. If the audience for the tables understands the issue then there is no problem. So OK for an audience of physicists or mathematicians etc. but maybe a problem for a non-scientific audience who says: 'you have made a mistake as 3+3+3 should be 10 and not 9'. So here the total is 100% and there are three equal percentages adding up to 100%. So does one tell the 'truth' and just write 33%, 33% and 33% for the three items? One could get closer to the truth by adding more decimal places but there will still be a 1 deficit in the last decimal place quoted. Or does one fiddle to make it add to 100% just to make it look neater and/or avoid someone querying your arithmetic? I remember an amateur art lesson where it was said that 'an artist lies to tell the truth'; that was among other gems such as 'an artist always kills his father'. So does a statistician sometimes need to lie to tell the truth?

27. This is the mathematical equivalent of playing word games. "+" (addition on the reals) and "+" (modular addition on {0,..,n-1}) are two *different* operations for which the same symbol happens to have been used, just as "post" (the verb) and "post" (the noun) are two different words for which the same spelling and proununciation are used.

This "2 + 2 is not always 4" business is like the old joke that a ham sandwich is better than eternal happinesss: obviously, nothing is better than eternal happiness, but a ham sandwich is better than nothing, therefore...

28. Steven Evans,
Science, sociology, and sex. you can't separate the three and get a true picture. There is always more or less some human component to anything we do, but especially with sex.

1. Steve, E=mc^2 is an empirical fact of the universe, regardless of sex or society, and biologists tell us there are 2 types of gamete involved in human sexual production so there are 2 and only 2 human sexes. It is not the job of natural science to accommodate people's "lived experience"; that's the job of society.

2. Steven Evans,

Science does accommodate lived experience. Endocrinology and psychology are two examples of how science is used to improve transgender people's lives.
Biological sex isn't neatly quantised in everyone, either.
It's also got nothing to do with physics, so science and sex can intersect without affecting the facts of the universe.
The dictionary is a very incomplete source of information about sex and gender.

3. C Thompson8:49 AM, June 16, 2021

I didn't say it doesn't, I said it's not its job to do so. Psychology is not a hard science, so yes probably can accommodate your ideology.
There are two biological sexes in humans, by the definition of sex based on sexual reproduction. Biological sex may not cover all humans, e.g. intersex people, but it is not required to. It is simply required to express the roles in sexual reproduction of which there are two.
The word gender means the traits associated with a sex. If you are talking about attributes of a human being not related to their sex, then it is a misnomer to call it gender.

Clearly, you are not interested in addressing these facts, and just want to repeat your ideology. Fine. But natural science isn't interested in your ideology.

4. Okay, Bracketed Numerals Evans... I consider myself bisexual because Homo Sapiens are mostly sexually dimorphic and everyone has some variation of those 2 forms which are not neatly quantised but are varied in many different ways so all people have potential to attract me sexually and/or romantically, gametes produced are not what I'm attracted by
Other people refer to themselves as pansexual because their orientation embraces all forms and expressions of sex, gender, romance interest and libido levels of humanity. There are gender-fluid and non-binary people who don't fit dimorphic gender expectations to be attracted to. Your insistence that in hard science being the arbiter is not how things work.

It is not solely my ideology, which is to understand people, try not to be a lousy human, and to work to leave society better than I found it.
Physics is irrelevant unless one can provably link the physics of conscious experience to human sexual and social experience.
Your definitions that you're adhering to are not how others define gender. Natural Science is not the arbiter of people's experiences.
Re-read what Dr. Hossenfelder wrote, and think about it. If one person alone define things according to your perceptions and beliefs, nobody else will agree with them, they are incorrect.

29. This has caused a firestorm of comments.

I'd like to bring up another similar subject:

1 x 1 = 0, thus bringing up Grassmannians and Fermions,
which of course is very very physicsy.

30. This comment has been removed by the author.

1. There were black women who calculated the mathematics needed for space travel.

2. This comment has been removed by the author.

3. By "black women", I presume you are referring to the King of Norway and the Emperor of Japan.

4. I assume you're being sarcastic, Incomprehensible Gender. Evans.

31. Steven,
There's hard sciences like physics and there's squishy sciences like biology and sociology. With hard science there's some times a few nuts that spice things up. With the squishy sciences, well thats just a morass best avoided in this context but they do slop over in to the hard sciences more than the hard sciences drill down into the squishy sciences.

1. Steve, There are 2 types of gamete in sexual reproduction, therefore 2 human sexes. Gender means traits associated with a particular sex so there can only be 2 genders as there are only 2 sexes.
The situation is clear and not "squishy". Unless you have a definition of a 3rd sex or a 3rd gender....

2. 3 other people besides Steve and myself have said otherwise regarding sex and gender by now, it might be time to consider that we know at least a bit more than you do, Unwieldly Gender Designation.

3. I have clearly defined what is meant by human sex and gender. Biology agrees with my definition of human sex and the OED agrees with my definition of gender. You have not provided any clear definitions. You are welcome to do so.

4. Steven,

"Gender means traits associated with a particular sex so there can only be 2 genders as there are only 2 sexes."

You have yourself explained above why this conclusion is wrong, though maybe this wasn't your intention. The traits society associates with a certain sex change over time, and they are also culturally different, to say the obvious. The reason gender is being discussed so much at the moment is, in a nutshell, that the younger generation doesn't want to be squeezed into the gender ideas of the older generation. This is of course a broad-brush summary, there are older people who are very progressive and younger people who are very 18th century, etc, but by and large that's what's going on.

So when someone tells you they're gender-diverse, I suggest you take that to mean that today's genders aren't the same as those you know from your childhood. I think you shouldn't be so dismissive about people's "lived experience". You seem to be overly certain of yourself.

5. Yes, thank you Sabine!

6. Steven,

Should have added but forgot, though you certainly know that, if you sample over a group no single member of the group actually has to represent the average.

7. Dr. H.,

I'm not denying anyone's lived experience, I'm just saying it doesn't change science or definitions of words.
The word gender still means traits associated with a particular biological sex. It may be more and more problematic to identify what those traits are, but the word gender still means that. What the Woke are referring to with their use of gender is collections of various random human attributes. Sexuality for example is not sex or gender, it is sexuality. That's why we have all these separate words with separate particular meanings so we don't end up saying "a woman with a penis can still be a lesbian". Which according to Woke "definitions" could actually be referring to a man who is gay, because his "lesbian" partner could also be another man identifying as a woman. This is the post-modernist rabbit hole you tumble down when you accept that words can mean absolutely anything. Nobody's individuality is being denied by the standard definitions of sex and gender.

8. Dr. H.,

So you know of no mansplainers? This is not a trait of men?

"Sabine Hossenfelder4:37 AM, June 13, 2021
I find it stunning how many men who know nothing about social media think they can tell me how do my job. "

I rest my case ;)

9. Steven,

Well, I agree with you that a lot of people use the words "sex", "sexual orientation", and "gender" in a non-standard and inconsistent way and in many cases end up confusing themselves together with everyone else. I am all in favor of clear definitions and share much of your frustration.

However I think you're throwing out the baby with the bathwater. As long as no one's being harmed, someone's gender or sexual preferences aren't the business of the government. As I already said above, biological sex matters in some cases, not least of which for medical trials. And for what the whole issue of transgender people in professional sport is concerned, this beautifully highlights that dividing up people by biological sex is little more than a first-order approximation, one that crudely fails some people, and that needs to be replace with something that makes more sense. (Or maybe we should discontinue professional sports competitions which personally I think would be the better solution.)

In any case, the point is that some of the concerns that are being brought up by the people you call "woke" are imo perfectly valid.

I don't really expect to change your opinion, but if you think of it as a generational conflict it might make a little more sense?

10. And I don't understand your comment about mansplaining, joking or not. It's not like I denied the existence of the male gender or something.

11. @Christine
AM, June 17, 2021
“3 other people besides Steve and myself have said otherwise regarding sex and gender by now, it might be time to consider that we know at least a bit more than you do”
Maybe the rest of us are just staying quiet. Steven Evans is absolutely correct.
Your problem is that you are arguing from emotion, a bias. Evans’ argument is unbiased. It’s his bombast that has driven your replies. You haven’t responded to the science behind his remarks.

12. Steven,

'...that's why we have all these separate words with separate particular meanings so we don't end up saying "a woman with a penis can still be a lesbian". Which according to Woke "definitions" could actually be referring to a man who is gay, because his "lesbian" partner could also be another man identifying as a woman. This is the post-modernist rabbit hole you tumble down when you accept that words can mean absolutely anything. Nobody's individuality is being denied by the standard definitions of sex and gender.'

No, only people who are too lazy to try understanding how people actually describe themselves would say something that ridiculous.

A gay cisgender or trans man wouldn't have a woman of any sort as his partner because THEY ARE HOMOSEXUAL.
A woman who was assigned male at birth and who still has a penis, can also be a homosexual. Pronouns matter; use them properly, as you do for cisgender people. Being respectful shouldn't be hard, except you seem to struggle, maybe you should practice.

It's actually quite sensible once you stop fixating on gametes and rather simplistic dictionary definitions.
All the explanations and definitions are online to enlighten people, if you can be bothered to find them.

Also, Dr. Hossenfelder knows what she's on about. You could take advantage of her willingness to discuss this with you and actually learn something.

13. My bias is empathy and respect for transgender, non-binary and gender-fluid people, including friends and acquaintances.

Steven is appealing to science and The dictionary to justify his/their/whatevers own transphobic bias, and much of science has nothing to say on the matter. His citing natural science doesn't bolster his arguments.
If others who agree with Steven are staying quiet, then they ought to check their ignorance on the matter too.
Read what Dr. Hossenfelder is saying, if you don't agree with me.

14. And Brad, my name isn't Christine.

Bee, This subject has gone far enough, we come here to learn science

16. I would argue that appealing to the dictionary is not going to work much longer ;)
based on Oxford, gender is "Either of the two sexes (male and female), especially when considered with reference to social and cultural differences rather than biological ones. The term is also used more broadly to denote a range of identities that do not correspond to established ideas of male and female."
Arguably, the other influential ones will eventually catch up.
Regardless of the specific argument here, words do change their meaning all the time, so insisting on "this means that" when it is clear that there are groups of people who see this different is kind of missing the point.

The gist of the matter, in my opinion, is - as is often the case with social issues - that it is really all up to finding some consensus or equilibrium state the majority of members of a society can live with, regardless of any objective or subjective opinions, clear or not so clear morals etc. Even though we tend to think of basic human rights as guaranteed and granted, they are only granted for as long as the majority of humanity (or a sufficiently powerful subset) wishes it so. If one of the nuclear powers decides to wipe out most lifeforms on the planet, there's little anyone else could do about it. That is to say, cultural and sociological achievements are not laws of nature and should not be treated as such.
Rather, they are agreements between human beings how life together should be organised. As such, again just my personal opinion, both sides of these arguments should strive to be more open minded, empathic and consensus oriented. One side is asking for a change to behaviour/norms/rules that have persisted for a long time, the other side is wed to keeping them. It would be beneficial for both sides to explain what exactly they see as the problem, why it is a problem and why things should change / remain the same. Arguing semantics is the opposite of that.

A side note: I'm of course aware of the standpoint that sometimes, good things only come about through violence and upheaval, but in my opinion, in those circumstances, the world may just as likely change for the worse, so I do not subscribe to that.

17. Sabine Hossenfelder11:21 AM, June 17, 2021

Dr. H, I agree with you. I have no problem with how people identify themselves. But the words sex and gender are clearly defined. While Woke gender just means individual human attributes. And the definition of a Woke gender can be completely subjective. Woke gender is certainly not a concept that could ever be put into science or law successfully.

" joking or not. It's not like I denied the existence of the male gender or something."
Joking. No, you didn't deny the existence of male gender. It was in response to this comment:
" if you sample over a group no single member of the group actually has to represent the average. "

You identified mansplaining as a male trait and clearly it has been attained by some males. Gender traits are fuzzy rather than a calculated mean or median which as you say might not actually be attained.

18. C Thompson12:09 PM, June 17, 2021

"A gay cisgender or trans man wouldn't have a woman of any sort as his partner because THEY ARE HOMOSEXUAL. "

But a lesbian transgender woman who is biologically a man could have a lesbian sexual relationship with another lesbian transgender woman who is biologically a man. I would refer to this as 2 men having a homosexual relationship, while you would be forced to refer to it as 2 lesbian women having a lesbian sexual relationship.

"If others who agree ... are staying quiet, then they ought to check their ignorance on the matter too "

The Thought Police have arrived.

19. Steven,

That the mean does not *have to be* represented by any member of the ensemble does not mean it isn't.

I was feeling pretty tired and irritated last night, so I apologise for attacking you. It looks like we agree more than disagree. I agree that if one describes people in strictly reproductive terms that there are 2 sexes producing 2 gametes. The point I've been trying to make is that nobody is represented by just their reproductive sex. I think everyone agrees with that.

@Steven Evans:
Your insistence on refering to the 'Woke' and such is an irritant.
'But a lesbian transgender woman who is biologically a man could have a lesbian sexual relationship with another lesbian transgender woman who is biologically a man. I would refer to this as 2 men having a homosexual relationship, while you would be forced to refer to it 'as 2 lesbian women having a lesbian sexual relationship.'
Because I'm going on how people identify themselves, I'm not checking on their genitals and what gametes they should be producing.

@ All in this conversation;
The terms that many trans and non-binary people prefer to use are 'assigned male at birth' and 'assigned male at birth' to describe themselves and other non-cisgender people. I'm not demanding that anyone use these terms, just sharing them as information. Again, my apologies if people are already aware of this.
I am thankful to Dr. Hossenfelder for facilitating and participating in these not-so-scientific discussions.

21. Sabine Hossenfelder1:06 AM, June 18, 2021
Dr. H.,
Sure, the mean could be attained by anywhere from none to all of the members of the sample. But we are talking about traits not means, so the traits are attained (by "most" of the sample). It may have been my fault for writing "average trait" at some point.

22. Edit: I meant to say 'assigned male' and 'assigned female at birth.'
I am not as good at saying things online as I aim to be.

23. "Because I'm going on how people identify themselves,"

But you are denying biological and physical reality. They are behaving as biological men, which as trans women is what they are, and physically they are performing homosexual sex. Yet you describe it as 2 women performing lesbian sex.

You are failing to recognise the reality that a trans woman, as well as being of female gender, is biologically male. And this biological maleness may be relevant in certain circumstances. Which takes all the way back to the initial point about trans women in female prisons.

24. Rather tangential, but isn't lesbian sex homosexual? Am I missing something?

"They are behaving as biological men, which as trans women is what they are"

It seems to me you have made an unjustified jump from biological sex to behavior. But maybe more importantly, I don't get your point. If a biological man wants to identify as woman and then ends up being lesbian, what's the problem?

25. Perhaps, it's admitting that there's a whole paradigm he's mistaken about is too hard and it's easier to keep arguing maybe? Steven Evans is perhaps fighting to keep things as he/she/they/string are comfortable with because shifting into a new paradigm might mean giving up one's sense of comfortable certainty?

I mean, I didn't grow up with views centred on others' lived experiences, I grew up with transgender, queer etc.people being seen as immoral and a joke. I had major shifts in how I saw society in general, it's disconcerting to have had it wrong. I'm a long way from where I started, especially since a large portion of people I know aren't 'cis-het'.

26. Also, Steven Evans, your point is missing the point and goes back to why I disagreed with that article about prisons that Dr. H tweeted.
Transgender women are not actually men. If there are cisgender men pretending to be transgender women, that's not the women's fault.

27. C Thompson,

If have a prison full of straight biological men who suddenly find out they can be transferred to a building full with biological females by simply declaring themselves to be female, some of them won't care at all whether they'll be joked about but simply use the opportunity. Pointing out that this is an undesired consequence of a recently passed law does not deny that some biological men in men's prisons might be better off in a woman's prisons. I hence fail to see what your problem is with the article.

28. Dr. Hossenfelder,

I don't disagree with what you've just said, but the point Steven Evans is making. He seems to insist that the most important thing we need to be aware of is someone's biological gender, because biological men in women's prisons is something that relates to transgender women in general somwhow? Is how he is landing with me.

My problem with the article was with what I percieved to be the writer's framing of the issue. I've made my points badly sorry.

29. C Thompson,

Sorry for the misunderstanding. I guess we largely agree.

30. Dr. Hossenfelder,

All good. :)
You're better at discussing and arguing about contentious topics than I am; I usually end up giving myself a headache.

31. C Thompson7:49 AM, June 18, 2021

You are blatantly misrepresenting what I wrote.

"Transgender women are not actually men. "
I wrote quite clearly that transgender women can be biological men, which is true.

You write:
"He seems to insist that the most important thing we need to be aware of is someone's biological gender"

But I wrote:
"And this biological maleness may be relevant in certain circumstances"

"it's admitting that there's a whole paradigm he's mistaken about is too hard and it's easier to keep arguing maybe?"

You are simply making this up. I have not denied the existence of transgender men and women.

32. Dr. H.,

"It seems to me you have made an unjustified jump from biological sex to behavior. "

Lesbian sex and male gay sex are not physically the same thing. But ok, we can define sex as in biology as simply sexual reproduction between a male (provider of small gametes) and female (provider of large gametes).

Problem solved.

33. Steven,

To describe transgender people as 'biological men/women' is actually transphobic, especially as you're using the terms. And you're insisting on rationalising your transphobia by talking about genitals and gametes.

I understand exactly what views like yours are; they're de-humanising, no matter what else you say.

You're the one who insists there's a problem. Fixating on biology is transphobic.
Are you going to insist on identifying everyone else by genitalia and what gametes they produce?

Transgender people, or anyone for that matter, are not defined by gametes. Defining people by reproductive function is ridiculous. We're not in an IVF clinic here.

You could stop being wrong any time you like, even if you only take what Dr. Hossenfelder has said into account.

34. I want to clarify: Not everyone here is using 'biological/biologically' in a transphobic context.

35. Steven Evans,

"But ok, we can define sex as in biology as simply sexual reproduction between a male (provider of small gametes) and female (provider of large gametes).

Problem solved."

Well, it is a definition, alright. Leaving aside that I am not sure what "sexual reproduction in between" might mean, it seems that most of what people presently call sex isn't actually sex-according-to-Steven, whereas some things that aren't sex (like IVF) suddenly become sex.

36. C Thompson1:04 AM, June 19, 2021

"To describe transgender people as 'biological men/women' is actually transphobic"

No, it is not transphobic as it does not deny their transgender nature. If they are biological men/women then that is simply a biological fact.

* You don't even understand the terms you are bandying around *

"Fixating on biology is transphobic. "

I'm not "fixating" on biology. I am simply pointing out that there are biological facts that don't disappear whatever identities people might assume or have.

"Are you going to insist on identifying everyone else by genitalia and what gametes they produce?"

I have not done so. That biological sex is determined by what gametes are produced is a fact. That certain genitalia are associated with each biological sex is also a fact.

"You could stop being wrong any time you like"

There are two biological sexes determined by the type of gamete produced. The word gender means traits associated with a particular sex. Concepts like cisgender and transgender fit with these definitions.

You are now welcome to show how wrong I am by providing your definition of sex and gender, which in several dozen posts you have failed to do because you don't even know yourself what you mean.

**** Let's now hear your definition of sex and gender ;))))) ****

37. Sabine Hossenfelder4:43 AM, June 19, 2021

Well, there are no problems with my definitions of sexual reproduction, human sex and gender. These are clearly defined in biology and the dictionary and can accommodate concepts like cisgender and transgender.

But you claim that two lesbian transgender women who are biologically male who perform sex which is physically indistinguishable from male gay sex, are actually having lesbian sex. This is denying physical reality.

Also, by these definitions, a lesbian transgender woman (who is biologically a man) could have lesbian sex with a lesbian cis woman, and the cis woman could become pregnant via lesbian sex. Or, a man (a transgender man who is biologically a woman) could have heterosexual sex with a woman (a transgender woman who is biologically a man) and the man could get pregnant.

I would claim in both cases that a man and a woman had heterosexual sex and the woman became pregnant. Because at the level of biological sex that is what has happened, whatever their genders. The Woke definitions are denying reality.

Lesbian sex cannot produce a child.
A man cannot become pregnant.

38. Steven,

"But you claim that two lesbian transgender women who are biologically male who perform sex which is physically indistinguishable from male gay sex, are actually having lesbian sex. This is denying physical reality."

Oh I think we entirely agree on the "physical reality". I am just telling you that the word "lesbian" doesn't mean what you think it does. You seem to think that sexual orientation implies a statement about biological sex.

"Also, by these definitions, a lesbian transgender woman (who is biologically a man) could have lesbian sex with a lesbian cis woman, and the cis woman could become pregnant via lesbian sex. Or, a man (a transgender man who is biologically a woman) could have heterosexual sex with a woman (a transgender woman who is biologically a man) and the man could get pregnant."

That's largely correct though I don't know why you insist on calling it "lesbian sex" or "heterosexual sex" rather than just "sex" when that's implicit in the gender.

"I would claim in both cases that a man and a woman had heterosexual sex and the woman became pregnant. Because at the level of biological sex that is what has happened, whatever their genders. The Woke definitions are denying reality."

I guess I was cheering too soon. You insist on using the terms "man" and "woman" to refer to biological sex which, I have to agree with C Thompson, is indeed transphobic. Why can't you just let people be what they want to be if no one's being harmed.

39. Steven Evans

I can't be arsed arguing with someone who uses a dictionary definition to back themselves up, vs. paying attention to an actual scientist, Dr. H, who understands far more than you, even.

I know exactly what I mean, you're committed to being wrong. Not my deal anymore.
If being knowledgeable about human diversity makes me part of the Woke Brigade, I'll take it as a compliment.
I would rather be woke than a bigot.

40. Sabine Hossenfelder7:10 AM, June 19, 2021

Sure, people's private lives are none of our business. But the Woke Guards are calling everyone bigots and transphobes and people are losing their jobs over it, so that part is our business.

The question is whether the gender theory definitions are contradictory or not. Even the great gender theorists of our time are debating the question of whether a transgender woman can be a lesbian, the great question of our time.

There are 2 channels as it were, biological sex (with associated phenotypes) and gender. Sex is a physical act along the biological/physical channel (obviously emotions are also involved). If you try to define sex via the gender channel alone, then you end up denying reality. A lesbian transgender woman who is biologically a man cannot experience "lesbian sex" with a cis woman whether straight or lesbian because he isn't performing lesbian sex, he's performing heterosexual sex proven by the fact that a child could result.

Referring to heterosexual sex between a biological man and a biological woman as "lesbian sex" is not an attempt to provide a more nuanced reality in alignment with people's lived experience, it is an ideological attempt to try to deny the physical reality of biological sex. It's self-contradictory.

I am not being transphobic because I don't deny the gender of trans people. I do deny that a transgender woman who is biologically a man is a woman though - she's a man. Prioritising biological sex does not constitute transphobia as I don't deny the trans gender. If we are so laid back about all this, why is it a problem to admit the reality that a transgender woman can have gay male sex?

41. C Thompson7:12 AM, June 19, 2021

You have admitted that you can't get these ideas straight in your own head, yet you call me a bigot.

"Bigot definition is - a person who is obstinately or intolerantly devoted to his or her own opinions and prejudices; especially : one who regards or treats the members of a group (such as a racial or ethnic group) with hatred and intolerance."

I accept the reality of trans gender so I am not prejudiced. I do not hate nor am I intolerant of trans people. We were simply discussing definitions of terms, and all the while I stated clearly that people are free to do whatever they want in their private lives.

You claimed that "a woman with a penis can be a lesbian", but this is an ongoing debate amongst gender theorists and the people affected. I have provided an argument that this claim contradicts physical reality. You just accuse people of being transphobic or a bigot and provide no argument.

42. Dr. H.

Definition of transphobia
: irrational fear of, aversion to, or discrimination against transgender people

How does this definition apply to what I have written? It doesn't.

43. Steven,

"But the Woke Guards are calling everyone bigots and transphobes and people are losing their jobs over it, so that part is our business."

The problem isn't crazy folks saying crazy things, but that employers can, and do, fire people for it. But since it makes no financial sense it'll fall out of fashion soon enough.

": irrational fear of, aversion to, or discrimination against transgender people

How does this definition apply to what I have written? It doesn't."

This whole thread is document to your aversion to refer to people by their gender. Transgender women are women. Transgender men are men. Why is that so hard to understand? What gametes someone produces isn't your business, unless maybe you're an urologist looking for a prostate or a gynecologist checking for an ectopic pregnancy. I hope I have made clear above that I entirely acknowledge biological sex matters in some cases and that I think it's rather silly to pretend it never plays a role, but for most of our lives it's irrelevant -- or at least it should be.

44. Stephen Evans:
Trashbag transphobes citing 'the Woke Brigade' vs. people being treated like shit by society and wanting to kill themselves and succeeding at times.
Eat a dick-tionary.
It's also ridiculous and hilarious watching you imply without saying that Dr. Hossenfelder is in danger of being manipulated by Woke ideology. OH NOES!

45. It's bigotry that casts laws making transgender and queer people's lives harder and more dangerous so any gender theorist who supports transphobia can go mine an asteroid without a spacesuit as far as I'm concerned.

46. Sabine Hossenfelder1:22 AM, June 20, 2021

"But since it makes no financial sense it'll fall out of fashion soon enough."

The sacking of gender-critical employees is already being challenged in law, so, yes, it will be stopped.

"This whole thread is document to your aversion to refer to people by their gender. "

That's simply untrue. I accept the transgender of transgender people and am happy to use their pronouns socially. A transgender woman is a man though biologically. The definition of male and female based on sex is based in fact and is consistent. The gender ideological definition is not consistent, as it ends up denying reality by claiming that a man and a woman performing heterosexual sex can be described as 2 women performing lesbian sex.

A transgender woman is a man, and it is not transphobic to state so as I accept her gender and use her pronouns.

47. C Thompson1:38 AM, June 20, 2021

The gender-critical view is consistent and based on biological fact. You have not shown it to be transphobic. The tactic of randomly calling people a transphobe or a bigot will not win. No-platforming of gender-critical speakers and sacking employees for expressing gender-critical views is going to prove legally problematic.

At which point you will have to start making some factual arguments or you can bleat "transphobe" into the wind with complete impotence.

48. Sabine Hossenfelder1:22 AM, June 20, 2021

Dr. H.,

"This whole thread is document to your aversion to refer to people by their gender. "

Just to be clear the definition of transphobia states an "aversion to.. transgender people" not an "aversion to refer to people by their gender".

I have no aversion to transgender people and you have not demonstrated one, and I have stated clearly that I am happy to recognise transgender people's gender via their pronouns.

So you have not demonstrated transphobia.

The only 2 points I have made is that self-declaration of gender is clearly open to abuse, and that the definitions of terms in gender ideology contradict reality.

49. I'll also note that I've managed to communicate sufficiently well that one of the premier physicists-cum-science communicators of our time understands and even agrees more or less with me suggests that I'm sufficiently clear and accurate in my comments, so not exactly bleating...

50. It occurred to me a moment ago that Steven Evans and some lousy Christians have similar erroneosly simplistic views, except Evans cites Gender Critical Theory (the proponents of which ought to be fired into active volcanoes, not just from their jobs, in my rather biased opinion) and the God-botherers cite the Book of Genesis: 'God made them male and female'. Ironic.

51. C Thompson5:12 AM, June 20, 2021

"I'll also note that I've managed to communicate sufficiently well"

You have not defined what you mean by human sex nor what you mean by human gender in 50 comments.

And you will not, because you have absolutely no idea what you yourself mean by those terms.

52. C Thompson8:30 AM, June 20, 2021

"similar erroneosly simplistic views"

So point out the error if I have erroneous views. You can't.

"the proponents of which ought to be fired into active volcanoes,"

And as with Luke Barnes, the ideologue without a supporting argument begins the "jokes" about killing people who refuse to adhere to the ideology.

"not just from their jobs"

This month, a UK employment tribunal found that gender-critical views are protected under the law. So it is illegal for an employer to sack an employee for simply writing "a transgender woman is not a woman". Which is fair enough because it's true.

Bad luck. In the UK at least, your gender-ideological-fascist state will not be implemented.

53. "So it is illegal for an employer to sack an employee for simply writing "a transgender woman is not a woman". Which is fair enough because it's true."

It's impossible to decide whether the statement is true or false without a definition of the terms. Just in case anyone's forgotten about it, this was the entire point of my video. I agree of course entirely that it shouldn't be possible to fire people for this, and I don't find it funny to make jokes about killing someone you disagree with.

54. I think transgender people deserve protection from being fired more than the people who de-humanise them. The U.K. also isn't going well in accepting transgender people thanks to the likes of the gender critical theorists.

I'm grateful that in Australia, despite our conservative, prejudiced leadership, trans and queer people cannot be fired for gender or orientation but not everyone on Earth has that protection.

If someone makes it their stance to de-humanise anyone by reducing them to genitals and gametes, I tend to think pretty poorly of them. I'm not going to try and do anything about having them removed from the population, just make hyperbolic statements to express my loathing.
I'll refrain from using such language here in the future.

55. Steven Evans

'Gender-ideological-fascist-state' That's a good one.
Thing is, those people aren't just writing that transgender women are men (tell me, who has been fired for exactly that, and not for something else related but worse), but for them to be denied access into women's spaces, and they bring up the 'trans panic' of men pretending to be women, and trans women assaulting cis women and children.

I already said that there are, broadly speaking, 2 forms that humans have for reproduction. That's my defenition, for mere reproductive purposes.

56. C Thompson8:02 PM, June 20, 2021

You1: "tell me, who has been fired for exactly that"

You2: "Gender Critical Theory (the proponents of which ought to be fired .. from their jobs"

You1: "'Gender-ideological-fascist-state' That's a good one."

You2: "Gender Critical Theory (the proponents of which ought to be fired into active volcanoes,"

"I think transgender people deserve protection from being fired more than the people who de-humanise them."

It's illegal to fire employees for being trans. The gender critical view does not claim trans people are not human nor suggest that they shouldn't be treated as humans.

It is not transphobic per se to define "man" and "woman" in terms of biology, as it does not deny the trans nature of trans people. And it is the only definition of "man" and "woman" that is based in scientific fact and consistent.

Also, it is not transphobic per se for women to defend women-only spaces as the criterion for entry is the objective fact of biological sex, so no comment is being made on the subjective experience of trans women.

32. I've never been entirely happy with 2+2=4
If the 2's are the same, then we have only one 2. So 2+2=1, i.e. 1*2

33. Don't you know there's a secret integer between three and four?
It's all explained in the movie "The Secret Number" (2012) from director Colin Levy.

34. By the way, what mean CRT, as seen in some comments here?
Please enlighten the reader by not using obscure acronyms, or at the very least spell them out.

1. critical race theory

35. How can you find out that a person is gay, vegan, transgender, etc. without asking?
Do not worry. The person will tell you.
What can I do more than not asking?

1. Henning,

You can stop making assumptions based on stereotypes, and you can accept that those things aren't your business unless they tell you.

2. This comment has been removed by the author.

3. Mr. Camp,

Because they're sick of being expected to behave according to outdated, oppressive social norms so everyone else can feel comfortable.
We've all got cisgender heterosexuality in our faces almost everywhere already.
I'm sick of that being the expected norm, myself.

4. Now you are making assumptions based on stereotypes.

36. I deleted my last comment because it wasn’t very succinct.
Here is another try at it:
I have an example of cyclic groups of numbers from a pilot’s point of view.
A compass rose is divided into 360°. 360 for north, 180 for south, 90 for east, etc.
If a pilot is flying a heading of 270° and makes a 180° turn to the north, he doesn’t end up on a heading of 270+180=450° which doesn’t exist. You need to subtract the 360 cyclic group from that to arrive at a true heading of 90°. So in this case, 270+180=90.
Of course, pilots don’t really do this in the cockpit, but it is one easy example of routine navigation in the cockpit while you enjoy the inflight movie.

37. Steven,
Words change their meanings and implications over time. That is happening now in some places with the word ' gender'. You may not like it but you can't stop it. All you can do is fight a rear guard action.

38. I though a transgender woman was a person whose birth gender was recorded as male. You know, this is getting confusing and frankly, a bit icky.

1. Steve,

That is the case.
That's why people use 'cisgender' and 'transgender' as descriptors, and refer to being 'assigned male/female at birth.'
I'm not trying to patronise you sorry, just that there seems to be varying degrees of awareness of these terms.

39. Sorry for the comment-spamming, but I think this is an important piece of where I'm coming from, taken from Wikipedia:

The paradox of tolerance states that if a society is tolerant without limit, its ability to be tolerant is eventually seized or destroyed by the intolerant. Karl Popper described it as the seemingly paradoxical idea that in order to maintain a tolerant society, the society must be intolerant of intolerance.

Insisting on being 'correct' over empowering people to define how they're known and referred to fits into intolerance, as far as I'm concerned.